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The Invisible Hand of Dynamic Market Pricing

2016-07-21
Vincent Cohen-Addad , Alon Eden , Michal Feldman +1 more
Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, … Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.
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A Simple and Approximately Optimal Mechanism for a Buyer with Complements

2017-06-20
Alon Eden , Michal Feldman , Ophir Friedler +2 more
We consider a revenue-maximizing seller with m heterogeneous items and a single buyer whose valuation v for the items may exhibit both substitutes (i.e., for some S, T, v(S ∪ … We consider a revenue-maximizing seller with m heterogeneous items and a single buyer whose valuation v for the items may exhibit both substitutes (i.e., for some S, T, v(S ∪ T) < v(S) + v(T)) and complements (i.e., for some S, T, v(S ∪ T) > v(S) + v(T)). We show that the mechanism first proposed by Babaioff et al. [2014] -- the better of selling the items separately and bundling them together -- guarantees a Θ(d) fraction of the optimal revenue, where $d$ is a measure on the degree of complementarity. Note that this is the first approximately optimal mechanism for a buyer whose valuation exhibits any kind of complementarity. It extends the work of Rubinstein and Weinberg [2015], which proved that the same simple mechanisms achieve a constant factor approximation when buyer valuations are subadditive, the most general class of complement-free valuations.
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Interdependent Values without Single-Crossing

2018-06-11
Alon Eden , Michal Feldman , Amos Fiat +1 more
We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation … We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation of each agent is a known function of all n private signals. This captures settings such as valuations for oil drilling rights, broadcast rights, pieces of art, and many more.
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An Economic-Based Analysis of RANKING for Online Bipartite Matching

2018-01-01
Alon Eden , Michal Feldman , Amos Fiat +1 more
In their seminal paper, Karp, Vazirani and Vazirani (STOC'90) introduce the online bipartite matching problem, and the RANKING algorithm, which admits a tight $1-\frac{1}{e}$ competitive ratio. Since its publication, the … In their seminal paper, Karp, Vazirani and Vazirani (STOC'90) introduce the online bipartite matching problem, and the RANKING algorithm, which admits a tight $1-\frac{1}{e}$ competitive ratio. Since its publication, the problem has received considerable attention, including a sequence of simplified proofs. In this paper we present a new proof that gives an economic interpretation of the RANKING algorithm -- further simplifying the proof and avoiding arguments such as duality. The new proof gives a new perspective on previous proofs.

Private Interdependent Valuations

2022-01-01
Alon Eden , Kira Goldner , Shuran Zheng
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Private Interdependent ValuationsAlon Eden, Kira Goldner, and Shuran ZhengAlon Eden, Kira Goldner, and Shuran … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Private Interdependent ValuationsAlon Eden, Kira Goldner, and Shuran ZhengAlon Eden, Kira Goldner, and Shuran Zhengpp.2920 - 2939Chapter DOI:https://doi.org/10.1137/1.9781611977073.113PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider the single-item interdependent value setting, where there is a single item sold by a monopolist, n buyers, and each buyer has a private signal si describing a piece of information about the item. Additionally, each bidder i has a valuation function vi(s1, …, sn) mapping the (private) signals of all buyers into a positive real number representing their value for the item. This setting captures scenarios where the item's information is asymmetric or dispersed among agents, such as in competitions for oil drilling rights, or in auctions for art pieces. Due to the increased complexity of this model compared to the standard private values model, it is generally assumed that each bidder's valuation function vi is public knowledge to the seller or all other buyers. But in many situations, the seller may not know the bidders' valuation functions—how a bidder aggregates signals into a valuation is often their private information. In this paper, we design mechanisms that guarantee approximately-optimal social welfare while satisfying ex-post incentive compatibility and individually rationality for the case where the valuation functions are private to the bidders, and thus may be strategically misreported to the seller. When the valuations are public, it is possible for optimal social welfare to be attained by a deterministic mechanism when the valuations satisfy a single-crossing condition. In contrast, when the valuations are the bidders' private information, we show that no finite bound on the social welfare can be achieved by any deterministic mechanism even under single-crossing. Moreover, no randomized mechanism can guarantee better than n-approximation. We thus consider valuation functions that are submodular over signals (SOS), introduced in the context of combinatorial auctions in a recent breakthrough paper by Eden et al. [EC'19]. Our main result is an O(log2 n)-approximation randomized mechanism for buyers with private signals and valuations under the SOS condition. We also give a tight Θ(k)-approximation mechanism for the case each agent's valuation depends on at most k other signals even for unknown k. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771
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Platform Behavior under Market Shocks: A Simulation Framework and Reinforcement-Learning Based Study

2023-04-26
Xintong Wang , Gary Qiurui , Alon Eden +4 more
We study the behavior of an economic platform (e.g., Amazon, Uber Eats, Instacart) under shocks, such as COVID-19 lockdowns, and the effect of different regulation considerations. To this end, we … We study the behavior of an economic platform (e.g., Amazon, Uber Eats, Instacart) under shocks, such as COVID-19 lockdowns, and the effect of different regulation considerations. To this end, we develop a multi-agent simulation environment of a platform economy in a multi-period setting where shocks may occur and disrupt the economy. Buyers and sellers are heterogeneous and modeled as economically-motivated agents, choosing whether or not to pay fees to access the platform. We use deep reinforcement learning to model the fee-setting and matching behavior of the platform, and consider two major types of regulation frameworks: (1) taxation policies and (2) platform fee restrictions. We offer a number of simulated experiments that cover different market settings and shed light on regulatory tradeoffs. Our results show that while many interventions are ineffective with a sophisticated platform actor, we identify a particular kind of regulation—fixing fees to the optimal, no-shock fees while still allowing a platform to choose how to match buyers and sellers—as holding promise for promoting the efficiency and resilience of the economic system.
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Pricing Online Decisions: Beyond Auctions

2014-12-22
Ilan Reuven Cohen , Alon Eden , Amos Fiat +1 more
We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare … We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare or revenue in an auction settings. This paper deals with quite general settings and minimizing social costs. We show that appropriately computed posted prices allow one to achieve essentially the same performance as the best online algorithm. This holds in a wide variety of settings. Unlike online algorithms that learn about the event, and then make enforcable decisions, prices are posted without knowing the future events or even the current event, and are thus inherently dominant strategy incentive compatible.In particular we show that one can give efficient posted price mechanisms for metrical task systems, some instances of the k-server problem, and metrical matching problems. We give both deterministic and randomized algorithms. Such posted price mechanisms decrease the social cost dramatically over selfish behavior where no decision incurs a charge. One alluring application of this is reducing the social cost of free parking exponentially.
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A Simple and Approximately Optimal Mechanism for a Buyer with Complements

2020-12-15
Alon Eden , Michal Feldman , Ophir Friedler +2 more
Recent literature on approximately optimal revenue maximization has shown that in settings where agent valuations for items are complement free, the better of selling the items separately and bundling them … Recent literature on approximately optimal revenue maximization has shown that in settings where agent valuations for items are complement free, the better of selling the items separately and bundling them together guarantees a constant fraction of the optimal revenue. However, most real-world settings involve some degree of complementarity among items. The role that complementarity plays in the trade-off of simplicity versus optimality has been an obvious missing piece of the puzzle. In “A Simple and Approximately Optimal Mechanism for a Buyer with Complements,” the authors show that the same simple selling mechanism—the better of selling separately and as a grand bundle—guarantees a $\Theta(d)$ fraction of the optimal revenue, where $d$ is a measure of the degree of complementarity. One key modeling contribution is a tractable notion of “degree of complementarity” that admits meaningful results and insights—they demonstrate that previous definitions fall short in this regard.
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On the Power and Limits of Dynamic Pricing in Combinatorial Markets

2020-01-01
Ben Berger , Alon Eden , Michal Feldman
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Reinforcement Learning of Sequential Price Mechanisms

2021-05-18
Gianluca Brero , Alon Eden , Matthias Gerstgrasser +2 more
We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of sequential price mechanisms, which generalizes both serial dictatorship and posted price mechanisms and essentially … We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of sequential price mechanisms, which generalizes both serial dictatorship and posted price mechanisms and essentially characterizes all strongly obviously strategyproof mechanisms. Learning an optimal mechanism within this class forms a partially-observable Markov decision process. We provide rigorous conditions for when this class of mechanisms is more powerful than simpler static mechanisms, for sufficiency or insufficiency of observation statistics for learning, and for the necessity of complex (deep) policies. We show that our approach can learn optimal or near-optimal mechanisms in several experimental settings.
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Price of Anarchy of Simple Auctions with Interdependent Values

2020-01-01
Alon Eden , Michal Feldman , Inbal Talgam-Cohen +1 more
We expand the literature on the price of anarchy (PoA) of simultaneous item auctions by considering settings with correlated values; we do this via the fundamental economic model of interdependent … We expand the literature on the price of anarchy (PoA) of simultaneous item auctions by considering settings with correlated values; we do this via the fundamental economic model of interdependent values (IDV). It is well-known that in multi-item settings with private values, correlated values can lead to bad PoA, which can be polynomially large in the number of agents $n$. In the more general model of IDV, we show that the PoA can be polynomially large even in single-item settings. On the positive side, we identify a natural condition on information dispersion in the market, termed $γ$-heterogeneity, which enables good PoA guarantees. Under this condition, we show that for single-item settings, the PoA of standard mechanisms degrades gracefully with $γ$. For settings with $m&gt;1$ items we show a separation between two domains: If $n \geq m$, we devise a new simultaneous item auction with good PoA (with respect to $γ$), under limited information asymmetry. To the best of our knowledge, this is the first positive PoA result for correlated values in multi-item settings. The main technical difficulty in establishing this result is that the standard tool for establishing PoA results -- the smoothness framework -- is unsuitable for IDV settings, and so we must introduce new techniques to address the unique challenges imposed by such settings. In the domain of $n \ll m$, we establish impossibility results even for surprisingly simple scenarios.

Cursed yet Satisfied Agents

2021-01-01
Yiling Chen , Alon Eden , Juntao Wang
In real life auctions, a widely observed phenomenon is the winner's curse -- the winner's high bid implies that the winner often over-estimates the value of the good for sale, … In real life auctions, a widely observed phenomenon is the winner's curse -- the winner's high bid implies that the winner often over-estimates the value of the good for sale, resulting in an incurred negative utility. The seminal work of Eyster and Rabin [Econometrica'05] introduced a behavioral model aimed to explain this observed anomaly. We term agents who display this bias "cursed agents". We adopt their model in the interdependent value setting, and aim to devise mechanisms that prevent the cursed agents from obtaining negative utility. We design mechanisms that are cursed ex-post IC, that is, incentivize agents to bid their true signal even though they are cursed, while ensuring that the outcome is individually rational -- the price the agents pay is no more than the agents' true value. Since the agents might over-estimate the good's value, such mechanisms might require the seller to make positive transfers to the agents to prevent agents from over-paying. For revenue maximization, we give the optimal deterministic and anonymous mechanism. For welfare maximization, we require ex-post budget balance (EPBB), as positive transfers might lead to negative revenue. We propose a masking operation that takes any deterministic mechanism, and imposes that the seller would not make positive transfers, enforcing EPBB. We show that in typical settings, EPBB implies that the mechanism cannot make any positive transfers, implying that applying the masking operation on the fully efficient mechanism results in a socially optimal EPBB mechanism. This further implies that if the valuation function is the maximum of agents' signals, the optimal EPBB mechanism obtains zero welfare. In contrast, we show that for sum-concave valuations, which include weighted-sum valuations and l_p-norms, the welfare optimal EPBB mechanism obtains half of the optimal welfare as the number of agents grows large.
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Combinatorial Auctions with Interdependent Valuations: SOS to the Rescue

2019-01-01
Alon Eden , Michal Feldman , Amos Fiat +2 more
We study combinatorial auctions with interdependent valuations. In such settings, each agent $i$ has a private signal $s_i$ that captures her private information, and the valuation function of every agent … We study combinatorial auctions with interdependent valuations. In such settings, each agent $i$ has a private signal $s_i$ that captures her private information, and the valuation function of every agent depends on the entire signal profile, ${\bf s}=(s_1,\ldots,s_n)$. The literature in economics shows that the interdependent model gives rise to strong impossibility results, and identifies assumptions under which optimal solutions can be attained. The computer science literature provides approximation results for simple single-parameter settings (mostly single item auctions, or matroid feasibility constraints). Both bodies of literature focus largely on valuations satisfying a technical condition termed {\em single crossing} (or variants thereof). We consider the class of {\em submodular over signals} (SOS) valuations (without imposing any single-crossing type assumption), and provide the first welfare approximation guarantees for multi-dimensional combinatorial auctions, achieved by universally ex-post IC-IR mechanisms. Our main results are: $(i)$ 4-approximation for any single-parameter downward-closed setting with single-dimensional signals and SOS valuations; $(ii)$ 4-approximation for any combinatorial auction with multi-dimensional signals and {\em separable}-SOS valuations; and $(iii)$ $(k+3)$- and $(2\log(k)+4)$-approximation for any combinatorial auction with single-dimensional signals, with $k$-sized signal space, for SOS and strong-SOS valuations, respectively. All of our results extend to a parameterized version of SOS, $d$-SOS, while losing a factor that depends on $d$.
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Constant Approximation for Private Interdependent Valuations

2023-11-06
Alon Eden , Michal Feldman , Kira Goldner +2 more
The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $s_{1}, \ldots, s_{n}$ of the n bidders … The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $s_{1}, \ldots, s_{n}$ of the n bidders and public valuation functions $v_{i}\left(s_{1}, \ldots, s_{n}\right)$. Recent work in TCS has shown that this setting admits a constant approximation to the optimal social welfare if the valuations satisfy a natural property called submodularity over signals (SOS). More recently, Eden et al. (2022) have extended the analysis of interdependent valuations to include settings with private signals and private valuations, and established $O\left(\log ^{2} n\right)$-approximation for SOS valuations. In this paper we show that this setting admits a constant factor approximation, settling the open question raised by Eden et al. (2022).
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The Invisible Hand of Dynamic Market Pricing

2016-07-24
Vincent Cohen-Addad , Alon Eden , Michal Feldman +1 more
Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, … Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.
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On the Power and Limits of Dynamic Pricing in Combinatorial Markets

2020-01-01
Ben Berger , Alon Eden , Michal Feldman
We study the power and limits of optimal dynamic pricing in combinatorial markets; i.e., dynamic pricing that leads to optimal social welfare. Previous work by Cohen-Addad et al. [EC'16] demonstrated … We study the power and limits of optimal dynamic pricing in combinatorial markets; i.e., dynamic pricing that leads to optimal social welfare. Previous work by Cohen-Addad et al. [EC'16] demonstrated the existence of optimal dynamic prices for unit-demand buyers, and showed a market with coverage valuations that admits no such prices. However, finding the frontier of markets (i.e., valuation functions) that admit optimal dynamic prices remains an open problem. In this work we establish positive and negative results that narrow the existing gap. On the positive side, we provide tools for handling markets beyond unit-demand valuations. In particular, we characterize all optimal allocations in multi-demand markets. This characterization allows us to partition the items into equivalence classes according to the role they play in achieving optimality. Using these tools, we provide a poly-time optimal dynamic pricing algorithm for up to $3$ multi-demand buyers. On the negative side, we establish a maximal domain theorem, showing that for every non-gross substitutes valuation, there exist unit-demand valuations such that adding them yields a market that does not admit an optimal dynamic pricing. This result is the dynamic pricing equivalent of the seminal maximal domain theorem by Gul and Stacchetti [JET'99] for Walrasian equilibrium. Yang [JET'17] discovered an error in their original proof, and established a different, incomparable version of their maximal domain theorem. En route to our maximal domain theorem for optimal dynamic pricing, we provide the first complete proof of the original theorem by Gul and Stacchetti.
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Reinforcement Learning of Simple Indirect Mechanisms.

2020-10-02
Gianluca Brero , Alon Eden , Matthias Gerstgrasser +2 more
We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of {\em sequential price mechanisms}, which generalizes both serial dictatorship and posted price mechanisms and … We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of {\em sequential price mechanisms}, which generalizes both serial dictatorship and posted price mechanisms and essentially characterizes all strongly obviously strategyproof mechanisms. Learning an optimal mechanism within this class forms a partially-observable Markov decision process. We provide rigorous conditions for when this class of mechanisms is more powerful than simpler static mechanisms, for sufficiency or insufficiency of observation statistics for learning, and for the necessity of complex (deep) policies. We show that our approach can learn optimal or near-optimal mechanisms in several experimental settings.

