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Mechanism design for social good

2018-10-19
Rediet Abebe , Kira Goldner
Across various domains---such as health, education, and housing---improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives ā€¦ Across various domains---such as health, education, and housing---improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives of individuals, whether in the form of access to quality healthcare, labor market outcomes, or how votes are accounted for in a democratic society. Problems that can have an outsized impact on individuals whose opportunities have historically been limited often pose conceptual and technical challenges, requiring insights from many disciplines. Conversely, the lack of inter-disciplinary approach can leave these urgent needs unaddressed and can even exacerbate underlying socioeconomic inequalities.
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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 ā€¦ 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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Optimal Mechanism Design for Single-Minded Agents

2020-07-09
Nikhil R. Devanur , Kira Goldner , Raghuvansh R. Saxena +2 more
We consider optimal (revenue maximizing) mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and the other is a 'type' that captures some auxiliary ā€¦ We consider optimal (revenue maximizing) mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and the other is a 'type' that captures some auxiliary information. A prototypical example of this is the FedEx Problem, for which Fiat et al. [2016] characterize the optimal mechanism for a single agent. Another example of this is when the type encodes the buyer's budget [DW17]. The question we address is how far can such characterizations goIn particular, we consider the setting of single-minded agents. A seller has heterogenous items. A buyer has a valuation vfor a specific subset of items S, and obtains value vif and only if he gets all the items in S(and potentially some others too).
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Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided Markets

2019-12-23
Moshe Babaioff , Kira Goldner , Yannai A. Gonczarowski
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided MarketsMoshe Babaioff, Kira Goldner, and Yannai A. GonczarowskiMoshe ā€¦ Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided MarketsMoshe Babaioff, Kira Goldner, and Yannai A. GonczarowskiMoshe Babaioff, Kira Goldner, and Yannai A. Gonczarowskipp.2452 - 2471Chapter DOI:https://doi.org/10.1137/1.9781611975994.150PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider the problem of welfare (and gains-from-trade) maximization in two-sided markets using simple mechanisms that are prior-independent. The seminal impossibility result of Myerson and Satterthwaite [1983] shows that even for bilateral trade, there is no feasible (individually rational, truthful, and budget balanced) mechanism that has welfare as high as the optimal-yet-infeasible VCG mechanism, which attains maximal welfare but runs a deficit. On the other hand, the optimal feasible mechanism needs to be carefully tailored to the Bayesian prior, and even worse, it is known to be extremely complex, eluding a precise description. In this paper we present Bulow-Klemperer-style results to circumvent these hurdles in double-auction market settings. We suggest using the Buyer Trade Reduction (BTR) mechanism, a variant of McAfee's mechanism, which is feasible and simple (in particular, it is deterministic, truthful, prior-independent, and anonymous). First, in the setting in which the values of the buyers and of the sellers are sampled independently and identically from the same distribution, we show that for any such market of any size, BTR with one additional buyer whose value is sampled from the same distribution has expected welfare at least as high as the optimal-yet-infeasible VCG mechanism in the original market. We then move to a more general setting in which the values of the buyers are sampled from one distribution, and those of the sellers from another, focusing on the case where the buyers' distribution first-order stochastically dominates the sellers' distribution. We present both upper bounds and lower bounds on the number of buyers that, when added, guarantees that BTR in the augmented market achieve welfare at least as high as the optimal in the original market. Our lower bounds extend to a large class of mechanisms, and all of our positive and negative results extend to adding sellers instead of buyers. In addition, we present positive results about the usefulness of pricing at a sample for welfare maximization (and more precisely, for gains-from-trade approximation) in two-sided markets under the above two settings, which to the best of our knowledge are the first sampling results in this context. Previous chapter Next chapter RelatedDetails Published:2020eISBN:978-1-61197-599-4 https://doi.org/10.1137/1.9781611975994Book Series Name:ProceedingsBook Code:PRDA20Book Pages:xxii + 3011
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On Multi-Dimensional Gains from Trade Maximization

2021-01-01
Yang Cai , Kira Goldner , Steven Ma +1 more
We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with n heterogeneous items, where each item is owned by a different seller i, and ā€¦ We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with n heterogeneous items, where each item is owned by a different seller i, and there is a constrained-additive buyer with feasibility constraint ā„±. Multi-dimensional settings in one-sided markets, e.g. where a seller owns multiple heterogeneous items but also is the mechanism designer, are well-understood. In addition, single-dimensional settings in two-sided markets, e.g. where a buyer and seller each seek or own a single item, are also well-understood. Multi-dimensional two-sided markets, however, encapsulate the major challenges of both lines of work: optimizing the sale of heterogeneous items, ensuring incentive-compatibility among both sides of the market, and enforcing budget balance. We present, to the best of our knowledge, the first worst-case approximation guarantee for gains from trade in a multi-dimensional two-sided market.Our first result provides an O(log(1/r))-approximation to the first-best gains from trade for a broad class of downward-closed feasibility constraints (such as matroid, matching, knapsack, or the intersection of these). Here r is the minimum probability over all items that a buyer's value for the item exceeds the seller's cost. Our second result removes the dependence on r and provides an unconditional O(log n)-approximation to the second-best gains from trade. We extend both results for a general constrained-additive buyer, losing another O(log n)-factor en-route. The first result is achieved using a fixed posted price mechanism, and the analysis involves a novel application of the prophet inequality or a new concentration inequality. Our second result follows from a stitching lemma that allows us to upper bound the second-best gains from trade by the first-best gains from trade from the "likely to trade" items (items with trade probability at least 1/n) and the optimal profit from selling the "unlikely to trade" items. We can obtain an O(log n)-approximation to the first term by invoking our O(log(1/r))-approximation on the "likely to trade" items. We introduce a generalization of the fixed posted price mechanismā€”seller adjusted posted priceā€”to obtain an O(log n)-approximation to the optimal profit for the "unlikely to trade" items. Unlike fixed posted price mechanisms, not all seller adjusted posted price mechanisms are incentive compatible and budget balanced. We develop a new argument based on "allocation coupling" to show the seller adjusted posted price mechanism used in our approximation is indeed budget balanced and incentive-compatible.
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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 ā€¦ 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, ā€¦, 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 functionsā€”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 Ī˜(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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Revenue Maximization with an Uncertainty-Averse Buyer

2018-01-01
Shuchi Chawla , Kira Goldner , J. Benjamin Miller +1 more
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real ā€¦ Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability x < 1 is worth strictly less to the agent than x times the value of the event when it occurs with certainty.We present three main results. First, we characterize optimal mechanisms as menus of two-outcome lotteries. Second, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation to the optimal revenue. Notably, this result is "risk-robust" in that it does not depend on the details of the buyer's risk attitude. Third, we consider dynamic settings in which the buyer's uncertainty about his future value may allow the seller to extract more revenue. In contrast to the positive result above, here we show it is not possible to achieve any constant-factor approximation to revenue using deterministic mechanisms in a risk-robust manner.
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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 ā€¦ 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 ā€¦ 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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Reducing Inefficiency in Carbon Auctions with Imperfect Competition

2019-01-01
Kira Goldner , Nicole Immorlica , Brendan Lucier
We study auctions for carbon licenses, a policy tool used to control the social cost of pollution. Each identical license grants the right to produce a unit of pollution. Each ā€¦ We study auctions for carbon licenses, a policy tool used to control the social cost of pollution. Each identical license grants the right to produce a unit of pollution. Each buyer (i.e., firm that pollutes during the manufacturing process) enjoys a decreasing marginal value for licenses, but society suffers an increasing marginal cost for each license distributed. The seller (i.e., the government) can choose a number of licenses to put up for auction, and wishes to maximize the societal welfare: the total economic value of the buyers minus the social cost. Motivated by emission license markets deployed in practice, we focus on uniform price auctions with a price floor and/or price ceiling. The seller has distributional information about the market, and their goal is to tune the auction parameters to maximize expected welfare. The target benchmark is the maximum expected welfare achievable by any such auction under truth-telling behavior. Unfortunately, the uniform price auction is not truthful, and strategic behavior can significantly reduce (even below zero) the welfare of a given auction configuration. We describe a subclass of "safe-price'" auctions for which the welfare at any Bayes-Nash equilibrium will approximate the welfare under truth-telling behavior. We then show that the better of a safe-price auction, or a truthful auction that allocates licenses to only a single buyer, will approximate the target benchmark. In particular, we show how to choose a number of licenses and a price floor so that the worst-case welfare, at any equilibrium, is a constant approximation to the best achievable welfare under truth-telling after excluding the welfare contribution of a single buyer.
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Optimal Mechanism Design for Single-Minded Agents

2020-02-15
Nikhil R. Devanur , Kira Goldner , Raghuvansh R. Saxena +2 more
We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting ā€¦ We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting is the FedEx Problem, for which FGKK [2016] characterize the optimal mechanism for a single agent. We ask: how far can such characterizations go? In particular, we consider single-minded agents. A seller has heterogenous items. A buyer has a value v for a specific subset of items S, and obtains value v iff he gets (at least) all the items in S. We show: 1. Deterministic mechanisms are optimal for distributions that satisfy the declining marginal revenue (DMR) property; we give an explicit construction of the optimal mechanism. 2. Without DMR, the result depends on the structure of the directed acyclic graph (DAG) representing the partial order among types. When the DAG has out-degree at most 1, we characterize the optimal mechanism a la FedEx. 3. Without DMR, when the DAG has some node with out-degree at least 2, we show that in this case the menu complexity is unbounded: for any M, there exist distributions over (v,S) pairs such that the menu complexity of the optimal mechanism is at least M. 4. For the case of 3 types, we show that for all distributions there exists an optimal mechanism of finite menu complexity. This is in contrast to 2 additive heterogenous items or which the menu complexity could be uncountable [MV07; DDT15]. In addition, we prove that optimal mechanisms for Multi-Unit Pricing (without DMR) can have unbounded menu complexity. We also propose an extension where the menu complexity of optimal mechanisms can be countable but not uncountable. Together these results establish that optimal mechanisms in interdimensional settings are both much richer than single-dimensional settings, yet also vastly more structured than multi-dimensional settings.

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 ā€¦ 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 ā€¦ 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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Simple and Approximately Optimal Pricing for Proportional Complementarities

2019-06-17
Yang Cai , Nikhil R. Devanur , Kira Goldner +1 more
We study a new model of complementary valuations, which we call "proportional complementarities.'' In contrast to common models, such as hypergraphic valuations, in our model, we do not assume that ā€¦ We study a new model of complementary valuations, which we call "proportional complementarities.'' In contrast to common models, such as hypergraphic valuations, in our model, we do not assume that the extra value derived from owning a set of items is independent of the buyer's base valuations for the items. Instead, we model the complementarities as proportional to the buyer's base valuations, and these proportionalities are known market parameters.
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On Multi-Dimensional Gains from Trade Maximization

2020-01-01
Yang Cai , Kira Goldner , Steven Ma +1 more
We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and ā€¦ We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and there is a constrained-additive buyer with feasibility constraint $\mathcal{F}$. Multi-dimensional settings in one-sided markets, e.g. where a seller owns multiple heterogeneous items but also is the mechanism designer, are well-understood. In addition, single-dimensional settings in two-sided markets, e.g. where a buyer and seller each seek or own a single item, are also well-understood. Multi-dimensional two-sided markets, however, encapsulate the major challenges of both lines of work: optimizing the sale of heterogeneous items, ensuring incentive-compatibility among both sides of the market, and enforcing budget balance. We present, to the best of our knowledge, the first worst-case approximation guarantee for gains from trade in a multi-dimensional two-sided market. Our first result provides an $O(\log (1/r))$-approximation to the first-best gains from trade for a broad class of downward-closed feasibility constraints (such as matroid, matching, knapsack, or the intersection of these). Here $r$ is the minimum probability over all items that a buyer's value for the item exceeds the seller's cost. Our second result removes the dependence on $r$ and provides an unconditional $O(\log n)$-approximation to the second-best gains from trade. We extend both results for a general constrained-additive buyer, losing another $O(\log n)$-factor en-route.
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Non-Adaptive Matroid Prophet Inequalities

2020-01-01
Shuchi Chawla , Kira Goldner , Anna R. Karlin +1 more
We investigate non-adaptive algorithms for matroid prophet inequalities. Matroid prophet inequalities have been considered resolved since 2012 when [KW12] introduced thresholds that guarantee a tight 2-approximation to the prophet; however, ā€¦ We investigate non-adaptive algorithms for matroid prophet inequalities. Matroid prophet inequalities have been considered resolved since 2012 when [KW12] introduced thresholds that guarantee a tight 2-approximation to the prophet; however, this algorithm is adaptive. Other approaches of [CHMS10] and [FSZ16] have used non-adaptive thresholds with a feasibility restriction; however, this translates to adaptively changing an item's threshold to infinity when it cannot be taken with respect to the additional feasibility constraint, hence the algorithm is not truly non-adaptive. A major application of prophet inequalities is in auction design, where non-adaptive prices possess a significant advantage: they convert to order-oblivious posted pricings, and are essential for translating a prophet inequality into a truthful mechanism for multi-dimensional buyers. The existing matroid prophet inequalities do not suffice for this application. We present the first non-adaptive constant-factor prophet inequality for graphic matroids.

