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A Little Charity Guarantees Almost Envy-Freeness

2019-12-23
Bhaskar Ray Chaudhury , Telikepalli Kavitha , Kurt Mehlhorn +1 more
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)A Little Charity Guarantees Almost Envy-FreenessBhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini SgouritsaBhaskar … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)A Little Charity Guarantees Almost Envy-FreenessBhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini SgouritsaBhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini Sgouritsapp.2658 - 2672Chapter DOI:https://doi.org/10.1137/1.9781611975994.162PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute m goods to n agents in a “fair” manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is “envy-freeness up to any good” (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when n = 3. We show there is always a partition of the set of goods into n + 1 subsets (X1, …, Xn, P) where for i ϵ [n], Xi is the bundle allocated to agent i and the set P is unallocated (or donated to charity) such that we have: (1)envy-freeness up to any good,(2)no agent values P higher than her own bundle, and(3)fewer than n goods go to charity, i.e., |P| < n (typically m ≫ n). Our proof is constructive. When agents have additive valuations and |P| is large (i.e., when |P| is close to n), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation. 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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A Little Charity Guarantees Almost Envy-Freeness

2021-01-01
Bhaskar Ray Chaudhury , Telikepalli Kavitha , Kurt Mehlhorn +1 more
The fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every … The fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent has a valuation for each subset of goods. We assume monotone valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good," EFX, where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$, where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have (1) envy-freeness up to any good, (2) no agent values $P$ higher than her own bundle, and (3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive and leads to a pseudopolynomial time algorithm to find such an allocation. When agents have additive valuations and $|{P}|$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation that is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation. (Very recently and independently, Amanatidis, Ntokos, and Markakis [Theoret. Comput. Sci., 841 (2020), pp. 94--109], also showed the existence of a 4/7-GMMS allocation.)
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Tight Bounds for the Price of Anarchy of Simultaneous First-Price Auctions

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

2015-01-01
George Christodoulou , Alkmini Sgouritsa , Bo Tang
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Designing Networks with Good Equilibria under Uncertainty

2015-12-21
George Christodoulou , Alkmini Sgouritsa
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Designing Networks with Good Equilibria under UncertaintyGeorge Christodoulou and Alkmini SgouritsaGeorge Christodoulou and Alkmini … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Designing Networks with Good Equilibria under UncertaintyGeorge Christodoulou and Alkmini SgouritsaGeorge Christodoulou and Alkmini Sgouritsapp.72 - 89Chapter DOI:https://doi.org/10.1137/1.9781611974331.ch6PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the goal of the designer is to choose a single, universal cost-sharing protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist classes of graphs where knowledge of the underlying metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than Ω(log k). Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of Ω(log k), which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated according to some probability distribution that is known to the designer. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark, that the first result holds also for the black-box model, where the probabilities are not known to the designer, but is allowed to draw independent (polynomially many) samples. Previous chapter Next chapter RelatedDetails Published:2016eISBN:978-1-61197-433-1 https://doi.org/10.1137/1.9781611974331Book Series Name:ProceedingsBook Code:PRDA16Book Pages:viii + 2106
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Improved Price of Anarchy via Predictions

2022-07-12
Vasilis Gkatzelis , Kostas Kollias , Alkmini Sgouritsa +1 more
A central goal in algorithmic game theory is to analyze the performance of decentralized multiagent systems, like communication and information networks. In the absence of a central planner who can … A central goal in algorithmic game theory is to analyze the performance of decentralized multiagent systems, like communication and information networks. In the absence of a central planner who can enforce how these systems are utilized, the users can strategically interact with the system, aiming to maximize their own utility, possibly leading to very inefficient outcomes, and thus a high price of anarchy. To alleviate this issue, the system designer can use decentralized mechanisms that regulate the use of each resource (e.g., using local queuing protocols or scheduling mechanisms), but with only limited information regarding the state of the system. These information limitations have a severe impact on what such decentralized mechanisms can achieve, so most of the success stories in this literature have had to make restrictive assumptions (e.g., by either restricting the structure of the networks or the types of cost functions).
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Designing Networks with Good Equilibria under Uncertainty

2019-01-01
George Christodoulou , Alkmini Sgouritsa
We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. … We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of $k$ terminal vertices or players needs to establish connectivity with the root. The social optimum is the minimum Steiner tree. We study situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying graph metric, but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the goal of the designer is to choose a single, universal cost-sharing protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is, to what extent can prior knowledge of the underlying graph metric help in the design? We first demonstrate that there exist classes of graphs where knowledge of the underlying graph metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than $\Omega(\log k)$. Then, in our main technical result, we show that there exist graph metrics for which knowing the underlying graph metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey theory in high-dimensional hypercubes. Then we switch to the stochastic model, where the players are activated according to some probability distribution that is known to the designer. We show that there exists a randomized ordered protocol that achieves constant PoA. If, further, each player is activated independently with some probability, by using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark that the first result holds also for the black-box model, where the probabilities are not known to the designer, but she is allowed to draw independent (polynomially many) samples.
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Tight Bounds for the Price of Anarchy of Simultaneous First Price Auctions

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

2015-08-05
George Christodoulou , Alkmini Sgouritsa , Bo Tang
We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial … We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial auctions where m all-pay auctions run in parallel, one for each good. For fractionally subadditive valuations, we strengthen the upper bound from 2 [Syrgkanis and Tardos STOC'13] to 1.82 by proving some structural properties that characterize the mixed Nash equilibria of the game. Next, we design an all-pay mechanism with a randomized allocation rule for the multi- unit auction. We show that, for bidders with submodular valuations, the mechanism admits a unique, 75% efficient, pure Nash equilibrium. The efficiency of this mechanism outperforms all the known bounds on the price of anarchy of mechanisms used for multi-unit auctions. Finally, we analyze single-item all-pay auctions motivated by their connection to contests and show tight bounds on the price of anarchy of social welfare, revenue and maximum bid.

On the Efficiency of All-Pay Mechanisms

2015-01-01
George Christodoulou , Alkmini Sgouritsa , Bo Tang
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Resource-Aware Protocols for Network Cost-Sharing Games

2020-07-09
George Christodoulou , Vasilis Gkatzelis , Mohamad Latifian +1 more
We study the extent to which decentralized cost-sharing protocols can achieve good price of anarchy (PoA) bounds in network cost-sharing games with $n$ agents. We focus on the model of … We study the extent to which decentralized cost-sharing protocols can achieve good price of anarchy (PoA) bounds in network cost-sharing games with $n$ agents. We focus on the model of resource-aware protocols, where the designer has prior access to the network structure and can also increase the total cost of an edge(overcharging), and we study classes of games with concave or convex cost functions. We first consider concave cost functions and our main result is a cost-sharing protocol for symmetric games on directed acyclic graphs that achieves a PoA of $2+\varepsilon$ for some arbitrary small positive $\varepsilon$, which improves to $1+\varepsilon$ for games with at least two players. We also achieve a PoA of 1 for series-parallel graphs and show that no protocol can achieve a PoA better than $\Omega(\sqrt{n})$ for multicast games. We then also consider convex cost functions and prove analogous results for series-parallel networks and multicast games, as well as a lower bound of $\Omega(n)$ for the PoA on directed acyclic graphs without the use of overcharging.
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On the Efficiency of All-Pay Mechanisms

2015-01-01
George Christodoulou , Alkmini Sgouritsa , Bo Tang
We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial … We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial auctions where m all-pay auctions run in parallel, one for each good. For fractionally subadditive valuations, we strengthen the upper bound from 2 [Syrgkanis and Tardos STOC'13] to 1.82 by proving some structural properties that characterize the mixed Nash equilibria of the game. Next, we design an all-pay mechanism with a randomized allocation rule for the multi- unit auction. We show that, for bidders with submodular valuations, the mechanism admits a unique, 75% efficient, pure Nash equilibrium. The efficiency of this mechanism outperforms all the known bounds on the price of anarchy of mechanisms used for multi-unit auctions. Finally, we analyze single-item all-pay auctions motivated by their connection to contests and show tight bounds on the price of anarchy of social welfare, revenue and maximum bid.

A Little Charity Guarantees Almost Envy-Freeness

2019-07-10
Bhaskar Ray Chaudhury , T. Kavitha , Kurt Mehlhorn +1 more
Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a fair manner, where every agent … Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a fair manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is up to any (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when $n=3$. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$ where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have$\colon$ 1) envy-freeness up to any good, 2) no agent values $P$ higher than her own bundle, and 3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive. When agents have additive valuations and $\lvert P \rvert$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is $4/7$ groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of $1/2$ known for an approximate GMMS allocation.

A Little Charity Guarantees Almost Envy-Freeness

2019-01-01
Bhaskar Ray Chaudhury , T. Kavitha , Kurt Mehlhorn +1 more
Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent … Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good" (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when $n=3$. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$ where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have$\colon$ 1) envy-freeness up to any good, 2) no agent values $P$ higher than her own bundle, and 3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive. When agents have additive valuations and $\lvert P \rvert$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is $4/7$ groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of $1/2$ known for an approximate GMMS allocation.
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Pushing the Frontier on Approximate EFX Allocations

2024-06-18
Georgios Amanatidis , Aris Filos-Ratsikas , Alkmini Sgouritsa
We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($\alpha$-EFX). … We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($\alpha$-EFX). The state-of-the-art results on the problem include that (exact) EFX allocations exist when (a) there are at most three agents, or (b) the agents' valuation functions can take at most two values, or (c) the agents' valuation functions can be represented via a graph. For $\alpha$-EFX, it is known that a $0.618$-EFX allocation exists for any number of agents with additive valuation functions. In this paper, we show that $2/3$-EFX allocations exist when (a) there are at most \emph{seven agents}, (b) the agents' valuation functions can take at most \emph{three values}, or (c) the agents' valuation functions can be represented via a \emph{multigraph}. Our results can be interpreted in two ways. First, by relaxing the notion of EFX to $2/3$-EFX, we obtain existence results for strict generalizations of the settings for which exact EFX allocations are known to exist. Secondly, by imposing restrictions on the setting, we manage to beat the barrier of $0.618$ and achieve an approximation guarantee of $2/3$. Therefore, our results push the \emph{frontier} of existence and computation of approximate EFX allocations, and provide insights into the challenges of settling the existence of exact EFX allocations.
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On the Efficiency of the Proportional Allocation Mechanism for Divisible Resources

2015-01-01
George Christodoulou , Alkmini Sgouritsa , Bo Tang
We study the efficiency of the proportional allocation mechanism, that is widely used to allocate divisible resources. Each agent submits a bid for each divisible resource and receives a fraction … We study the efficiency of the proportional allocation mechanism, that is widely used to allocate divisible resources. Each agent submits a bid for each divisible resource and receives a fraction proportional to her bids. We quantify the inefficiency of Nash equilibria by studying the Price of Anarchy (PoA) of the induced game under complete and incomplete information. When agents' valuations are concave, we show that the Bayesian Nash equilibria can be arbitrarily inefficient, in contrast to the well-known 4/3 bound for pure equilibria. Next, we upper bound the PoA over Bayesian equilibria by 2 when agents' valuations are subadditive, generalizing and strengthening previous bounds on lattice submodular valuations. Furthermore, we show that this bound is tight and cannot be improved by any simple or scale-free mechanism. Then we switch to settings with budget constraints, and we show an improved upper bound on the PoA over coarse-correlated equilibria. Finally, we prove that the PoA is exactly 2 for pure equilibria in the polyhedral environment.