The Competition Complexity of Auctions: A Bulow-Klemperer Result for Multi-Dimensional Bidders

2016-01-01
Alon Eden , Michal Feldman , Ophir Friedler +2 more
A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to $n$ bidders whose values are drawn i.i.d. from … A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to $n$ bidders whose values are drawn i.i.d. from a regular distribution, the simple welfare-maximizing VCG mechanism (in this case, a second price-auction) with one additional bidder extracts at least as much revenue in expectation as the optimal mechanism. The beauty of this theorem stems from the fact that VCG is a {\em prior-independent} mechanism, where the seller possesses no information about the distribution, and yet, by recruiting one additional bidder it performs better than any prior-dependent mechanism tailored exactly to the distribution at hand (without the additional bidder). In this work, we establish the first {\em full Bulow-Klemperer} results in {\em multi-dimensional} environments, proving that by recruiting additional bidders, the revenue of the VCG mechanism surpasses that of the optimal (possibly randomized, Bayesian incentive compatible) mechanism. For a given environment with i.i.d. bidders, we term the number of additional bidders needed to achieve this guarantee the environment's {\em competition complexity}. Using the recent duality-based framework of Cai et al. [2016] for reasoning about optimal revenue, we show that the competition complexity of $n$ bidders with additive valuations over $m$ independent, regular items is at most $n+2m-2$ and at least $\log(m)$. We extend our results to bidders with additive valuations subject to downward-closed constraints, showing that these significantly more general valuations increase the competition complexity by at most an additive $m-1$ factor. We further improve this bound for the special case of matroid constraints, and provide additional extensions as well.
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Interdependent Values without Single-Crossing

2018-01-01
Alon Eden , Michal Feldman , Amos Fiat +1 more
We consider a setting where an auctioneer sells a single item to $n$ potential agents with {\em interdependent values}. That is, each agent has her own private signal, and the … We consider a setting where an auctioneer sells a single item to $n$ potential agents with {\em interdependent values}. That is, each agent has her own private signal, and the valuation of each agent is a known function of all $n$ private signals. This captures settings such as valuations for artwork, oil drilling rights, broadcast rights, and many more. In the interdependent value setting, all previous work has assumed a so-called {\sl single-crossing condition}. Single-crossing means that the impact of agent $i$'s private signal, $s_i$, on her own valuation is greater than the impact of $s_i$ on the valuation of any other agent. It is known that without the single-crossing condition an efficient outcome cannot be obtained. We study welfare maximization for interdependent valuations through the lens of approximation. We show that, in general, without the single-crossing condition, one cannot hope to approximate the optimal social welfare any better than the approximation given by assigning the item to a random bidder. Consequently, we introduce a relaxed version of single-crossing, {\sl $c$-single-crossing}, parameterized by $c\geq 1$, which means that the impact of $s_i$ on the valuation of agent $i$ is at least $1/c$ times the impact of $s_i$ on the valuation of any other agent ($c=1$ is single-crossing). Using this parameterized notion, we obtain a host of positive results. We propose a prior-free deterministic mechanism that gives an $(n-1)c$-approximation guarantee to welfare. We then show that a random version of the proposed mechanism gives a prior-free universally truthful $2c$-approximation to the optimal welfare for any concave $c$-single crossing setting (and a $2\sqrt{n}c^{3/2}$-approximation in the absence of concavity). We extend this mechanism to a universally truthful mechanism that gives $O(c^2)$-approximation to the optimal revenue.

Combinatorial Auctions with Interdependent Valuations: SOS to the Rescue

2023-05-15
Alon Eden , Michal Feldman , Amos Fiat +2 more
We study combinatorial auctions with interdependent valuations, where each agent i has a private signal s i that captures her private information and the valuation function of every agent depends … We study combinatorial auctions with interdependent valuations, where each agent i has a private signal s i that captures her private information and the valuation function of every agent depends on the entire signal profile, [Formula: see text]. The literature in economics shows that the interdependent model gives rise to strong impossibility results and identifies assumptions under which optimal solutions can be attained. The computer science literature provides approximation results for simple single-parameter settings (mostly single-item auctions or matroid feasibility constraints). Both bodies of literature focus largely on valuations satisfying a technical condition termed single crossing (or variants thereof). We consider the class of submodular over signals (SOS) valuations (without imposing any single crossing-type assumption) and provide the first welfare approximation guarantees for multidimensional combinatorial auctions achieved by universally ex post incentive-compatible, individually rational mechanisms. Our main results are (i) four approximation for any single-parameter downward-closed setting with single-dimensional signals and SOS valuations; (ii) four approximation for any combinatorial auction with multidimensional signals and separable-SOS valuations; and (iii) (k + 3) and (2 log(k) + 4) approximation for any combinatorial auction with single-dimensional signals, with k-sized signal space, for SOS and strong-SOS valuations, respectively. All of our results extend to a parameterized version of SOS, d-approximate SOS, while losing a factor that depends on d. Funding: A. Eden was partially supported by NSF Award IIS-2007887, the European Research Council (ERC) under the European Union's Seventh Framework Programme [FP7/2007-2013]/ERC Grant Agreement 337122, by the Israel Science Foundation [Grant 317/17], and by an Amazon research award. M. Feldman received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program [Grant Agreement 866132], by the Israel Science Foundation [Grant 317/17], by an Amazon research award, and by the NSF-BSF [Grant 2020788]. The work of K. Goldner was supported partially by NSF awards DMS-1903037 and CNS-2228610 and a Shibulal Family Career Development Professorship. A. R. Karlin was supported by the NSF-CCF [Grant 1813135].
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The Competition Complexity of Auctions: A Bulow-Klemperer Result for Multi-Dimensional Bidders

2016-12-28
Alon Eden , Michal Feldman , Ophir Friedler +2 more
A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to $n$ bidders whose values are drawn i.i.d. from … A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to $n$ bidders whose values are drawn i.i.d. from a regular distribution, the simple welfare-maximizing VCG mechanism (in this case, a second price-auction) with one additional bidder extracts at least as much revenue in expectation as the optimal mechanism. The beauty of this theorem stems from the fact that VCG is a {\em prior-independent} mechanism, where the seller possesses no information about the distribution, and yet, by recruiting one additional bidder it performs better than any prior-dependent mechanism tailored exactly to the distribution at hand (without the additional bidder). In this work, we establish the first {\em full Bulow-Klemperer} results in {\em multi-dimensional} environments, proving that by recruiting additional bidders, the revenue of the VCG mechanism surpasses that of the optimal (possibly randomized, Bayesian incentive compatible) mechanism. For a given environment with i.i.d. bidders, we term the number of additional bidders needed to achieve this guarantee the environment's {\em competition complexity}. Using the recent duality-based framework of Cai et al. [2016] for reasoning about optimal revenue, we show that the competition complexity of $n$ bidders with additive valuations over $m$ independent, regular items is at most $n+2m-2$ and at least $\log(m)$. We extend our results to bidders with additive valuations subject to downward-closed constraints, showing that these significantly more general valuations increase the competition complexity by at most an additive $m-1$ factor. We further improve this bound for the special case of matroid constraints, and provide additional extensions as well.

Private Interdependent Valuations: New Bounds for Single-Item Auctions and Matroids

2024-02-19
Alon Eden , Michal Feldman , Simon Mauras +1 more
We study auction design within the widely acclaimed model of interdependent values, introduced by Milgrom and Weber [1982]. In this model, every bidder $i$ has a private signal $s_i$ for … We study auction design within the widely acclaimed model of interdependent values, introduced by Milgrom and Weber [1982]. In this model, every bidder $i$ has a private signal $s_i$ for the item for sale, and a public valuation function $v_i(s_1,\ldots,s_n)$ which maps every vector of private signals (of all bidders) into a real value. A recent line of work established the existence of approximately-optimal mechanisms within this framework, even in the more challenging scenario where each bidder's valuation function $v_i$ is also private. This body of work has primarily focused on single-item auctions with two natural classes of valuations: those exhibiting submodularity over signals (SOS) and $d$-critical valuations. In this work we advance the state of the art on interdependent values with private valuation functions, with respect to both SOS and $d$-critical valuations. For SOS valuations, we devise a new mechanism that gives an improved approximation bound of $5$ for single-item auctions. This mechanism employs a novel variant of an "eating mechanism", leveraging LP-duality to achieve feasibility with reduced welfare loss. For $d$-critical valuations, we broaden the scope of existing results beyond single-item auctions, introducing a mechanism that gives a $(d+1)$-approximation for any environment with matroid feasibility constraints on the set of agents that can be simultaneously served. Notably, this approximation bound is tight, even with respect to single-item auctions.
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Truthful Secretaries with Budgets

2015-01-01
Alon Eden , Michal Feldman , Adi Vardi
We study online auction settings in which agents arrive and depart dynamically in a random (secretary) order, and each agent's private type consists of the agent's arrival and departure times, … We study online auction settings in which agents arrive and depart dynamically in a random (secretary) order, and each agent's private type consists of the agent's arrival and departure times, value and budget. We consider multi-unit auctions with additive agents for the allocation of both divisible and indivisible items. For both settings, we devise truthful mechanisms that give a constant approximation with respect to the auctioneer's revenue, under a large market assumption. For divisible items, we devise in addition a truthful mechanism that gives a constant approximation with respect to the liquid welfare --- a natural efficiency measure for budgeted settings introduced by Dobzinski and Paes Leme [ICALP'14]. Our techniques provide high-level principles for transforming offline truthful mechanisms into online ones, with or without budget constraints. To the best of our knowledge, this is the first work that addresses the non-trivial challenge of combining online settings with budgeted agents.

Pricing Social Goods

2017-06-30
Alon Eden , Tomer Ezra , Michal Feldman
Social goods are goods that grant value not only to their owners but also to the owners' surroundings, be it their families, friends or office mates. The benefit a non-owner … Social goods are goods that grant value not only to their owners but also to the owners' surroundings, be it their families, friends or office mates. The benefit a non-owner derives from the good is affected by many factors, including the type of the good, its availability, and the social status of the non-owner. Depending on the magnitude of the benefit and on the price of the good, a potential buyer might stay away from purchasing the good, hoping to free ride on others' purchases. A revenue-maximizing seller who sells social goods must take these considerations into account when setting prices for the good. The literature on optimal pricing has advanced considerably over the last decade, but little is known about optimal pricing schemes for selling social goods. In this paper, we conduct a systematic study of revenue-maximizing pricing schemes for social goods: we introduce a Bayesian model for this scenario, and devise nearly-optimal pricing schemes for various types of externalities, both for simultaneous sales and for sequential sales.

Max-Min Greedy Matching.

2018-03-14
Alon Eden , Uriel Feige , Michal Feldman
A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $\pi$ over $V$, the other player imposes a permutation $\sigma$ over $U$. In the … A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $\pi$ over $V$, the other player imposes a permutation $\sigma$ over $U$. In the greedy matching algorithm, vertices of $U$ arrive in order $\sigma$ and each vertex is matched to the lowest (under $\pi$) yet unmatched neighbor in $V$ (or left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals $\pi$, and the second (min) player responds with the worst possible $\sigma$ for $\pi$, does there exist a permutation $\pi$ ensuring to match strictly more than a half of the vertices? Can such a permutation be computed in polynomial time? The main result of this paper is an affirmative answer for this question: we show that there exists a polytime algorithm to compute $\pi$ for which for every $\sigma$ at least $\rho > 0.51$ fraction of the vertices of $V$ are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian. Interestingly, even for regular graphs with arbitrarily large degree (implying a large number of disjoint perfect matchings), there is no $\pi$ ensuring to match more than a fraction $8/9$ of the vertices. The max-min greedy matching problem solves an open problem regarding the welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations. In addition, it has implications for the size of the unique stable matching in markets with global preferences, subject to the graph structure.

Prompt Scheduling for Selfish Agents.

2018-04-09
Alon Eden , Michal Feldman , Amos Fiat +1 more
We give a prompt online mechanism for minimizing the sum of [weighted] completion times. This is the first prompt online algorithm for the problem. When such jobs are strategic agents, … We give a prompt online mechanism for minimizing the sum of [weighted] completion times. This is the first prompt online algorithm for the problem. When such jobs are strategic agents, delaying scheduling decisions makes little sense. Moreover, the mechanism has a particularly simple form of an anonymous menu of options.

Private Interdependent Valuations

2021-01-01
Alon Eden , Kira Goldner , Shuran Zheng
We consider the single-item interdependent value setting, where there is a monopolist, $n$ buyers, and each buyer has a private signal $s_i$ describing a piece of information about the item. … We consider the single-item interdependent value setting, where there is a monopolist, $n$ buyers, and each buyer has a private signal $s_i$ describing a piece of information about the item. Each bidder $i$ also has a valuation function $v_i(s_1,\ldots,s_n)$ mapping the (private) signals of all buyers to a positive real number representing their value for the item. This setting captures scenarios where the item's information is asymmetric or dispersed among agents, such as in competitions for oil drilling rights, or in auctions for art pieces. Due to the increased complexity of this model compared to standard private values, it is generally assumed that each bidder's valuation function $v_i$ is public knowledge. But in many situations, the seller may not know how a bidder aggregates signals into a valuation. In this paper, we design mechanisms that guarantee approximately-optimal social welfare while satisfying ex-post incentive compatibility and individual rationality for the case where the valuation functions are private to the bidders. When the valuations are public, it is possible for optimal social welfare to be attained by a deterministic mechanism under a single-crossing condition. In contrast, when the valuations are the bidders' private information, we show that no finite bound can be achieved by any deterministic mechanism even under single-crossing. Moreover, no randomized mechanism can guarantee better than an $n$-approximation. We thus consider valuation functions that are submodular over signals (SOS), introduced in the context of combinatorial auctions in a recent breakthrough paper by Eden et al. [EC'19]. Our main result is an $O(\log^2 n)$-approximation for buyers with private signals and valuations under the SOS condition. We also give a tight $\Theta(k)$-approximation for the case each agent's valuation depends on at most $k$ other signals even for unknown $k$.
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Platform Behavior under Market Shocks: A Simulation Framework and Reinforcement-Learning Based Study

2022-01-01
Xintong Wang , Gary Qiurui Ma , Alon Eden +4 more
We study the behavior of an economic platform (e.g., Amazon, Uber Eats, Instacart) under shocks, such as COVID-19 lockdowns, and the effect of different regulation considerations imposed on a platform. … We study the behavior of an economic platform (e.g., Amazon, Uber Eats, Instacart) under shocks, such as COVID-19 lockdowns, and the effect of different regulation considerations imposed on a platform. To this end, we develop a multi-agent Gym environment of a platform economy in a dynamic, multi-period setting, with the possible occurrence of economic shocks. Buyers and sellers are modeled as economically-motivated agents, choosing whether or not to pay corresponding fees to use the platform. We formulate the platform's problem as a partially observable Markov decision process, and use deep reinforcement learning to model its fee setting and matching behavior. We consider two major types of regulation frameworks: (1) taxation policies and (2) platform fee restrictions, and offer extensive simulated experiments to characterize regulatory tradeoffs under optimal platform responses. Our results show that while many interventions are ineffective with a sophisticated platform actor, we identify a particular kind of regulation -- fixing fees to optimal, pre-shock fees while still allowing a platform to choose how to match buyer demands to sellers -- as promoting the efficiency, seller diversity, and resilience of the overall economic system.

Max-Min Greedy Matching

2018-01-01
Alon Eden , Uriel Feige , Michal Feldman
A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $\pi$ over $V$, the other player imposes a permutation $\sigma$ over $U$. In the … A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $\pi$ over $V$, the other player imposes a permutation $\sigma$ over $U$. In the greedy matching algorithm, vertices of $U$ arrive in order $\sigma$ and each vertex is matched to the lowest (under $\pi$) yet unmatched neighbor in $V$ (or left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals $\pi$, and the second (min) player responds with the worst possible $\sigma$ for $\pi$, does there exist a permutation $\pi$ ensuring to match strictly more than a half of the vertices? Can such a permutation be computed in polynomial time? The main result of this paper is an affirmative answer for this question: we show that there exists a polytime algorithm to compute $\pi$ for which for every $\sigma$ at least $\rho > 0.51$ fraction of the vertices of $V$ are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian. Interestingly, even for regular graphs with arbitrarily large degree (implying a large number of disjoint perfect matchings), there is no $\pi$ ensuring to match more than a fraction $8/9$ of the vertices. The max-min greedy matching problem solves an open problem regarding the welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations. In addition, it has implications for the size of the unique stable matching in markets with global preferences, subject to the graph structure.