Aversion to Uncertainty and Its Implications for Revenue Maximization.

2017-03-24
Shuchi Chawla , Kira Goldner , J. Benjamin Miller +1 more
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real ā€¦ Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is worth strictly less to the agent than $x$ times the value of the event when it occurs with certainty. In contrast to the risk-neutral setting, the optimal mechanism may be randomized and appears challenging to find, even for a single buyer and a single item for sale. Nevertheless, we give a characterization of the optimal mechanism which enables positive approximation results. In particular, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation. Notably, this result is in that it does not depend on the details of the buyer's risk attitude. Finally, we examine a dynamic setting in which the buyer is uncertain about his future value. In contrast to positive results for a risk-neutral buyer, we show that the buyer's risk aversion may prevent the seller from approximating the optimal revenue in a risk-robust manner.

Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided Markets

2019-03-15
Moshe Babaioff , Kira Goldner , Yannai A. Gonczarowski
We consider the problem of welfare maximization in two-sided markets using simple mechanisms that are prior-independent. The Myerson-Satterthwaite impossibility theorem shows that even for bilateral trade, there is no feasible ā€¦ We consider the problem of welfare maximization in two-sided markets using simple mechanisms that are prior-independent. The Myerson-Satterthwaite impossibility theorem shows that even for bilateral trade, there is no feasible (IR, truthful, budget balanced) mechanism that has welfare as high as the optimal-yet-infeasible VCG mechanism, which attains maximal welfare but runs a deficit. On the other hand, the optimal feasible mechanism needs to be carefully tailored to the Bayesian prior, and is extremely complex, eluding a precise description. We present Bulow-Klemperer-style results to circumvent these hurdles in double-auction markets. We suggest using the Buyer Trade Reduction (BTR) mechanism, a variant of McAfee's mechanism, which is feasible and simple (in particular, deterministic, truthful, prior-independent, anonymous). First, in the setting where buyers' and sellers' values are sampled i.i.d. from the same distribution, we show that for any such market of any size, BTR with one additional buyer whose value is sampled from the same distribution has expected welfare at least as high as the optimal in the original market. We then move to a more general setting where buyers' values are sampled from one distribution and sellers' from another, focusing on the case where the buyers' distribution first-order stochastically dominates the sellers'. We present bounds on the number of buyers that, when added, guarantees that BTR in the augmented market have welfare at least as high as the optimal in the original market. Our lower bounds extend to a large class of mechanisms, and all of our results extend to adding sellers instead of buyers. In addition, we present positive results about the usefulness of pricing at a sample for welfare maximization in two-sided markets under the above two settings, which to the best of our knowledge are the first sampling results in this context.

Mechanism Design for Social Good

2018-01-01
Rediet Abebe , Kira Goldner
Across various domains--such as health, education, and housing--improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives ā€¦ Across various domains--such as health, education, and housing--improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives of individuals, whether in the form of access to quality healthcare, labor market outcomes, or how votes are accounted for in a democratic society. Problems that can have an out-sized impact on individuals whose opportunities have historically been limited often pose conceptual and technical challenges, requiring insights from many disciplines. Conversely, the lack of interdisciplinary approach can leave these urgent needs unaddressed and can even exacerbate underlying socioeconomic inequalities. To realize the opportunities in these domains, we need to correctly set objectives and reason about human behavior and actions. Doing so requires a deep grounding in the field of interest and collaboration with domain experts who understand the societal implications and feasibility of proposed solutions. These insights can play an instrumental role in proposing algorithmically-informed policies. In this article, we describe the Mechanism Design for Social Good (MD4SG) research agenda, which involves using insights from algorithms, optimization, and mechanism design to improve access to opportunity. The MD4SG research community takes an interdisciplinary, multi-stakeholder approach to improve societal welfare. We discuss three exciting research avenues within MD4SG related to improving access to opportunity in the developing world, labor markets and discrimination, and housing. For each of these, we showcase ongoing work, underline new directions, and discuss potential for implementing existing work in practice.
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Revenue Maximization with an Uncertainty-Averse Buyer

2017-03-24
Shuchi Chawla , Kira Goldner , J. Benjamin Miller +1 more
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real ā€¦ Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is worth strictly less to the agent than $x$ times the value of the event when it occurs with certainty. In contrast to the risk-neutral setting, the optimal mechanism may be randomized and appears challenging to find, even for a single buyer and a single item for sale. Nevertheless, we give a characterization of the optimal mechanism which enables positive approximation results. In particular, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation. Notably, this result is in that it does not depend on the details of the buyer's risk attitude. Finally, we examine a dynamic setting in which the buyer is uncertain about his future value. In contrast to positive results for a risk-neutral buyer, we show that the buyer's risk aversion may prevent the seller from approximating the optimal revenue in a risk-robust manner.

When to Limit Market Entry under Mandatory Purchase.

2020-02-15
Meryem Essaidi , Kira Goldner , S. Matthew Weinberg
We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with ā€¦ We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with private insurers to provide plans for their employees. The market regulator can choose to do nothing, running a Free Market, or can exercise her regulatory power by limiting the entry of providers (decreasing consumer welfare by limiting options, but also decreasing revenue via enhanced competition). We investigate whether limiting entry increases or decreases the utility (welfare minus revenue) of the consumers who purchase from the providers, specifically in settings where the outside option of purchasing nothing is prohibitively undesirable. We focus primarily on the case where providers are symmetric. We propose a sufficient condition on the distribution of consumer values for (a) a unique symmetric equilibrium to exist in both markets and (b) utility to be higher with limited entry. (We also establish that these conclusions do not necessarily hold for all distributions, and therefore some condition is necessary.) Our techniques are primarily based on tools from revenue maximization, and in particular Myerson's virtual value theory. We also consider extensions to settings where providers have identical costs for providing plans, and to two providers with an asymmetric distribution.

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. ā€¦ 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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Mechanism Design for Social Good

2021-09-28
Rediet Abebe , Kira Goldner
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On Multi-Dimensional Gains from Trade Maximization

2020-07-27
Yang Cai , Kira Goldner , Steven Ma +1 more
We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and ā€¦ We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and there is a constrained-additive buyer with feasibility constraint $\mathcal{F}$. Multi-dimensional settings in one-sided markets, e.g. where a seller owns multiple heterogeneous items but also is the mechanism designer, are well-understood. In addition, single-dimensional settings in two-sided markets, e.g. where a buyer and seller each seek or own a single item, are also well-understood. Multi-dimensional two-sided markets, however, encapsulate the major challenges of both lines of work: optimizing the sale of heterogeneous items, ensuring incentive-compatibility among both sides of the market, and enforcing budget balance. We present, to the best of our knowledge, the first worst-case approximation guarantee for gains from trade in a multi-dimensional two-sided market. Our first result provides an $O(\log (1/r))$-approximation to the first-best gains from trade for a broad class of downward-closed feasibility constraints (such as matroid, matching, knapsack, or the intersection of these). Here $r$ is the minimum probability over all items that a buyer's value for the item exceeds the seller's cost. Our second result removes the dependence on $r$ and provides an unconditional $O(\log n)$-approximation to the second-best gains from trade. We extend both results for a general constrained-additive buyer, losing another $O(\log n)$-factor en-route.
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When to Limit Market Entry under Mandatory Purchase

2020-02-15
Meryem Essaidi , Kira Goldner , S. Matthew Weinberg
We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with ā€¦ We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with private insurers to provide plans for their employees. The market regulator can choose to do nothing, running a Free Market, or can exercise her regulatory power by limiting the entry of providers (decreasing consumer welfare by limiting options, but also decreasing revenue via enhanced competition). We investigate whether limiting entry increases or decreases the utility (welfare minus revenue) of the consumers who purchase from the providers, specifically in settings where the outside option of "purchasing nothing" is prohibitively undesirable. We focus primarily on the case where providers are symmetric. We propose a sufficient condition on the distribution of consumer values for (a) a unique symmetric equilibrium to exist in both markets and (b) utility to be higher with limited entry. (We also establish that these conclusions do not necessarily hold for all distributions, and therefore some condition is necessary.) Our techniques are primarily based on tools from revenue maximization, and in particular Myerson's virtual value theory. We also consider extensions to settings where providers have identical costs for providing plans, and to two providers with an asymmetric distribution.
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Optimal Mechanism Design for Single-Minded Agents

2020-01-01
Nikhil R. Devanur , Kira Goldner , Raghuvansh R. Saxena +2 more
We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting ā€¦ We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting is the FedEx Problem, for which FGKK [2016] characterize the optimal mechanism for a single agent. We ask: how far can such characterizations go? In particular, we consider single-minded agents. A seller has heterogenous items. A buyer has a value v for a specific subset of items S, and obtains value v iff he gets (at least) all the items in S. We show: 1. Deterministic mechanisms are optimal for distributions that satisfy the "declining marginal revenue" (DMR) property; we give an explicit construction of the optimal mechanism. 2. Without DMR, the result depends on the structure of the directed acyclic graph (DAG) representing the partial order among types. When the DAG has out-degree at most 1, we characterize the optimal mechanism a la FedEx. 3. Without DMR, when the DAG has some node with out-degree at least 2, we show that in this case the menu complexity is unbounded: for any M, there exist distributions over (v,S) pairs such that the menu complexity of the optimal mechanism is at least M. 4. For the case of 3 types, we show that for all distributions there exists an optimal mechanism of finite menu complexity. This is in contrast to 2 additive heterogenous items or which the menu complexity could be uncountable [MV07; DDT15]. In addition, we prove that optimal mechanisms for Multi-Unit Pricing (without DMR) can have unbounded menu complexity. We also propose an extension where the menu complexity of optimal mechanisms can be countable but not uncountable. Together these results establish that optimal mechanisms in interdimensional settings are both much richer than single-dimensional settings, yet also vastly more structured than multi-dimensional settings.

Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided Markets

2019-01-01
Moshe Babaioff , Kira Goldner , Yannai A. Gonczarowski
We consider the problem of welfare maximization in two-sided markets using simple mechanisms that are prior-independent. The Myerson-Satterthwaite impossibility theorem shows that even for bilateral trade, there is no feasible ā€¦ We consider the problem of welfare maximization in two-sided markets using simple mechanisms that are prior-independent. The Myerson-Satterthwaite impossibility theorem shows that even for bilateral trade, there is no feasible (IR, truthful, budget balanced) mechanism that has welfare as high as the optimal-yet-infeasible VCG mechanism, which attains maximal welfare but runs a deficit. On the other hand, the optimal feasible mechanism needs to be carefully tailored to the Bayesian prior, and is extremely complex, eluding a precise description. We present Bulow-Klemperer-style results to circumvent these hurdles in double-auction markets. We suggest using the Buyer Trade Reduction (BTR) mechanism, a variant of McAfee's mechanism, which is feasible and simple (in particular, deterministic, truthful, prior-independent, anonymous). First, in the setting where buyers' and sellers' values are sampled i.i.d. from the same distribution, we show that for any such market of any size, BTR with one additional buyer whose value is sampled from the same distribution has expected welfare at least as high as the optimal in the original market. We then move to a more general setting where buyers' values are sampled from one distribution and sellers' from another, focusing on the case where the buyers' distribution first-order stochastically dominates the sellers'. We present bounds on the number of buyers that, when added, guarantees that BTR in the augmented market have welfare at least as high as the optimal in the original market. Our lower bounds extend to a large class of mechanisms, and all of our results extend to adding sellers instead of buyers. In addition, we present positive results about the usefulness of pricing at a sample for welfare maximization in two-sided markets under the above two settings, which to the best of our knowledge are the first sampling results in this context.
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Revenue Maximization with an Uncertainty-Averse Buyer

2017-01-01
Shuchi Chawla , Kira Goldner , J. Benjamin Miller +1 more
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real ā€¦ Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is worth strictly less to the agent than $x$ times the value of the event when it occurs with certainty. In contrast to the risk-neutral setting, the optimal mechanism may be randomized and appears challenging to find, even for a single buyer and a single item for sale. Nevertheless, we give a characterization of the optimal mechanism which enables positive approximation results. In particular, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation. Notably, this result is "risk-robust" in that it does not depend on the details of the buyer's risk attitude. Finally, we examine a dynamic setting in which the buyer is uncertain about his future value. In contrast to positive results for a risk-neutral buyer, we show that the buyer's risk aversion may prevent the seller from approximating the optimal revenue in a risk-robust manner.
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When to Limit Market Entry under Mandatory Purchase

2020-01-01
Meryem Essaidi , Kira Goldner , S. Matthew Weinberg
We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with ā€¦ We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with private insurers to provide plans for their employees. The market regulator can choose to do nothing, running a Free Market, or can exercise her regulatory power by limiting the entry of providers (decreasing consumer welfare by limiting options, but also decreasing revenue via enhanced competition). We investigate whether limiting entry increases or decreases the utility (welfare minus revenue) of the consumers who purchase from the providers, specifically in settings where the outside option of "purchasing nothing" is prohibitively undesirable. We focus primarily on the case where providers are symmetric. We propose a sufficient condition on the distribution of consumer values for (a) a unique symmetric equilibrium to exist in both markets and (b) utility to be higher with limited entry. (We also establish that these conclusions do not necessarily hold for all distributions, and therefore some condition is necessary.) Our techniques are primarily based on tools from revenue maximization, and in particular Myerson's virtual value theory. We also consider extensions to settings where providers have identical costs for providing plans, and to two providers with an asymmetric distribution.

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 ā€¦ 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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Simple Mechanisms for Utility Maximization: Approximating Welfare in the I.I.D. Unit-Demand Setting

2024-02-19
Kira Goldner , Taylor Lundy
We investigate the objective of utility maximization from the perspective of Bayesian mechanism design, initiating this direction, and focus on the unit-demand setting where values are i.i.d. across both items ā€¦ We investigate the objective of utility maximization from the perspective of Bayesian mechanism design, initiating this direction, and focus on the unit-demand setting where values are i.i.d. across both items and buyers. We take the approach of developing simple, approximately optimal mechanisms, targeting the simplest benchmark of optimal welfare. We give a $(1-1/e)$-approximation when there are more items than buyers, and an $O(\log(n/m))$-approximation when there are more buyers than items, which is tight up to constant factors. We also characterize complexities in this setting that defy our intuition from the welfare and revenue literature, and motivate why coming up with a better benchmark than welfare is a hard problem itself.
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Simple Mechanisms for Utility Maximization: Approximating Welfare in the I.I.D. Unit-Demand Setting

2024-02-19
Kira Goldner , Taylor Lundy
We investigate the objective of utility maximization from the perspective of Bayesian mechanism design, initiating this direction, and focus on the unit-demand setting where values are i.i.d. across both items ā€¦ We investigate the objective of utility maximization from the perspective of Bayesian mechanism design, initiating this direction, and focus on the unit-demand setting where values are i.i.d. across both items and buyers. We take the approach of developing simple, approximately optimal mechanisms, targeting the simplest benchmark of optimal welfare. We give a $(1-1/e)$-approximation when there are more items than buyers, and an $O(\log(n/m))$-approximation when there are more buyers than items, which is tight up to constant factors. We also characterize complexities in this setting that defy our intuition from the welfare and revenue literature, and motivate why coming up with a better benchmark than welfare is a hard problem itself.
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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 ā€¦ 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 ā€¦ 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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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 ā€¦ 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 ā€¦ 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, ā€¦, 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 functionsā€”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 Ī˜(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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Mechanism Design for Social Good

2021-09-28
Rediet Abebe , Kira Goldner
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On Multi-Dimensional Gains from Trade Maximization

2021-01-01
Yang Cai , Kira Goldner , Steven Ma +1 more
We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with n heterogeneous items, where each item is owned by a different seller i, and ā€¦ We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with n heterogeneous items, where each item is owned by a different seller i, and there is a constrained-additive buyer with feasibility constraint ā„±. Multi-dimensional settings in one-sided markets, e.g. where a seller owns multiple heterogeneous items but also is the mechanism designer, are well-understood. In addition, single-dimensional settings in two-sided markets, e.g. where a buyer and seller each seek or own a single item, are also well-understood. Multi-dimensional two-sided markets, however, encapsulate the major challenges of both lines of work: optimizing the sale of heterogeneous items, ensuring incentive-compatibility among both sides of the market, and enforcing budget balance. We present, to the best of our knowledge, the first worst-case approximation guarantee for gains from trade in a multi-dimensional two-sided market.Our first result provides an O(log(1/r))-approximation to the first-best gains from trade for a broad class of downward-closed feasibility constraints (such as matroid, matching, knapsack, or the intersection of these). Here r is the minimum probability over all items that a buyer's value for the item exceeds the seller's cost. Our second result removes the dependence on r and provides an unconditional O(log n)-approximation to the second-best gains from trade. We extend both results for a general constrained-additive buyer, losing another O(log n)-factor en-route. The first result is achieved using a fixed posted price mechanism, and the analysis involves a novel application of the prophet inequality or a new concentration inequality. Our second result follows from a stitching lemma that allows us to upper bound the second-best gains from trade by the first-best gains from trade from the "likely to trade" items (items with trade probability at least 1/n) and the optimal profit from selling the "unlikely to trade" items. We can obtain an O(log n)-approximation to the first term by invoking our O(log(1/r))-approximation on the "likely to trade" items. We introduce a generalization of the fixed posted price mechanismā€”seller adjusted posted priceā€”to obtain an O(log n)-approximation to the optimal profit for the "unlikely to trade" items. Unlike fixed posted price mechanisms, not all seller adjusted posted price mechanisms are incentive compatible and budget balanced. We develop a new argument based on "allocation coupling" to show the seller adjusted posted price mechanism used in our approximation is indeed budget balanced and incentive-compatible.
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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. ā€¦ 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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On Multi-Dimensional Gains from Trade Maximization

2020-07-27
Yang Cai , Kira Goldner , Steven Ma +1 more
We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and ā€¦ We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and there is a constrained-additive buyer with feasibility constraint $\mathcal{F}$. Multi-dimensional settings in one-sided markets, e.g. where a seller owns multiple heterogeneous items but also is the mechanism designer, are well-understood. In addition, single-dimensional settings in two-sided markets, e.g. where a buyer and seller each seek or own a single item, are also well-understood. Multi-dimensional two-sided markets, however, encapsulate the major challenges of both lines of work: optimizing the sale of heterogeneous items, ensuring incentive-compatibility among both sides of the market, and enforcing budget balance. We present, to the best of our knowledge, the first worst-case approximation guarantee for gains from trade in a multi-dimensional two-sided market. Our first result provides an $O(\log (1/r))$-approximation to the first-best gains from trade for a broad class of downward-closed feasibility constraints (such as matroid, matching, knapsack, or the intersection of these). Here $r$ is the minimum probability over all items that a buyer's value for the item exceeds the seller's cost. Our second result removes the dependence on $r$ and provides an unconditional $O(\log n)$-approximation to the second-best gains from trade. We extend both results for a general constrained-additive buyer, losing another $O(\log n)$-factor en-route.
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Optimal Mechanism Design for Single-Minded Agents

2020-07-09
Nikhil R. Devanur , Kira Goldner , Raghuvansh R. Saxena +2 more
We consider optimal (revenue maximizing) mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and the other is a 'type' that captures some auxiliary ā€¦ We consider optimal (revenue maximizing) mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and the other is a 'type' that captures some auxiliary information. A prototypical example of this is the FedEx Problem, for which Fiat et al. [2016] characterize the optimal mechanism for a single agent. Another example of this is when the type encodes the buyer's budget [DW17]. The question we address is how far can such characterizations goIn particular, we consider the setting of single-minded agents. A seller has heterogenous items. A buyer has a valuation vfor a specific subset of items S, and obtains value vif and only if he gets all the items in S(and potentially some others too).
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Optimal Mechanism Design for Single-Minded Agents

2020-02-15
Nikhil R. Devanur , Kira Goldner , Raghuvansh R. Saxena +2 more
We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting ā€¦ We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting is the FedEx Problem, for which FGKK [2016] characterize the optimal mechanism for a single agent. We ask: how far can such characterizations go? In particular, we consider single-minded agents. A seller has heterogenous items. A buyer has a value v for a specific subset of items S, and obtains value v iff he gets (at least) all the items in S. We show: 1. Deterministic mechanisms are optimal for distributions that satisfy the declining marginal revenue (DMR) property; we give an explicit construction of the optimal mechanism. 2. Without DMR, the result depends on the structure of the directed acyclic graph (DAG) representing the partial order among types. When the DAG has out-degree at most 1, we characterize the optimal mechanism a la FedEx. 3. Without DMR, when the DAG has some node with out-degree at least 2, we show that in this case the menu complexity is unbounded: for any M, there exist distributions over (v,S) pairs such that the menu complexity of the optimal mechanism is at least M. 4. For the case of 3 types, we show that for all distributions there exists an optimal mechanism of finite menu complexity. This is in contrast to 2 additive heterogenous items or which the menu complexity could be uncountable [MV07; DDT15]. In addition, we prove that optimal mechanisms for Multi-Unit Pricing (without DMR) can have unbounded menu complexity. We also propose an extension where the menu complexity of optimal mechanisms can be countable but not uncountable. Together these results establish that optimal mechanisms in interdimensional settings are both much richer than single-dimensional settings, yet also vastly more structured than multi-dimensional settings.

When to Limit Market Entry under Mandatory Purchase.

2020-02-15
Meryem Essaidi , Kira Goldner , S. Matthew Weinberg
We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with ā€¦ We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with private insurers to provide plans for their employees. The market regulator can choose to do nothing, running a Free Market, or can exercise her regulatory power by limiting the entry of providers (decreasing consumer welfare by limiting options, but also decreasing revenue via enhanced competition). We investigate whether limiting entry increases or decreases the utility (welfare minus revenue) of the consumers who purchase from the providers, specifically in settings where the outside option of purchasing nothing is prohibitively undesirable. We focus primarily on the case where providers are symmetric. We propose a sufficient condition on the distribution of consumer values for (a) a unique symmetric equilibrium to exist in both markets and (b) utility to be higher with limited entry. (We also establish that these conclusions do not necessarily hold for all distributions, and therefore some condition is necessary.) Our techniques are primarily based on tools from revenue maximization, and in particular Myerson's virtual value theory. We also consider extensions to settings where providers have identical costs for providing plans, and to two providers with an asymmetric distribution.