Resource-Aware Cost-Sharing Mechanisms with Priors

2021-06-03
Vasilis Gkatzelis , Emmanouil Pountourakis , Alkmini Sgouritsa
In a decentralized system with $m$ machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a … In a decentralized system with $m$ machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a function of the total load assigned to it, and some cost-sharing mechanism distributes this cost among the machine's users. The users choose a machine aiming to minimize their own share of the cost, so the cost-sharing mechanism induces a game among them. We approach this problem from the perspective of a designer who can select which cost-sharing mechanism to use, aiming to minimize the price of anarchy (PoA) of the induced games. Recent work introduced the class of \emph{resource-aware} cost-sharing mechanisms, whose decisions can depend on the set of machines in the system, but are oblivious to the total number of users. These mechanisms can guarantee low PoA bounds for instances where the cost functions of the machines are all convex or concave, but can suffer from very high PoA for cost functions that deviate from these families. In this paper we show that if we enhance the class of resource-aware mechanisms with some prior information regarding the users, then they can achieve low PoA for a much more general family of cost functions. We first show that, as long as the mechanism knows just two of the participating users, then it can assign special roles to them and ensure a constant PoA. We then extend this idea to settings where the mechanism has access to the probability with which each user is present in the system. For all these instances, we provide a mechanism that achieves an expected PoA that is logarithmic in the expected number of users.

Resource-Aware Cost-Sharing Mechanisms with Priors

2021-07-18
Vasilis Gkatzelis , Emmanouil Pountourakis , Alkmini Sgouritsa
In a decentralized system with m machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a … In a decentralized system with m machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a function of the total load assigned to it, and some cost-sharing mechanism distributes this cost among the machine's users. The users choose a machine aiming to minimize their own share of the cost, so the cost-sharing mechanism induces a game among them. We approach this problem from the perspective of a designer who can select which cost-sharing mechanism to use, aiming to minimize the price of anarchy (PoA) of the induced games. Recent work introduced the class of resource-aware cost-sharing mechanisms, whose decisions can depend on the set of machines in the system, but are oblivious to the total number of users. These mechanisms can guarantee low PoA bounds for instances where the cost functions of the machines are all convex or concave, but can suffer from very high PoA for cost functions that deviate from these families.
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Resource-Aware Cost-Sharing Mechanisms with Priors

2021-01-01
Vasilis Gkatzelis , Emmanouil Pountourakis , Alkmini Sgouritsa
In a decentralized system with $m$ machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a … In a decentralized system with $m$ machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a function of the total load assigned to it, and some cost-sharing mechanism distributes this cost among the machine's users. The users choose a machine aiming to minimize their own share of the cost, so the cost-sharing mechanism induces a game among them. We approach this problem from the perspective of a designer who can select which cost-sharing mechanism to use, aiming to minimize the price of anarchy (PoA) of the induced games. Recent work introduced the class of \emph{resource-aware} cost-sharing mechanisms, whose decisions can depend on the set of machines in the system, but are oblivious to the total number of users. These mechanisms can guarantee low PoA bounds for instances where the cost functions of the machines are all convex or concave, but can suffer from very high PoA for cost functions that deviate from these families. In this paper we show that if we enhance the class of resource-aware mechanisms with some prior information regarding the users, then they can achieve low PoA for a much more general family of cost functions. We first show that, as long as the mechanism knows just two of the participating users, then it can assign special roles to them and ensure a constant PoA. We then extend this idea to settings where the mechanism has access to the probability with which each user is present in the system. For all these instances, we provide a mechanism that achieves an expected PoA that is logarithmic in the expected number of users.

Designing Networks with Good Equilibria under Uncertainty

2015-01-01
George Christodoulou , Alkmini Sgouritsa
We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. … We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the designer's goal is to choose a single, universal protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist graphs (outerplanar) where knowledge of the underlying metric can dramatically improve the performance of good network design. Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol with constant PoA.
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Improved Price of Anarchy via Predictions

2022-01-01
Vasilis Gkatzelis , Kostas Kollias , Alkmini Sgouritsa +1 more
A central goal in algorithmic game theory is to analyze the performance of decentralized multiagent systems, like communication and information networks. In the absence of a central planner who can … A central goal in algorithmic game theory is to analyze the performance of decentralized multiagent systems, like communication and information networks. In the absence of a central planner who can enforce how these systems are utilized, the users can strategically interact with the system, aiming to maximize their own utility, possibly leading to very inefficient outcomes, and thus a high price of anarchy. To alleviate this issue, the system designer can use decentralized mechanisms that regulate the use of each resource (e.g., using local queuing protocols or scheduling mechanisms), but with only limited information regarding the state of the system. These information limitations have a severe impact on what such decentralized mechanisms can achieve, so most of the success stories in this literature have had to make restrictive assumptions (e.g., by either restricting the structure of the networks or the types of cost functions). In this paper, we overcome some of the obstacles that the literature has imposed on decentralized mechanisms, by designing mechanisms that are enhanced with predictions regarding the missing information. Specifically, inspired by the big success of the literature on "algorithms with predictions", we design decentralized mechanisms with predictions and evaluate their price of anarchy as a function of the prediction error, focusing on two very well-studied classes of games: scheduling games and multicast network formation games.
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Mechanism design augmented with output advice

2024-06-20
George Christodoulou , Alkmini Sgouritsa , Ioannis Vlachos
Our work revisits the design of mechanisms via the learning-augmented framework. In this model, the algorithm is enhanced with imperfect (machine-learned) information concerning the input, usually referred to as prediction. … Our work revisits the design of mechanisms via the learning-augmented framework. In this model, the algorithm is enhanced with imperfect (machine-learned) information concerning the input, usually referred to as prediction. The goal is to design algorithms whose performance degrades gently as a function of the prediction error and, in particular, perform well if the prediction is accurate, but also provide a worst-case guarantee under any possible error. This framework has been successfully applied recently to various mechanism design settings, where in most cases the mechanism is provided with a prediction about the types of the players. We adopt a perspective in which the mechanism is provided with an output recommendation. We make no assumptions about the quality of the suggested outcome, and the goal is to use the recommendation to design mechanisms with low approximation guarantees whenever the recommended outcome is reasonable, but at the same time to provide worst-case guarantees whenever the recommendation significantly deviates from the optimal one. We propose a generic, universal measure, which we call quality of recommendation, to evaluate mechanisms across various information settings. We demonstrate how this new metric can provide refined analysis in existing results. This model introduces new challenges, as the mechanism receives limited information comparing to settings that use predictions about the types of the agents. We study, through this lens, several well-studied mechanism design paradigms, devising new mechanisms, but also providing refined analysis for existing ones, using as a metric the quality of recommendation. We complement our positive results, by exploring the limitations of known classes of strategyproof mechanisms that can be devised using output recommendation.
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EF2X Exists For Four Agents

2024-11-29
Arash Ashuri , Vasilis Gkatzelis , Alkmini Sgouritsa
We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which … We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which has proven to be extremely challenging, is regarding the existence of an EFX allocation, i.e., an allocation such that any envy from some agent i toward another agent j would vanish if we were to remove any single good from the bundle allocated to j. When the agents' valuations are additive, which has been the main focus of prior works, Chaudhury et al. [2024] showed that an EFX allocation is guaranteed to exist for all instances involving up to three agents. Subsequently, Berger et al. [2022] extended this guarantee to nice-cancelable valuations and Akrami et al. [2023] to MMS-feasible valuations. However, the existence of EFX allocations for instances involving four agents remains open, even for additive valuations. We contribute to this literature by focusing on EF2X, a relaxation of EFX which requires that any envy toward some agent vanishes if any two of the goods allocated to that agent were to be removed. Our main result shows that EF2X allocations are guaranteed to exist for any instance with four agents, even for the class of cancelable valuations, which is more general than additive. Our proof is constructive, proposing an algorithm that computes such an allocation in pseudopolynomial time. Furthermore, for instances involving three agents we provide an algorithm that computes an EF2X allocation in polynomial time, in contrast to EFX, for which the fastest known algorithm for three agents is only pseudopolynomial.
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Pushing the Frontier on Approximate EFX Allocations

2024-07-08
Georgios Amanatidis , Aris Filos-Ratsikas , Alkmini Sgouritsa
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Maximin Share Guarantees for Few Agents with Subadditive Valuations

2025-02-07
George Christodoulou , Vasilis Christoforidis , Symeon Mastrakoulis +1 more
We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an … We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an improved lower bound of $1/2$ (which is tight) for the case of subadditive valuations when the number of agents is at most four. We also provide a tight lower bound for the case of multiple agents, when they are equipped with one of two possible types of valuations. Moreover, we propose a new model that extends previously studied models in the area of fair division, which will hopefully give rise to further research. We demonstrate the usefulness of this model by employing it as a technical tool to derive our main result, and we provide a thorough analysis for this model for the case of three agents. Finally, we provide an improved impossibility result for the case of three submodular agents.
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On the existence of EFX allocations in multigraphs