Reinforcement Learning of Sequential Price Mechanisms

2020-01-01
Gianluca Brero , Alon Eden , Matthias Gerstgrasser +2 more
We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of sequential price mechanisms, which generalizes both serial dictatorship and posted price mechanisms and essentially … We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of sequential price mechanisms, which generalizes both serial dictatorship and posted price mechanisms and essentially characterizes all strongly obviously strategyproof mechanisms. Learning an optimal mechanism within this class forms a partially-observable Markov decision process. We provide rigorous conditions for when this class of mechanisms is more powerful than simpler static mechanisms, for sufficiency or insufficiency of observation statistics for learning, and for the necessity of complex (deep) policies. We show that our approach can learn optimal or near-optimal mechanisms in several experimental settings.
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The Invisible Hand of Dynamic Market Pricing

2015-01-01
Vincent Cohen-Addad , Alon Eden , Michal Feldman +1 more
Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, … Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.
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A Simple and Approximately Optimal Mechanism for a Buyer with Complements

2016-01-01
Alon Eden , Michal Feldman , Ophir Friedler +2 more
We consider a revenue-maximizing seller with $m$ heterogeneous items and a single buyer whose valuation $v$ for the items may exhibit both substitutes (i.e., for some $S, T$, $v(S \cup … We consider a revenue-maximizing seller with $m$ heterogeneous items and a single buyer whose valuation $v$ for the items may exhibit both substitutes (i.e., for some $S, T$, $v(S \cup T) < v(S) + v(T)$) and complements (i.e., for some $S, T$, $v(S \cup T) > v(S) + v(T)$). We show that the mechanism first proposed by Babaioff et al. [2014] - the better of selling the items separately and bundling them together - guarantees a $\Theta(d)$ fraction of the optimal revenue, where $d$ is a measure on the degree of complementarity. Note that this is the first approximately optimal mechanism for a buyer whose valuation exhibits any kind of complementarity, and extends the work of Rubinstein and Weinberg [2015], which proved that the same simple mechanisms achieve a constant factor approximation when buyer valuations are subadditive, the most general class of complement-free valuations. Our proof is enabled by the recent duality framework developed in Cai et al. [2016], which we use to obtain a bound on the optimal revenue in this setting. Our main technical contributions are specialized to handle the intricacies of settings with complements, and include an algorithm for partitioning edges in a hypergraph. Even nailing down the right model and notion of "degree of complementarity" to obtain meaningful results is of interest, as the natural extensions of previous definitions provably fail.
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Prompt Scheduling for Selfish Agents

2018-01-01
Alon Eden , Michal Feldman , Amos Fiat +1 more
We give a prompt online mechanism for minimizing the sum of [weighted] completion times. This is the first prompt online algorithm for the problem. When such jobs are strategic agents, … We give a prompt online mechanism for minimizing the sum of [weighted] completion times. This is the first prompt online algorithm for the problem. When such jobs are strategic agents, delaying scheduling decisions makes little sense. Moreover, the mechanism has a particularly simple form of an anonymous menu of options.
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Pricing Online Decisions: Beyond Auctions

2015-01-01
Ilan Reuven Cohen , Alon Eden , Amos Fiat +1 more
We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare … We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare or revenue in an auction settings. This paper deals with quite general settings and minimizing social costs. We show that appropriately computed posted prices allow one to achieve essentially the same performance as the best online algorithm. This holds in a wide variety of settings. Unlike online algorithms that learn about the event, and then make enforceable decisions, prices are posted without knowing the future events or even the current event, and are thus inherently dominant strategy incentive compatible. In particular we show that one can give efficient posted price mechanisms for metrical task systems, some instances of the $k$-server problem, and metrical matching problems. We give both deterministic and randomized algorithms. Such posted price mechanisms decrease the social cost dramatically over selfish behavior where no decision incurs a charge. One alluring application of this is reducing the social cost of free parking exponentially.
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Stackelberg POMDP: A Reinforcement Learning Approach for Economic Design

2022-01-01
Gianluca Brero , Alon Eden , Darshan Chakrabarti +3 more
We introduce a reinforcement learning framework for economic design where the interaction between the environment designer and the participants is modeled as a Stackelberg game. In this game, the designer … We introduce a reinforcement learning framework for economic design where the interaction between the environment designer and the participants is modeled as a Stackelberg game. In this game, the designer (leader) sets up the rules of the economic system, while the participants (followers) respond strategically. We integrate algorithms for determining followers' response strategies into the leader's learning environment, providing a formulation of the leader's learning problem as a POMDP that we call the Stackelberg POMDP. We prove that the optimal leader's strategy in the Stackelberg game is the optimal policy in our Stackelberg POMDP under a limited set of possible policies, establishing a connection between solving POMDPs and Stackelberg games. We solve our POMDP under a limited set of policy options via the centralized training with decentralized execution framework. For the specific case of followers that are modeled as no-regret learners, we solve an array of increasingly complex settings, including problems of indirect mechanism design where there is turn-taking and limited communication by agents. We demonstrate the effectiveness of our training framework through ablation studies. We also give convergence results for no-regret learners to a Bayesian version of a coarse-correlated equilibrium, extending known results to the case of correlated types.
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Platform Equilibrium: Analayzing Social Welfare in Online Market Places

2023-01-01
Alon Eden , Gary Qiurui Ma , David C. Parkes
We introduce the theoretical study of a Platform Equilibrium. There are unit-demand buyers and unit-supply sellers. Each seller chooses to join or remain off a trading platform, and each buyer … We introduce the theoretical study of a Platform Equilibrium. There are unit-demand buyers and unit-supply sellers. Each seller chooses to join or remain off a trading platform, and each buyer can transact with any on-platform seller but only a subset of different off-platform sellers. Given the choices of sellers, prices form a competitive equilibrium, and clear the market considering constraints on trade. Further, the platform charges a transaction fee to all on-platform sellers, in the form of a fraction of on-platform sellers' price. The platform chooses the fraction $\alpha \in [0, 1]$ to maximize revenue. A Platform Equilibrium is a Nash equilibrium of the game where each seller decides whether or not to join the platform, balancing the effect of a possibly larger trading network against the imposition of a transaction fee. Our main insights are:(i) In homogeneous (identical) good markets, pure equilibria always exist and can be found by a polynomial time algorithm; (ii) When the platform is unregulated, the resulting Platform Equilibrium guarantees a tight $\Theta(log(min(m, n)))$-approximation of the ideal welfare in homogeneous good markets; (iii) Even light regulation helps. When the platform's fee is capped at $\alpha$, the price of anarchy is 2-$\alpha$/1-$\alpha$ for general unit demand valuations. For example, if the platform takes 30 percent of the seller's revenue, a rather high fee, our analysis implies the welfare in a Platform Equilibrium is still a 0.412-fraction of the ideal welfare. Some of our analysis extends beyond unit-demand buyers as well as to markets where sellers have production costs.
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Constant Approximation for Private Interdependent Valuations

2023-01-01
Alon Eden , Michal Feldman , Kira Goldner +2 more
The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $s_1, \ldots, s_n$ of the $n$ bidders … The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $s_1, \ldots, s_n$ of the $n$ bidders and public valuation functions $v_i(s_1, \ldots, s_n)$. Recent work in TCS has shown that this setting admits a constant approximation to the optimal social welfare if the valuations satisfy a natural property called submodularity over signals (SOS). More recently, Eden et al. (2022) have extended the analysis of interdependent valuations to include settings with private signals and private valuations, and established $O(\log^2 n)$-approximation for SOS valuations. In this paper we show that this setting admits a constant factor approximation, settling the open question raised by Eden et al. (2022).
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Pricing Social Goods

2017-01-01
Alon Eden , Tomer Ezra , Michal Feldman
Social goods are goods that grant value not only to their owners but also to the owners' surroundings, be it their families, friends or office mates. The benefit a non-owner … Social goods are goods that grant value not only to their owners but also to the owners' surroundings, be it their families, friends or office mates. The benefit a non-owner derives from the good is affected by many factors, including the type of the good, its availability, and the social status of the non-owner. Depending on the magnitude of the benefit and on the price of the good, a potential buyer might stay away from purchasing the good, hoping to free ride on others' purchases. A revenue-maximizing seller who sells social goods must take these considerations into account when setting prices for the good. The literature on optimal pricing has advanced considerably over the last decade, but little is known about optimal pricing schemes for selling social goods. In this paper, we conduct a systematic study of revenue-maximizing pricing schemes for social goods: we introduce a Bayesian model for this scenario, and devise nearly-optimal pricing schemes for various types of externalities, both for simultaneous sales and for sequential sales.

Plant-and-Steal: Truthful Fair Allocations via Predictions

2024-06-11
Ilan Reuven Cohen , Alon Eden , Talya Eden +1 more
We study truthful mechanisms for approximating the Maximin-Share (MMS) allocation of agents with additive valuations for indivisible goods. Algorithmically, constant factor approximations exist for the problem for any number of … We study truthful mechanisms for approximating the Maximin-Share (MMS) allocation of agents with additive valuations for indivisible goods. Algorithmically, constant factor approximations exist for the problem for any number of agents. When adding incentives to the mix, a jarring result by Amanatidis, Birmpas, Christodoulou, and Markakis [EC 2017] shows that the best possible approximation for two agents and $m$ items is $\lfloor \frac{m}{2} \rfloor$. We adopt a learning-augmented framework to investigate what is possible when some prediction on the input is given. For two agents, we give a truthful mechanism that takes agents' ordering over items as prediction. When the prediction is accurate, we give a $2$-approximation to the MMS (consistency), and when the prediction is off, we still get an $\lceil \frac{m}{2} \rceil$-approximation to the MMS (robustness). We further show that the mechanism's performance degrades gracefully in the number of ``mistakes" in the prediction; i.e., we interpolate (up to constant factors) between the two extremes: when there are no mistakes, and when there is a maximum number of mistakes. We also show an impossibility result on the obtainable consistency for mechanisms with finite robustness. For the general case of $n\ge 2$ agents, we give a 2-approximation mechanism for accurate predictions, with relaxed fallback guarantees. Finally, we give experimental results which illustrate when different components of our framework, made to insure consistency and robustness, come into play.
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Private Interdependent Valuations: New Bounds for Single-Item Auctions and Matroids

2024-07-08
Alon Eden , Michal Feldman , Simon Mauras +1 more
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Stable Marriage: Loyalty vs. Competition

2025-01-30
Amit Ronen , James D. Hess , Yael Belfer +2 more
We consider the stable matching problem (e.g. between doctors and hospitals) in a one-to-one matching setting, where preferences are drawn uniformly at random. It is known that when doctors propose … We consider the stable matching problem (e.g. between doctors and hospitals) in a one-to-one matching setting, where preferences are drawn uniformly at random. It is known that when doctors propose and the number of doctors equals the number of hospitals, then the expected rank of doctors for their match is $\Theta(\log n)$, while the expected rank of the hospitals for their match is $\Theta(n/\log n)$, where $n$ is the size of each side of the market. However, when adding even a single doctor, [Ashlagi, Kanoria and Leshno, 2017] show that the tables have turned: doctors have expected rank of $\Theta(n/\log n)$ while hospitals have expected rank of $\Theta(\log n)$. That is, (slight) competition has a much more dramatically harmful effect than the benefit of being on the proposing side. Motivated by settings where agents inflate their value for an item if it is already allocated to them (termed endowment effect), we study the case where hospitals exhibit ``loyalty". We model loyalty as a parameter $k$, where a hospital currently matched to their $\ell$th most preferred doctor accepts proposals from their $\ell-k-1$th most preferred doctors. Hospital loyalty should help doctors mitigate the harmful effect of competition, as many more outcomes are now stable. However, we show that the effect of competition is so dramatic that, even in settings with extremely high loyalty, in unbalanced markets, the expected rank of doctors already becomes $\tilde{\Theta}(\sqrt{n})$ for loyalty $k=n-\sqrt{n}\log n=n(1-o(1))$.
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Fair Division via Resource Augmentation

2025-02-13
Hannaneh Akrami , Alon Eden , Michal Feldman +2 more
We introduce and formalize the notion of resource augmentation for maximin share allocations -- an idea that can be traced back to the seminal work of Budish [JPE 2011]. Specifically, … We introduce and formalize the notion of resource augmentation for maximin share allocations -- an idea that can be traced back to the seminal work of Budish [JPE 2011]. Specifically, given a fair division instance with $m$ goods and $n$ agents, we ask how many copies of the goods should be added in order to guarantee that each agent receives at least their original maximin share, or an approximation thereof. We establish a tight bound of $m/e$ copies for arbitrary monotone valuations. For additive valuations, we show that at most $\min\{n-2,\lfloor \frac{m}{3}\rfloor (1+o(1))\}$ copies suffice. For approximate-MMS in ordered instances, we give a tradeoff between the number of copies needed and the approximation guarantee. In particular, we prove that $\lfloor n/2 \rfloor$ copies suffice to guarantee a $6/7$-approximation to the original MMS, and $\lfloor n/3 \rfloor$ copies suffice for a $4/5$-approximation. Both results improve upon the best known approximation guarantees for additive valuations in the absence of copies.
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Fair Division via Resource Augmentation

2025-02-13
Hannaneh Akrami , Alon Eden , Michal Feldman +2 more
We introduce and formalize the notion of resource augmentation for maximin share allocations -- an idea that can be traced back to the seminal work of Budish [JPE 2011]. Specifically, … We introduce and formalize the notion of resource augmentation for maximin share allocations -- an idea that can be traced back to the seminal work of Budish [JPE 2011]. Specifically, given a fair division instance with $m$ goods and $n$ agents, we ask how many copies of the goods should be added in order to guarantee that each agent receives at least their original maximin share, or an approximation thereof. We establish a tight bound of $m/e$ copies for arbitrary monotone valuations. For additive valuations, we show that at most $\min\{n-2,\lfloor \frac{m}{3}\rfloor (1+o(1))\}$ copies suffice. For approximate-MMS in ordered instances, we give a tradeoff between the number of copies needed and the approximation guarantee. In particular, we prove that $\lfloor n/2 \rfloor$ copies suffice to guarantee a $6/7$-approximation to the original MMS, and $\lfloor n/3 \rfloor$ copies suffice for a $4/5$-approximation. Both results improve upon the best known approximation guarantees for additive valuations in the absence of copies.
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Stable Marriage: Loyalty vs. Competition

2025-01-30
Amit Ronen , James D. Hess , Yael Belfer +2 more
We consider the stable matching problem (e.g. between doctors and hospitals) in a one-to-one matching setting, where preferences are drawn uniformly at random. It is known that when doctors propose … We consider the stable matching problem (e.g. between doctors and hospitals) in a one-to-one matching setting, where preferences are drawn uniformly at random. It is known that when doctors propose and the number of doctors equals the number of hospitals, then the expected rank of doctors for their match is $\Theta(\log n)$, while the expected rank of the hospitals for their match is $\Theta(n/\log n)$, where $n$ is the size of each side of the market. However, when adding even a single doctor, [Ashlagi, Kanoria and Leshno, 2017] show that the tables have turned: doctors have expected rank of $\Theta(n/\log n)$ while hospitals have expected rank of $\Theta(\log n)$. That is, (slight) competition has a much more dramatically harmful effect than the benefit of being on the proposing side. Motivated by settings where agents inflate their value for an item if it is already allocated to them (termed endowment effect), we study the case where hospitals exhibit ``loyalty". We model loyalty as a parameter $k$, where a hospital currently matched to their $\ell$th most preferred doctor accepts proposals from their $\ell-k-1$th most preferred doctors. Hospital loyalty should help doctors mitigate the harmful effect of competition, as many more outcomes are now stable. However, we show that the effect of competition is so dramatic that, even in settings with extremely high loyalty, in unbalanced markets, the expected rank of doctors already becomes $\tilde{\Theta}(\sqrt{n})$ for loyalty $k=n-\sqrt{n}\log n=n(1-o(1))$.
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Private Interdependent Valuations: New Bounds for Single-Item Auctions and Matroids

2024-07-08
Alon Eden , Michal Feldman , Simon Mauras +1 more
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Plant-and-Steal: Truthful Fair Allocations via Predictions

2024-06-11
Ilan Reuven Cohen , Alon Eden , Talya Eden +1 more
We study truthful mechanisms for approximating the Maximin-Share (MMS) allocation of agents with additive valuations for indivisible goods. Algorithmically, constant factor approximations exist for the problem for any number of … We study truthful mechanisms for approximating the Maximin-Share (MMS) allocation of agents with additive valuations for indivisible goods. Algorithmically, constant factor approximations exist for the problem for any number of agents. When adding incentives to the mix, a jarring result by Amanatidis, Birmpas, Christodoulou, and Markakis [EC 2017] shows that the best possible approximation for two agents and $m$ items is $\lfloor \frac{m}{2} \rfloor$. We adopt a learning-augmented framework to investigate what is possible when some prediction on the input is given. For two agents, we give a truthful mechanism that takes agents' ordering over items as prediction. When the prediction is accurate, we give a $2$-approximation to the MMS (consistency), and when the prediction is off, we still get an $\lceil \frac{m}{2} \rceil$-approximation to the MMS (robustness). We further show that the mechanism's performance degrades gracefully in the number of ``mistakes" in the prediction; i.e., we interpolate (up to constant factors) between the two extremes: when there are no mistakes, and when there is a maximum number of mistakes. We also show an impossibility result on the obtainable consistency for mechanisms with finite robustness. For the general case of $n\ge 2$ agents, we give a 2-approximation mechanism for accurate predictions, with relaxed fallback guarantees. Finally, we give experimental results which illustrate when different components of our framework, made to insure consistency and robustness, come into play.
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Private Interdependent Valuations: New Bounds for Single-Item Auctions and Matroids

