Featured Researches

Computer Science And Game Theory

(Almost Full) EFX Exists for Four Agents (and Beyond)

The existence of EFX allocations is a major open problem in fair division, even for additive valuations. The current state of the art is that no setting where EFX allocations are impossible is known, and EFX is known to exist for ( i ) agents with identical valuations, ( ii ) 2 agents, ( iii ) 3 agents with additive valuations, ( iv ) agents with one of two additive valuations and ( v ) agents with two-valued instances. It is also known that EFX exists if one can leave n?? items unallocated, where n is the number of agents. We develop new techniques that allow us to push the boundaries of the enigmatic EFX problem beyond these known results, and, arguably, to simplify proofs of earlier results. Our main results are ( i ) every setting with 4 additive agents admits an EFX allocation that leaves at most a single item unallocated, ( ii ) every setting with n additive valuations has an EFX allocation with at most n?? unallocated items. Moreover, all of our results extend beyond additive valuations to all nice cancelable valuations (a new class, including additive, unit-demand, budget-additive and multiplicative valuations, among others). Furthermore, using our new techniques, we show that previous results for additive valuations extend to nice cancelable valuations.

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Computer Science And Game Theory

(In)Existence of Equilibria for 2-Players, 2-Values Games with Concave Valuations

We consider 2-players, 2-values minimization games where the players' costs take on two values, a,b , a<b . The players play mixed strategies and their costs are evaluated by unimodal valuations. This broad class of valuations includes all concave, one-parameter functions F:[0,1]→R with a unique maximum point. Our main result is an impossibility result stating that: If the maximum is obtained in (0,1) and F( 1 2 )≠b , then there exists a 2-players, 2-values game without F -equilibrium. The counterexample game used for the impossibility result belongs to a new class of very sparse 2-players, 2-values bimatrix games which we call normal games. In an attempt to investigate the remaining case F( 1 2 )=b , we show that: - Every normal, n -strategies game has an F -equilibrium when F( 1 2 )=b . We present a linear time algorithm for computing such an equilibrium. - For 2-players, 2-values games with 3 strategies we have that if F( 1 2 )≤b , then every 2-players, 2-values, 3-strategies game has an F -equilibrium; if F( 1 2 )>b , then there exists a normal 2-players, 2-values, 3-strategies game without F -equilibrium. To the best of our knowledge, this work is the first to provide an (almost complete) answer on whether there is, for a given concave function F , a counterexample game without F -equilibrium.

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Computer Science And Game Theory

A Balance for Fairness: Fair Distribution Utilising Physics in Games of Characteristic Function Form

In chaotic modern society, there is an increasing demand for the realization of true 'fairness'. In Greek mythology, Themis, the 'goddess of justice', has a sword in her right hand to protect society from vices, and a 'balance of judgment' in her left hand that measures good and evil. In this study, we propose a fair distribution method 'utilising physics' for the profit in games of characteristic function form. Specifically, we show that the linear programming problem for calculating 'nucleolus' can be efficiently solved by considering it as a physical system in which gravity works. In addition to being able to significantly reduce computational complexity thereby, we believe that this system could have flexibility necessary to respond to real-time changes in the parameter.

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Computer Science And Game Theory

A Branch and Bound Algorithm for Coalition Structure Generation over Graphs

We give a column generation based branch and bound algorithm for coalition structure generation over graphs problem using valuation functions for which this problem is proven to be NP-complete. For a given graph G = (V;E) and a valuation function w : 2^V -> R, the problem is to find the most valuable coalition structure (or partition) of V. We consider two cases: first when the value of a coalition is the sum of the weights of its edges which can be positive or negative, second when the value of a coalition takes account of both inter- and intra-coalitional disagreements and agreements, respectively. For both valuations we give experimental results which cover for the first time sets of more than forty agents. For another valuation function (coordination) we give only the theoretical considerations in the appendix.

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Computer Science And Game Theory

A Complete Characterization of Infinitely Repeated Two-Player Games having Computable Strategies with no Computable Best Response under Limit-of-Means Payoff

It is well-known that for infinitely repeated games, there are computable strategies that have best responses, but no computable best responses. These results were originally proved for either specific games (e.g., Prisoner's dilemma), or for classes of games satisfying certain conditions not known to be both necessary and sufficient. We derive a complete characterization in the form of simple necessary and sufficient conditions for the existence of a computable strategy without a computable best response under limit-of-means payoff. We further refine the characterization by requiring the strategy profiles to be Nash equilibria or subgame-perfect equilibria, and we show how the characterizations entail that it is efficiently decidable whether an infinitely repeated game has a computable strategy without a computable best response.