2025-02-13
Alkmini Sgouritsa , Minas Marios Sotiriou
We study the problem of "fairly" dividing indivisible goods to several agents that have valuation set functions over the sets of goods. As fair we consider the allocations that are … We study the problem of "fairly" dividing indivisible goods to several agents that have valuation set functions over the sets of goods. As fair we consider the allocations that are envy-free up to any good (EFX), i.e., no agent envies any proper subset of the goods given to any other agent. The existence or not of EFX allocations is a major open problem in Fair Division, and there are only positive results for special cases. [George Christodoulou, Amos Fiat, Elias Koutsoupias, Alkmini Sgouritsa 2023] introduced a restriction on the agents' valuations according to a graph structure: the vertices correspond to agents and the edges to goods, and each vertex/agent has zero marginal value (or in other words, they are indifferent) for the edges/goods that are not adjacent to them. The existence of EFX allocations has been shown for simple graphs with general monotone valuations [George Christodoulou, Amos Fiat, Elias Koutsoupias, Alkmini Sgouritsa 2023], and for multigraphs for restricted additive valuations [Alireza Kaviani, Masoud Seddighin, Amir Mohammad Shahrezaei 2024]. In this work, we push the state-of-the-art further, and show that the EFX allocations always exists in multigraphs and general monotone valuations if any of the following three conditions hold: either (a) the multigraph is bipartite, or (b) each agent has at most $\lceil \frac{n}{4} \rceil -1$ neighbors, where $n$ is the total number of agents, or (c) the shortest cycle with non-parallel edges has length at least 6.
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On the existence of EFX allocations in multigraphs

2025-02-13
Alkmini Sgouritsa , Minas Marios Sotiriou
We study the problem of "fairly" dividing indivisible goods to several agents that have valuation set functions over the sets of goods. As fair we consider the allocations that are … We study the problem of "fairly" dividing indivisible goods to several agents that have valuation set functions over the sets of goods. As fair we consider the allocations that are envy-free up to any good (EFX), i.e., no agent envies any proper subset of the goods given to any other agent. The existence or not of EFX allocations is a major open problem in Fair Division, and there are only positive results for special cases. [George Christodoulou, Amos Fiat, Elias Koutsoupias, Alkmini Sgouritsa 2023] introduced a restriction on the agents' valuations according to a graph structure: the vertices correspond to agents and the edges to goods, and each vertex/agent has zero marginal value (or in other words, they are indifferent) for the edges/goods that are not adjacent to them. The existence of EFX allocations has been shown for simple graphs with general monotone valuations [George Christodoulou, Amos Fiat, Elias Koutsoupias, Alkmini Sgouritsa 2023], and for multigraphs for restricted additive valuations [Alireza Kaviani, Masoud Seddighin, Amir Mohammad Shahrezaei 2024]. In this work, we push the state-of-the-art further, and show that the EFX allocations always exists in multigraphs and general monotone valuations if any of the following three conditions hold: either (a) the multigraph is bipartite, or (b) each agent has at most $\lceil \frac{n}{4} \rceil -1$ neighbors, where $n$ is the total number of agents, or (c) the shortest cycle with non-parallel edges has length at least 6.
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Maximin Share Guarantees for Few Agents with Subadditive Valuations

2025-02-07
George Christodoulou , Vasilis Christoforidis , Symeon Mastrakoulis +1 more
We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an … We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an improved lower bound of $1/2$ (which is tight) for the case of subadditive valuations when the number of agents is at most four. We also provide a tight lower bound for the case of multiple agents, when they are equipped with one of two possible types of valuations. Moreover, we propose a new model that extends previously studied models in the area of fair division, which will hopefully give rise to further research. We demonstrate the usefulness of this model by employing it as a technical tool to derive our main result, and we provide a thorough analysis for this model for the case of three agents. Finally, we provide an improved impossibility result for the case of three submodular agents.
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EF2X Exists For Four Agents

2024-11-29
Arash Ashuri , Vasilis Gkatzelis , Alkmini Sgouritsa
We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which … We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which has proven to be extremely challenging, is regarding the existence of an EFX allocation, i.e., an allocation such that any envy from some agent i toward another agent j would vanish if we were to remove any single good from the bundle allocated to j. When the agents' valuations are additive, which has been the main focus of prior works, Chaudhury et al. [2024] showed that an EFX allocation is guaranteed to exist for all instances involving up to three agents. Subsequently, Berger et al. [2022] extended this guarantee to nice-cancelable valuations and Akrami et al. [2023] to MMS-feasible valuations. However, the existence of EFX allocations for instances involving four agents remains open, even for additive valuations. We contribute to this literature by focusing on EF2X, a relaxation of EFX which requires that any envy toward some agent vanishes if any two of the goods allocated to that agent were to be removed. Our main result shows that EF2X allocations are guaranteed to exist for any instance with four agents, even for the class of cancelable valuations, which is more general than additive. Our proof is constructive, proposing an algorithm that computes such an allocation in pseudopolynomial time. Furthermore, for instances involving three agents we provide an algorithm that computes an EF2X allocation in polynomial time, in contrast to EFX, for which the fastest known algorithm for three agents is only pseudopolynomial.
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Pushing the Frontier on Approximate EFX Allocations

2024-07-08
Georgios Amanatidis , Aris Filos-Ratsikas , Alkmini Sgouritsa
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Mechanism design augmented with output advice

2024-06-20
George Christodoulou , Alkmini Sgouritsa , Ioannis Vlachos
Our work revisits the design of mechanisms via the learning-augmented framework. In this model, the algorithm is enhanced with imperfect (machine-learned) information concerning the input, usually referred to as prediction. … Our work revisits the design of mechanisms via the learning-augmented framework. In this model, the algorithm is enhanced with imperfect (machine-learned) information concerning the input, usually referred to as prediction. The goal is to design algorithms whose performance degrades gently as a function of the prediction error and, in particular, perform well if the prediction is accurate, but also provide a worst-case guarantee under any possible error. This framework has been successfully applied recently to various mechanism design settings, where in most cases the mechanism is provided with a prediction about the types of the players. We adopt a perspective in which the mechanism is provided with an output recommendation. We make no assumptions about the quality of the suggested outcome, and the goal is to use the recommendation to design mechanisms with low approximation guarantees whenever the recommended outcome is reasonable, but at the same time to provide worst-case guarantees whenever the recommendation significantly deviates from the optimal one. We propose a generic, universal measure, which we call quality of recommendation, to evaluate mechanisms across various information settings. We demonstrate how this new metric can provide refined analysis in existing results. This model introduces new challenges, as the mechanism receives limited information comparing to settings that use predictions about the types of the agents. We study, through this lens, several well-studied mechanism design paradigms, devising new mechanisms, but also providing refined analysis for existing ones, using as a metric the quality of recommendation. We complement our positive results, by exploring the limitations of known classes of strategyproof mechanisms that can be devised using output recommendation.
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Pushing the Frontier on Approximate EFX Allocations

2024-06-18
Georgios Amanatidis , Aris Filos-Ratsikas , Alkmini Sgouritsa
We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($\alpha$-EFX). … We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($\alpha$-EFX). The state-of-the-art results on the problem include that (exact) EFX allocations exist when (a) there are at most three agents, or (b) the agents' valuation functions can take at most two values, or (c) the agents' valuation functions can be represented via a graph. For $\alpha$-EFX, it is known that a $0.618$-EFX allocation exists for any number of agents with additive valuation functions. In this paper, we show that $2/3$-EFX allocations exist when (a) there are at most \emph{seven agents}, (b) the agents' valuation functions can take at most \emph{three values}, or (c) the agents' valuation functions can be represented via a \emph{multigraph}. Our results can be interpreted in two ways. First, by relaxing the notion of EFX to $2/3$-EFX, we obtain existence results for strict generalizations of the settings for which exact EFX allocations are known to exist. Secondly, by imposing restrictions on the setting, we manage to beat the barrier of $0.618$ and achieve an approximation guarantee of $2/3$. Therefore, our results push the \emph{frontier} of existence and computation of approximate EFX allocations, and provide insights into the challenges of settling the existence of exact EFX allocations.
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Improved Price of Anarchy via Predictions

2022-07-12
Vasilis Gkatzelis , Kostas Kollias , Alkmini Sgouritsa +1 more
A central goal in algorithmic game theory is to analyze the performance of decentralized multiagent systems, like communication and information networks. In the absence of a central planner who can … A central goal in algorithmic game theory is to analyze the performance of decentralized multiagent systems, like communication and information networks. In the absence of a central planner who can enforce how these systems are utilized, the users can strategically interact with the system, aiming to maximize their own utility, possibly leading to very inefficient outcomes, and thus a high price of anarchy. To alleviate this issue, the system designer can use decentralized mechanisms that regulate the use of each resource (e.g., using local queuing protocols or scheduling mechanisms), but with only limited information regarding the state of the system. These information limitations have a severe impact on what such decentralized mechanisms can achieve, so most of the success stories in this literature have had to make restrictive assumptions (e.g., by either restricting the structure of the networks or the types of cost functions).
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Improved Price of Anarchy via Predictions

2022-01-01
Vasilis Gkatzelis , Kostas Kollias , Alkmini Sgouritsa +1 more
A central goal in algorithmic game theory is to analyze the performance of decentralized multiagent systems, like communication and information networks. In the absence of a central planner who can … A central goal in algorithmic game theory is to analyze the performance of decentralized multiagent systems, like communication and information networks. In the absence of a central planner who can enforce how these systems are utilized, the users can strategically interact with the system, aiming to maximize their own utility, possibly leading to very inefficient outcomes, and thus a high price of anarchy. To alleviate this issue, the system designer can use decentralized mechanisms that regulate the use of each resource (e.g., using local queuing protocols or scheduling mechanisms), but with only limited information regarding the state of the system. These information limitations have a severe impact on what such decentralized mechanisms can achieve, so most of the success stories in this literature have had to make restrictive assumptions (e.g., by either restricting the structure of the networks or the types of cost functions). In this paper, we overcome some of the obstacles that the literature has imposed on decentralized mechanisms, by designing mechanisms that are enhanced with predictions regarding the missing information. Specifically, inspired by the big success of the literature on "algorithms with predictions", we design decentralized mechanisms with predictions and evaluate their price of anarchy as a function of the prediction error, focusing on two very well-studied classes of games: scheduling games and multicast network formation games.
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Resource-Aware Cost-Sharing Mechanisms with Priors