2024-02-19
Alon Eden , Michal Feldman , Simon Mauras +1 more
We study auction design within the widely acclaimed model of interdependent values, introduced by Milgrom and Weber [1982]. In this model, every bidder $i$ has a private signal $s_i$ for … We study auction design within the widely acclaimed model of interdependent values, introduced by Milgrom and Weber [1982]. In this model, every bidder $i$ has a private signal $s_i$ for the item for sale, and a public valuation function $v_i(s_1,\ldots,s_n)$ which maps every vector of private signals (of all bidders) into a real value. A recent line of work established the existence of approximately-optimal mechanisms within this framework, even in the more challenging scenario where each bidder's valuation function $v_i$ is also private. This body of work has primarily focused on single-item auctions with two natural classes of valuations: those exhibiting submodularity over signals (SOS) and $d$-critical valuations. In this work we advance the state of the art on interdependent values with private valuation functions, with respect to both SOS and $d$-critical valuations. For SOS valuations, we devise a new mechanism that gives an improved approximation bound of $5$ for single-item auctions. This mechanism employs a novel variant of an "eating mechanism", leveraging LP-duality to achieve feasibility with reduced welfare loss. For $d$-critical valuations, we broaden the scope of existing results beyond single-item auctions, introducing a mechanism that gives a $(d+1)$-approximation for any environment with matroid feasibility constraints on the set of agents that can be simultaneously served. Notably, this approximation bound is tight, even with respect to single-item auctions.
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Constant Approximation for Private Interdependent Valuations

2023-11-06
Alon Eden , Michal Feldman , Kira Goldner +2 more
The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $s_{1}, \ldots, s_{n}$ of the n bidders … The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $s_{1}, \ldots, s_{n}$ of the n bidders and public valuation functions $v_{i}\left(s_{1}, \ldots, s_{n}\right)$. Recent work in TCS has shown that this setting admits a constant approximation to the optimal social welfare if the valuations satisfy a natural property called submodularity over signals (SOS). More recently, Eden et al. (2022) have extended the analysis of interdependent valuations to include settings with private signals and private valuations, and established $O\left(\log ^{2} n\right)$-approximation for SOS valuations. In this paper we show that this setting admits a constant factor approximation, settling the open question raised by Eden et al. (2022).
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Combinatorial Auctions with Interdependent Valuations: SOS to the Rescue

2023-05-15
Alon Eden , Michal Feldman , Amos Fiat +2 more
We study combinatorial auctions with interdependent valuations, where each agent i has a private signal s i that captures her private information and the valuation function of every agent depends … We study combinatorial auctions with interdependent valuations, where each agent i has a private signal s i that captures her private information and the valuation function of every agent depends on the entire signal profile, [Formula: see text]. The literature in economics shows that the interdependent model gives rise to strong impossibility results and identifies assumptions under which optimal solutions can be attained. The computer science literature provides approximation results for simple single-parameter settings (mostly single-item auctions or matroid feasibility constraints). Both bodies of literature focus largely on valuations satisfying a technical condition termed single crossing (or variants thereof). We consider the class of submodular over signals (SOS) valuations (without imposing any single crossing-type assumption) and provide the first welfare approximation guarantees for multidimensional combinatorial auctions achieved by universally ex post incentive-compatible, individually rational mechanisms. Our main results are (i) four approximation for any single-parameter downward-closed setting with single-dimensional signals and SOS valuations; (ii) four approximation for any combinatorial auction with multidimensional signals and separable-SOS valuations; and (iii) (k + 3) and (2 log(k) + 4) approximation for any combinatorial auction with single-dimensional signals, with k-sized signal space, for SOS and strong-SOS valuations, respectively. All of our results extend to a parameterized version of SOS, d-approximate SOS, while losing a factor that depends on d. Funding: A. Eden was partially supported by NSF Award IIS-2007887, the European Research Council (ERC) under the European Union's Seventh Framework Programme [FP7/2007-2013]/ERC Grant Agreement 337122, by the Israel Science Foundation [Grant 317/17], and by an Amazon research award. M. Feldman received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program [Grant Agreement 866132], by the Israel Science Foundation [Grant 317/17], by an Amazon research award, and by the NSF-BSF [Grant 2020788]. The work of K. Goldner was supported partially by NSF awards DMS-1903037 and CNS-2228610 and a Shibulal Family Career Development Professorship. A. R. Karlin was supported by the NSF-CCF [Grant 1813135].
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Platform Behavior under Market Shocks: A Simulation Framework and Reinforcement-Learning Based Study

2023-04-26
Xintong Wang , Gary Qiurui , Alon Eden +4 more
We study the behavior of an economic platform (e.g., Amazon, Uber Eats, Instacart) under shocks, such as COVID-19 lockdowns, and the effect of different regulation considerations. To this end, we … We study the behavior of an economic platform (e.g., Amazon, Uber Eats, Instacart) under shocks, such as COVID-19 lockdowns, and the effect of different regulation considerations. To this end, we develop a multi-agent simulation environment of a platform economy in a multi-period setting where shocks may occur and disrupt the economy. Buyers and sellers are heterogeneous and modeled as economically-motivated agents, choosing whether or not to pay fees to access the platform. We use deep reinforcement learning to model the fee-setting and matching behavior of the platform, and consider two major types of regulation frameworks: (1) taxation policies and (2) platform fee restrictions. We offer a number of simulated experiments that cover different market settings and shed light on regulatory tradeoffs. Our results show that while many interventions are ineffective with a sophisticated platform actor, we identify a particular kind of regulation—fixing fees to the optimal, no-shock fees while still allowing a platform to choose how to match buyers and sellers—as holding promise for promoting the efficiency and resilience of the economic system.
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Platform Equilibrium: Analayzing Social Welfare in Online Market Places

2023-01-01
Alon Eden , Gary Qiurui Ma , David C. Parkes
We introduce the theoretical study of a Platform Equilibrium. There are unit-demand buyers and unit-supply sellers. Each seller chooses to join or remain off a trading platform, and each buyer … We introduce the theoretical study of a Platform Equilibrium. There are unit-demand buyers and unit-supply sellers. Each seller chooses to join or remain off a trading platform, and each buyer can transact with any on-platform seller but only a subset of different off-platform sellers. Given the choices of sellers, prices form a competitive equilibrium, and clear the market considering constraints on trade. Further, the platform charges a transaction fee to all on-platform sellers, in the form of a fraction of on-platform sellers' price. The platform chooses the fraction $\alpha \in [0, 1]$ to maximize revenue. A Platform Equilibrium is a Nash equilibrium of the game where each seller decides whether or not to join the platform, balancing the effect of a possibly larger trading network against the imposition of a transaction fee. Our main insights are:(i) In homogeneous (identical) good markets, pure equilibria always exist and can be found by a polynomial time algorithm; (ii) When the platform is unregulated, the resulting Platform Equilibrium guarantees a tight $\Theta(log(min(m, n)))$-approximation of the ideal welfare in homogeneous good markets; (iii) Even light regulation helps. When the platform's fee is capped at $\alpha$, the price of anarchy is 2-$\alpha$/1-$\alpha$ for general unit demand valuations. For example, if the platform takes 30 percent of the seller's revenue, a rather high fee, our analysis implies the welfare in a Platform Equilibrium is still a 0.412-fraction of the ideal welfare. Some of our analysis extends beyond unit-demand buyers as well as to markets where sellers have production costs.
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Constant Approximation for Private Interdependent Valuations

2023-01-01
Alon Eden , Michal Feldman , Kira Goldner +2 more
The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $s_1, \ldots, s_n$ of the $n$ bidders … The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $s_1, \ldots, s_n$ of the $n$ bidders and public valuation functions $v_i(s_1, \ldots, s_n)$. Recent work in TCS has shown that this setting admits a constant approximation to the optimal social welfare if the valuations satisfy a natural property called submodularity over signals (SOS). More recently, Eden et al. (2022) have extended the analysis of interdependent valuations to include settings with private signals and private valuations, and established $O(\log^2 n)$-approximation for SOS valuations. In this paper we show that this setting admits a constant factor approximation, settling the open question raised by Eden et al. (2022).
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Private Interdependent Valuations

2022-01-01
Alon Eden , Kira Goldner , Shuran Zheng
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Private Interdependent ValuationsAlon Eden, Kira Goldner, and Shuran ZhengAlon Eden, Kira Goldner, and Shuran … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Private Interdependent ValuationsAlon Eden, Kira Goldner, and Shuran ZhengAlon Eden, Kira Goldner, and Shuran Zhengpp.2920 - 2939Chapter DOI:https://doi.org/10.1137/1.9781611977073.113PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider the single-item interdependent value setting, where there is a single item sold by a monopolist, n buyers, and each buyer has a private signal si describing a piece of information about the item. Additionally, each bidder i has a valuation function vi(s1, …, sn) mapping the (private) signals of all buyers into a positive real number representing their value for the item. This setting captures scenarios where the item's information is asymmetric or dispersed among agents, such as in competitions for oil drilling rights, or in auctions for art pieces. Due to the increased complexity of this model compared to the standard private values model, it is generally assumed that each bidder's valuation function vi is public knowledge to the seller or all other buyers. But in many situations, the seller may not know the bidders' valuation functions—how a bidder aggregates signals into a valuation is often their private information. In this paper, we design mechanisms that guarantee approximately-optimal social welfare while satisfying ex-post incentive compatibility and individually rationality for the case where the valuation functions are private to the bidders, and thus may be strategically misreported to the seller. When the valuations are public, it is possible for optimal social welfare to be attained by a deterministic mechanism when the valuations satisfy a single-crossing condition. In contrast, when the valuations are the bidders' private information, we show that no finite bound on the social welfare can be achieved by any deterministic mechanism even under single-crossing. Moreover, no randomized mechanism can guarantee better than n-approximation. We thus consider valuation functions that are submodular over signals (SOS), introduced in the context of combinatorial auctions in a recent breakthrough paper by Eden et al. [EC'19]. Our main result is an O(log2 n)-approximation randomized mechanism for buyers with private signals and valuations under the SOS condition. We also give a tight Θ(k)-approximation mechanism for the case each agent's valuation depends on at most k other signals even for unknown k. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771
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Platform Behavior under Market Shocks: A Simulation Framework and Reinforcement-Learning Based Study

2022-01-01
Xintong Wang , Gary Qiurui Ma , Alon Eden +4 more
We study the behavior of an economic platform (e.g., Amazon, Uber Eats, Instacart) under shocks, such as COVID-19 lockdowns, and the effect of different regulation considerations imposed on a platform. … We study the behavior of an economic platform (e.g., Amazon, Uber Eats, Instacart) under shocks, such as COVID-19 lockdowns, and the effect of different regulation considerations imposed on a platform. To this end, we develop a multi-agent Gym environment of a platform economy in a dynamic, multi-period setting, with the possible occurrence of economic shocks. Buyers and sellers are modeled as economically-motivated agents, choosing whether or not to pay corresponding fees to use the platform. We formulate the platform's problem as a partially observable Markov decision process, and use deep reinforcement learning to model its fee setting and matching behavior. We consider two major types of regulation frameworks: (1) taxation policies and (2) platform fee restrictions, and offer extensive simulated experiments to characterize regulatory tradeoffs under optimal platform responses. Our results show that while many interventions are ineffective with a sophisticated platform actor, we identify a particular kind of regulation -- fixing fees to optimal, pre-shock fees while still allowing a platform to choose how to match buyer demands to sellers -- as promoting the efficiency, seller diversity, and resilience of the overall economic system.

Stackelberg POMDP: A Reinforcement Learning Approach for Economic Design

2022-01-01
Gianluca Brero , Alon Eden , Darshan Chakrabarti +3 more
We introduce a reinforcement learning framework for economic design where the interaction between the environment designer and the participants is modeled as a Stackelberg game. In this game, the designer … We introduce a reinforcement learning framework for economic design where the interaction between the environment designer and the participants is modeled as a Stackelberg game. In this game, the designer (leader) sets up the rules of the economic system, while the participants (followers) respond strategically. We integrate algorithms for determining followers' response strategies into the leader's learning environment, providing a formulation of the leader's learning problem as a POMDP that we call the Stackelberg POMDP. We prove that the optimal leader's strategy in the Stackelberg game is the optimal policy in our Stackelberg POMDP under a limited set of possible policies, establishing a connection between solving POMDPs and Stackelberg games. We solve our POMDP under a limited set of policy options via the centralized training with decentralized execution framework. For the specific case of followers that are modeled as no-regret learners, we solve an array of increasingly complex settings, including problems of indirect mechanism design where there is turn-taking and limited communication by agents. We demonstrate the effectiveness of our training framework through ablation studies. We also give convergence results for no-regret learners to a Bayesian version of a coarse-correlated equilibrium, extending known results to the case of correlated types.
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Reinforcement Learning of Sequential Price Mechanisms

2021-05-18
Gianluca Brero , Alon Eden , Matthias Gerstgrasser +2 more
We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of sequential price mechanisms, which generalizes both serial dictatorship and posted price mechanisms and essentially … We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of sequential price mechanisms, which generalizes both serial dictatorship and posted price mechanisms and essentially characterizes all strongly obviously strategyproof mechanisms. Learning an optimal mechanism within this class forms a partially-observable Markov decision process. We provide rigorous conditions for when this class of mechanisms is more powerful than simpler static mechanisms, for sufficiency or insufficiency of observation statistics for learning, and for the necessity of complex (deep) policies. We show that our approach can learn optimal or near-optimal mechanisms in several experimental settings.
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Cursed yet Satisfied Agents

2021-01-01
Yiling Chen , Alon Eden , Juntao Wang
In real life auctions, a widely observed phenomenon is the winner's curse -- the winner's high bid implies that the winner often over-estimates the value of the good for sale, … In real life auctions, a widely observed phenomenon is the winner's curse -- the winner's high bid implies that the winner often over-estimates the value of the good for sale, resulting in an incurred negative utility. The seminal work of Eyster and Rabin [Econometrica'05] introduced a behavioral model aimed to explain this observed anomaly. We term agents who display this bias "cursed agents". We adopt their model in the interdependent value setting, and aim to devise mechanisms that prevent the cursed agents from obtaining negative utility. We design mechanisms that are cursed ex-post IC, that is, incentivize agents to bid their true signal even though they are cursed, while ensuring that the outcome is individually rational -- the price the agents pay is no more than the agents' true value. Since the agents might over-estimate the good's value, such mechanisms might require the seller to make positive transfers to the agents to prevent agents from over-paying. For revenue maximization, we give the optimal deterministic and anonymous mechanism. For welfare maximization, we require ex-post budget balance (EPBB), as positive transfers might lead to negative revenue. We propose a masking operation that takes any deterministic mechanism, and imposes that the seller would not make positive transfers, enforcing EPBB. We show that in typical settings, EPBB implies that the mechanism cannot make any positive transfers, implying that applying the masking operation on the fully efficient mechanism results in a socially optimal EPBB mechanism. This further implies that if the valuation function is the maximum of agents' signals, the optimal EPBB mechanism obtains zero welfare. In contrast, we show that for sum-concave valuations, which include weighted-sum valuations and l_p-norms, the welfare optimal EPBB mechanism obtains half of the optimal welfare as the number of agents grows large.
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Private Interdependent Valuations

2021-01-01
Alon Eden , Kira Goldner , Shuran Zheng
We consider the single-item interdependent value setting, where there is a monopolist, $n$ buyers, and each buyer has a private signal $s_i$ describing a piece of information about the item. … We consider the single-item interdependent value setting, where there is a monopolist, $n$ buyers, and each buyer has a private signal $s_i$ describing a piece of information about the item. Each bidder $i$ also has a valuation function $v_i(s_1,\ldots,s_n)$ mapping the (private) signals of all buyers to a positive real number representing their value for the item. This setting captures scenarios where the item's information is asymmetric or dispersed among agents, such as in competitions for oil drilling rights, or in auctions for art pieces. Due to the increased complexity of this model compared to standard private values, it is generally assumed that each bidder's valuation function $v_i$ is public knowledge. But in many situations, the seller may not know how a bidder aggregates signals into a valuation. In this paper, we design mechanisms that guarantee approximately-optimal social welfare while satisfying ex-post incentive compatibility and individual rationality for the case where the valuation functions are private to the bidders. When the valuations are public, it is possible for optimal social welfare to be attained by a deterministic mechanism under a single-crossing condition. In contrast, when the valuations are the bidders' private information, we show that no finite bound can be achieved by any deterministic mechanism even under single-crossing. Moreover, no randomized mechanism can guarantee better than an $n$-approximation. We thus consider valuation functions that are submodular over signals (SOS), introduced in the context of combinatorial auctions in a recent breakthrough paper by Eden et al. [EC'19]. Our main result is an $O(\log^2 n)$-approximation for buyers with private signals and valuations under the SOS condition. We also give a tight $\Theta(k)$-approximation for the case each agent's valuation depends on at most $k$ other signals even for unknown $k$.
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A Simple and Approximately Optimal Mechanism for a Buyer with Complements

2020-12-15
Alon Eden , Michal Feldman , Ophir Friedler +2 more
Recent literature on approximately optimal revenue maximization has shown that in settings where agent valuations for items are complement free, the better of selling the items separately and bundling them … Recent literature on approximately optimal revenue maximization has shown that in settings where agent valuations for items are complement free, the better of selling the items separately and bundling them together guarantees a constant fraction of the optimal revenue. However, most real-world settings involve some degree of complementarity among items. The role that complementarity plays in the trade-off of simplicity versus optimality has been an obvious missing piece of the puzzle. In “A Simple and Approximately Optimal Mechanism for a Buyer with Complements,” the authors show that the same simple selling mechanism—the better of selling separately and as a grand bundle—guarantees a $\Theta(d)$ fraction of the optimal revenue, where $d$ is a measure of the degree of complementarity. One key modeling contribution is a tractable notion of “degree of complementarity” that admits meaningful results and insights—they demonstrate that previous definitions fall short in this regard.
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Reinforcement Learning of Simple Indirect Mechanisms.