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Computer Science And Game Theory

A Difficulty in Controlling Blockchain Mining Costs via Cryptopuzzle Difficulty

Blockchain systems often employ proof-of-work consensus protocols to validate and add transactions into hashchains. These protocols stimulate competition among miners in solving cryptopuzzles (e.g. SHA-256 hash computation in Bitcoin) in exchange for a monetary reward. Here, we model mining as an all-pay auction, where miners' computational efforts are interpreted as bids, and the allocation function is the probability of solving the cryptopuzzle in a single attempt with unit (normalized) computational capability. Such an allocation function captures how blockchain systems control the difficulty of the cryptopuzzle as a function of miners' computational abilities (bids). In an attempt to reduce mining costs, we investigate designing a mining auction mechanism which induces a logit equilibrium amongst the miners with choice distributions that are unilaterally decreasing with costs at each miner. We show it is impossible to design a lenient allocation function that does this. Specifically, we show that there exists no allocation function that discourages miners to bid higher costs at logit equilibrium, if the rate of change of difficulty with respect to each miner's cost is bounded by the inverse of the sum of costs at all the miners.

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Computer Science And Game Theory

A Few Queries Go a Long Way: Information-Distortion Tradeoffs in Matching

We consider the one-sided matching problem, where n agents have preferences over n items, and these preferences are induced by underlying cardinal valuation functions. The goal is to match every agent to a single item so as to maximize the social welfare. Most of the related literature, however, assumes that the values of the agents are not a priori known, and only access to the ordinal preferences of the agents over the items is provided. Consequently, this incomplete information leads to loss of efficiency, which is measured by the notion of distortion. In this paper, we further assume that the agents can answer a small number of queries, allowing us partial access to their values. We study the interplay between elicited cardinal information (measured by the number of queries per agent) and distortion for one-sided matching, as well as a wide range of well-studied related problems. Qualitatively, our results show that with a limited number of queries, it is possible to obtain significant improvements over the classic setting, where only access to ordinal information is given.

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Computer Science And Game Theory

A Fine-Grained View on Stable Many-To-One Matching Problems with Lower and Upper Quotas

In the Hospital Residents problem with lower and upper quotas ( HR− Q U L ), the goal is to find a stable matching of residents to hospitals where the number of residents matched to a hospital is either between its lower and upper quota or zero [Biró et al., TCS 2010]. We analyze this problem from a parameterized perspective using several natural parameters such as the number of hospitals and the number of residents. Moreover, we present a polynomial-time algorithm that finds a stable matching if it exists on instances with maximum lower quota two. Alongside HR− Q U L , we also consider two closely related models of independent interest, namely, the special case of HR− Q U L where each hospital has only a lower quota but no upper quota and the variation of HR− Q U L where hospitals do not have preferences over residents, which is also known as the House Allocation problem with lower and upper quotas. Lastly, we investigate how the parameterized complexity of these three models changes if preferences may contain ties.

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Computer Science And Game Theory

A Finite Time Combinatorial Algorithm for Instantaneous Dynamic Equilibrium Flows

Instantaneous dynamic equilibrium (IDE) is a standard game-theoretic concept in dynamic traffic assignment in which individual flow particles myopically select en route currently shortest paths towards their destination. We analyze IDE within the Vickrey bottleneck model, where current travel times along a path consist of the physical travel times plus the sum of waiting times in all the queues along a path. Although IDE have been studied for decades, several fundamental questions regarding equilibrium computation and complexity are not well understood. In particular, all existence results and computational methods are based on fixed-point theorems and numerical discretization schemes and no exact finite time algorithm for equilibrium computation is known to date. As our main result we show that a natural extension algorithm needs only finitely many phases to converge leading to the first finite time combinatorial algorithm computing an IDE. We complement this result by several hardness results showing that computing IDE with natural properties is NP-hard.

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Computer Science And Game Theory

A Fragile multi-CPR Game

A Fragile CPR Game is an instance of a resource sharing game where a common-pool resource, which is prone to failure due to overuse, is shared among several players. Each player has a fixed initial endowment and is faced with the task of investing in the common-pool resource without forcing it to fail. The return from the common-pool resource is subject to uncertainty and is perceived by the players in a prospect-theoretic manner. It is shown in [A.~R.~Hota, S.~Garg, S.~Sundaram, \textit{Fragility of the commons under prospect-theoretic risk attitudes}, Games and Economic Behavior \textbf{98} (2016) 135--164.] that, under some mild assumptions, a Fragile CPR Game admits a unique Nash equilibrium. In this article we investigate an extended version of a Fragile CPR Game, in which players are allowed to share multiple common-pool resources that are also prone to failure due to overuse. We refer to this game as a Fragile multi-CPR Game. Our main result states that, under some mild assumptions, a Fragile multi-CPR Game admits a Generalized Nash equilibrium. Moreover, we show that, when there are more players than common-pool resources, the set consisting of all Generalized Nash equilibria of a Fragile multi-CPR Game is of Lebesgue measure zero.

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