2021-07-18
Vasilis Gkatzelis , Emmanouil Pountourakis , Alkmini Sgouritsa
In a decentralized system with m machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a … In a decentralized system with m machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a function of the total load assigned to it, and some cost-sharing mechanism distributes this cost among the machine's users. The users choose a machine aiming to minimize their own share of the cost, so the cost-sharing mechanism induces a game among them. We approach this problem from the perspective of a designer who can select which cost-sharing mechanism to use, aiming to minimize the price of anarchy (PoA) of the induced games. Recent work introduced the class of resource-aware cost-sharing mechanisms, whose decisions can depend on the set of machines in the system, but are oblivious to the total number of users. These mechanisms can guarantee low PoA bounds for instances where the cost functions of the machines are all convex or concave, but can suffer from very high PoA for cost functions that deviate from these families.
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Resource-Aware Cost-Sharing Mechanisms with Priors

2021-06-03
Vasilis Gkatzelis , Emmanouil Pountourakis , Alkmini Sgouritsa
In a decentralized system with $m$ machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a … In a decentralized system with $m$ machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a function of the total load assigned to it, and some cost-sharing mechanism distributes this cost among the machine's users. The users choose a machine aiming to minimize their own share of the cost, so the cost-sharing mechanism induces a game among them. We approach this problem from the perspective of a designer who can select which cost-sharing mechanism to use, aiming to minimize the price of anarchy (PoA) of the induced games. Recent work introduced the class of \emph{resource-aware} cost-sharing mechanisms, whose decisions can depend on the set of machines in the system, but are oblivious to the total number of users. These mechanisms can guarantee low PoA bounds for instances where the cost functions of the machines are all convex or concave, but can suffer from very high PoA for cost functions that deviate from these families. In this paper we show that if we enhance the class of resource-aware mechanisms with some prior information regarding the users, then they can achieve low PoA for a much more general family of cost functions. We first show that, as long as the mechanism knows just two of the participating users, then it can assign special roles to them and ensure a constant PoA. We then extend this idea to settings where the mechanism has access to the probability with which each user is present in the system. For all these instances, we provide a mechanism that achieves an expected PoA that is logarithmic in the expected number of users.

A Little Charity Guarantees Almost Envy-Freeness

2021-01-01
Bhaskar Ray Chaudhury , Telikepalli Kavitha , Kurt Mehlhorn +1 more
The fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every … The fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent has a valuation for each subset of goods. We assume monotone valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good," EFX, where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$, where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have (1) envy-freeness up to any good, (2) no agent values $P$ higher than her own bundle, and (3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive and leads to a pseudopolynomial time algorithm to find such an allocation. When agents have additive valuations and $|{P}|$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation that is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation. (Very recently and independently, Amanatidis, Ntokos, and Markakis [Theoret. Comput. Sci., 841 (2020), pp. 94--109], also showed the existence of a 4/7-GMMS allocation.)
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Resource-Aware Cost-Sharing Mechanisms with Priors

2021-01-01
Vasilis Gkatzelis , Emmanouil Pountourakis , Alkmini Sgouritsa
In a decentralized system with $m$ machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a … In a decentralized system with $m$ machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a function of the total load assigned to it, and some cost-sharing mechanism distributes this cost among the machine's users. The users choose a machine aiming to minimize their own share of the cost, so the cost-sharing mechanism induces a game among them. We approach this problem from the perspective of a designer who can select which cost-sharing mechanism to use, aiming to minimize the price of anarchy (PoA) of the induced games. Recent work introduced the class of \emph{resource-aware} cost-sharing mechanisms, whose decisions can depend on the set of machines in the system, but are oblivious to the total number of users. These mechanisms can guarantee low PoA bounds for instances where the cost functions of the machines are all convex or concave, but can suffer from very high PoA for cost functions that deviate from these families. In this paper we show that if we enhance the class of resource-aware mechanisms with some prior information regarding the users, then they can achieve low PoA for a much more general family of cost functions. We first show that, as long as the mechanism knows just two of the participating users, then it can assign special roles to them and ensure a constant PoA. We then extend this idea to settings where the mechanism has access to the probability with which each user is present in the system. For all these instances, we provide a mechanism that achieves an expected PoA that is logarithmic in the expected number of users.

Resource-Aware Protocols for Network Cost-Sharing Games

2020-07-09
George Christodoulou , Vasilis Gkatzelis , Mohamad Latifian +1 more
We study the extent to which decentralized cost-sharing protocols can achieve good price of anarchy (PoA) bounds in network cost-sharing games with $n$ agents. We focus on the model of … We study the extent to which decentralized cost-sharing protocols can achieve good price of anarchy (PoA) bounds in network cost-sharing games with $n$ agents. We focus on the model of resource-aware protocols, where the designer has prior access to the network structure and can also increase the total cost of an edge(overcharging), and we study classes of games with concave or convex cost functions. We first consider concave cost functions and our main result is a cost-sharing protocol for symmetric games on directed acyclic graphs that achieves a PoA of $2+\varepsilon$ for some arbitrary small positive $\varepsilon$, which improves to $1+\varepsilon$ for games with at least two players. We also achieve a PoA of 1 for series-parallel graphs and show that no protocol can achieve a PoA better than $\Omega(\sqrt{n})$ for multicast games. We then also consider convex cost functions and prove analogous results for series-parallel networks and multicast games, as well as a lower bound of $\Omega(n)$ for the PoA on directed acyclic graphs without the use of overcharging.
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A Little Charity Guarantees Almost Envy-Freeness

2019-12-23
Bhaskar Ray Chaudhury , Telikepalli Kavitha , Kurt Mehlhorn +1 more
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)A Little Charity Guarantees Almost Envy-FreenessBhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini SgouritsaBhaskar … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)A Little Charity Guarantees Almost Envy-FreenessBhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini SgouritsaBhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini Sgouritsapp.2658 - 2672Chapter DOI:https://doi.org/10.1137/1.9781611975994.162PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute m goods to n agents in a “fair” manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is “envy-freeness up to any good” (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when n = 3. We show there is always a partition of the set of goods into n + 1 subsets (X1, …, Xn, P) where for i ϵ [n], Xi is the bundle allocated to agent i and the set P is unallocated (or donated to charity) such that we have: (1)envy-freeness up to any good,(2)no agent values P higher than her own bundle, and(3)fewer than n goods go to charity, i.e., |P| < n (typically m ≫ n). Our proof is constructive. When agents have additive valuations and |P| is large (i.e., when |P| is close to n), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation. 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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A Little Charity Guarantees Almost Envy-Freeness

2019-07-10
Bhaskar Ray Chaudhury , T. Kavitha , Kurt Mehlhorn +1 more
Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a fair manner, where every agent … Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a fair manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is up to any (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when $n=3$. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$ where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have$\colon$ 1) envy-freeness up to any good, 2) no agent values $P$ higher than her own bundle, and 3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive. When agents have additive valuations and $\lvert P \rvert$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is $4/7$ groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of $1/2$ known for an approximate GMMS allocation.

Designing Networks with Good Equilibria under Uncertainty

2019-01-01
George Christodoulou , Alkmini Sgouritsa
We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. … We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of $k$ terminal vertices or players needs to establish connectivity with the root. The social optimum is the minimum Steiner tree. We study situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying graph metric, but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the goal of the designer is to choose a single, universal cost-sharing protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is, to what extent can prior knowledge of the underlying graph metric help in the design? We first demonstrate that there exist classes of graphs where knowledge of the underlying graph metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than $\Omega(\log k)$. Then, in our main technical result, we show that there exist graph metrics for which knowing the underlying graph metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey theory in high-dimensional hypercubes. Then we switch to the stochastic model, where the players are activated according to some probability distribution that is known to the designer. We show that there exists a randomized ordered protocol that achieves constant PoA. If, further, each player is activated independently with some probability, by using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark that the first result holds also for the black-box model, where the probabilities are not known to the designer, but she is allowed to draw independent (polynomially many) samples.
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A Little Charity Guarantees Almost Envy-Freeness

2019-01-01
Bhaskar Ray Chaudhury , T. Kavitha , Kurt Mehlhorn +1 more
Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent … Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good" (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when $n=3$. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$ where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have$\colon$ 1) envy-freeness up to any good, 2) no agent values $P$ higher than her own bundle, and 3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive. When agents have additive valuations and $\lvert P \rvert$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is $4/7$ groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of $1/2$ known for an approximate GMMS allocation.
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Tight Bounds for the Price of Anarchy of Simultaneous First-Price Auctions

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

2015-12-21
George Christodoulou , Alkmini Sgouritsa
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Designing Networks with Good Equilibria under UncertaintyGeorge Christodoulou and Alkmini SgouritsaGeorge Christodoulou and Alkmini … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Designing Networks with Good Equilibria under UncertaintyGeorge Christodoulou and Alkmini SgouritsaGeorge Christodoulou and Alkmini Sgouritsapp.72 - 89Chapter DOI:https://doi.org/10.1137/1.9781611974331.ch6PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the goal of the designer is to choose a single, universal cost-sharing protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist classes of graphs where knowledge of the underlying metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than Ω(log k). Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of Ω(log k), which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated according to some probability distribution that is known to the designer. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark, that the first result holds also for the black-box model, where the probabilities are not known to the designer, but is allowed to draw independent (polynomially many) samples. Previous chapter Next chapter RelatedDetails Published:2016eISBN:978-1-61197-433-1 https://doi.org/10.1137/1.9781611974331Book Series Name:ProceedingsBook Code:PRDA16Book Pages:viii + 2106
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On the Efficiency of All-Pay Mechanisms

2015-08-05
George Christodoulou , Alkmini Sgouritsa , Bo Tang
We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial … We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial auctions where m all-pay auctions run in parallel, one for each good. For fractionally subadditive valuations, we strengthen the upper bound from 2 [Syrgkanis and Tardos STOC'13] to 1.82 by proving some structural properties that characterize the mixed Nash equilibria of the game. Next, we design an all-pay mechanism with a randomized allocation rule for the multi- unit auction. We show that, for bidders with submodular valuations, the mechanism admits a unique, 75% efficient, pure Nash equilibrium. The efficiency of this mechanism outperforms all the known bounds on the price of anarchy of mechanisms used for multi-unit auctions. Finally, we analyze single-item all-pay auctions motivated by their connection to contests and show tight bounds on the price of anarchy of social welfare, revenue and maximum bid.