2020-10-02
Gianluca Brero , Alon Eden , Matthias Gerstgrasser +2 more
We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of {\em sequential price mechanisms}, which generalizes both serial dictatorship and posted price mechanisms and … We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of {\em sequential price mechanisms}, which generalizes both serial dictatorship and posted price mechanisms and essentially characterizes all strongly obviously strategyproof mechanisms. Learning an optimal mechanism within this class forms a partially-observable Markov decision process. We provide rigorous conditions for when this class of mechanisms is more powerful than simpler static mechanisms, for sufficiency or insufficiency of observation statistics for learning, and for the necessity of complex (deep) policies. We show that our approach can learn optimal or near-optimal mechanisms in several experimental settings.

On the Power and Limits of Dynamic Pricing in Combinatorial Markets

2020-01-01
Ben Berger , Alon Eden , Michal Feldman
We study the power and limits of optimal dynamic pricing in combinatorial markets; i.e., dynamic pricing that leads to optimal social welfare. Previous work by Cohen-Addad et al. [EC'16] demonstrated … We study the power and limits of optimal dynamic pricing in combinatorial markets; i.e., dynamic pricing that leads to optimal social welfare. Previous work by Cohen-Addad et al. [EC'16] demonstrated the existence of optimal dynamic prices for unit-demand buyers, and showed a market with coverage valuations that admits no such prices. However, finding the frontier of markets (i.e., valuation functions) that admit optimal dynamic prices remains an open problem. In this work we establish positive and negative results that narrow the existing gap. On the positive side, we provide tools for handling markets beyond unit-demand valuations. In particular, we characterize all optimal allocations in multi-demand markets. This characterization allows us to partition the items into equivalence classes according to the role they play in achieving optimality. Using these tools, we provide a poly-time optimal dynamic pricing algorithm for up to $3$ multi-demand buyers. On the negative side, we establish a maximal domain theorem, showing that for every non-gross substitutes valuation, there exist unit-demand valuations such that adding them yields a market that does not admit an optimal dynamic pricing. This result is the dynamic pricing equivalent of the seminal maximal domain theorem by Gul and Stacchetti [JET'99] for Walrasian equilibrium. Yang [JET'17] discovered an error in their original proof, and established a different, incomparable version of their maximal domain theorem. En route to our maximal domain theorem for optimal dynamic pricing, we provide the first complete proof of the original theorem by Gul and Stacchetti.
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Price of Anarchy of Simple Auctions with Interdependent Values

2020-01-01
Alon Eden , Michal Feldman , Inbal Talgam-Cohen +1 more
We expand the literature on the price of anarchy (PoA) of simultaneous item auctions by considering settings with correlated values; we do this via the fundamental economic model of interdependent … We expand the literature on the price of anarchy (PoA) of simultaneous item auctions by considering settings with correlated values; we do this via the fundamental economic model of interdependent values (IDV). It is well-known that in multi-item settings with private values, correlated values can lead to bad PoA, which can be polynomially large in the number of agents $n$. In the more general model of IDV, we show that the PoA can be polynomially large even in single-item settings. On the positive side, we identify a natural condition on information dispersion in the market, termed $γ$-heterogeneity, which enables good PoA guarantees. Under this condition, we show that for single-item settings, the PoA of standard mechanisms degrades gracefully with $γ$. For settings with $m&gt;1$ items we show a separation between two domains: If $n \geq m$, we devise a new simultaneous item auction with good PoA (with respect to $γ$), under limited information asymmetry. To the best of our knowledge, this is the first positive PoA result for correlated values in multi-item settings. The main technical difficulty in establishing this result is that the standard tool for establishing PoA results -- the smoothness framework -- is unsuitable for IDV settings, and so we must introduce new techniques to address the unique challenges imposed by such settings. In the domain of $n \ll m$, we establish impossibility results even for surprisingly simple scenarios.

On the Power and Limits of Dynamic Pricing in Combinatorial Markets

2020-01-01
Ben Berger , Alon Eden , Michal Feldman
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Reinforcement Learning of Sequential Price Mechanisms

2020-01-01
Gianluca Brero , Alon Eden , Matthias Gerstgrasser +2 more
We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of sequential price mechanisms, which generalizes both serial dictatorship and posted price mechanisms and essentially … We introduce the use of reinforcement learning for indirect mechanisms, working with the existing class of sequential price mechanisms, which generalizes both serial dictatorship and posted price mechanisms and essentially characterizes all strongly obviously strategyproof mechanisms. Learning an optimal mechanism within this class forms a partially-observable Markov decision process. We provide rigorous conditions for when this class of mechanisms is more powerful than simpler static mechanisms, for sufficiency or insufficiency of observation statistics for learning, and for the necessity of complex (deep) policies. We show that our approach can learn optimal or near-optimal mechanisms in several experimental settings.
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Combinatorial Auctions with Interdependent Valuations: SOS to the Rescue

2019-01-01
Alon Eden , Michal Feldman , Amos Fiat +2 more
We study combinatorial auctions with interdependent valuations. In such settings, each agent $i$ has a private signal $s_i$ that captures her private information, and the valuation function of every agent … We study combinatorial auctions with interdependent valuations. In such settings, each agent $i$ has a private signal $s_i$ that captures her private information, and the valuation function of every agent depends on the entire signal profile, ${\bf s}=(s_1,\ldots,s_n)$. The literature in economics shows that the interdependent model gives rise to strong impossibility results, and identifies assumptions under which optimal solutions can be attained. The computer science literature provides approximation results for simple single-parameter settings (mostly single item auctions, or matroid feasibility constraints). Both bodies of literature focus largely on valuations satisfying a technical condition termed {\em single crossing} (or variants thereof). We consider the class of {\em submodular over signals} (SOS) valuations (without imposing any single-crossing type assumption), and provide the first welfare approximation guarantees for multi-dimensional combinatorial auctions, achieved by universally ex-post IC-IR mechanisms. Our main results are: $(i)$ 4-approximation for any single-parameter downward-closed setting with single-dimensional signals and SOS valuations; $(ii)$ 4-approximation for any combinatorial auction with multi-dimensional signals and {\em separable}-SOS valuations; and $(iii)$ $(k+3)$- and $(2\log(k)+4)$-approximation for any combinatorial auction with single-dimensional signals, with $k$-sized signal space, for SOS and strong-SOS valuations, respectively. All of our results extend to a parameterized version of SOS, $d$-SOS, while losing a factor that depends on $d$.
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Interdependent Values without Single-Crossing

2018-06-11
Alon Eden , Michal Feldman , Amos Fiat +1 more
We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation … We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation of each agent is a known function of all n private signals. This captures settings such as valuations for oil drilling rights, broadcast rights, pieces of art, and many more.
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Prompt Scheduling for Selfish Agents.

2018-04-09
Alon Eden , Michal Feldman , Amos Fiat +1 more
We give a prompt online mechanism for minimizing the sum of [weighted] completion times. This is the first prompt online algorithm for the problem. When such jobs are strategic agents, … We give a prompt online mechanism for minimizing the sum of [weighted] completion times. This is the first prompt online algorithm for the problem. When such jobs are strategic agents, delaying scheduling decisions makes little sense. Moreover, the mechanism has a particularly simple form of an anonymous menu of options.

Max-Min Greedy Matching.

2018-03-14
Alon Eden , Uriel Feige , Michal Feldman
A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $\pi$ over $V$, the other player imposes a permutation $\sigma$ over $U$. In the … A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $\pi$ over $V$, the other player imposes a permutation $\sigma$ over $U$. In the greedy matching algorithm, vertices of $U$ arrive in order $\sigma$ and each vertex is matched to the lowest (under $\pi$) yet unmatched neighbor in $V$ (or left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals $\pi$, and the second (min) player responds with the worst possible $\sigma$ for $\pi$, does there exist a permutation $\pi$ ensuring to match strictly more than a half of the vertices? Can such a permutation be computed in polynomial time? The main result of this paper is an affirmative answer for this question: we show that there exists a polytime algorithm to compute $\pi$ for which for every $\sigma$ at least $\rho > 0.51$ fraction of the vertices of $V$ are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian. Interestingly, even for regular graphs with arbitrarily large degree (implying a large number of disjoint perfect matchings), there is no $\pi$ ensuring to match more than a fraction $8/9$ of the vertices. The max-min greedy matching problem solves an open problem regarding the welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations. In addition, it has implications for the size of the unique stable matching in markets with global preferences, subject to the graph structure.

An Economic-Based Analysis of RANKING for Online Bipartite Matching

2018-01-01
Alon Eden , Michal Feldman , Amos Fiat +1 more
In their seminal paper, Karp, Vazirani and Vazirani (STOC'90) introduce the online bipartite matching problem, and the RANKING algorithm, which admits a tight $1-\frac{1}{e}$ competitive ratio. Since its publication, the … In their seminal paper, Karp, Vazirani and Vazirani (STOC'90) introduce the online bipartite matching problem, and the RANKING algorithm, which admits a tight $1-\frac{1}{e}$ competitive ratio. Since its publication, the problem has received considerable attention, including a sequence of simplified proofs. In this paper we present a new proof that gives an economic interpretation of the RANKING algorithm -- further simplifying the proof and avoiding arguments such as duality. The new proof gives a new perspective on previous proofs.

Interdependent Values without Single-Crossing

2018-01-01
Alon Eden , Michal Feldman , Amos Fiat +1 more
We consider a setting where an auctioneer sells a single item to $n$ potential agents with {\em interdependent values}. That is, each agent has her own private signal, and the … We consider a setting where an auctioneer sells a single item to $n$ potential agents with {\em interdependent values}. That is, each agent has her own private signal, and the valuation of each agent is a known function of all $n$ private signals. This captures settings such as valuations for artwork, oil drilling rights, broadcast rights, and many more. In the interdependent value setting, all previous work has assumed a so-called {\sl single-crossing condition}. Single-crossing means that the impact of agent $i$'s private signal, $s_i$, on her own valuation is greater than the impact of $s_i$ on the valuation of any other agent. It is known that without the single-crossing condition an efficient outcome cannot be obtained. We study welfare maximization for interdependent valuations through the lens of approximation. We show that, in general, without the single-crossing condition, one cannot hope to approximate the optimal social welfare any better than the approximation given by assigning the item to a random bidder. Consequently, we introduce a relaxed version of single-crossing, {\sl $c$-single-crossing}, parameterized by $c\geq 1$, which means that the impact of $s_i$ on the valuation of agent $i$ is at least $1/c$ times the impact of $s_i$ on the valuation of any other agent ($c=1$ is single-crossing). Using this parameterized notion, we obtain a host of positive results. We propose a prior-free deterministic mechanism that gives an $(n-1)c$-approximation guarantee to welfare. We then show that a random version of the proposed mechanism gives a prior-free universally truthful $2c$-approximation to the optimal welfare for any concave $c$-single crossing setting (and a $2\sqrt{n}c^{3/2}$-approximation in the absence of concavity). We extend this mechanism to a universally truthful mechanism that gives $O(c^2)$-approximation to the optimal revenue.

Max-Min Greedy Matching

2018-01-01
Alon Eden , Uriel Feige , Michal Feldman
A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $\pi$ over $V$, the other player imposes a permutation $\sigma$ over $U$. In the … A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $\pi$ over $V$, the other player imposes a permutation $\sigma$ over $U$. In the greedy matching algorithm, vertices of $U$ arrive in order $\sigma$ and each vertex is matched to the lowest (under $\pi$) yet unmatched neighbor in $V$ (or left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals $\pi$, and the second (min) player responds with the worst possible $\sigma$ for $\pi$, does there exist a permutation $\pi$ ensuring to match strictly more than a half of the vertices? Can such a permutation be computed in polynomial time? The main result of this paper is an affirmative answer for this question: we show that there exists a polytime algorithm to compute $\pi$ for which for every $\sigma$ at least $\rho > 0.51$ fraction of the vertices of $V$ are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian. Interestingly, even for regular graphs with arbitrarily large degree (implying a large number of disjoint perfect matchings), there is no $\pi$ ensuring to match more than a fraction $8/9$ of the vertices. The max-min greedy matching problem solves an open problem regarding the welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations. In addition, it has implications for the size of the unique stable matching in markets with global preferences, subject to the graph structure.

Prompt Scheduling for Selfish Agents

2018-01-01
Alon Eden , Michal Feldman , Amos Fiat +1 more
We give a prompt online mechanism for minimizing the sum of [weighted] completion times. This is the first prompt online algorithm for the problem. When such jobs are strategic agents, … We give a prompt online mechanism for minimizing the sum of [weighted] completion times. This is the first prompt online algorithm for the problem. When such jobs are strategic agents, delaying scheduling decisions makes little sense. Moreover, the mechanism has a particularly simple form of an anonymous menu of options.
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Pricing Social Goods

2017-06-30
Alon Eden , Tomer Ezra , Michal Feldman
Social goods are goods that grant value not only to their owners but also to the owners' surroundings, be it their families, friends or office mates. The benefit a non-owner … Social goods are goods that grant value not only to their owners but also to the owners' surroundings, be it their families, friends or office mates. The benefit a non-owner derives from the good is affected by many factors, including the type of the good, its availability, and the social status of the non-owner. Depending on the magnitude of the benefit and on the price of the good, a potential buyer might stay away from purchasing the good, hoping to free ride on others' purchases. A revenue-maximizing seller who sells social goods must take these considerations into account when setting prices for the good. The literature on optimal pricing has advanced considerably over the last decade, but little is known about optimal pricing schemes for selling social goods. In this paper, we conduct a systematic study of revenue-maximizing pricing schemes for social goods: we introduce a Bayesian model for this scenario, and devise nearly-optimal pricing schemes for various types of externalities, both for simultaneous sales and for sequential sales.

A Simple and Approximately Optimal Mechanism for a Buyer with Complements

2017-06-20
Alon Eden , Michal Feldman , Ophir Friedler +2 more
We consider a revenue-maximizing seller with m heterogeneous items and a single buyer whose valuation v for the items may exhibit both substitutes (i.e., for some S, T, v(S ∪ … We consider a revenue-maximizing seller with m heterogeneous items and a single buyer whose valuation v for the items may exhibit both substitutes (i.e., for some S, T, v(S ∪ T) < v(S) + v(T)) and complements (i.e., for some S, T, v(S ∪ T) > v(S) + v(T)). We show that the mechanism first proposed by Babaioff et al. [2014] -- the better of selling the items separately and bundling them together -- guarantees a Θ(d) fraction of the optimal revenue, where $d$ is a measure on the degree of complementarity. Note that this is the first approximately optimal mechanism for a buyer whose valuation exhibits any kind of complementarity. It extends the work of Rubinstein and Weinberg [2015], which proved that the same simple mechanisms achieve a constant factor approximation when buyer valuations are subadditive, the most general class of complement-free valuations.
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Pricing Social Goods

2017-01-01
Alon Eden , Tomer Ezra , Michal Feldman
Social goods are goods that grant value not only to their owners but also to the owners' surroundings, be it their families, friends or office mates. The benefit a non-owner … Social goods are goods that grant value not only to their owners but also to the owners' surroundings, be it their families, friends or office mates. The benefit a non-owner derives from the good is affected by many factors, including the type of the good, its availability, and the social status of the non-owner. Depending on the magnitude of the benefit and on the price of the good, a potential buyer might stay away from purchasing the good, hoping to free ride on others' purchases. A revenue-maximizing seller who sells social goods must take these considerations into account when setting prices for the good. The literature on optimal pricing has advanced considerably over the last decade, but little is known about optimal pricing schemes for selling social goods. In this paper, we conduct a systematic study of revenue-maximizing pricing schemes for social goods: we introduce a Bayesian model for this scenario, and devise nearly-optimal pricing schemes for various types of externalities, both for simultaneous sales and for sequential sales.