When to Limit Market Entry under Mandatory Purchase

2020-02-15
Meryem Essaidi , Kira Goldner , S. Matthew Weinberg
We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with ā€¦ We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with private insurers to provide plans for their employees. The market regulator can choose to do nothing, running a Free Market, or can exercise her regulatory power by limiting the entry of providers (decreasing consumer welfare by limiting options, but also decreasing revenue via enhanced competition). We investigate whether limiting entry increases or decreases the utility (welfare minus revenue) of the consumers who purchase from the providers, specifically in settings where the outside option of "purchasing nothing" is prohibitively undesirable. We focus primarily on the case where providers are symmetric. We propose a sufficient condition on the distribution of consumer values for (a) a unique symmetric equilibrium to exist in both markets and (b) utility to be higher with limited entry. (We also establish that these conclusions do not necessarily hold for all distributions, and therefore some condition is necessary.) Our techniques are primarily based on tools from revenue maximization, and in particular Myerson's virtual value theory. We also consider extensions to settings where providers have identical costs for providing plans, and to two providers with an asymmetric distribution.
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On Multi-Dimensional Gains from Trade Maximization

2020-01-01
Yang Cai , Kira Goldner , Steven Ma +1 more
We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and ā€¦ We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and there is a constrained-additive buyer with feasibility constraint $\mathcal{F}$. Multi-dimensional settings in one-sided markets, e.g. where a seller owns multiple heterogeneous items but also is the mechanism designer, are well-understood. In addition, single-dimensional settings in two-sided markets, e.g. where a buyer and seller each seek or own a single item, are also well-understood. Multi-dimensional two-sided markets, however, encapsulate the major challenges of both lines of work: optimizing the sale of heterogeneous items, ensuring incentive-compatibility among both sides of the market, and enforcing budget balance. We present, to the best of our knowledge, the first worst-case approximation guarantee for gains from trade in a multi-dimensional two-sided market. Our first result provides an $O(\log (1/r))$-approximation to the first-best gains from trade for a broad class of downward-closed feasibility constraints (such as matroid, matching, knapsack, or the intersection of these). Here $r$ is the minimum probability over all items that a buyer's value for the item exceeds the seller's cost. Our second result removes the dependence on $r$ and provides an unconditional $O(\log n)$-approximation to the second-best gains from trade. We extend both results for a general constrained-additive buyer, losing another $O(\log n)$-factor en-route.
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Non-Adaptive Matroid Prophet Inequalities

2020-01-01
Shuchi Chawla , Kira Goldner , Anna R. Karlin +1 more
We investigate non-adaptive algorithms for matroid prophet inequalities. Matroid prophet inequalities have been considered resolved since 2012 when [KW12] introduced thresholds that guarantee a tight 2-approximation to the prophet; however, ā€¦ We investigate non-adaptive algorithms for matroid prophet inequalities. Matroid prophet inequalities have been considered resolved since 2012 when [KW12] introduced thresholds that guarantee a tight 2-approximation to the prophet; however, this algorithm is adaptive. Other approaches of [CHMS10] and [FSZ16] have used non-adaptive thresholds with a feasibility restriction; however, this translates to adaptively changing an item's threshold to infinity when it cannot be taken with respect to the additional feasibility constraint, hence the algorithm is not truly non-adaptive. A major application of prophet inequalities is in auction design, where non-adaptive prices possess a significant advantage: they convert to order-oblivious posted pricings, and are essential for translating a prophet inequality into a truthful mechanism for multi-dimensional buyers. The existing matroid prophet inequalities do not suffice for this application. We present the first non-adaptive constant-factor prophet inequality for graphic matroids.

Optimal Mechanism Design for Single-Minded Agents

2020-01-01
Nikhil R. Devanur , Kira Goldner , Raghuvansh R. Saxena +2 more
We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting ā€¦ We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting is the FedEx Problem, for which FGKK [2016] characterize the optimal mechanism for a single agent. We ask: how far can such characterizations go? In particular, we consider single-minded agents. A seller has heterogenous items. A buyer has a value v for a specific subset of items S, and obtains value v iff he gets (at least) all the items in S. We show: 1. Deterministic mechanisms are optimal for distributions that satisfy the "declining marginal revenue" (DMR) property; we give an explicit construction of the optimal mechanism. 2. Without DMR, the result depends on the structure of the directed acyclic graph (DAG) representing the partial order among types. When the DAG has out-degree at most 1, we characterize the optimal mechanism a la FedEx. 3. Without DMR, when the DAG has some node with out-degree at least 2, we show that in this case the menu complexity is unbounded: for any M, there exist distributions over (v,S) pairs such that the menu complexity of the optimal mechanism is at least M. 4. For the case of 3 types, we show that for all distributions there exists an optimal mechanism of finite menu complexity. This is in contrast to 2 additive heterogenous items or which the menu complexity could be uncountable [MV07; DDT15]. In addition, we prove that optimal mechanisms for Multi-Unit Pricing (without DMR) can have unbounded menu complexity. We also propose an extension where the menu complexity of optimal mechanisms can be countable but not uncountable. Together these results establish that optimal mechanisms in interdimensional settings are both much richer than single-dimensional settings, yet also vastly more structured than multi-dimensional settings.

When to Limit Market Entry under Mandatory Purchase

2020-01-01
Meryem Essaidi , Kira Goldner , S. Matthew Weinberg
We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with ā€¦ We study a problem inspired by regulated health insurance markets, such as those created by the government in the Affordable Care Act Exchanges or by employers when they contract with private insurers to provide plans for their employees. The market regulator can choose to do nothing, running a Free Market, or can exercise her regulatory power by limiting the entry of providers (decreasing consumer welfare by limiting options, but also decreasing revenue via enhanced competition). We investigate whether limiting entry increases or decreases the utility (welfare minus revenue) of the consumers who purchase from the providers, specifically in settings where the outside option of "purchasing nothing" is prohibitively undesirable. We focus primarily on the case where providers are symmetric. We propose a sufficient condition on the distribution of consumer values for (a) a unique symmetric equilibrium to exist in both markets and (b) utility to be higher with limited entry. (We also establish that these conclusions do not necessarily hold for all distributions, and therefore some condition is necessary.) Our techniques are primarily based on tools from revenue maximization, and in particular Myerson's virtual value theory. We also consider extensions to settings where providers have identical costs for providing plans, and to two providers with an asymmetric distribution.

Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided Markets

2019-12-23
Moshe Babaioff , Kira Goldner , Yannai A. Gonczarowski
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided MarketsMoshe Babaioff, Kira Goldner, and Yannai A. GonczarowskiMoshe ā€¦ Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided MarketsMoshe Babaioff, Kira Goldner, and Yannai A. GonczarowskiMoshe Babaioff, Kira Goldner, and Yannai A. Gonczarowskipp.2452 - 2471Chapter DOI:https://doi.org/10.1137/1.9781611975994.150PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider the problem of welfare (and gains-from-trade) maximization in two-sided markets using simple mechanisms that are prior-independent. The seminal impossibility result of Myerson and Satterthwaite [1983] shows that even for bilateral trade, there is no feasible (individually rational, truthful, and budget balanced) mechanism that has welfare as high as the optimal-yet-infeasible VCG mechanism, which attains maximal welfare but runs a deficit. On the other hand, the optimal feasible mechanism needs to be carefully tailored to the Bayesian prior, and even worse, it is known to be extremely complex, eluding a precise description. In this paper we present Bulow-Klemperer-style results to circumvent these hurdles in double-auction market settings. We suggest using the Buyer Trade Reduction (BTR) mechanism, a variant of McAfee's mechanism, which is feasible and simple (in particular, it is deterministic, truthful, prior-independent, and anonymous). First, in the setting in which the values of the buyers and of the sellers are sampled independently and identically from the same distribution, we show that for any such market of any size, BTR with one additional buyer whose value is sampled from the same distribution has expected welfare at least as high as the optimal-yet-infeasible VCG mechanism in the original market. We then move to a more general setting in which the values of the buyers are sampled from one distribution, and those of the sellers from another, focusing on the case where the buyers' distribution first-order stochastically dominates the sellers' distribution. We present both upper bounds and lower bounds on the number of buyers that, when added, guarantees that BTR in the augmented market achieve welfare at least as high as the optimal in the original market. Our lower bounds extend to a large class of mechanisms, and all of our positive and negative results extend to adding sellers instead of buyers. In addition, we present positive results about the usefulness of pricing at a sample for welfare maximization (and more precisely, for gains-from-trade approximation) in two-sided markets under the above two settings, which to the best of our knowledge are the first sampling results in this context. Previous chapter Next chapter RelatedDetails Published:2020eISBN:978-1-61197-599-4 https://doi.org/10.1137/1.9781611975994Book Series Name:ProceedingsBook Code:PRDA20Book Pages:xxii + 3011
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Simple and Approximately Optimal Pricing for Proportional Complementarities

2019-06-17
Yang Cai , Nikhil R. Devanur , Kira Goldner +1 more
We study a new model of complementary valuations, which we call "proportional complementarities.'' In contrast to common models, such as hypergraphic valuations, in our model, we do not assume that ā€¦ We study a new model of complementary valuations, which we call "proportional complementarities.'' In contrast to common models, such as hypergraphic valuations, in our model, we do not assume that the extra value derived from owning a set of items is independent of the buyer's base valuations for the items. Instead, we model the complementarities as proportional to the buyer's base valuations, and these proportionalities are known market parameters.
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Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided Markets

2019-03-15
Moshe Babaioff , Kira Goldner , Yannai A. Gonczarowski
We consider the problem of welfare maximization in two-sided markets using simple mechanisms that are prior-independent. The Myerson-Satterthwaite impossibility theorem shows that even for bilateral trade, there is no feasible ā€¦ We consider the problem of welfare maximization in two-sided markets using simple mechanisms that are prior-independent. The Myerson-Satterthwaite impossibility theorem shows that even for bilateral trade, there is no feasible (IR, truthful, budget balanced) mechanism that has welfare as high as the optimal-yet-infeasible VCG mechanism, which attains maximal welfare but runs a deficit. On the other hand, the optimal feasible mechanism needs to be carefully tailored to the Bayesian prior, and is extremely complex, eluding a precise description. We present Bulow-Klemperer-style results to circumvent these hurdles in double-auction markets. We suggest using the Buyer Trade Reduction (BTR) mechanism, a variant of McAfee's mechanism, which is feasible and simple (in particular, deterministic, truthful, prior-independent, anonymous). First, in the setting where buyers' and sellers' values are sampled i.i.d. from the same distribution, we show that for any such market of any size, BTR with one additional buyer whose value is sampled from the same distribution has expected welfare at least as high as the optimal in the original market. We then move to a more general setting where buyers' values are sampled from one distribution and sellers' from another, focusing on the case where the buyers' distribution first-order stochastically dominates the sellers'. We present bounds on the number of buyers that, when added, guarantees that BTR in the augmented market have welfare at least as high as the optimal in the original market. Our lower bounds extend to a large class of mechanisms, and all of our results extend to adding sellers instead of buyers. In addition, we present positive results about the usefulness of pricing at a sample for welfare maximization in two-sided markets under the above two settings, which to the best of our knowledge are the first sampling results in this context.