On the Efficiency of the Proportional Allocation Mechanism for Divisible Resources

2015-01-01
George Christodoulou , Alkmini Sgouritsa , Bo Tang
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On the Efficiency of All-Pay Mechanisms

2015-01-01
George Christodoulou , Alkmini Sgouritsa , Bo Tang
We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial … We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial auctions where m all-pay auctions run in parallel, one for each good. For fractionally subadditive valuations, we strengthen the upper bound from 2 [Syrgkanis and Tardos STOC'13] to 1.82 by proving some structural properties that characterize the mixed Nash equilibria of the game. Next, we design an all-pay mechanism with a randomized allocation rule for the multi- unit auction. We show that, for bidders with submodular valuations, the mechanism admits a unique, 75% efficient, pure Nash equilibrium. The efficiency of this mechanism outperforms all the known bounds on the price of anarchy of mechanisms used for multi-unit auctions. Finally, we analyze single-item all-pay auctions motivated by their connection to contests and show tight bounds on the price of anarchy of social welfare, revenue and maximum bid.

On the Efficiency of the Proportional Allocation Mechanism for Divisible Resources

2015-01-01
George Christodoulou , Alkmini Sgouritsa , Bo Tang
We study the efficiency of the proportional allocation mechanism, that is widely used to allocate divisible resources. Each agent submits a bid for each divisible resource and receives a fraction … We study the efficiency of the proportional allocation mechanism, that is widely used to allocate divisible resources. Each agent submits a bid for each divisible resource and receives a fraction proportional to her bids. We quantify the inefficiency of Nash equilibria by studying the Price of Anarchy (PoA) of the induced game under complete and incomplete information. When agents' valuations are concave, we show that the Bayesian Nash equilibria can be arbitrarily inefficient, in contrast to the well-known 4/3 bound for pure equilibria. Next, we upper bound the PoA over Bayesian equilibria by 2 when agents' valuations are subadditive, generalizing and strengthening previous bounds on lattice submodular valuations. Furthermore, we show that this bound is tight and cannot be improved by any simple or scale-free mechanism. Then we switch to settings with budget constraints, and we show an improved upper bound on the PoA over coarse-correlated equilibria. Finally, we prove that the PoA is exactly 2 for pure equilibria in the polyhedral environment.

On the Efficiency of All-Pay Mechanisms

2015-01-01
George Christodoulou , Alkmini Sgouritsa , Bo Tang
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Designing Networks with Good Equilibria under Uncertainty

2015-01-01
George Christodoulou , Alkmini Sgouritsa
We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. … We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the designer's goal is to choose a single, universal protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist graphs (outerplanar) where knowledge of the underlying metric can dramatically improve the performance of good network design. Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol with constant PoA.
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Tight Bounds for the Price of Anarchy of Simultaneous First Price Auctions

2013-01-01
George Christodoulou , Annamária Kovács , Alkmini Sgouritsa +1 more
We study the Price of Anarchy of simultaneous first-price auctions for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian Price of Anarchy of these … We study the Price of Anarchy of simultaneous first-price auctions for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian Price of Anarchy of these auctions are e/(e-1) [Syrgkanis and Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching lower bounds for both cases even for the case of full information and for mixed Nash equilibria via an explicit construction. We present an alternative proof of the upper bound of e/(e-1) for first-price auctions with fractionally subadditive valuations which reveals the worst-case price distribution, that is used as a building block for the matching lower bound construction. We generalize our results to a general class of item bidding auctions that we call bid-dependent auctions (including first-price auctions and all-pay auctions) where the winner is always the highest bidder and each bidder's payment depends only on his own bid. Finally, we apply our techniques to discriminatory price multi-unit auctions. We complement the results of [de Keijzer et al. 2013] for the case of subadditive valuations, by providing a matching lower bound of 2. For the case of submodular valuations, we provide a lower bound of 1.109. For the same class of valuations, we were able to reproduce the upper bound of e/(e-1) using our non-smooth approach.
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Coauthor Papers Together
George Christodoulou 12
Bo Tang 7
Vasilis Gkatzelis 7
Bhaskar Ray Chaudhury 4
Kurt Mehlhorn 4
Emmanouil Pountourakis 3
Telikepalli Kavitha 2
Georgios Amanatidis 2
Kostas Kollias 2
Xizhi Tan 2
T. Kavitha 2
Aris Filos-Ratsikas 2
Annamária Kovács 2
Vasilis Christoforidis 1
Ioannis Vlachos 1
George Christodoulou 1
Minas Marios Sotiriou 1
Symeon Mastrakoulis 1
Mohamad Latifian 1
Arash Ashuri 1

Non-price equilibria in markets of discrete goods

2011-06-05
Avinatan Hassidim , Haim Kaplan , Yishay Mansour +1 more
No abstract available. No abstract available.
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Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

2013-02-14
George Christodoulou , Kurt Mehlhorn , Evangelia Pyrga
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Efficient Coordination Mechanisms for Unrelated Machine Scheduling

2012-04-24
Ioannis Caragiannis
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Efficiency Loss in a Network Resource Allocation Game

2004-08-01
Ramesh Johari , John N. Tsitsiklis
We explore the properties of a congestion game in which users of a congested resource anticipate the effect of their actions on the price of the resource. When users are … We explore the properties of a congestion game in which users of a congested resource anticipate the effect of their actions on the price of the resource. When users are sharing a single resource, we establish that the aggregate utility received by the users is at least 3/4 of the maximum possible aggregate utility. We also consider extensions to a network context, where users submit individual payments for each link in the network they may wish to use. In this network model, we again show that the selfish behavior of the users leads to an aggregate utility that is no worse than 3/4 of the maximum possible aggregate utility. We also show that the same analysis extends to a wide class of resource allocation systems where end users simultaneously require multiple scarce resources. These results form part of a growing literature on the “price of anarchy,” i.e., the extent to which selfish behavior affects system efficiency.
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Simultaneous auctions are (almost) efficient

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

2019-01-01
George Christodoulou , Alkmini Sgouritsa
We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. … We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of $k$ terminal vertices or players needs to establish connectivity with the root. The social optimum is the minimum Steiner tree. We study situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying graph metric, but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the goal of the designer is to choose a single, universal cost-sharing protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is, to what extent can prior knowledge of the underlying graph metric help in the design? We first demonstrate that there exist classes of graphs where knowledge of the underlying graph metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than $\Omega(\log k)$. Then, in our main technical result, we show that there exist graph metrics for which knowing the underlying graph metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey theory in high-dimensional hypercubes. Then we switch to the stochastic model, where the players are activated according to some probability distribution that is known to the designer. We show that there exists a randomized ordered protocol that achieves constant PoA. If, further, each player is activated independently with some probability, by using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark that the first result holds also for the black-box model, where the probabilities are not known to the designer, but she is allowed to draw independent (polynomially many) samples.
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Efficient Black-Box Reductions for Separable Cost Sharing

2020-08-25
Tobias Harks , Martin Hoefer , Anja Schedel +1 more
In cost-sharing games with delays, a set of agents jointly uses a subset of resources. Each resource has a fixed cost that has to be shared by the players, and … In cost-sharing games with delays, a set of agents jointly uses a subset of resources. Each resource has a fixed cost that has to be shared by the players, and each agent has a nonshareable player-specific delay for each resource. A separable cost-sharing protocol determines cost shares that are budget-balanced, separable, and guarantee existence of pure Nash equilibria (PNE). We provide black-box reductions reducing the design of such a protocol to the design of an approximation algorithm for the underlying cost-minimization problem. In this way, we obtain separable cost-sharing protocols in matroid games, single-source connection games, and connection games on n-series-parallel graphs. All these reductions are efficiently computable - given an initial allocation profile, we obtain a cheaper profile and separable cost shares turning the profile into a PNE. Hence, in these domains, any approximation algorithm yields a separable cost-sharing protocol with price of stability bounded by the approximation factor.
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Potential Games Are <i>Necessary</i> to Ensure Pure Nash Equilibria in Cost Sharing Games

2014-05-27
Ragavendran Gopalakrishnan , Jason R. Marden , Adam Wierman
We consider the problem of designing distribution rules to share “welfare” (cost or revenue) among individually strategic agents. There are many known distribution rules that guarantee the existence of a … We consider the problem of designing distribution rules to share “welfare” (cost or revenue) among individually strategic agents. There are many known distribution rules that guarantee the existence of a (pure) Nash equilibrium in this setting, e.g., the Shapley value and its weighted variants; however, a characterization of the space of distribution rules that guarantees the existence of a Nash equilibrium is unknown. Our work provides an exact characterization of this space for a specific class of scalable and separable games that includes a variety of applications such as facility location, routing, network formation, and coverage games. Given arbitrary local welfare functions 𝕨, we prove that a distribution rule guarantees equilibrium existence for all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is equivalent to a generalized weighted Shapley value on some “ground” welfare functions 𝕨′, which can be distinct from 𝕨. However, if budget-balance is required in addition to the existence of a Nash equilibrium, then 𝕨′ must be the same as 𝕨. We also provide an alternate characterization of this space in terms of “generalized” marginal contributions, which is more appealing from the point of view of computational tractability. A possibly surprising consequence of our result is that, in order to guarantee equilibrium existence in all games with any fixed local welfare functions, it is necessary to work within the class of potential games.
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Composable and efficient mechanisms