The Competition Complexity of Auctions: A Bulow-Klemperer Result for Multi-Dimensional Bidders

2016-12-28
Alon Eden , Michal Feldman , Ophir Friedler +2 more
A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to $n$ bidders whose values are drawn i.i.d. from … A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to $n$ bidders whose values are drawn i.i.d. from a regular distribution, the simple welfare-maximizing VCG mechanism (in this case, a second price-auction) with one additional bidder extracts at least as much revenue in expectation as the optimal mechanism. The beauty of this theorem stems from the fact that VCG is a {\em prior-independent} mechanism, where the seller possesses no information about the distribution, and yet, by recruiting one additional bidder it performs better than any prior-dependent mechanism tailored exactly to the distribution at hand (without the additional bidder). In this work, we establish the first {\em full Bulow-Klemperer} results in {\em multi-dimensional} environments, proving that by recruiting additional bidders, the revenue of the VCG mechanism surpasses that of the optimal (possibly randomized, Bayesian incentive compatible) mechanism. For a given environment with i.i.d. bidders, we term the number of additional bidders needed to achieve this guarantee the environment's {\em competition complexity}. Using the recent duality-based framework of Cai et al. [2016] for reasoning about optimal revenue, we show that the competition complexity of $n$ bidders with additive valuations over $m$ independent, regular items is at most $n+2m-2$ and at least $\log(m)$. We extend our results to bidders with additive valuations subject to downward-closed constraints, showing that these significantly more general valuations increase the competition complexity by at most an additive $m-1$ factor. We further improve this bound for the special case of matroid constraints, and provide additional extensions as well.

The Invisible Hand of Dynamic Market Pricing

2016-07-24
Vincent Cohen-Addad , Alon Eden , Michal Feldman +1 more
Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, … Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.
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The Invisible Hand of Dynamic Market Pricing

2016-07-21
Vincent Cohen-Addad , Alon Eden , Michal Feldman +1 more
Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, … Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.
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The Competition Complexity of Auctions: A Bulow-Klemperer Result for Multi-Dimensional Bidders

2016-01-01
Alon Eden , Michal Feldman , Ophir Friedler +2 more
A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to $n$ bidders whose values are drawn i.i.d. from … A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to $n$ bidders whose values are drawn i.i.d. from a regular distribution, the simple welfare-maximizing VCG mechanism (in this case, a second price-auction) with one additional bidder extracts at least as much revenue in expectation as the optimal mechanism. The beauty of this theorem stems from the fact that VCG is a {\em prior-independent} mechanism, where the seller possesses no information about the distribution, and yet, by recruiting one additional bidder it performs better than any prior-dependent mechanism tailored exactly to the distribution at hand (without the additional bidder). In this work, we establish the first {\em full Bulow-Klemperer} results in {\em multi-dimensional} environments, proving that by recruiting additional bidders, the revenue of the VCG mechanism surpasses that of the optimal (possibly randomized, Bayesian incentive compatible) mechanism. For a given environment with i.i.d. bidders, we term the number of additional bidders needed to achieve this guarantee the environment's {\em competition complexity}. Using the recent duality-based framework of Cai et al. [2016] for reasoning about optimal revenue, we show that the competition complexity of $n$ bidders with additive valuations over $m$ independent, regular items is at most $n+2m-2$ and at least $\log(m)$. We extend our results to bidders with additive valuations subject to downward-closed constraints, showing that these significantly more general valuations increase the competition complexity by at most an additive $m-1$ factor. We further improve this bound for the special case of matroid constraints, and provide additional extensions as well.
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A Simple and Approximately Optimal Mechanism for a Buyer with Complements

2016-01-01
Alon Eden , Michal Feldman , Ophir Friedler +2 more
We consider a revenue-maximizing seller with $m$ heterogeneous items and a single buyer whose valuation $v$ for the items may exhibit both substitutes (i.e., for some $S, T$, $v(S \cup … We consider a revenue-maximizing seller with $m$ heterogeneous items and a single buyer whose valuation $v$ for the items may exhibit both substitutes (i.e., for some $S, T$, $v(S \cup T) < v(S) + v(T)$) and complements (i.e., for some $S, T$, $v(S \cup T) > v(S) + v(T)$). We show that the mechanism first proposed by Babaioff et al. [2014] - the better of selling the items separately and bundling them together - guarantees a $\Theta(d)$ fraction of the optimal revenue, where $d$ is a measure on the degree of complementarity. Note that this is the first approximately optimal mechanism for a buyer whose valuation exhibits any kind of complementarity, and extends the work of Rubinstein and Weinberg [2015], which proved that the same simple mechanisms achieve a constant factor approximation when buyer valuations are subadditive, the most general class of complement-free valuations. Our proof is enabled by the recent duality framework developed in Cai et al. [2016], which we use to obtain a bound on the optimal revenue in this setting. Our main technical contributions are specialized to handle the intricacies of settings with complements, and include an algorithm for partitioning edges in a hypergraph. Even nailing down the right model and notion of "degree of complementarity" to obtain meaningful results is of interest, as the natural extensions of previous definitions provably fail.
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Truthful Secretaries with Budgets

2015-01-01
Alon Eden , Michal Feldman , Adi Vardi
We study online auction settings in which agents arrive and depart dynamically in a random (secretary) order, and each agent's private type consists of the agent's arrival and departure times, … We study online auction settings in which agents arrive and depart dynamically in a random (secretary) order, and each agent's private type consists of the agent's arrival and departure times, value and budget. We consider multi-unit auctions with additive agents for the allocation of both divisible and indivisible items. For both settings, we devise truthful mechanisms that give a constant approximation with respect to the auctioneer's revenue, under a large market assumption. For divisible items, we devise in addition a truthful mechanism that gives a constant approximation with respect to the liquid welfare --- a natural efficiency measure for budgeted settings introduced by Dobzinski and Paes Leme [ICALP'14]. Our techniques provide high-level principles for transforming offline truthful mechanisms into online ones, with or without budget constraints. To the best of our knowledge, this is the first work that addresses the non-trivial challenge of combining online settings with budgeted agents.

The Invisible Hand of Dynamic Market Pricing

2015-01-01
Vincent Cohen-Addad , Alon Eden , Michal Feldman +1 more
Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, … Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.
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Pricing Online Decisions: Beyond Auctions

2015-01-01
Ilan Reuven Cohen , Alon Eden , Amos Fiat +1 more
We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare … We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare or revenue in an auction settings. This paper deals with quite general settings and minimizing social costs. We show that appropriately computed posted prices allow one to achieve essentially the same performance as the best online algorithm. This holds in a wide variety of settings. Unlike online algorithms that learn about the event, and then make enforceable decisions, prices are posted without knowing the future events or even the current event, and are thus inherently dominant strategy incentive compatible. In particular we show that one can give efficient posted price mechanisms for metrical task systems, some instances of the $k$-server problem, and metrical matching problems. We give both deterministic and randomized algorithms. Such posted price mechanisms decrease the social cost dramatically over selfish behavior where no decision incurs a charge. One alluring application of this is reducing the social cost of free parking exponentially.
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Pricing Online Decisions: Beyond Auctions

2014-12-22
Ilan Reuven Cohen , Alon Eden , Amos Fiat +1 more
We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare … We consider dynamic pricing schemes in online settings where selfish agents generate online events. Previous work on online mechanisms has dealt almost entirely with the goal of maximizing social welfare or revenue in an auction settings. This paper deals with quite general settings and minimizing social costs. We show that appropriately computed posted prices allow one to achieve essentially the same performance as the best online algorithm. This holds in a wide variety of settings. Unlike online algorithms that learn about the event, and then make enforcable decisions, prices are posted without knowing the future events or even the current event, and are thus inherently dominant strategy incentive compatible.In particular we show that one can give efficient posted price mechanisms for metrical task systems, some instances of the k-server problem, and metrical matching problems. We give both deterministic and randomized algorithms. Such posted price mechanisms decrease the social cost dramatically over selfish behavior where no decision incurs a charge. One alluring application of this is reducing the social cost of free parking exponentially.
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Coauthor Papers Together
Michal Feldman 28
Amos Fiat 13
Kira Goldner 8
David C. Parkes 6
Inbal Talgam-Cohen 6
S. Matthew Weinberg 5
Ophir Friedler 5
Gianluca Brero 4
Matthias Gerstgrasser 4
Simon Mauras 4
Divyarthi Mohan 4
Duncan Rheingans-Yoo 3
Ilan Reuven Cohen 3
Vincent Cohen-Addad 3
Łukasz Jeż 2
Tomer Ezra 2
Gary Qiurui Ma 2
Anna R. Karlin 2
Clara Li 2
Shuran Zheng 2
Uriel Feige 2
Tzahi Taub 2
Stephan Zheng 2
Ben Berger 2
Talya Eden 1
Arsen Vasilyan 1
Juntao Wang 1
Yiling Chen 1
Xintong Wang 1
Xintong Wang 1
Vincent Li 1
Alexander T. Trott 1
James D. Hess 1
Amit Ronen 1
Kineret Segal 1
Adi Vardi 1
Yael Belfer 1
Alexander Trott 1
David Parkes 1
Gary Qiurui 1
Simon Mauras 1
Ori Zviran 1
Darshan Chakrabarti 1
Hannaneh Akrami 1
Yoav Gal-Tzur 1

Approximate revenue maximization in interdependent value settings

2014-05-30
Shuchi Chawla , Hu Fu , Anna R. Karlin
We study revenue maximization in settings where agents' values are interdependent: each agent receives a signal drawn from a correlated distribution and agents' values are functions of all of the … We study revenue maximization in settings where agents' values are interdependent: each agent receives a signal drawn from a correlated distribution and agents' values are functions of all of the signals. We introduce a variant of the generalized VCG auction with reserve prices and random admission, and show that this auction gives a constant approximation to the optimal expected revenue in matroid environments. Our results do not require any assumptions on the signal distributions, however, they require the value functions to satisfy a standard single-crossing property and a concavity-type condition.
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Interdependent Values without Single-Crossing

2018-06-11
Alon Eden , Michal Feldman , Amos Fiat +1 more
We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation … We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation of each agent is a known function of all n private signals. This captures settings such as valuations for oil drilling rights, broadcast rights, pieces of art, and many more.
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Algorithmic pricing via virtual valuations

2007-06-11
Shuchi Chawla , Jason D. Hartline , Robert Kleinberg
Algorithmic pricing is the computational problem that sellers (e.g.,in supermarkets) face when trying to set prices for their items to maximize their profit in the presence of a known demand. … Algorithmic pricing is the computational problem that sellers (e.g.,in supermarkets) face when trying to set prices for their items to maximize their profit in the presence of a known demand. Guruswami etal. (SODA, 2005) proposed this problem and gave logarithmic approximations (in the number of consumers) for the unit-demand and single-parameter cases where there is a specific set of consumers and their valuations for bundles are known precisely. Subsequently several versions of the problem have been shown to have poly-logarithmic in approximability. This problem has direct ties to the important open question of better understanding the Bayesian optimal mechanism in multi-parameter agent settings; however, for this purpose approximation factors logarithmic in the number of agents are inadequate. It is therefore of vital interest to consider special cases where constant approximations are possible. We consider the unit-demand variant of this pricing problem. Here a consumer has a valuation for each different item and their value for aset of items is simply the maximum value they have for any item in the set. Instead of considering a set of consumers with precisely known preferences, like the prior algorithmic pricing literature, we assume that the preferences of the consumers are drawn from a distribution. This is the standard assumption in economics; furthermore, the setting of a specific set of customers with specific preferences, which is employed in all of the prior work in algorithmic pricing, is a special case of this general Bayesian pricing problem, where there is a discrete Bayesian distribution for preferences specified by picking one consumer uniformly from the given set of consumers. Notice that the distribution over the valuations for the individual items that this generates is obviously correlated. Our work complements these existing works by considering the case where the consumer's valuations for the different items are independent random variables. Our main result is a constant approximation algorithm for this problem that makes use of an interesting connection between this problem and the concept of virtual valuations from the single-parameter Bayesian optimal mechanism design literature.
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Do prices coordinate markets?

2016-06-10
Justin Hsu , Jamie Morgenstern , Ryan Rogers +2 more
Walrasian equilibrium prices can be said to coordinate markets: They support a welfare optimal allocation in which each buyer is buying bundle of goods that is individually most preferred. However, … Walrasian equilibrium prices can be said to coordinate markets: They support a welfare optimal allocation in which each buyer is buying bundle of goods that is individually most preferred. However, this clean story has two caveats. First, the prices alone are not sufficient to coordinate the market, and buyers may need to select among their most preferred bundles in a coordinated way to find a feasible allocation. Second, we don't in practice expect to encounter exact equilibrium prices tailored to the market, but instead only approximate prices, somehow encoding "distributional" information about the market. How well do prices work to coordinate markets when tie-breaking is not coordinated, and they encode only distributional information? We answer this question. First, we provide a genericity condition such that for buyers with Matroid Based Valuations, overdemand with respect to equilibrium prices is at most 1, independent of the supply of goods, even when tie-breaking is done in an uncoordinated fashion. Second, we provide learning-theoretic results that show that such prices are robust to changing the buyers in the market, so long as all buyers are sampled from the same (unknown) distribution.
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Combinatorial Auctions via Posted Prices

2014-12-22
Michal Feldman , Nick Gravin , Brendan Lucier
We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in … We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC).We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity.
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Simultaneous auctions are (almost) efficient

2013-05-28
Michal Feldman , Hu Fu , Nick Gravin +1 more
Simultaneous item auctions are simple and practical procedures for allocating items to bidders with potentially complex preferences. In a simultaneous auction, every bidder submits independent bids on all items simultaneously. … Simultaneous item auctions are simple and practical procedures for allocating items to bidders with potentially complex preferences. In a simultaneous auction, every bidder submits independent bids on all items simultaneously. The allocation and prices are then resolved for each item separately, based solely on the bids submitted on that item. We study the efficiency of Bayes-Nash equilibrium (BNE) outcomes of simultaneous first- and second-price auctions when bidders have complement-free (a.k.a. subadditive) valuations. While it is known that the social welfare of every pure Nash equilibrium (NE) constitutes a constant fraction of the optimal social welfare, a pure NE rarely exists, and moreover, the full information assumption is often unrealistic. Therefore, quantifying the welfare loss in Bayes-Nash equilibria is of particular interest. Previous work established a logarithmic bound on the ratio between the social welfare of a BNE and the expected optimal social welfare in both first-price auctions (Hassidim et al., 2011) and second-price auctions (Bhawalkar and Roughgarden, 2011), leaving a large gap between a constant and a logarithmic ratio. We introduce a new proof technique and use it to resolve both of these gaps in a unified way. Specifically, we show that the expected social welfare of any BNE is at least 1/2 of the optimal social welfare in the case of first-price auctions, and at least 1/4 in the case of second-price auctions.
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Auctions with Interdependence and SOS: Improved Approximation

2021-01-01
Ameer Amer , Inbal Talgam-Cohen
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Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

2017-10-01
Paul Dütting , Michal Feldman , Thomas Keßelheim +1 more
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold … We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.
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Matroid prophet inequalities

2012-05-19
Robert Kleinberg , S. Matthew Weinberg
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent … Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of $p$ matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most $O(p)$, and this factor is also tight.
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The power of randomness in bayesian optimal mechanism design

2010-06-07
Shuchi Chawla , David L. Malec , Balasubramanian Sivan
We investigate the power of randomness in the context of a fundamental Bayesian optimal mechanism design problem - a single seller aims to maximize expected revenue by allocating multiple kinds … We investigate the power of randomness in the context of a fundamental Bayesian optimal mechanism design problem - a single seller aims to maximize expected revenue by allocating multiple kinds of resources to "unit-demand" agents with preferences drawn from a known distribution. When the agents' preferences are single-dimensional Myerson's seminal work [14] shows that randomness offers no benefit - the optimal mechanism is always deterministic. In the multi-dimensional case, where each agent's preferences are given by different values for each of the available services, Briest et al.[6] recently showed that the gap between the expected revenue obtained by an optimal randomized mechanism and an optimal deterministic mechanism can be unbounded even when a single agent is offered only 4 services. However, this large gap is attained through unnatural instances where values of the agent for different services are correlated in a specific way. We show that when the agent's values involve no correlation or a specific kind of positive correlation, the benefit of randomness is only a small constant factor (4 and 8 respectively). Our model of positively correlated values (that we call the common base value model) is a natural model for unit-demand agents and items that are substitutes. Our results extend to multiple agent settings as well.
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The power of randomness in Bayesian optimal mechanism design