Reducing Inefficiency in Carbon Auctions with Imperfect Competition

2019-01-01
Kira Goldner , Nicole Immorlica , Brendan Lucier
We study auctions for carbon licenses, a policy tool used to control the social cost of pollution. Each identical license grants the right to produce a unit of pollution. Each ā€¦ We study auctions for carbon licenses, a policy tool used to control the social cost of pollution. Each identical license grants the right to produce a unit of pollution. Each buyer (i.e., firm that pollutes during the manufacturing process) enjoys a decreasing marginal value for licenses, but society suffers an increasing marginal cost for each license distributed. The seller (i.e., the government) can choose a number of licenses to put up for auction, and wishes to maximize the societal welfare: the total economic value of the buyers minus the social cost. Motivated by emission license markets deployed in practice, we focus on uniform price auctions with a price floor and/or price ceiling. The seller has distributional information about the market, and their goal is to tune the auction parameters to maximize expected welfare. The target benchmark is the maximum expected welfare achievable by any such auction under truth-telling behavior. Unfortunately, the uniform price auction is not truthful, and strategic behavior can significantly reduce (even below zero) the welfare of a given auction configuration. We describe a subclass of "safe-price'" auctions for which the welfare at any Bayes-Nash equilibrium will approximate the welfare under truth-telling behavior. We then show that the better of a safe-price auction, or a truthful auction that allocates licenses to only a single buyer, will approximate the target benchmark. In particular, we show how to choose a number of licenses and a price floor so that the worst-case welfare, at any equilibrium, is a constant approximation to the best achievable welfare under truth-telling after excluding the welfare contribution of a single buyer.
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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 ā€¦ 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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Bulow-Klemperer-Style Results for Welfare Maximization in Two-Sided Markets

2019-01-01
Moshe Babaioff , Kira Goldner , Yannai A. Gonczarowski
We consider the problem of welfare maximization in two-sided markets using simple mechanisms that are prior-independent. The Myerson-Satterthwaite impossibility theorem shows that even for bilateral trade, there is no feasible ā€¦ We consider the problem of welfare maximization in two-sided markets using simple mechanisms that are prior-independent. The Myerson-Satterthwaite impossibility theorem shows that even for bilateral trade, there is no feasible (IR, truthful, budget balanced) mechanism that has welfare as high as the optimal-yet-infeasible VCG mechanism, which attains maximal welfare but runs a deficit. On the other hand, the optimal feasible mechanism needs to be carefully tailored to the Bayesian prior, and is extremely complex, eluding a precise description. We present Bulow-Klemperer-style results to circumvent these hurdles in double-auction markets. We suggest using the Buyer Trade Reduction (BTR) mechanism, a variant of McAfee's mechanism, which is feasible and simple (in particular, deterministic, truthful, prior-independent, anonymous). First, in the setting where buyers' and sellers' values are sampled i.i.d. from the same distribution, we show that for any such market of any size, BTR with one additional buyer whose value is sampled from the same distribution has expected welfare at least as high as the optimal in the original market. We then move to a more general setting where buyers' values are sampled from one distribution and sellers' from another, focusing on the case where the buyers' distribution first-order stochastically dominates the sellers'. We present bounds on the number of buyers that, when added, guarantees that BTR in the augmented market have welfare at least as high as the optimal in the original market. Our lower bounds extend to a large class of mechanisms, and all of our results extend to adding sellers instead of buyers. In addition, we present positive results about the usefulness of pricing at a sample for welfare maximization in two-sided markets under the above two settings, which to the best of our knowledge are the first sampling results in this context.
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Mechanism design for social good

2018-10-19
Rediet Abebe , Kira Goldner
Across various domains---such as health, education, and housing---improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives ā€¦ Across various domains---such as health, education, and housing---improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives of individuals, whether in the form of access to quality healthcare, labor market outcomes, or how votes are accounted for in a democratic society. Problems that can have an outsized impact on individuals whose opportunities have historically been limited often pose conceptual and technical challenges, requiring insights from many disciplines. Conversely, the lack of inter-disciplinary approach can leave these urgent needs unaddressed and can even exacerbate underlying socioeconomic inequalities.
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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 ā€¦ 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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Revenue Maximization with an Uncertainty-Averse Buyer

2018-01-01
Shuchi Chawla , Kira Goldner , J. Benjamin Miller +1 more
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real ā€¦ Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability x < 1 is worth strictly less to the agent than x times the value of the event when it occurs with certainty.We present three main results. First, we characterize optimal mechanisms as menus of two-outcome lotteries. Second, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation to the optimal revenue. Notably, this result is "risk-robust" in that it does not depend on the details of the buyer's risk attitude. Third, we consider dynamic settings in which the buyer's uncertainty about his future value may allow the seller to extract more revenue. In contrast to the positive result above, here we show it is not possible to achieve any constant-factor approximation to revenue using deterministic mechanisms in a risk-robust manner.
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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 ā€¦ 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.

Mechanism Design for Social Good

2018-01-01
Rediet Abebe , Kira Goldner
Across various domains--such as health, education, and housing--improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives ā€¦ Across various domains--such as health, education, and housing--improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives of individuals, whether in the form of access to quality healthcare, labor market outcomes, or how votes are accounted for in a democratic society. Problems that can have an out-sized impact on individuals whose opportunities have historically been limited often pose conceptual and technical challenges, requiring insights from many disciplines. Conversely, the lack of interdisciplinary approach can leave these urgent needs unaddressed and can even exacerbate underlying socioeconomic inequalities. To realize the opportunities in these domains, we need to correctly set objectives and reason about human behavior and actions. Doing so requires a deep grounding in the field of interest and collaboration with domain experts who understand the societal implications and feasibility of proposed solutions. These insights can play an instrumental role in proposing algorithmically-informed policies. In this article, we describe the Mechanism Design for Social Good (MD4SG) research agenda, which involves using insights from algorithms, optimization, and mechanism design to improve access to opportunity. The MD4SG research community takes an interdisciplinary, multi-stakeholder approach to improve societal welfare. We discuss three exciting research avenues within MD4SG related to improving access to opportunity in the developing world, labor markets and discrimination, and housing. For each of these, we showcase ongoing work, underline new directions, and discuss potential for implementing existing work in practice.
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Aversion to Uncertainty and Its Implications for Revenue Maximization.

2017-03-24
Shuchi Chawla , Kira Goldner , J. Benjamin Miller +1 more
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real ā€¦ Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is worth strictly less to the agent than $x$ times the value of the event when it occurs with certainty. In contrast to the risk-neutral setting, the optimal mechanism may be randomized and appears challenging to find, even for a single buyer and a single item for sale. Nevertheless, we give a characterization of the optimal mechanism which enables positive approximation results. In particular, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation. Notably, this result is in that it does not depend on the details of the buyer's risk attitude. Finally, we examine a dynamic setting in which the buyer is uncertain about his future value. In contrast to positive results for a risk-neutral buyer, we show that the buyer's risk aversion may prevent the seller from approximating the optimal revenue in a risk-robust manner.

Revenue Maximization with an Uncertainty-Averse Buyer

2017-03-24
Shuchi Chawla , Kira Goldner , J. Benjamin Miller +1 more
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real ā€¦ Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is worth strictly less to the agent than $x$ times the value of the event when it occurs with certainty. In contrast to the risk-neutral setting, the optimal mechanism may be randomized and appears challenging to find, even for a single buyer and a single item for sale. Nevertheless, we give a characterization of the optimal mechanism which enables positive approximation results. In particular, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation. Notably, this result is in that it does not depend on the details of the buyer's risk attitude. Finally, we examine a dynamic setting in which the buyer is uncertain about his future value. In contrast to positive results for a risk-neutral buyer, we show that the buyer's risk aversion may prevent the seller from approximating the optimal revenue in a risk-robust manner.

Revenue Maximization with an Uncertainty-Averse Buyer

2017-01-01
Shuchi Chawla , Kira Goldner , J. Benjamin Miller +1 more
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real ā€¦ Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is worth strictly less to the agent than $x$ times the value of the event when it occurs with certainty. In contrast to the risk-neutral setting, the optimal mechanism may be randomized and appears challenging to find, even for a single buyer and a single item for sale. Nevertheless, we give a characterization of the optimal mechanism which enables positive approximation results. In particular, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation. Notably, this result is "risk-robust" in that it does not depend on the details of the buyer's risk attitude. Finally, we examine a dynamic setting in which the buyer is uncertain about his future value. In contrast to positive results for a risk-neutral buyer, we show that the buyer's risk aversion may prevent the seller from approximating the optimal revenue in a risk-robust manner.
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Coauthor Papers Together
Alon Eden 8
S. Matthew Weinberg 6
Michal Feldman 6
Shuchi Chawla 5
J. Benjamin Miller 4
Amos Fiat 4
Emmanouil Pountourakis 4
Nikhil R. Devanur 4
Yang Cai 4
Raghuvansh R. Saxena 3
Anna R. Karlin 3
Steven Ma 3
Ariel Schvartzman 3
Yannai A. Gonczarowski 3
Meryem Essaidi 3
Rediet Abebe 3
Moshe Babaioff 3
Simon Mauras 2
Mingfei Zhao 2
Divyarthi Mohan 2
Shuran Zheng 2
Mingfei Zhao 1
J. Benjamin Miller 1
Taylor Lundy 1
Brendan Lucier 1
R. Preston McAfee 1
Nicole Immorlica 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 ā€¦ 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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Approximating Gains from Trade in Two-sided Markets via Simple Mechanisms

2017-06-20
Johannes Brustle , Yang Cai , Fa Wu +1 more
We design simple mechanisms to approximate the Gains from Trade (GFT) in two-sided markets with multiple unit-supply sellers and multiple unit-demand buyers. A classical impossibility result by Myerson and Satterthwaite ā€¦ We design simple mechanisms to approximate the Gains from Trade (GFT) in two-sided markets with multiple unit-supply sellers and multiple unit-demand buyers. A classical impossibility result by Myerson and Satterthwaite showed that even with only one seller and one buyer, no Bayesian Incentive Compatible (BIC), Individually Rational (IR), and Budget-Balanced (BB) mechanism can achieve full GFT (trade whenever buyer's value is higher than the seller's cost). The same paper also proposed the ``second-best'' mechanism that maximizes the GFT subject to BIC, IR, and BB constraints, which is unfortunately rather complex for even the single-seller single-buyer case. Our mechanism is simple, BIC, IR, and BB and achieves 1/2 of the optimal GFT among all BIC, IR, and BB mechanisms. The result holds for arbitrary distributions of the buyers' and sellers' values and can accommodate any downward-closed feasibility constraints over the allocations. The analysis of our mechanism is facilitated by extending the Cai-Weinberg-Devanur duality framework to two-sided markets.
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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 ā€¦ 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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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 ā€¦ 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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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, ā€¦ 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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Auctions with Interdependence and SOS: Improved Approximation

2021-01-01
Ameer Amer , Inbal Talgam-Cohen
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Strong Duality for a Multiple-Good Monopolist

2015-06-12
Constantinos Daskalakis , Alan Deckelbaum , Christos Tzamos
We provide a duality-based framework for revenue maximization in a multiple-good monopoly. Our framework shows that every optimal mechanism has a certificate of optimality, taking the form of an optimal ā€¦ We provide a duality-based framework for revenue maximization in a multiple-good monopoly. Our framework shows that every optimal mechanism has a certificate of optimality, taking the form of an optimal transportation map between measures. Using our framework, we prove that grand-bundling mechanisms are optimal if and only if two stochastic dominance conditions hold between specific measures induced by the buyer's type distribution. This result strengthens several results in the literature, where only sufficient conditions for grand-bundling optimality have been provided. As a corollary of our tight characterization of grand-bundling optimality, we show that the optimal mechanism for n independent uniform items each supported on [c; c + 1] is a grand-bundling mechanism, as long as c is sufficiently large, extending Pavlov's result for 2 items [Pavlov 2011]. Surprisingly, our characterization also implies that, for all c and for all sufficiently large n, the optimal mechanism for n independent uniform items supported on [c; c + 1] is not a grand bundling mechanism. The necessary and sufficient condition for grand bundling optimality is a special case of our more general characterization result that provides necessary and sufficient conditions for the optimality of an arbitrary mechanism for an arbitrary type distribution.
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Optimal mechanisms with simple menus