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

2009-04-24
Jian Li
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Multiple birds with one stone: Beating 1/2 for EFX and GMMS via envy cycle elimination

2020-07-16
Georgios Amanatidis , Evangelos Markakis , Apostolos Ntokos
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Welfare Guarantees for Proportional Allocations

2014-01-01
Ioannis Caragiannis , Alexandros A. Voudouris
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Finding Fair and Efficient Allocations

2018-06-11
Siddharth Barman , Sanath Kumar Krishnamurthy , Rohit Vaish
We study the problem of allocating a set of indivisible goods among a set of agents in a fair and efficient manner. An allocation is said to be fair if … We study the problem of allocating a set of indivisible goods among a set of agents in a fair and efficient manner. An allocation is said to be fair if it is envy-free up to one good (EF1), which means that each agent prefers its own bundle over the bundle of any other agent up to the removal of one good. In addition, an allocation is deemed efficient if it satisfies Pareto efficiency. While each of these well-studied properties is easy to achieve separately, achieving them together is far from obvious. Recently, Caragiannis et al. (2016) established the surprising result that when agents have additive valuations for the goods, there always exists an allocation that simultaneously satisfies these two seemingly incompatible properties. Specifically, they showed that an allocation that maximizes the Nash social welfare objective is both EF1 and Pareto efficient. However, the problem of maximizing Nash social welfare is NP-hard. As a result, this approach does not provide an efficient algorithm for finding a fair and efficient allocation. In this paper, we bypass this barrier, and develop a pseudopolynomial time algorithm for finding allocations that are EF1 and Pareto efficient; in particular, when the valuations are bounded, our algorithm finds such an allocation in polynomial time. Furthermore, we establish a stronger existence result compared to Caragiannis et al. (2016): For additive valuations, there always exists an allocation that is EF1 and fractionally Pareto efficient. Another key contribution of our work is to show that our algorithm provides a polynomial-time 1.45-approximation to the Nash social welfare objective. This improves upon the best known approximation ratio for this problem (namely, the 2-approximation algorithm of Cole et al., 2017), and also matches the lower bound on the integrality gap of the convex program of Cole et al. (2017). Unlike many of the existing approaches, our algorithm is completely combinatorial, and relies on constructing integral Fisher markets wherein specific equilibria are not only efficient, but also fair.
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Comparing Approximate Relaxations of Envy-Freeness

2018-07-01
Georgios Amanatidis , Georgios Birmpas , Vangelis Markakis
In fair division problems with indivisible goods it is well known that one cannot have any guarantees for the classic fairness notions of envy-freeness and proportionality. As a result, several … In fair division problems with indivisible goods it is well known that one cannot have any guarantees for the classic fairness notions of envy-freeness and proportionality. As a result, several relaxations have been introduced, most of which in quite recent works. We focus on four such notions, namely envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin share fairness (MMS), and pairwise maximin share fairness (PMMS). Since obtaining these relaxations also turns out to be problematic in several scenarios, approximate versions of them have also been considered. In this work, we investigate further the connections between the four notions mentioned above and their approximate versions. We establish several tight or almost tight results concerning the approximation quality that any of these notions guarantees for the others, providing an almost complete picture of this landscape. Some of our findings reveal interesting and surprising consequences regarding the power of these notions, e.g., PMMS and EFX provide the same worst-case guarantee for MMS, despite PMMS being a strictly stronger notion than EFX. We believe such implications provide further insight on the quality of approximately fair solutions.
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Potential games are necessary to ensure pure nash equilibria in cost sharing games

2013-06-16
Ragavendran Gopalakrishnan , Jason R. Marden , Adam Wierman
We consider the problem of designing distribution rules to share `welfare' (cost or revenue) among individually strategic agents. There are many known distribution rules that guarantee the existence of a … We consider the problem of designing distribution rules to share `welfare' (cost or revenue) among individually strategic agents. There are many known distribution rules that guarantee the existence of a (pure) Nash equilibrium in this setting, e.g., the Shapley value and its weighted variants; however, a characterization of the space of distribution rules that guarantee the existence of a Nash equilibrium is unknown. Our work provides an exact characterization of this space for a specific class of scalable and separable games, which includes a variety of applications such as facility location, routing, network formation, and coverage games. Given arbitrary local welfare functions $W$, we prove that a distribution rule guarantees equilibrium existence for all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is equivalent to a generalized weighted Shapley value on some `ground' welfare functions $W$, which can be distinct from W. However, if budget-balance is required in addition to the existence of a Nash equilibrium, then $W$ must be the same as $W$. We also provide an alternate characterization of this space in terms of `generalized' marginal contributions, which is more appealing from the point of view of computational tractability. A possibly surprising consequence of our result is that, in order to guarantee equilibrium existence in all games with any fixed local welfare functions, it is \textitnecessary to work within the class of potential games.
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Inefficiency of Standard Multi-unit Auctions

2013-01-01
Bart de Keijzer , Evangelos Markakis , Guido Schäfer +1 more
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Almost Envy-Freeness with General Valuations

2020-01-01
Benjamin Plaut , Tim Roughgarden
The goal of fair division is to distribute resources among competing players in a “fair" way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do … The goal of fair division is to distribute resources among competing players in a “fair" way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.
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A price-anticipating resource allocation mechanism for distributed shared clusters

2005-06-05
Michal Feldman , Kevin Lai , Li Zhang
In this paper we formulate the fixed budget resource allocation game to understand the performance of a distributed market-based resource allocation system. Multiple users decide how to distribute their budget … In this paper we formulate the fixed budget resource allocation game to understand the performance of a distributed market-based resource allocation system. Multiple users decide how to distribute their budget bids) among multiple machines according to their individual preferences to maximize their individual utility. We look at both the efficiency and the fairness of the allocation at the equilibrium, where fairness is evaluated through the measures of utility uniformity and envy-freeness. We show analytically and through simulations that despite being highly decentralized, such a system converges quickly to an equilibrium and unlike the social optimum that achieves high efficiency but poor fairness, the proposed allocation scheme achieves a nice balance of high degrees of efficiency and fairness at the equilibrium.
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Approximation Algorithms for Maximin Fair Division

2017-06-20
Siddharth Barman , Sanath Kumar Krishna Murthy
We consider the problem of dividing indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin … We consider the problem of dividing indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, that is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focussed on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee.
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Sub-additive functions

1950-09-01
R. A. Rosenbaum

Optimal auctions with correlated bidders are easy

2011-06-06
Shahar Dobzinski , Hu Fu , Robert Kleinberg
We consider the problem of designing a revenue-maximizing auction for a single item, when the values of the bidders are drawn from a correlated distribution. We observe that there exists … We consider the problem of designing a revenue-maximizing auction for a single item, when the values of the bidders are drawn from a correlated distribution. We observe that there exists an algorithm that finds the optimal randomized mechanism that runs in time polynomial in the size of the support. We leverage this result to show that in the oracle model introduced by Ronen and Saberi [FOCS'02], there exists a polynomial time truthful in expectation mechanism that provides a (1.5+ε)-approximation to the revenue achievable by an optimal truthful-in-expectation mechanism, and a polynomial time deterministic truthful mechanism that guarantees 5/3 approximation to the revenue achievable by an optimal deterministic truthful mechanism.
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Groupwise Maximin Fair Allocation of Indivisible Goods

2018-04-25
Siddharth Barman , Arpita Biswas , Sanath Kumar Krishnamurthy +1 more
We study the problem of allocating indivisible goods among n agents in a fair manner. For this problem, maximin share (MMS) is a well-studied solution concept which provides a fairness … We study the problem of allocating indivisible goods among n agents in a fair manner. For this problem, maximin share (MMS) is a well-studied solution concept which provides a fairness threshold. Specifically, maximin share is defined as the minimum utility that an agent can guarantee for herself when asked to partition the set of goods into n bundles such that the remaining (n-1) agents pick their bundles adversarially. An allocation is deemed to be fair if every agent gets a bundle whose valuation is at least her maximin share. Even though maximin shares provide a natural benchmark for fairness, it has its own drawbacks and, in particular, it is not sufficient to rule out unsatisfactory allocations. Motivated by these considerations, in this work we define a stronger notion of fairness, called groupwise maximin share guarantee (GMMS). In GMMS, we require that the maximin share guarantee is achieved not just with respect to the grand bundle, but also among all the subgroups of agents. Hence, this solution concept strengthens MMS and provides an ex-post fairness guarantee. We show that in specific settings, GMMS allocations always exist. We also establish the existence of approximate GMMS allocations under additive valuations, and develop a polynomial-time algorithm to find such allocations. Moreover, we establish a scale of fairness wherein we show that GMMS implies approximate envy freeness. Finally, we empirically demonstrate the existence of GMMS allocations in a large set of randomly generated instances. For the same set of instances, we additionally show that our algorithm achieves an approximation factor better than the established, worst-case bound.
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A Little Charity Guarantees Almost Envy-Freeness

2021-01-01
Bhaskar Ray Chaudhury , Telikepalli Kavitha , Kurt Mehlhorn +1 more
The fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every … The fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent has a valuation for each subset of goods. We assume monotone valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good," EFX, where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$, where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have (1) envy-freeness up to any good, (2) no agent values $P$ higher than her own bundle, and (3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive and leads to a pseudopolynomial time algorithm to find such an allocation. When agents have additive valuations and $|{P}|$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation that is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation. (Very recently and independently, Amanatidis, Ntokos, and Markakis [Theoret. Comput. Sci., 841 (2020), pp. 94--109], also showed the existence of a 4/7-GMMS allocation.)
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Online Graph Algorithms with Predictions