2012-08-28
Shuchi Chawla , David Malec , Balasubramanian Sivan
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On Optimal Ordering in the Optimal Stopping Problem

2020-07-09
Shipra Agrawal , Jay Sethuraman , Xingyu Zhang
Consider a player who can probe a sequence of n independent random variables X1, . . . , Xn with known distributions. After observing (the realized value of) Xi, the … Consider a player who can probe a sequence of n independent random variables X1, . . . , Xn with known distributions. After observing (the realized value of) Xi, the player needs to decide whether to stop and earn reward Xi, or reject the reward and probe the next variable Xi+1. The goal is to maximize the expected reward at the stopping time. This is an instance of the optimal stopping problem, which is a fundamental problem studied from many different aspects in mathematics, statistics, and computer science, and has found a wide variety of applications in sequential decision making and mechanism design.
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Payment rules through discriminant-based classifiers

2012-06-04
Paul Dütting , Felix Fischer , Pichayut Jirapinyo +3 more
In mechanism design it is typical to impose incentive compatibility and then derive an optimal mechanism subject to this constraint. By replacing the incentive compatibility requirement with the goal of … In mechanism design it is typical to impose incentive compatibility and then derive an optimal mechanism subject to this constraint. By replacing the incentive compatibility requirement with the goal of minimizing expected ex post regret, we are able to adapt statistical machine learning techniques to the design of payment rules. This computational approach to mechanism design is applicable to domains with multi-dimensional types and situations where computational efficiency is a concern. Specifically, given an outcome rule and access to a type distribution, we train a support vector machine with a special discriminant function structure such that it implicitly establishes a payment rule with desirable incentive properties. We discuss applications to a multi-minded combinatorial auction with a greedy winner-determination algorithm and to an assignment problem with egalitarian outcome rule. Experimental results demonstrate both that the construction produces payment rules with low ex post regret, and that penalizing classification errors is effective in preventing failures of ex post individual rationality.
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Asynchronous Methods for Deep Reinforcement Learning

2016-01-01
Volodymyr Mnih , Adrià Puigdomènech Badia , Mehdi Mirza +5 more
We propose a conceptually simple and lightweight framework for deep reinforcement learning that uses asynchronous gradient descent for optimization of deep neural network controllers. We present asynchronous variants of four … We propose a conceptually simple and lightweight framework for deep reinforcement learning that uses asynchronous gradient descent for optimization of deep neural network controllers. We present asynchronous variants of four standard reinforcement learning algorithms and show that parallel actor-learners have a stabilizing effect on training allowing all four methods to successfully train neural network controllers. The best performing method, an asynchronous variant of actor-critic, surpasses the current state-of-the-art on the Atari domain while training for half the time on a single multi-core CPU instead of a GPU. Furthermore, we show that asynchronous actor-critic succeeds on a wide variety of continuous motor control problems as well as on a new task of navigating random 3D mazes using a visual input.

Scalable trust-region method for deep reinforcement learning using Kronecker-factored approximation

2017-01-01
Yuhuai Wu , Elman Mansimov , S. Matthew Liao +2 more
In this work, we propose to apply trust region optimization to deep reinforcement learning using a recently proposed Kronecker-factored approximation to the curvature. We extend the framework of natural policy … In this work, we propose to apply trust region optimization to deep reinforcement learning using a recently proposed Kronecker-factored approximation to the curvature. We extend the framework of natural policy gradient and propose to optimize both the actor and the critic using Kronecker-factored approximate curvature (K-FAC) with trust region; hence we call our method Actor Critic using Kronecker-Factored Trust Region (ACKTR). To the best of our knowledge, this is the first scalable trust region natural gradient method for actor-critic methods. It is also a method that learns non-trivial tasks in continuous control as well as discrete control policies directly from raw pixel inputs. We tested our approach across discrete domains in Atari games as well as continuous domains in the MuJoCo environment. With the proposed methods, we are able to achieve higher rewards and a 2- to 3-fold improvement in sample efficiency on average, compared to previous state-of-the-art on-policy actor-critic methods. Code is available at https://github.com/openai/baselines

Simple mechanisms for subadditive buyers via duality

2017-06-15
Yang Cai , Mingfei Zhao
We provide simple and approximately revenue-optimal mechanisms in the multi-item multi-bidder settings. We unify and improve all previous results, as well as generalize the results to broader cases. In particular, … We provide simple and approximately revenue-optimal mechanisms in the multi-item multi-bidder settings. We unify and improve all previous results, as well as generalize the results to broader cases. In particular, we prove that the better of the following two simple, deterministic and Dominant Strategy Incentive Compatible mechanisms, a sequential posted price mechanism or an anonymous sequential posted price mechanism with entry fee, achieves a constant fraction of the optimal revenue among all randomized, Bayesian Incentive Compatible mechanisms, when buyers' valuations are XOS over independent items. If the buyers' valuations are subadditive over independent items, the approximation factor degrades to O(logm), where m is the number of items. We obtain our results by first extending the Cai-Devanur-Weinberg duality framework to derive an effective benchmark of the optimal revenue for subadditive bidders, and then analyzing this upper bound with new techniques.
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Interdependent Public Projects

2023-01-01
Avi J. Cohen , Michal Feldman , Divyarthi Mohan +1 more
In the interdependent values (IDV) model introduced by Milgrom and Weber [1982], agents have private signals that capture their information about different social alternatives, and the valuation of every agent … In the interdependent values (IDV) model introduced by Milgrom and Weber [1982], agents have private signals that capture their information about different social alternatives, and the valuation of every agent is a function of all agent signals. While interdependence has been mainly studied for auctions, it is extremely relevant for a large variety of social choice settings, including the canonical and practically important setting of public projects. The IDV model is much more realistic but also very challenging relative to the standard independent private values model. Welfare guarantees for IDV have been achieved mainly through two alternative conditions known as single-crossing and submodularity over signals (SOS). In either case, the existing theory falls short of solving the public projects setting.Our contribution is twofold: (i) We give a useful characterization of truthfulness for IDV public projects, parallel to the known characterization for independent private values, and identify the domain frontier for which this characterization applies; (ii) Using this characterization, we provide possibility and impossibility results for welfare approximation in public projects with SOS valuations. Our main impossibility result is that, in contrast to auctions, no universally truthful mechanism performs better for public projects with SOS valuations than choosing a project at random. Our main positive result applies to excludable public projects with SOS, for which we establish a constant factor approximation similar to auctions. Our results suggest that exclusion may be a key tool for achieving welfare guarantees in the IDV model.* The full version of the paper can be accessed at https://arxiv.org/abs/2204.08044. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement no. 866132), by the Israel Science Foundation (ISF grant nos. 317/17 and 336/18), by an Amazon Research Award, and by the NSF-BSF (grant no. 2020788).
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Prior-Free Clock Auctions for Bidders with Interdependent Values

2021-01-01
Vasilis Gkatzelis , R. Patel , Emmanouil Pountourakis +1 more
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Complexity of mechanism design

2002-08-01
Vincent Conitzer , Tüomas Sandholm
The aggregation of conflicting preferences is a central problem in multiagent systems. The key difficulty is that the agents may report their preferences insincerely. Mechanism design is the art of … The aggregation of conflicting preferences is a central problem in multiagent systems. The key difficulty is that the agents may report their preferences insincerely. Mechanism design is the art of designing the rules of the game so that the agents are motivated to report their preferences truthfully and a (socially) desirable outcome is chosen. We propose an approach where a mechanism is automatically created for the preference aggregation setting at hand. This has several advantages, but the downside is that the mechanism design optimization problem needs to be solved anew each time. Focusing-on settings where side payments are not possible, we show that the mechanism design problem is NP-complete for deterministic mechanisms. This holds both for dominantstrategy implementation and for Bayes-Nash implementation. We then show that if we allow randomized mechanisms, the mechanism design problem becomes tractable. In other words, the coordinator can tackle the computational complexity introduced by its uncertainty the agents face additional uncertainty. This comes at no loss, and in some cases at a gain, in the (social) objective.

A Markovian Decision Process

1957-01-01
Richard Bellman
Abstract : The purpose of this paper is to discuss the asymptotic behavior of the sequence (f sub n(i)) generated by a nonlinear recurrence relation. This problem arises in connection … Abstract : The purpose of this paper is to discuss the asymptotic behavior of the sequence (f sub n(i)) generated by a nonlinear recurrence relation. This problem arises in connection with an equipment replacement problem, cf. S. Dreyfus, A Note on an Industrial Replacement Process.
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Reinforcement Mechanism Design: With Applications to Dynamic Pricing in Sponsored Search Auctions

2020-04-03
Weiran Shen , Binghui Peng , Hanpeng Liu +7 more
In many social systems in which individuals and organizations interact with each other, there can be no easy laws to govern the rules of the environment, and agents' payoffs are … In many social systems in which individuals and organizations interact with each other, there can be no easy laws to govern the rules of the environment, and agents' payoffs are often influenced by other agents' actions. We examine such a social system in the setting of sponsored search auctions and tackle the search engine's dynamic pricing problem by combining the tools from both mechanism design and the AI domain. In this setting, the environment not only changes over time, but also behaves strategically. Over repeated interactions with bidders, the search engine can dynamically change the reserve prices and determine the optimal strategy that maximizes the profit. We first train a buyer behavior model, with a real bidding data set from a major search engine, that predicts bids given information disclosed by the search engine and the bidders' performance data from previous rounds. We then formulate the dynamic pricing problem as an MDP and apply a reinforcement-based algorithm that optimizes reserve prices over time. Experiments demonstrate that our model outperforms static optimization strategies including the ones that are currently in use as well as several other dynamic ones.
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Private Interdependent Valuations

2022-01-01
Alon Eden , Kira Goldner , Shuran Zheng
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Private Interdependent ValuationsAlon Eden, Kira Goldner, and Shuran ZhengAlon Eden, Kira Goldner, and Shuran … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Private Interdependent ValuationsAlon Eden, Kira Goldner, and Shuran ZhengAlon Eden, Kira Goldner, and Shuran Zhengpp.2920 - 2939Chapter DOI:https://doi.org/10.1137/1.9781611977073.113PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider the single-item interdependent value setting, where there is a single item sold by a monopolist, n buyers, and each buyer has a private signal si describing a piece of information about the item. Additionally, each bidder i has a valuation function vi(s1, …, sn) mapping the (private) signals of all buyers into a positive real number representing their value for the item. This setting captures scenarios where the item's information is asymmetric or dispersed among agents, such as in competitions for oil drilling rights, or in auctions for art pieces. Due to the increased complexity of this model compared to the standard private values model, it is generally assumed that each bidder's valuation function vi is public knowledge to the seller or all other buyers. But in many situations, the seller may not know the bidders' valuation functions—how a bidder aggregates signals into a valuation is often their private information. In this paper, we design mechanisms that guarantee approximately-optimal social welfare while satisfying ex-post incentive compatibility and individually rationality for the case where the valuation functions are private to the bidders, and thus may be strategically misreported to the seller. When the valuations are public, it is possible for optimal social welfare to be attained by a deterministic mechanism when the valuations satisfy a single-crossing condition. In contrast, when the valuations are the bidders' private information, we show that no finite bound on the social welfare can be achieved by any deterministic mechanism even under single-crossing. Moreover, no randomized mechanism can guarantee better than n-approximation. We thus consider valuation functions that are submodular over signals (SOS), introduced in the context of combinatorial auctions in a recent breakthrough paper by Eden et al. [EC'19]. Our main result is an O(log2 n)-approximation randomized mechanism for buyers with private signals and valuations under the SOS condition. We also give a tight Θ(k)-approximation mechanism for the case each agent's valuation depends on at most k other signals even for unknown k. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771
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Understanding Incentives: Mechanism Design Becomes Algorithm Design

2013-10-01
Yang Cai , Constantinos Daskalakis , S. Matthew Weinberg
We provide a computationally efficient black-box reduction from mechanism design to algorithm design in very general settings. Specifically, we give an approximation-preserving reduction from truthfully maximizing any objective under arbitrary … We provide a computationally efficient black-box reduction from mechanism design to algorithm design in very general settings. Specifically, we give an approximation-preserving reduction from truthfully maximizing any objective under arbitrary feasibility constraints with arbitrary bidder types to (not necessarily truthfully) maximizing the same objective plus virtual welfare (under the same feasibility constraints). Our reduction is based on a fundamentally new approach: we describe a mechanism's behavior indirectly only in terms of the expected value it awards bidders for certain behavior, and never directly access the allocation rule at all. Applying our new approach to revenue, we exhibit settings where our reduction holds both ways. That is, we also provide an approximation-sensitive reduction from (non-truthfully) maximizing virtual welfare to (truthfully) maximizing revenue, and therefore the two problems are computationally equivalent. With this equivalence in hand, we show that both problems are NP-hard to approximate within any polynomial factor, even for a single monotone sub modular bidder. We further demonstrate the applicability of our reduction by providing a truthful mechanism maximizing fractional max-min fairness.
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The Invisible Hand of Dynamic Market Pricing

2016-07-21
Vincent Cohen-Addad , Alon Eden , Michal Feldman +1 more
Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, … Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.
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Combinatorial Walrasian Equilibrium

2013-04-08
Michal Feldman , Nick Gravin , Brendan Lucier
We study a combinatorial market design problem, where a collection of indivisible objects is to be priced and sold to potential buyers subject to equilibrium constraints.The classic solution concept for … We study a combinatorial market design problem, where a collection of indivisible objects is to be priced and sold to potential buyers subject to equilibrium constraints.The classic solution concept for such problems is Walrasian Equilibrium (WE), which provides a simple and transparent pricing structure that achieves optimal social welfare. The main weakness of the WE notion is that it exists only in very restrictive cases. To overcome this limitation, we introduce the notion of a Combinatorial Walrasian equilibium (CWE), a natural relaxation of WE. The difference between a CWE and a (non-combinatorial) WE is that the seller can package the items into indivisible bundles prior to sale, and the market does not necessarily clear. We show that every valuation profile admits a CWE that obtains at least half of the optimal (unconstrained) social welfare. Moreover, we devise a poly-time algorithm that, given an arbitrary allocation X, computes a CWE that achieves at least half of the welfare of X. Thus, the economic problem of finding a CWE with high social welfare reduces to the algorithmic problem of social-welfare approximation. In addition, we show that every valuation profile admits a CWE that extracts a logarithmic fraction of the optimal welfare as revenue. Finally, these results are complemented by strong lower bounds when the seller is restricted to using item prices only, which motivates the use of bundles. The strength of our results derives partly from their generality - our results hold for arbitrary valuations that may exhibit complex combinations of substitutes and complements.

Better Approximation for Interdependent SOS Valuations

2022-01-01
Pinyan Lu , Enze Sun , Chenghan Zhou
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A unifying hierarchy of valuations with complements and substitutes

2015-01-25
Uriel Feige , Michal Feldman , Nicole Immorlica +3 more
We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hyper-graphs). Levels of the hierarchy correspond to the degree of complementarity in … We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hyper-graphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank κ (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-κ. Every monotone function that has supermodular degree κ (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(κ+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-κ. One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of κ + 1 if all players hold valuation functions in MPH-κ. The other is an upper bound of 2κ on the price of anarchy of simultaneous first price auctions.