2014-05-30
Zihe Wang , Pingzhong Tang
We consider revenue-optimal mechanism design for the case with one buyer and two items. The buyer's valuations towards the two items are independent and additive. In this setting, optimal mechanism ā€¦ We consider revenue-optimal mechanism design for the case with one buyer and two items. The buyer's valuations towards the two items are independent and additive. In this setting, optimal mechanism is unknown for general valuation distributions. We obtain two categories of structural results that shed light on the optimal mechanisms. These results can be summarized into one conclusion: under certain conditions, the optimal mechanisms have simple menus.
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The menu-size complexity of revenue approximation

2017-06-15
Moshe Babaioff , Yannai A. Gonczarowski , Noam Nisan
We consider a monopolist that is selling n items to a single additive buyer, where the buyer's values for the items are drawn according to independent distributions F1,F2,ā€¦,Fn that possibly ā€¦ We consider a monopolist that is selling n items to a single additive buyer, where the buyer's values for the items are drawn according to independent distributions F1,F2,ā€¦,Fn that possibly have unbounded support. It is well known that - unlike in the single item case - the revenue-optimal auction (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. It is also known that simple auctions with a finite bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, the question of the possibility of extracting an arbitrarily high fraction of the optimal revenue via a finite menu size remained open.
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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 ā€¦ 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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Strategic Classification

2016-01-05
Moritz Hardt , Nimrod Megiddo , Christos H. Papadimitriou +1 more
Machine learning relies on the assumption that unseen test instances of a classification problem follow the same distribution as observed training data. However, this principle can break down when machine ā€¦ Machine learning relies on the assumption that unseen test instances of a classification problem follow the same distribution as observed training data. However, this principle can break down when machine learning is used to make important decisions about the welfare (employment, education, health) of strategic individuals. Knowing information about the classifier, such individuals may manipulate their attributes in order to obtain a better classification outcome. As a result of this behavior -- often referred to as gaming -- the performance of the classifier may deteriorate sharply. Indeed, gaming is a well-known obstacle for using machine learning methods in practice; in financial policy-making, the problem is widely known as Goodhart's law. In this paper, we formalize the problem, and pursue algorithms for learning classifiers that are robust to gaming.
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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. ā€¦ 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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Approximately Efficient Two-Sided Combinatorial Auctions

2017-06-20
Riccardo Colini-Baldeschi , Paul W. Goldberg , Bart de Keijzer +3 more
We develop and extend a line of recent work on the design of mechanisms for two-sided markets. The markets we consider consist of buyers and sellers of a number of ā€¦ We develop and extend a line of recent work on the design of mechanisms for two-sided markets. The markets we consider consist of buyers and sellers of a number of items, and the aim of a mechanism is to improve the social welfare by arranging purchases and sales of the items. A mechanism is given prior distributions on the agents' valuations of the items, but not the actual valuations; thus the aim is to maximise the expected social welfare over these distributions. As in previous work, we are interested in the worst-case ratio between the social welfare achieved by a truthful mechanism, and the best social welfare possible.
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A Simple and Approximately Optimal Mechanism for an Additive Buyer

2014-10-01
Moshe Babaioff , Nicole Immorlica , Brendan Lucier +1 more
We consider a monopolist seller with n heterogeneous items, facing a single buyer. The buyer hasa value for each item drawn independently according to(non-identical) distributions, and his value for a ā€¦ We consider a monopolist seller with n heterogeneous items, facing a single buyer. The buyer hasa value for each item drawn independently according to(non-identical) distributions, and his value for a set ofitems is additive. The seller aims to maximize his revenue.It is known that an optimal mechanism in this setting maybe quite complex, requiring randomization [19] and menusof infinite size [15]. Hart and Nisan [17] have initiated astudy of two very simple pricing schemes for this setting:item pricing, in which each item is priced at its monopolyreserve; and bundle pricing, in which the entire set ofitems is priced and sold as one bundle. Hart and Nisan [17]have shown that neither scheme can guarantee more thana vanishingly small fraction of the optimal revenue. Insharp contrast, we show that for any distributions, thebetter of item and bundle pricing is a constant-factorapproximation to the optimal revenue. We further discussextensions to multiple buyers and to valuations that arecorrelated across items.
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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 ā€¦ 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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Comparison of Threshold Stop Rules and Maximum for Independent Nonnegative Random Variables

1984-11-01
Ester Samuelā€Cahn
Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined ā€¦ Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined by $t(c) = \text{smallest} i$ for which $X_i \geq c(s(c) = \text{smallest} i$ for which $X_i > c), = n$ otherwise. Let $m$ be a median of the distribution of $X^\ast_n$. It is shown that for every $n$ and $\underline{X}$ either $EX^\ast_n \leq 2EX_{t(m)}$ or $EX^\ast_n \leq 2EX_{s(m)}$. This improves previously known results, [1], [4]. Some results for i.i.d. $X_i$ are also included.
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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 ā€¦ 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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The sample complexity of auctions with side information

2016-06-10
Nikhil R. Devanur , Zhiyi Huang , Alexandros Psomas
Traditionally, the Bayesian optimal auction design problem has been considered either when the bidder values are i.i.d, or when each bidder is individually identifiable via her value distribution. The latter ā€¦ Traditionally, the Bayesian optimal auction design problem has been considered either when the bidder values are i.i.d, or when each bidder is individually identifiable via her value distribution. The latter is a reasonable approach when the bidders can be classified into a few categories, but there are many instances where the classification of bidders is a continuum. For example, the classification of the bidders may be based on their annual income, their propensity to buy an item based on past behavior, or in the case of ad auctions, the click through rate of their ads. We introduce an alternate model that captures this aspect, where bidders are a priori identical, but can be distinguished based (only) on some side information the auctioneer obtains at the time of the auction. We extend the sample complexity approach of Dhangwatnotai et al. and Cole and Roughgarden to this model and obtain almost matching upper and lower bounds. As an aside, we obtain a revenue monotonicity lemma which may be of independent interest. We also show how to use Empirical Risk Minimization techniques to improve the sample complexity bound of Cole and Roughgarden for the non-identical but independent value distribution case.
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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 ā€¦ 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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Reaping the Benefits of Bundling under High Production Costs

2015-12-08
Will Ma , David Simchiā€Levi
It is well-known that selling different goods in a single bundle can significantly increase revenue, even when the valuations for the goods are independent. However, bundling is no longer profitable ā€¦ It is well-known that selling different goods in a single bundle can significantly increase revenue, even when the valuations for the goods are independent. However, bundling is no longer profitable if the goods have high production costs. To overcome this challenge, we introduce a new mechanism, Pure Bundling with Disposal for Cost (PBDC), where after buying the bundle, the customer is allowed to return any subset of goods for their production cost. We derive both distribution-dependent and distribution-free guarantees on its profitability, which improve previous techniques. Our distribution-dependent bound suggests that the firm should never price the bundle such that the profit margin is less than 1/3 of the expected welfare, while also showing that PBDC is optimal for a large number of independent goods. Our distribution-free bound suggests that on the distributions where PBDC performs worst, individual sales perform well. Finally, we conduct extensive simulations which confirm that PBDC outperforms other simple pricing schemes overall.

Beyond matroids: secretary problem and prophet inequality with general constraints

2016-06-10
Aviad Rubinstein
We study generalizations of the ``Prophet Inequality'' and ``Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both ā€¦ We study generalizations of the ``Prophet Inequality'' and ``Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both problems. This is close to the Omega(n/log n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O(log(n) log(r))-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O(loglog(n) log(r)) factor) an open question posed to us by Bobby Kleinberg.
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Almost) Efficient Mechanisms for Bilateral Trading

2016-04-17
Liad Blumrosen , Shahar Dobzinski
We study the bilateral trade problem: one seller, one buyer and a single, indivisible item for sale. It is well known that there is no fully-efficient and incentive compatible mechanism ā€¦ We study the bilateral trade problem: one seller, one buyer and a single, indivisible item for sale. It is well known that there is no fully-efficient and incentive compatible mechanism for this problem that maintains a balanced budget. We design simple and robust mechanisms that obtain approximate efficiency with these properties. We show that even minimal use of statistical data can yield good approximation results. Finally, we demonstrate how a mechanism for this simple bilateral-trade problem can be used as a black-box for constructing mechanisms in more general environments.

Non-Revelation Mechanism Design.

2016-08-05
Jason D. Hartline , Samuel Taggart

Efficient empirical revenue maximization in single-parameter auction environments

2017-06-15
Yannai A. Gonczarowski , Noam Nisan
We present a polynomial-time algorithm that, given samples from the unknown valuation distribution of each bidder, learns an auction that approximately maximizes the auctioneer's revenue in a variety of single-parameter ā€¦ We present a polynomial-time algorithm that, given samples from the unknown valuation distribution of each bidder, learns an auction that approximately maximizes the auctioneer's revenue in a variety of single-parameter auction environments including matroid environments, position environments, and the public project environment. The valuation distributions may be arbitrary bounded distributions (in particular, they may be irregular, and may differ for the various bidders), thus resolving a problem left open by previous papers. The analysis uses basic tools, is performed in its entirety in value-space, and simplifies the analysis of previously known results for special cases. Furthermore, the analysis extends to certain single-parameter auction environments where precise revenue maximization is known to be intractable, such as knapsack environments.
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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 āˆŖ ā€¦ 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 āˆŖ T) < v(S) + v(T)) and complements (i.e., for some S, T, v(S āˆŖ 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 Ī˜(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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Fairness at Equilibrium in the Labor Market

2017-01-01
Lily Hu , Yiā€Ling Chen
Recent literature on computational notions of fairness has been broadly divided into two distinct camps, supporting interventions that address either individual-based or group-based fairness. Rather than privilege a single definition, ā€¦ Recent literature on computational notions of fairness has been broadly divided into two distinct camps, supporting interventions that address either individual-based or group-based fairness. Rather than privilege a single definition, we seek to resolve both within the particular domain of employment discrimination. To this end, we construct a dual labor market model composed of a Temporary Labor Market, in which firm strategies are constrained to ensure group-level fairness, and a Permanent Labor Market, in which individual worker fairness is guaranteed. We show that such restrictions on hiring practices induces an equilibrium that Pareto-dominates those arising from strategies that employ statistical discrimination or a "group-blind" criterion. Individual worker reputations produce externalities for collective reputation, generating a feedback loop termed a "self-fulfilling prophecy." Our model produces its own feedback loop, raising the collective reputation of an initially disadvantaged group via a fairness intervention that need not be permanent. Moreover, we show that, contrary to popular assumption, the asymmetric equilibria resulting from hiring practices that disregard group-fairness may be immovable without targeted intervention. The enduring nature of such equilibria that are both inequitable and Pareto inefficient suggest that fairness interventions are of critical importance in moving the labor market to be more socially just and efficient.
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The Value of Information Concealment

2018-01-01
Hu Fu , Christopher Liaw , Pinyan Lu +1 more
We consider a revenue optimizing seller selling a single item to a buyer, on whose private value the seller has a noisy signal. We show that, when the signal is ā€¦ We consider a revenue optimizing seller selling a single item to a buyer, on whose private value the seller has a noisy signal. We show that, when the signal is kept private, arbitrarily more revenue could potentially be extracted than if the signal is leaked or revealed. We then show that, if the seller is not allowed to make payments to the buyer and if the value distribution conditioning on each signal is regular, the gap between the two is bounded by a multiplicative factor of 3. We give examples showing that both conditions are necessary for a constant bound on the gap to hold.We connect this scenario to multi-bidder single-item auctions where bidders' values are correlated. Similarly to the setting above, we show that the revenue of a Bayesian incentive compatible, ex post individually rational auction can be arbitrarily larger than that of a dominant strategy incentive compatible auction, whereas the two are no more than a factor of 5 apart if the auctioneer never pays the bidders and if the distribution is jointly regular. The upper bounds in both settings degrade gracefully when the distribution is a mixture of a small number of regular distributions.
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Fixed Price Approximability of the Optimal Gain from Trade

2017-01-01
Riccardo Colini-Baldeschi , Paul W. Goldberg , Bart de Keijzer +2 more
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MUDA: A Truthful Multi-Unit Double-Auction Mechanism

2017-12-19
Erel Segal-Halevi , Avinatan Hassidim , Yonatan Aumann
In a seminal paper, McAfee (1992) presented a truthful mechanism for double auctions, attaining asymptotically-optimal gain-from-trade without any prior information on the valuations of the traders. McAfee's mechanism handles single-parametric ā€¦ In a seminal paper, McAfee (1992) presented a truthful mechanism for double auctions, attaining asymptotically-optimal gain-from-trade without any prior information on the valuations of the traders. McAfee's mechanism handles single-parametric agents, allowing each seller to sell a single unit and each buyer to buy a single unit. This paper presents a double-auction mechanism that handles multi-parametric agents and allows multiple units per trader, as long as the valuation functions of all traders have decreasing marginal returns. The mechanism is prior-free, ex-post individually-rational, dominant-strategy truthful and strongly-budget-balanced. Its gain-from-trade approaches the optimum when the market size is sufficiently large.

Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization.

2018-05-24
Rad Niazadeh , Tim Roughgarden , Joshua R. Wang
In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has ā€¦ In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first $\frac{1}{2}$-approximation algorithm for continuous submodular function maximization; this approximation factor of $\frac{1}{2}$ is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular functions is also coordinate wise concave along all coordinates, we provide a different $\frac{1}{2}$-approximation algorithm that runs in quasilinear time. Both of these results improve upon prior work [Bian et al, 2017, Soma and Yoshida, 2017]. Our first algorithm uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications.

Using Search Queries to Understand Health Information Needs in Africa

2018-06-14
Rediet Abebe , Shawndra Hill , Jennifer Wortman Vaughan +2 more
The lack of comprehensive, high-quality health data in developing nations creates a roadblock for combating the impacts of disease. One key challenge is understanding the health information needs of people ā€¦ The lack of comprehensive, high-quality health data in developing nations creates a roadblock for combating the impacts of disease. One key challenge is understanding the health information needs of people in these nations. Without understanding people's everyday needs, concerns, and misconceptions, health organizations and policymakers lack the ability to effectively target education and programming efforts. In this paper, we propose a bottom-up approach that uses search data from individuals to uncover and gain insight into health information needs in Africa. We analyze Bing searches related to HIV/AIDS, malaria, and tuberculosis from all 54 African nations. For each disease, we automatically derive a set of common search themes or topics, revealing a wide-spread interest in various types of information, including disease symptoms, drugs, concerns about breastfeeding, as well as stigma, beliefs in natural cures, and other topics that may be hard to uncover through traditional surveys. We expose the different patterns that emerge in health information needs by demographic groups (age and sex) and country. We also uncover discrepancies in the quality of content returned by search engines to users by topic. Combined, our results suggest that search data can help illuminate health information needs in Africa and inform discussions on health policy and targeted education efforts both on- and offline.

Mechanism design for social good

2018-10-19
Rediet Abebe , Kira Goldner
Across various domains---such as health, education, and housing---improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives ā€¦ Across various domains---such as health, education, and housing---improving societal welfare involves allocating resources, setting policies, targeting interventions, and regulating activities. These solutions have an immense impact on the day-to-day lives of individuals, whether in the form of access to quality healthcare, labor market outcomes, or how votes are accounted for in a democratic society. Problems that can have an outsized impact on individuals whose opportunities have historically been limited often pose conceptual and technical challenges, requiring insights from many disciplines. Conversely, the lack of inter-disciplinary approach can leave these urgent needs unaddressed and can even exacerbate underlying socioeconomic inequalities.
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Optimal (and benchmark-optimal) competition complexity for additive buyers over independent items

2019-06-20
Hedyeh Beyhaghi , S. Matthew Weinberg
The Competition Complexity of an auction setting refers to the number of additional bidders necessary in order for the (deterministic, prior-independent, dominant strategy truthful) Vickrey-Clarke-Groves mechanism to achieve greater revenue ā€¦ The Competition Complexity of an auction setting refers to the number of additional bidders necessary in order for the (deterministic, prior-independent, dominant strategy truthful) Vickrey-Clarke-Groves mechanism to achieve greater revenue than the (randomized, prior-dependent, Bayesian-truthful) optimal mechanism without the additional bidders.
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Allocating Interventions Based on Predicted Outcomes: A Case Study on Homelessness Services

2019-07-17
Amanda Kube , Sanmay Das , Patrick J. Fowler
Modern statistical and machine learning methods are increasingly capable of modeling individual or personalized treatment effects. These predictions could be used to allocate different interventions across populations based on individual ā€¦ Modern statistical and machine learning methods are increasingly capable of modeling individual or personalized treatment effects. These predictions could be used to allocate different interventions across populations based on individual characteristics. In many domains, like social services, the availability of different possible interventions can be severely resource limited. This paper considers possible improvements to the allocation of such services in the context of homelessness service provision in a major metropolitan area. Using data from the homeless system, we use a counterfactual approach to show potential for substantial benefits in terms of reducing the number of families who experience repeat episodes of homelessness by choosing optimal allocations (based on predicted outcomes) to a fixed number of beds in different types of homelessness service facilities. Such changes in the allocation mechanism would not be without tradeoffs, however; a significant fraction of households are predicted to have a higher probability of re-entry in the optimal allocation than in the original one. We discuss the efficiency, equity and fairness issues that arise and consider potential implications for policy.

The Vickrey Auction with a Single Duplicate Bidder Approximates the Optimal Revenue

2019-01-01
Hu Fu , Christopher Liaw , Sikander Randhawa
Bulow and Klemperer's well-known result states that, in a single-item auction where the $n$ bidders' values are independently and identically drawn from a regular distribution, the Vickrey auction with one ā€¦ Bulow and Klemperer's well-known result states that, in a single-item auction where the $n$ bidders' values are independently and identically drawn from a regular distribution, the Vickrey auction with one additional bidder (a duplicate) extracts at least as much revenue as the optimal auction without the duplicate. Hartline and Roughgarden, in their influential 2009 paper, removed the requirement that the distributions be identical, at the cost of allowing the Vickrey auction to recruit $n$ duplicates, one from each distribution, and relaxing its revenue advantage to a $2$-approximation. In this work we restore Bulow and Klemperer's number of duplicates in Hartline and Roughgarden's more general setting with a worse approximation ratio. We show that recruiting a duplicate from one of the distributions suffices for the Vickrey auction to $10$-approximate the optimal revenue. We also show that in a $k$-items unit demand auction, recruiting $k$ duplicates suffices for the VCG auction to $O(1)$-approximate the optimal revenue. As another result, we tighten the analysis for Hartline and Roughgarden's Vickrey auction with $n$ duplicates for the case with two bidders in the auction. We show that in this case the Vickrey auction with two duplicates obtains at least $3/4$ of the optimal revenue. This is tight by meeting a lower bound by Hartline and Roughgarden. En route, we obtain a transparent analysis of their $2$-approximation for $n$~bidders, via a natural connection to Ronen's lookahead auction.

Simple Mechanisms for Profit Maximization in Multi-item Auctions

2019-06-17
Yang Cai , Mingfei Zhao
We study a classical Bayesian mechanism design problem where a seller is selling multiple items to a buyer. We consider the case where the seller has costs to produce the ā€¦ We study a classical Bayesian mechanism design problem where a seller is selling multiple items to a buyer. We consider the case where the seller has costs to produce the items, and these costs are private information to the seller. How can the seller design a mechanism to maximize her profit? Two well-studied problems, revenue maximization in multi-item auctions and signaling in ad auctions, are special cases of our problem. We show that there exists a simple mechanism whose profit is at least 1/11 the optimal profit, when the buyer has a constraint-additive valuation over independent items. The approximation factor becomes 6 when the buyer is additive. Our result holds even when the seller's costs are correlated across items.
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Mechanism Design for Subadditive Agents via an Ex-Ante Relaxation

2016-03-11
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 ā€¦ 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 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. Our work expands upon and unifies long lines of work on unit-demand settings and additive settings. We employ the ex ante relaxation approach developed by Alaei (2011) for reducing a multiple-buyer mechanism design problem with an ex post supply constraint into single-buyer ones with ex ante supply constraints. Solving the single-agent problems requires us to significantly extend techniques developed in the context of additive values by Li and Yao (2013) and their extension to subadditive values by Rubinstein and Weinberg (2015).

On the Power of Randomization in Algorithmic Mechanism Design

2009-01-01
Shahar Dobzinski , Shaddin Dughmi
In many settings the power of truthful mechanisms is severely bounded. In this paper we use randomization to overcome this problem. In particular, we construct an FPTAS for multi-unit auctions ā€¦ In many settings the power of truthful mechanisms is severely bounded. In this paper we use randomization to overcome this problem. In particular, we construct an FPTAS for multi-unit auctions that is truthful in expectation, whereas there is evidence that no polynomial-time truthful deterministic mechanism provides an approximation ratio better than 2. We also show for the first time that truthful in expectation polynomial-time mechanisms are \emph{provably} stronger than polynomial-time universally truthful mechanisms. Specifically, we show that there is a setting in which: (1) there is a non-polynomial time truthful mechanism that always outputs the optimal solution, and that (2) no universally truthful randomized mechanism can provide an approximation ratio better than 2 in polynomial time, but (3) an FPTAS that is truthful in expectation exists.

Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

2018-10-01
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 ā€¦ 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 downward-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. In the second part of the article, we develop a connection between approximately optimal simple mechanisms and approximate revenue monotonicity with respect to buyersā€™ valuations. Revenue non-monotonicity is the phenomenon that sometimes strictly increasing buyersā€™ values for every set can strictly decrease the revenue of the optimal mechanism. Using our main result, we derive a bound on how bad this degradation can be (and dub such a bound a proof of approximate revenue monotonicity); we further show that better bounds on approximate monotonicity imply a better analysis of our simple mechanisms.
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Online Contention Resolution Schemes

2015-12-21
Moran Feldman , Ola Svensson , Rico Zenklusen
We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our ā€¦ We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our rounding technique, which we call online contention resolution schemes (OCRSs), is applicable to many online selection problems, including Bayesian online selection, oblivious posted pricing mechanisms, and stochastic probing models. It allows for handling a wide set of constraints, and shares many strong properties of offline contention resolution schemes. In particular, OCRSs for different constraint families can be combined to obtain an OCRS for their intersection. Moreover, we can approximately maximize submodular functions in the online settings we consider.We, thus, get a broadly applicable framework for several online selection problems, which improves on previous approaches in terms of the types of constraints that can be handled, the objective functions that can be dealt with, and the assumptions on the strength of the adversary. Furthermore, we resolve two open problems from the literature; namely, we present the first constant-factor constrained oblivious posted price mechanism for matroid constraints, and the first constant-factor algorithm for weighted stochastic probing with deadlines.
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Prior-independent auctions for risk-averse agents

2013-06-11
Hu Fu , Jason D. Hartline , Darrell Hoy
We study simple and approximately optimal auctions for agents with a particular form of risk-averse preferences. We show that, for symmetric agents, the optimal revenue (given a prior distribution over ā€¦ We study simple and approximately optimal auctions for agents with a particular form of risk-averse preferences. We show that, for symmetric agents, the optimal revenue (given a prior distribution over the agent preferences) can be approximated by the first-price auction (which is prior independent), and, for asymmetric agents, the optimal revenue can be approximated by an auction with simple form. These results are based on two technical methods. The first is for upper-bounding the revenue from a risk-averse agent. The second gives a payment identity for mechanisms with pay-your-bid semantics.
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Double Auctions in Markets for Multiple Kinds of Goods