2022-01-01
Yossi Azar , Debmalya Panigrahi , Noam Touitou
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Online Graph Algorithms with PredictionsYossi Azar, Debmalya Panigrahi, and Noam TouitouYossi Azar, Debmalya Panigrahi, … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Online Graph Algorithms with PredictionsYossi Azar, Debmalya Panigrahi, and Noam TouitouYossi Azar, Debmalya Panigrahi, and Noam Touitoupp.35 - 66Chapter DOI:https://doi.org/10.1137/1.9781611977073.3PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Online algorithms with predictions is a popular and elegant framework for bypassing pessimistic lower bounds in competitive analysis. In this model, online algorithms are supplied with future predictions and the goal is for the competitive ratio to smoothly interpolate between the best offline and online bounds as a function of the prediction error. In this paper, we study online graph problems with predictions. Our contributions are the following: The first question is defining prediction error. For graph/metric problems, there can be two types of error, locations that are not predicted, and locations that are predicted but the predicted and actual locations do not coincide exactly. We design a novel definition of prediction error called metric error with outliers to simultaneously capture both types of errors, which thereby generalizes previous definitions of error that only capture one of the two error types. We give a general framework for obtaining online algorithms with predictions that combines, in a "black box" fashion, existing online and offline algorithms, under certain technical conditions. To the best of our knowledge, this is the first general-purpose tool for obtaining online algorithms with predictions. Using our framework, we obtain tight bounds on the competitive ratio of several classical graph problems as a function of metric error with outliers: Steiner tree, Steiner forest, priority Steiner tree/forest, and uncapacitated/capacitated facility location. Both the definition of metric error with outliers and the general framework for combining offline and online algorithms are not specific to the problems that we consider in this paper. We hope that these will be useful for future work on other problems in this domain. 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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Almost Envy-Freeness with General Valuations

2018-01-01
Benjamin Plaut , Tim Roughgarde
The goal of fair division is to distribute resources among competing players in a “fair” way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do … The goal of fair division is to distribute resources among competing players in a “fair” way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.
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A Ramsey-type result for the hypercube

2006-01-01
Noga Alon , Radoš Radoičić , Benny Sudakov +1 more
We prove that for every fixed k and ℓ ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Qn with k colors contains a monochromatic cycle … We prove that for every fixed k and ℓ ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Qn with k colors contains a monochromatic cycle of length 2 ℓ. This answers an open question of Chung. Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, that is, have the property that for every k, any k-edge coloring of a sufficiently large Qn contains a monochromatic copy of H. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 196–208, 2006

Convex Program Duality, Fisher Markets, and Nash Social Welfare

2016-01-01
Richard Cole , Nikhil R. Devanur , Vasilis Gkatzelis +4 more
We study Fisher markets and the problem of maximizing the Nash social welfare (NSW), and show several closely related new results. In particular, we obtain: -- A new integer program … We study Fisher markets and the problem of maximizing the Nash social welfare (NSW), and show several closely related new results. In particular, we obtain: -- A new integer program for the NSW maximization problem whose fractional relaxation has a bounded integrality gap. In contrast, the natural integer program has an unbounded integrality gap. -- An improved, and tight, factor 2 analysis of the algorithm of [7]; in turn showing that the integrality gap of the above relaxation is at most 2. The approximation factor shown by [7] was $2e^{1/e} \approx 2.89$. -- A lower bound of $e^{1/e}\approx 1.44$ on the integrality gap of this relaxation. -- New convex programs for natural generalizations of linear Fisher markets and proofs that these markets admit rational equilibria. These results were obtained by establishing connections between previously known disparate results, and they help uncover their mathematical underpinnings. We show a formal connection between the convex programs of Eisenberg and Gale and that of Shmyrev, namely that their duals are equivalent up to a change of variables. Both programs capture equilibria of linear Fisher markets. By adding suitable constraints to Shmyrev's program, we obtain a convex program that captures equilibria of the spending-restricted market model defined by [7] in the context of the NSW maximization problem. Further, adding certain integral constraints to this program we get the integer program for the NSW mentioned above. The basic tool we use is convex programming duality. In the special case of convex programs with linear constraints (but convex objectives), we show a particularly simple way of obtaining dual programs, putting it almost at par with linear program duality. This simple way of finding duals has been used subsequently for many other applications.
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On Fair Division of Indivisible Items

2018-01-01
Bhaskar Chaudhury , Yun Kuen Cheung , Jugal Garg +3 more
We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to … We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of $e^{1/{e}} \approx 1.445$. The computed allocation is Pareto-optimal and approximates envy-freeness up to one item up to a factor of $2 + \eps$
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Online network design algorithms via hierarchical decompositions

2015-01-04
Seeun William Umboh
We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of … We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design.In a sense, our work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings --- embed the entire graph into a tree at the beginning and then solve the problem on the tree --- and results in randomized O(log n)-competitive algorithms, where n is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get O(log k) competitive ratios, where k is the number of terminals.In this paper, we apply our approach to obtain deterministic O(log k)-competitive online algorithms for the following problems.1. Steiner network with edge duplication. Previously, only a randomized O(log n)-competitive algorithm was known.2. Rent-or-buy. Previously, only deterministic O(log2k)-competitive and randomized O(log k)-competitive algorithms by Awerbuch, Azar and Bartal (Theoretical Computer Science 2004) were known.3. Connected facility location. Previously, only a randomized O(log2k)-competitive algorithm of San Felice, Williamson and Lee (LATIN 2014) was known.4. Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (ICALP 2011) and give a simpler analysis.Our competitive ratios are optimal up to constant factors as these problems capture the online Steiner tree problem which has a lower bound of Ω(log k).

Pushing the Frontier on Approximate EFX Allocations

2024-06-18
Georgios Amanatidis , Aris Filos-Ratsikas , Alkmini Sgouritsa
We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($\alpha$-EFX). … We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($\alpha$-EFX). The state-of-the-art results on the problem include that (exact) EFX allocations exist when (a) there are at most three agents, or (b) the agents' valuation functions can take at most two values, or (c) the agents' valuation functions can be represented via a graph. For $\alpha$-EFX, it is known that a $0.618$-EFX allocation exists for any number of agents with additive valuation functions. In this paper, we show that $2/3$-EFX allocations exist when (a) there are at most \emph{seven agents}, (b) the agents' valuation functions can take at most \emph{three values}, or (c) the agents' valuation functions can be represented via a \emph{multigraph}. Our results can be interpreted in two ways. First, by relaxing the notion of EFX to $2/3$-EFX, we obtain existence results for strict generalizations of the settings for which exact EFX allocations are known to exist. Secondly, by imposing restrictions on the setting, we manage to beat the barrier of $0.618$ and achieve an approximation guarantee of $2/3$. Therefore, our results push the \emph{frontier} of existence and computation of approximate EFX allocations, and provide insights into the challenges of settling the existence of exact EFX allocations.
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Approximation Algorithms for Computing Maximin Share Allocations

2017-10-31
Georgios Amanatidis , Evangelos Markakis , Afshin Nikzad +1 more
We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of $n$ agents and a set of goods, the maximin share of a … We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of $n$ agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into $n$ bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, so we resort to approximation algorithms. Our main result is a $2/3$-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang, which also produces a $2/3$-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of their algorithm. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances, with high probability there exists a maximin share allocation. This can be seen as a justification of the experimental evidence reported in relevant works. Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of $3$ agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of $2$ agents). We provide a $7/8$-approximation algorithm, improving the previously known result of $3/4$. The second case is when all item values belong to $\{0, 1, 2\}$, extending the $\{0, 1\}$ setting studied in Bouveret and Lema\^itre. We obtain an exact algorithm for any number of agents in this case.
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Welfare Guarantees for Proportional Allocations

2016-03-22
Ioannis Caragiannis , Alexandros A. Voudouris
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Improving the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math>-bound on the price of stability in undirected Shapley network design games

2014-10-24
Yann Disser , Andreas Emil Feldmann , Max Klimm +1 more
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Inner product spaces for MinSum coordination mechanisms

2011-06-06
Richard Cole , José R. Correa , Vasilis Gkatzelis +2 more
We study coordination mechanisms aiming to minimize the weighted sum of completion times of jobs in the context of selfish scheduling problems. Our goal is to design local policies that … We study coordination mechanisms aiming to minimize the weighted sum of completion times of jobs in the context of selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obtain these approximation bounds, we introduce a new technique that while conceptually simple, seems to be quite powerful. The method entails mapping strategy vectors into a carefully chosen inner product space; costs are shown to correspond to the norm in this space, and the Nash condition also has a simple description. With this structure in place, we are able to prove a number of results, as follows. First, we consider Smith's Rule, which orders the jobs on a machine in ascending processing time to weight ratio, and show that it achieves an approximation ratio of 4. We also demonstrate that this is the best possible for deterministic non-preemptive strongly local policies. Since Smith's Rule is always optimal for a given fixed assignment, this may seem unsurprising, but we then show that better approximation ratios can be obtained if either preemption or randomization is allowed.
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Tight Bounds for the Price of Anarchy of Simultaneous First-Price Auctions

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

2017-06-20
Richard Cole , Nikhil R. Devanur , Vasilis Gkatzelis +4 more
No abstract available. No abstract available.
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Nash Social Welfare, Matrix Permanent, and Stable Polynomials

2016-01-01
Nima Anari , Shayan Oveis Gharan , Amin Saberi +1 more
We study the problem of allocating $m$ items to $n$ agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, … We study the problem of allocating $m$ items to $n$ agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a $1/e$ approximation factor of the objective. Our main technical contribution is an extension of Gurvits's lower bound on the coefficient of the square-free monomial of a degree $m$-homogeneous stable polynomial on $m$ variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.