Combinatorial walrasian equilibrium

2013-05-28
Michal Feldman , Nick Gravin , Brendan Lucier
We study a combinatorial market design problem, where a collection of indivisible objects is to be priced and sold to potential buyers subject to equilibrium constraints. The classic solution concept … We study a combinatorial market design problem, where a collection of indivisible objects is to be priced and sold to potential buyers subject to equilibrium constraints. The classic solution concept for such problems is Walrasian Equilibrium (WE), which provides a simple and transparent pricing structure that achieves optimal social welfare. The main weakness of the WE notion is that it exists only in very restrictive cases. To overcome this limitation, we introduce the notion of a Combinatorial Walrasian equilibium (CWE), a natural relaxation of WE. The difference between a CWE and a (non-combinatorial) WE is that the seller can package the items into indivisible bundles prior to sale, and the market does not necessarily clear. We show that every valuation profile admits a CWE that obtains at least half of the optimal (unconstrained) social welfare. Moreover, we devise a poly-time algorithm that, given an arbitrary allocation X, computes a CWE that achieves at least half of the welfare of X. Thus, the economic problem of finding a CWE with high social welfare reduces to the algorithmic problem of social-welfare approximation. In addition, we show that every valuation profile admits a CWE that extracts a logarithmic fraction of the optimal welfare as revenue. Finally, these results are complemented by strong lower bounds when the seller is restricted to using item prices only, which motivates the use of bundles. The strength of our results derives partly from their generality --- our results hold for arbitrary valuations that may exhibit complex combinations of substitutes and complements.
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M-Convex Function on Generalized Polymatroid

1999-02-01
Kazuo Murota , Akiyoshi Shioura
The concept of M-convex function, introduced by Murota (1996), is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of … The concept of M-convex function, introduced by Murota (1996), is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of Dress and Wenzel (1990). In this paper, we extend this concept to functions on generalized polymatroids with a view to providing a unified framework for efficiently solvable nonlinear discrete optimization problems.
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Sampling and Representation Complexity of Revenue Maximization

2014-01-01
Shaddin Dughmi , Li Han , Noam Nisan
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Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

2015-06-12
Aviad Rubinstein , S. Matthew Weinberg
We study the revenue maximization problem of a seller with n heterogeneous items for sale to a single buyer whose valuation function for sets of items is unknown and drawn … We study the revenue maximization problem of a seller with n heterogeneous items for sale to a single buyer whose valuation function for sets of items is unknown and drawn from some distribution D. We show that if D is a distribution over subadditive valuations with independent items, then the better of pricing each item separately or pricing only the grand bundle achieves a constant-factor approximation to the revenue of the optimal mechanism. This includes buyers who are k-demand, additive up to a matroid constraint, or additive up to constraints of any downwards-closed set system (and whose values for the individual items are sampled independently), as well as buyers who are fractionally subadditive with item multipliers drawn independently. Our proof makes use of the core-tail decomposition framework developed in prior work showing similar results for the significantly simpler class of additive buyers [Li and Yao 2013; Babaioff et al.2014].
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Composable and efficient mechanisms

2013-05-28
Vasilis Syrgkanis , Éva Tardos
We initiate the study of efficient mechanism design with guaranteed good properties even when players participate in multiple mechanisms simultaneously or sequentially. We define the class of smooth mechanisms, related … We initiate the study of efficient mechanism design with guaranteed good properties even when players participate in multiple mechanisms simultaneously or sequentially. We define the class of smooth mechanisms, related to smooth games defined by Roughgarden, that can be thought of as mechanisms that generate approximately market clearing prices. We show that smooth mechanisms result in high quality outcome both in equilibrium and in learning outcomes in the full information setting, as well as in Bayesian equilibrium with uncertainty about participants. Our main result is to show that smooth mechanisms compose well: smoothness locally at each mechanism implies global efficiency.
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Tight Bounds for the Price of Anarchy of Simultaneous First-Price Auctions

2016-01-13
George Christodoulou , Annamária Kovács , Alkmini Sgouritsa +1 more
We study the price of anarchy (PoA) of simultaneous first-price auctions (FPAs) for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian price of anarchy … We study the price of anarchy (PoA) of simultaneous first-price auctions (FPAs) for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian price of anarchy (BPoA) of these auctions are e /( e − 1) [Syrgkanis and Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching lower bounds for both cases even for the case of full information and for mixed Nash equilibria via an explicit construction. We present an alternative proof of the upper bound of e /( e − 1) for FPAs with fractionally subadditive valuations that reveals the worst-case price distribution, which is used as a building block for the matching lower bound construction. We generalize our results to a general class of item bidding auctions that we call bid-dependent auctions (including FPAs and all-pay auctions) where the winner is always the highest bidder and each bidder’s payment depends only on his own bid. Finally, we apply our techniques to discriminatory price multiunit auctions. We complement the results of de Keijzer et al. [2013] for the case of subadditive valuations by providing a matching lower bound of 2. For the case of submodular valuations, we provide a lower bound of 1.109. For the same class of valuations, we were able to reproduce the upper bound of e /( e − 1) using our nonsmooth approach.
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Pricing Multi-unit Markets

2018-01-01
Tomer Ezra , Michal Feldman , Tim Roughgarden +1 more
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Proximal Policy Optimization Algorithms

2017-01-01
John Schulman , Filip Wolski , Prafulla Dhariwal +2 more
We propose a new family of policy gradient methods for reinforcement learning, which alternate between sampling data through interaction with the environment, and optimizing a "surrogate" objective function using stochastic … We propose a new family of policy gradient methods for reinforcement learning, which alternate between sampling data through interaction with the environment, and optimizing a "surrogate" objective function using stochastic gradient ascent. Whereas standard policy gradient methods perform one gradient update per data sample, we propose a novel objective function that enables multiple epochs of minibatch updates. The new methods, which we call proximal policy optimization (PPO), have some of the benefits of trust region policy optimization (TRPO), but they are much simpler to implement, more general, and have better sample complexity (empirically). Our experiments test PPO on a collection of benchmark tasks, including simulated robotic locomotion and Atari game playing, and we show that PPO outperforms other online policy gradient methods, and overall strikes a favorable balance between sample complexity, simplicity, and wall-time.

Non-Optimal Mechanism Design

2015-10-01
Jason D. Hartline , Brendan Lucier
The optimal allocation of resources in complex environments—like allocation of dynamic wireless spectrum, cloud computing services, and Internet advertising—is computationally challenging even given the true preferences of the participants. In … The optimal allocation of resources in complex environments—like allocation of dynamic wireless spectrum, cloud computing services, and Internet advertising—is computationally challenging even given the true preferences of the participants. In the theory and practice of optimization in complex environments, a wide variety of special and general purpose algorithms have been developed; these algorithms produce outcomes that are satisfactory but not generally optimal or incentive compatible. This paper develops a very simple approach for converting any, potentially non-optimal, algorithm for optimization given the true participant preferences, into a Bayesian incentive compatible mechanism that weakly improves social welfare and revenue. (JEL D82, H82, L82)
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Minimum partition of a matroid into independent subsets

1965-01-01
Jack Edmonds
A matroid M is a finit e se t M of e le me nts with a famil y of subsets, calle d inde pe nde nt, s uc h … A matroid M is a finit e se t M of e le me nts with a famil y of subsets, calle d inde pe nde nt, s uc h th a t (I) every su bset of an ind e pe nd e nt se t is indepe nde nt, and ( 2) for e ve ry s ubset A of M , all maximal inde pe nd e nt s ub sets of A have th e sa me ca rdinality , calle d th e rank r\A) o f A. It is proved that a matroid ca n be partitione d into as few as k sets, eac h ind e pe nd e nt , if and o nly if e ve ry s ub se t A has cardinality at mos t k .r(A ).

Optimal mechanisms for selling information

2012-06-04
Moshe Babaioff , Robert Kleinberg , Renato Paes Leme
The buying and selling of information is taking place at a scale unprecedented in the history of commerce, thanks to the formation of online marketplaces for user data. Data providing … The buying and selling of information is taking place at a scale unprecedented in the history of commerce, thanks to the formation of online marketplaces for user data. Data providing agencies sell user information to advertisers to allow them to match ads to viewers more effectively. In this paper we study the design of optimal mechanisms for a monopolistic data provider to sell information to a buyer, in a model where both parties have (possibly correlated) private signals about a state of the world, and the buyer uses information learned from the seller, along with his own signal, to choose an action (e.g., displaying an ad) whose payoff depends on the state of the world.
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On the efficiency of the walrasian mechanism

2014-05-30
Moshe Babaioff , Brendan Lucier , Noam Nisan +1 more
Central results in economics guarantee the existence of efficient equilibria for various classes of markets. An underlying assumption in early work is that agents are price-takers, i.e., agents honestly report … Central results in economics guarantee the existence of efficient equilibria for various classes of markets. An underlying assumption in early work is that agents are price-takers, i.e., agents honestly report their true demand in response to prices. A line of research in economics, initiated by Hurwicz (1972), is devoted to understanding how such markets perform when agents are strategic about their demands. This is captured by the Walrasian Mechanism that proceeds by collecting reported demands, finding clearing prices in the reported market via an ascending price tatonnement procedure, and returns the resulting allocation. Similar mechanisms are used, for example, in the daily opening of the New York Stock Exchange and the call market for copper and gold in London.
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A duality based unified approach to Bayesian mechanism design

2016-06-10
Yang Cai , Nikhil R. Devanur , S. Matthew Weinberg
We provide a unified view of many recent developments in Bayesian mechanism design, including the black-box reductions of Cai et. al., simple auctions for additive buyers, and posted-price mechanisms for … We provide a unified view of many recent developments in Bayesian mechanism design, including the black-box reductions of Cai et. al., simple auctions for additive buyers, and posted-price mechanisms for unit-demand buyers. Additionally, we show that viewing these three previously disjoint lines of work through the same lens leads to new developments as well. First, we provide a duality framework for Bayesian mechanism design, which naturally accommodates multiple agents and arbitrary objectives/feasibility constraints. Using this, we prove that either a posted-price mechanism or the VCG auction with per-bidder entry fees achieves a constant-factor of the optimal Bayesian IC revenue whenever buyers are unit-demand or additive, unifying previous breakthroughs of Chawla et. al. and Yao, and improving both approximation ratios (from 33.75 to 24 and 69 to 8). Finally, we show that this view also leads to improved structural characterizations in the Cai et. al. framework.
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An impossibility result for truthful combinatorial auctions with submodular valuations

2011-06-06
Shahar Dobzinski
We show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that provides an approximation ratio of m1/ 2 -ε must use exponentially many value queries, where … We show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that provides an approximation ratio of m1/ 2 -ε must use exponentially many value queries, where m is the number of items. In contrast, ignoring incentives there exist constant ratio approximation algorithms for this problem. Our approach is based on a novel direct hardness technique that completely skips the notoriously hard step of characterizing truthful mechanisms. The characterization step was the main obstacle for proving impossibility results in algorithmic mechanism design so far. We demonstrate two additional applications of our new technique: (1) an impossibility result for universally-truthful polynomial time flexible combinatorial public projects and (2) an impossibility result for truthful-in-expectation mechanisms for exact combinatorial public projects. The latter is the first result that bounds the power of polynomial-time truthful in expectation mechanisms in any setting.
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Mechanism Design for Subadditive Agents via an Ex Ante Relaxation

2016-07-21
Shuchi Chawla , J. Benjamin Miller
We consider the problem of maximizing revenue for a monopolist offering multiple items to multiple heterogeneous buyers. We develop a simple mechanism that obtains a constant factor approximation under the … We consider the problem of maximizing revenue for a monopolist offering multiple items to multiple heterogeneous buyers. We develop a simple mechanism that obtains a constant factor approximation under the assumption that the buyers' values are additive subject to a matroid feasibility constraint and independent across items. Importantly, different buyers in our setting can have different constraints on the sets of items they desire. Our mechanism is a sequential variant of two-part tariffs. Prior to our work, simple approximation mechanisms for such multi-buyer problems were known only for the special cases of all unit-demand or all additive value buyers.
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Mechanism design via optimal transport

2013-06-16
Constantinos Daskalakis , Alan Deckelbaum , Christos Tzamos
Optimal mechanisms have been provided in quite general multi-item settings [Cai et al. 2012b, as long as each bidder's type distribution is given explicitly by listing every type in the … Optimal mechanisms have been provided in quite general multi-item settings [Cai et al. 2012b, as long as each bidder's type distribution is given explicitly by listing every type in the support along with its associated probability. In the implicit setting, e.g. when the bidders have additive valuations with independent and/or continuous values for the items, these results do not apply, and it was recently shown that exact revenue optimization is intractable, even when there is only one bidder [Daskalakis et al. 2013]. Even for item distributions with special structure, optimal mechanisms have been surprisingly rare [Manelli and Vincent 2006] and the problem is challenging even in the two-item case [Hart and Nisan 2012]. In this paper, we provide a framework for designing optimal mechanisms using optimal transport theory and duality theory. We instantiate our framework to obtain conditions under which only pricing the grand bundle is optimal in multi-item settings (complementing the work of [Manelli and Vincent 2006]), as well as to characterize optimal two-item mechanisms. We use our results to derive closed-form descriptions of the optimal mechanism in several two-item settings, exhibiting also a setting where a continuum of lotteries is necessary for revenue optimization but a closed-form representation of the mechanism can still be found efficiently using our framework.
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Combinatorial Walrasian Equilibrium

2016-01-01
Michal Feldman , Nick Gravin , Brendan Lucier
We study a combinatorial market design problem, where a collection of indivisible objects is to be priced and sold to potential buyers subject to equilibrium constraints. The classic solution concept … We study a combinatorial market design problem, where a collection of indivisible objects is to be priced and sold to potential buyers subject to equilibrium constraints. The classic solution concept for such problems is Walrasian equilibrium (WE), which provides a simple and transparent pricing structure that achieves optimal social welfare. The main weakness of the WE notion is that it exists only in very restrictive cases. To overcome this limitation, we introduce the notion of a combinatorial Walrasian equilibium (CWE), a natural relaxation of WE. The difference between a CWE and a (noncombinatorial) WE is that the seller can package the items into indivisible bundles prior to sale, and the market does not necessarily clear. We show that every valuation profile admits a CWE that obtains at least half the optimal (unconstrained) social welfare. Moreover, we devise a polynomial time algorithm that, given an arbitrary allocation, computes a CWE that achieves at least half its welfare. Thus, the economic problem of finding a CWE with high social welfare reduces to the algorithmic problem of social-welfare approximation. In addition, we show that every valuation profile admits a CWE that extracts a logarithmic fraction of the optimal welfare as revenue. Finally, to motivate the use of bundles, we establish strong lower bounds when the seller is restricted to using item prices only. The strength of our results derives partly from their generality---our results hold for arbitrary valuations that may exhibit complex combinations of substitutes and complements.
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Makespan Minimization via Posted Prices

2017-06-20
Michal Feldman , Amos Fiat , Alan Roytman
We consider job scheduling settings, with multiple machines, where jobs arrive online and choose a machine selfishly so as to minimize their cost. Our objective is the classic makespan minimization … We consider job scheduling settings, with multiple machines, where jobs arrive online and choose a machine selfishly so as to minimize their cost. Our objective is the classic makespan minimization objective, which corresponds to the completion time of the last job to complete. The incentives of the selfish jobs may lead to poor performance. To reconcile the differing objectives, we introduce posted machine prices. The selfish job seeks to minimize the sum of its completion time on the machine and the posted price for the machine. Prices may be static (i.e., set once and for all before any arrival) or dynamic (i.e., change over time), but they are determined only by the past, assuming nothing about upcoming events. Obviously, such schemes are inherently truthful.
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Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

2016-12-09
Paul Dütting , Michal Feldman , Thomas Keßelheim +1 more
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold … We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.

The Invisible Hand of Dynamic Market Pricing

2016-07-24
Vincent Cohen-Addad , Alon Eden , Michal Feldman +1 more
Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, … Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.
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Bounding the inefficiency of outcomes in generalized second price auctions

2014-04-30
Ioannis Caragiannis , Christos Kaklamanis , Panagiotis Kanellopoulos +4 more
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Bayesian mechanism design for budget-constrained agents

2011-06-05
Shuchi Chawla , David L. Malec , Azarakhsh Malekian
We study Bayesian mechanism design problems in settings where agents have budgets. Specifically, an agent's utility for an outcome is given by his value for the outcome minus any payment … We study Bayesian mechanism design problems in settings where agents have budgets. Specifically, an agent's utility for an outcome is given by his value for the outcome minus any payment he makes to the mechanism, as long as the payment is below his budget, and is negative infinity otherwise. This discontinuity in the utility function presents a significant challenge in the design of good mechanisms, and classical mechanisms fail to work in settings with budgets. The goal of this paper is to develop general reductions from budget-constrained Bayesian MD to unconstrained Bayesian MD with small loss in performance. We consider this question in the context of the two most well-studied objectives in mechanism design---social welfare and revenue---and present constant factor approximations in a number of settings. Some of our results extend to settings where budgets are private and agents need to be incentivized to reveal them truthfully.
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Computing Walrasian Equilibria: Fast Algorithms and Economic Insights.

2015-11-12
Renato Paes Leme , Sam Chiu-wai Wong
We give the first polynomial algorithm to compute a Walrasian equilibrium in an economy with indivisible goods and general valuations having only access to an aggregate demand oracle, i.e., an … We give the first polynomial algorithm to compute a Walrasian equilibrium in an economy with indivisible goods and general valuations having only access to an aggregate demand oracle, i.e., an oracle that given a price vector, returns the aggregated demand over the entire population of buyers. Our algorithm queries the aggregate demand oracle $\tilde{O}(n)$ times and takes $\tilde{O}(n^3)$ time, where n is the number of items. We also give the fastest known algorithm for computing Walrasian equilibrium for gross substitute valuations in the value oracle model. Our algorithm has running time $\tilde{O}((mn+n^3 )T_V)$ where $T_V$ is the cost of querying the value oracle. En route, we give necessary and sufficient conditions for the existence of robust Walrasian prices, i.e., price vectors such that each agent has a unique demanded bundle and the demanded bundles clear the market. When such prices exist, the market can be perfectly coordinated by solely using prices.