2018-07-01
Erel Segal-Halevi , Avinatan Hassidim , Yonatan Aumann
Motivated by applications such as stock exchanges and spectrum auctions, there is a growing interest in mechanisms for arranging trade in two-sided markets. However, existing mechanisms are either not truthful, ā€¦ Motivated by applications such as stock exchanges and spectrum auctions, there is a growing interest in mechanisms for arranging trade in two-sided markets. However, existing mechanisms are either not truthful, do not guarantee an asymptotically-optimal gain-from-trade, rely on a prior on the traders' valuations, or operate in limited settings such as a single type of good. We extend the random-sampling technique used in earlier works to multi-good markets where traders have gross-substitute valuations. We show a prior free, truthful and strongly-budget-balanced mechanism which guarantees near-optimal gain from trade when the market sizes of all goods grow to infinity at a similar rate.
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Using Search Queries to Understand Health Information Needs in Africa

2019-07-06
Rediet Abebe , Shawndra Hill , Jennifer Wortman Vaughan +2 more
The lack of comprehensive, high-quality health data in developing nations creates a roadblock for combating the impacts of disease. One key challenge is understanding health information needs of people. Without ā€¦ The lack of comprehensive, high-quality health data in developing nations creates a roadblock for combating the impacts of disease. One key challenge is understanding health information needs of people. Without understanding peopleā€™s everyday concerns, health organizations and policymakers are less able to effectively target education and programming efforts. In this paper, we propose a bottom-up approach that uses search data to uncover and gain insight into health information needs of individuals in Africa. We analyze Bing searches related to HIV/AIDS, malaria, and tuberculosis from all 54 African nations. For each disease, we automatically derive a set of common topics, revealing a widespread interest in various types of information, including disease symptoms, drugs, concerns about breastfeeding, as well as stigma, beliefs in natural cures, and other topics that may be hard to uncover through traditional surveys. We expose the different patterns that emerge in health information needs by demographic groups (age and gender) and country. Using finergrained data, we also uncover discrepancies in the quality of content returned by search engines to users by topic and highlight differences in user behavior and satisfaction. Combined, our results suggest that search data can help illuminate health information needs in Africa and inform discussions on health policy and targeted education efforts both on- and off-line.
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Bounding the menu-size of approximately optimal auctions via optimal-transport duality

2018-06-20
Yannai A. Gonczarowski
The question of the minimum menu-size for approximate (i.e., up-to-Īµ) Bayesian revenue maximization when selling two goods to an additive risk-neutral quasilinear buyer was introduced by Hart and Nisan [2013], ā€¦ The question of the minimum menu-size for approximate (i.e., up-to-Īµ) Bayesian revenue maximization when selling two goods to an additive risk-neutral quasilinear buyer was introduced by Hart and Nisan [2013], who give an upper bound of O(1/Īµ4) for this problem. Using the optimal-transport duality framework of Daskalakis, Deckelbaum, and Tzamos [2013, 2015], we derive the first lower bound for this problem ā€” of Ī©(1/āˆœĪµ), even when the values for the two goods are drawn i.i.d. from "nice" distributions, establishing how to reason about approximately optimal mechanisms via this duality framework. This bound implies, for any fixed number of goods, a tight bound of Ī˜(log1/Īµ) on the minimum deterministic communication complexity guaranteed to suffice for running some approximately revenue-maximizing mechanism, thereby completely resolving this problem. As a secondary result, we show that under standard economic assumptions on distributions, the above upper bound of Hart and Nisan [2013] can be strengthened to O(1/Īµ2).
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Non-monotone Continuous DR-submodular Maximization: Structure and Algorithms

2017-01-01
An Bian , Kfir Y. Levy , Andreas Krause +1 more
DR-submodular continuous functions are important objectives with wide real-world applications spanning MAP inference in determinantal point processes (DPPs), and mean-field inference for probabilistic submodular models, amongst others. DR-submodularity captures a ā€¦ DR-submodular continuous functions are important objectives with wide real-world applications spanning MAP inference in determinantal point processes (DPPs), and mean-field inference for probabilistic submodular models, amongst others. DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time. In this work we study the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints. We start by investigating geometric properties that underlie such objectives, e.g., a strong relation between (approximately) stationary points and global optimum is proved. These properties are then used to devise two optimization algorithms with provable guarantees. Concretely, we first devise a two-phase'' algorithm with 1/4 approximation guarantee. This algorithm allows the use of existing methods for finding (approximately) stationary points as a subroutine, thus, harnessing recent progress in non-convex optimization. Then we present a non-monotone Frank-Wolfe variant with 1/e approximation guarantee and sublinear convergence rate. Finally, we extend our approach to a broader class of generalized DR-submodular continuous functions, which captures a wider spectrum of applications. Our theoretical findings are validated on synthetic and real-world problem instances.

Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

2017-10-01
Paul DĆ¼tting , Michal Feldman , Thomas KeƟelheim +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 ā€¦ 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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On the Competition Complexity of Dynamic Mechanism Design

2018-01-01
Siqi Liu , Alexandros Psomas
The Competition Complexity of an auction measures how much competition is needed for the revenue of a simple auction to surpass the optimal revenue. A classic result from auction theory ā€¦ The Competition Complexity of an auction measures how much competition is needed for the revenue of a simple auction to surpass the optimal revenue. A classic result from auction theory by Bulow and Klemperer [11], states that the Competition Complexity of VCG, in the case of n i.i.d. buyers and a single item, is 1. In other words, it is better to invest in recruiting one extra buyer and run a second price auction than to invest in learning exactly the buyers' underlying distribution and run the revenue-maximizing auction tailored to this distribution.In this paper we study the Competition Complexity of dynamic auctions. Consider the following problem: a monopolist is auctioning off m items in m consecutive stages to n interested buyers. A buyer realizes her value for item k in the beginning of stage k. How many additional buyers are necessary and sufficient for a second price auction at each stage to extract revenue at least that of the optimal dynamic auction? We prove that the Competition Complexity of dynamic auctions is at most 3n - and at least linear in n - even when the buyers' values are correlated across stages, under a monotone hazard rate assumption on the stage (marginal) distributions. This assumption can be relaxed if one settles for independent stages. We also prove results on the number of additional buyers necessary for VCG at every stage to be an Ī±-approximation of the optimal revenue; we term this number the Ī±-approximate Competition Complexity. For example, under the same mild assumptions on the stage distributions we prove that one extra buyer suffices for a -approximation. As a corollary we provide the first results on prior-independent dynamic auctions. This is, to the best of our knowledge, the first nontrivial positive guarantees for simple ex-post IR dynamic auctions for correlated stages.A key step towards proving bounds on the Competition Complexity is getting a good benchmark/upper bound to the optimal revenue. To this end, we extend the recent duality framework of [14] to dynamic settings. As an aside to our approach we obtain a revenue non-monotonicity lemma for dynamic auctions, which may be of independent interest.
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Learning Multi-Item Auctions with (or without) Samples

2017-10-01
Yang Cai , Constantinos Daskalakis
We provide algorithms that learn simple auctions whose revenue is approximately optimal in multi-item multi-bidder settings, for a wide range of bidder valuations including unit-demand, additive, constrained additive, XOS, and ā€¦ We provide algorithms that learn simple auctions whose revenue is approximately optimal in multi-item multi-bidder settings, for a wide range of bidder valuations including unit-demand, additive, constrained additive, XOS, and subadditive. We obtain our learning results in two settings. The first is the commonly studied setting where sample access to the bidders' distributions over valuations is given, for both regular distributions and arbitrary distributions with bounded support. Here, our algorithms require polynomially many samples in the number of items and bidders. The second is a more general max-min learning setting that we introduce, where we are given "approximate distributions," and we seek to compute a mechanism whose revenue is approximately optimal simultaneously for all "true distributions" that are close to the ones we were given. These results are more general in that they imply the sample-based results, and are also applicable in settings where we have no sample access to the underlying distributions but have estimated them indirectly via market research or by observation of bidder behavior in previously run, potentially non-truthful auctions. All our results hold for valuation distributions satisfying the standard (and necessary) independence-across-items property. They also generalize and improve upon recent works of Goldner and Karlin [25] and Morgenstern and Roughgarden [32], which have provided algorithms that learn approximately optimal multi-item mechanisms in more restricted settings with additive, subadditive and unit-demand valuations using sample access to distributions. We generalize these results to the complete unit-demand, additive, and XOS setting, to i.i.d. subadditive bidders, and to the max-min setting. Our results are enabled by new uniform convergence bounds for hypotheses classes under product measures. Our bounds result in exponential savings in sample complexity compared to bounds derived by bounding the VC dimension and are of independent interest.
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The menu complexity of ā€œone-and-a-half-dimensionalā€ mechanism design

2018-01-01
Raghuvansh R. Saxena , Ariel Schvartzman , S. Matthew Weinberg
We study the menu complexity of optimal and approximately-optimal auctions in the context of the "FedEx" problem, a so-called "one-and-a-half-dimensional" setting where a single bidder has both a value and ā€¦ We study the menu complexity of optimal and approximately-optimal auctions in the context of the "FedEx" problem, a so-called "one-and-a-half-dimensional" setting where a single bidder has both a value and a deadline for receiving an item [FGKK16]. The menu complexity of an auction is equal to the number of distinct (allocation, price) pairs that a bidder might receive [HN13]. We show the following when the bidder has n possible deadlines:ā€¢Exponential menu complexity is necessary to be exactly optimal: There exist instances where the optimal mechanism has menu complexity ā‰„ 2n ā€“ 1. This matches exactly the upper bound provided by Fiat et al.'s algorithm, and resolves one of their open questions [FGKK16].ā€¢Fully polynomial menu complexity is necessary and sufficient for approximation: For all instances, there exists a mechanism guaranteeing a multiplicative (1 ā€“ āˆŠ)-approximation to the optimal revenue with menu complexity , where Ļ…max denotes the largest value in the support of integral distributions.ā€¢There exist instances where any mechanism guaranteeing a multiplicative (1 ā€“ O(1/n2))-approximation to the optimal revenue requires menu complexity Ī©(n2).Our main technique is the polygon approximation of concave functions [Rot92], and our results here should be of independent interest. We further show how our techniques can be used to resolve an open question of [DW17] on the menu complexity of optimal auctions for a budget-constrained buyer.
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99% Revenue via Enhanced Competition

2018-06-11
Michal Feldman , Ophir Friedler , Aviad Rubinstein
A sequence of recent studies show that even in the simple setting of a single seller and a single buyer with additive, independent valuations over m items, the revenue-maximizing mechanism ā€¦ A sequence of recent studies show that even in the simple setting of a single seller and a single buyer with additive, independent valuations over m items, the revenue-maximizing mechanism is prohibitively complex. This problem has been addressed using two main approaches: Approximation: the best of two simple mechanisms (sell each item separately, or sell all the items as one bundle) gives 1/6 of the optimal revenue [1]. Enhanced competition: running the simple VCG mechanism with additional m buyers extracts at least the optimal revenue in the original market [17]. Both approaches, however, suffer from severe drawbacks: On the one hand, losing 83% of the revenue is hardly acceptable in any application. On the other hand, attracting a linear number of new buyers may be prohibitive. We show that by combining the two approaches one can achieve the best of both worlds. Specifically, for any constant Īµ one can obtain a (1-Īµ) fraction of the optimal revenue by running simple mechanisms --- either selling each item separately or selling all items as a single bundle --- with substantially fewer additional buyers: logarithmic, constant, or even none in some cases.
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