Optimal Crowdsourcing Contests

2012-01-17
Shuchi Chawla , Jason D. Hartline , Balasubramanian Sivan
We study the design and approximation of optimal crowdsourcing contests. Crowdsourcing contests can be modeled as all-pay auctions because entrants must exert effort up-front to enter. Unlike all-pay auctions where … We study the design and approximation of optimal crowdsourcing contests. Crowdsourcing contests can be modeled as all-pay auctions because entrants must exert effort up-front to enter. Unlike all-pay auctions where a usual design objective would be to maximize revenue, in crowdsourcing contests, the principal only benefits from the submission with the highest quality. We give a theory for optimal crowdsourcing contests that mirrors the theory of optimal auction design: the optimal crowdsourcing contest is a virtual valuation optimizer (the virtual valuation function depends on the distribution of contestant skills and the number of contestants). We also compare crowdsourcing contests with more conventional means of procurement. In this comparison, crowdsourcing contests are relatively disadvantaged because the effort of losing contestants is wasted. Nonetheless, we show that crowdsourcing contests are 2-approximations to conventional methods for a large family of “regular” distributions, and 4-approximations, otherwise.
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Envy-freeness up to any item with high Nash welfare: The virtue of donating items

2019-02-12
Ioannis Caragiannis , Nick Gravin , Xin Huang
Several fairness concepts have been proposed recently in attempts to approximate envy-freeness in settings with indivisible goods. Among them, the concept of envy-freeness up to any item (EFX) is arguably … Several fairness concepts have been proposed recently in attempts to approximate envy-freeness in settings with indivisible goods. Among them, the concept of envy-freeness up to any item (EFX) is arguably the closest to envy-freeness. Unfortunately, EFX allocations are not known to exist except in a few special cases. We make significant progress in this direction. We show that for every instance with additive valuations, there is an EFX allocation of a subset of items with a Nash welfare that is at least half of the maximum possible Nash welfare for the original set of items. That is, after donating some items to a charity, one can distribute the remaining items in a fair way with high efficiency. This bound is proved to be best possible. Our proof is constructive and highlights the importance of maximum Nash welfare allocation. Starting with such an allocation, our algorithm decides which items to donate and redistributes the initial bundles to the agents, eventually obtaining an allocation with the claimed efficiency guarantee. The application of our algorithm to large markets, where the valuations of an agent for every item is relatively small, yields EFX with almost optimal Nash welfare. To the best of our knowledge, this is the first use of large market assumptions in the fair division literature. We also show that our algorithm can be modified to compute, in polynomial-time, EFX allocations that approximate optimal Nash welfare within a factor of at most $2\rho$, using a $\rho$-approximate allocation on input instead of the maximum Nash welfare one.

An Improved Approximation Algorithm for Maximin Shares

2019-01-01
Jugal Garg , Setareh Taki
Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. We study the case … Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. We study the case where m indivisible items need to be divided among n agents with additive valuations using the popular fairness notion of maximin share (MMS). An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist, a series of work provided approximation algorithms for a 2/3-MMS allocation in which each agent receives a bundle worth at least 2/3 times her maximin share. More recently, Ghodsi et al. [EC'2018] showed the existence of a 3/4-MMS allocation and a PTAS to find a (3/4-\epsilon)-MMS allocation for an \epsilon > 0. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain. In this paper, we develop a new approach that gives a simple algorithm for showing the existence of a 3/4-MMS allocation. Furthermore, our approach is powerful enough to be easily extended in two directions: First, we get a strongly polynomial-time algorithm to find a 3/4-MMS allocation, where we do not need to approximate the MMS values at all. Second, we show that there always exists a (3/4 + 1/(12n))-MMS allocation, improving the best previous factor. This improves the approximation guarantee, most notably for small n. We note that 3/4 was the best factor known for n> 4.

Approximation Algorithms for Maximin Fair Division

2020-02-29
Siddharth Barman , Sanath Kumar Krishnamurthy
We consider the problem of allocating indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin … We consider the problem of allocating indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share , which is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focused on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations, a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations . Specifically, we show that when the valuations of the agents are nonnegative , monotone , and submodular, then a 0.21-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the article is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions .
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Designing Networks with Good Equilibria under Uncertainty

2015-12-21
George Christodoulou , Alkmini Sgouritsa
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Designing Networks with Good Equilibria under UncertaintyGeorge Christodoulou and Alkmini SgouritsaGeorge Christodoulou and Alkmini … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Designing Networks with Good Equilibria under UncertaintyGeorge Christodoulou and Alkmini SgouritsaGeorge Christodoulou and Alkmini Sgouritsapp.72 - 89Chapter DOI:https://doi.org/10.1137/1.9781611974331.ch6PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the goal of the designer is to choose a single, universal cost-sharing protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist classes of graphs where knowledge of the underlying metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than Ω(log k). Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of Ω(log k), which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated according to some probability distribution that is known to the designer. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark, that the first result holds also for the black-box model, where the probabilities are not known to the designer, but is allowed to draw independent (polynomially many) samples. Previous chapter Next chapter RelatedDetails Published:2016eISBN:978-1-61197-433-1 https://doi.org/10.1137/1.9781611974331Book Series Name:ProceedingsBook Code:PRDA16Book Pages:viii + 2106
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Efficient Black-Box Reductions for Separable Cost Sharing

2018-01-01
Tobias Harks , Martin Hoefer , Anja Huber +1 more
In cost sharing games with delays, a set of agents jointly uses a finite subset of resources. Each resource has a fixed cost that has to be shared by the … In cost sharing games with delays, a set of agents jointly uses a finite subset of resources. Each resource has a fixed cost that has to be shared by the players, and each agent has a non-shareable player-specific delay for each resource. A prominent example is uncapacitated facility location (UFL), where facilities need to be opened (at a shareable cost) and clients want to connect to opened facilities. Each client pays a cost share and his non-shareable physical connection cost. Given any profile of subsets used by the agents, a separable cost sharing protocol determines cost shares that satisfy budget balance on every resource and separability over the resources. Moreover, a separable protocol guarantees existence of pure Nash equilibria in the induced strategic game for the agents. In this paper, we study separable cost sharing protocols in several general combinatorial domains. We provide black-box reductions to reduce the design of a separable cost sharing protocol to the design of an approximation algorithm for the underlying cost minimization problem. In this way, we obtain new separable cost sharing protocols in games based on arbitrary player-specific matroids, single-source connection games without delays, and connection games on n-series-parallel graphs with delays. All these reductions are efficiently computable - given an initial allocation profile, we obtain a profile of no larger cost and separable cost shares turning the profile into a pure Nash equilibrium. Hence, in these domains any approximation algorithm can be used to obtain a separable cost sharing protocol with a price of stability bounded by the approximation factor.

Almost envy-freeness with general valuations

2018-01-07
Benjamin Plaut , Tim Roughgarden
The goal of division is to distribute resources among competing players in a fair way. Envy-freeness is the most extensively studied fairness notion in division. Envy-free allocations do not always … The goal of division is to distribute resources among competing players in a fair way. Envy-freeness is the most extensively studied fairness notion in division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the division of indivisible goods with different classes of player valuations.

Maximin-Aware Allocations of Indivisible Goods

2019-07-28
Hau Chan , Jing Chen , Bo Li +1 more
We study envy-free allocations of indivisible goods to agents in settings where each agent is unaware of the goods allocated to other agents. In particular, we propose the maximin aware … We study envy-free allocations of indivisible goods to agents in settings where each agent is unaware of the goods allocated to other agents. In particular, we propose the maximin aware (MMA) fairness measure, which guarantees that every agent, given the bundle allocated to her, is aware that she does not envy at least one other agent, even if she does not know how the other goods are distributed among other agents. We also introduce two of its relaxations, and discuss their egalitarian guarantee and existence. Finally, we present a polynomial-time algorithm, which computes an allocation that approximately satisfies MMA or its relaxations. Interestingly, the returned allocation is also 1/2-approximate EFX when all agents have sub- additive valuations, which improves the algorithm in [Plaut and Roughgarden, 2018].
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Multiple Birds with One Stone: Beating 1/2 for EFX and GMMS via Envy Cycle Elimination

2020-04-03
Georgios Amanatidis , Evangelos Markakis , Apostolos Ntokos
Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most … Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a (φ−1)-approximation of envy-freeness up to any good (EFX) and a 2/φ+2 -approximation of groupwise maximin share fairness (GMMS), where φ is the golden ratio. The best known approximation factor, in polynomial time, for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good (EF1) and a 2/3-approximation of pairwise maximin share fairness (PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our algorithm and improve further the guarantees for GMMS or PMMS.Finally, we show that GMMS—and thus PMMS and EFX—allocations always exist when the number of goods does not exceed the number of agents by more than two.
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EFX Exists for Three Agents

2020-01-01
Bhaskar Ray Chaudhury , Jugal Garg , Kurt Mehlhorn
We study the problem of distributing a set of indivisible items among agents with additive valuations in a $\mathit{fair}$ manner. The fairness notion under consideration is Envy-freeness up to any … We study the problem of distributing a set of indivisible items among agents with additive valuations in a $\mathit{fair}$ manner. The fairness notion under consideration is Envy-freeness up to any item (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this paper, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture by Caragiannis et al. by showing an instance with three agents for which there is a partial EFX allocation (some items are not allocated) with higher Nash welfare than that of any complete EFX allocation.

Maximum Nash welfare and other stories about EFX

2021-02-09
Georgios Amanatidis , Georgios Birmpas , Aris Filos-Ratsikas +2 more
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An Improved Approximation Algorithm for Maximin Shares

2020-07-09
Jugal Garg , Setareh Taki
We study the problem of fair allocation of m indivisible items among n agents with additive valuations using the popular notion of maximin share (MMS) as our measure of fairness. … We study the problem of fair allocation of m indivisible items among n agents with additive valuations using the popular notion of maximin share (MMS) as our measure of fairness. An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist [5, 7], a series of remarkable work [1-3, 6, 7] provided 2/3 approximation algorithms in which each agent receives a bundle worth at least 2/3 times her maximin share. More recently, [4] showed the existence of 3/4 MMS allocations and a PTAS to find a 3/4 - ε MMS allocation. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain.
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Almost Envy-freeness, Envy-rank, and Nash Social Welfare Matchings

2021-05-18
Alireza Farhadi , MohammadTaghi Hajiaghayi , Mohamad Latifian +2 more
Envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 … Envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents and additive valuations. In addition, the best approximation guarantee for EFX is (φ − 1) ≃ 0.61 by Amanitidis et al.. In order to find a middle ground to bridge this gap, in this paper we suggest another fairness criterion, namely envy-freeness up to a random good or EFR, which is weaker than EFX, yet stronger than EF1. For this notion, we provide a polynomial-time 0.73-approximation allocation algorithm. For our algorithm we introduce Nash Social Welfare Matching which makes a connection between Nash Social Welfare and envy freeness.
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