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Zero-Sum Polymatrix Games: A Generalization of Minmax

Yang Cai, Ozan Candogan, Constantinos Daskalakis, Christos Papadimitriou
2016 Mathematics of Operations Research  
We show that in zero-sum polymatrix games, a multiplayer generalization of two-person zerosum games, Nash equilibria can be found efficiently with linear programming.  ...  In contrast, other important properties of two-person zero-sum games are not preserved: Nash equilibrium payoffs need not be unique, and Nash equilibrium strategies need not be exchangeable or max-min.  ...  Discussion Our main result is a generalization of von Neumann's minmax theorem from two-person zero-sum games to zero-sum polymatrix games.  ... 
doi:10.1287/moor.2015.0745 fatcat:22yrig737bbu5kkuszqq3mgkna

On a Network Generalization of the Minmax Theorem [chapter]

Constantinos Daskalakis, Christos H. Papadimitriou
2009 Lecture Notes in Computer Science  
We consider graphical games in which the edges are zero-sum games between the endpoints/players; the payoff of a player is the sum of the payoffs from each incident edge.  ...  We give a simple reduction of such games to two-person zero-sum games; as a corollary, a mixed Nash equilibrium can be computed efficiently by solving a linear program and rounding off the results.  ...  by the column generation version of the simplex method in a couple of special cases, one of which is our zero-sum polymatrix games.  ... 
doi:10.1007/978-3-642-02930-1_35 fatcat:zlm63emfr5as5mm7pbudzps5ve

On Minmax Theorems for Multiplayer Games [chapter]

Yang Cai, Constantinos Daskalakis
2011 Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms  
We prove a generalization of von Neumann's minmax theorem to the class of separable multiplayer zerosum games, introduced in [Bregman and Fokin 1998 ].  ...  Surprisingly we show that a polymatrix game comprising of strictly competitive games on its edges is PPAD-complete to solve, proving a striking difference in the complexity of networks of zero-sum and  ...  Strictly Competitive Polymatrix Games Two-player strictly competitive games are a commonly used generalization of zero-sum games.  ... 
doi:10.1137/1.9781611973082.20 dblp:conf/soda/CaiD11 fatcat:kkpyegftf5bipjhxia2mcwrjye

Online Learning in Periodic Zero-Sum Games [article]

Tanner Fiez, Ryann Sim, Stratis Skoulakis, Georgios Piliouras, Lillian Ratliff
2021 arXiv   pre-print
A seminal result in game theory is von Neumann's minmax theorem, which states that zero-sum games admit an essentially unique equilibrium solution.  ...  We analyze the robustness of these online learning behaviors in the case of periodic zero-sum games with a time-invariant equilibrium.  ...  A20H6b0151) from the Agency for Science, Technology and Research (A*STAR). Tanner Fiez was supported by a National Defense Science and Engineering Graduate Fellowship.  ... 
arXiv:2111.03377v1 fatcat:uwnflltbozasto5u7bakvc34ou

A Stackelberg Game to Optimize the Distribution of Controls in Transportation Networks [chapter]

Ralf Borndörfer, Bertrand Omont, Guillaume Sagnol, Elmar Swarat
2012 Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering  
We propose a game theoretic model for the spatial distribution of inspectors on a transportation network. The problem is to spread out the controls so as to enforce the payment of a transit toll.  ...  Furthermore, we show that the problem of finding an optimal mixed strategy for a coalition of N inspectors can be solved efficiently by a column generation procedure.  ...  We point out a recent paper of Daskalakis and Papadimitriou [3] , who have generalized the Neumann's minmax theorem to the case of zero-sum polymatrix games.  ... 
doi:10.1007/978-3-642-35582-0_17 fatcat:eqp5xm6stnc3ddo3dow2gz33nm

Inspection Games [chapter]

Rudolf Avenhaus, Morton J. Canty
2009 Encyclopedia of Complexity and Systems Science  
The equilibrium of the general non-zero-sum game is found using an auxiliary zero-sum game in which the inspectee chooses a viola-tion procedure and the inspector chooses a statistical test with a given  ...  A minmax analysis leads to a zero-sum game with the operating unit as inspectee which cannot observe any inspections.  ... 
doi:10.1007/978-0-387-30440-3_287 fatcat:kqirpa7wunfqnkv2qiusanhc5m

A Survey of Decision Making in Adversarial Games [article]

Xiuxian Li, Min Meng, Yiguang Hong, Jie Chen
2022 arXiv   pre-print
Along this line, this paper provides a systematic survey on three main game models widely employed in adversarial games, i.e., zero-sum normal-form and extensive-form games, Stackelberg (security) games  ...  , zero-sum differential games, from an array of perspectives, including basic knowledge of game models, (approximate) equilibrium concepts, problem classifications, research frontiers, (approximate) optimal  ...  In the literature, most of works focus on multi-player zero-sum polymatrix games (also called network matrix games in some works), where the utility of each player is composed of the sum of utilities gained  ... 
arXiv:2207.07971v1 fatcat:egfm3ha6hrbijcu4rhpanspz2i

GANGs: Generative Adversarial Network Games [article]

Frans A. Oliehoek, Rahul Savani, Jose Gallego-Posada, Elise van der Pol, Edwin D. de Jong, Roderich Gross
2017 arXiv   pre-print
We introduce Generative Adversarial Network Games (GANGs), which explicitly model a finite zero-sum game between a generator (G) and classifier (C) that use mixed strategies.  ...  The size of these games precludes exact solution methods, therefore we define resource-bounded best responses (RBBRs), and a resource-bounded Nash Equilibrium (RB-NE) as a pair of mixed strategies such  ...  This could potentially be modelled by zero-sum polymatrix games [7] . ACKNOWLEDGMENTS This research made use of a GPU donated by Nvidia. RBBRs to s j Definition 13.  ... 
arXiv:1712.00679v2 fatcat:ld2pt6lffzcalcy7kh7i2h4qne

Spatial Coordination Games for Large-Scale Visualization [chapter]

Andre Ribeiro, Eiko Yoneki
2015 Lecture Notes in Computer Science  
We show that the problem can be framed as a game of coordination and solved with standard game-theoretic concepts.  ...  The solution is a very practical application of game-theory in an important problem, with fast and low-stress embeddings.  ...  Erdem and Pelillo [7] solved a generic polymatrix game using evolutionary game-theory (i.e., the replicator dynamics) to estimate a classification decision over partially-observed values in a set of  ... 
doi:10.1007/978-3-319-17130-2_22 fatcat:og6zablj2bdhdjjzxaovjn3lxu

Vortices Instead of Equilibria in MinMax Optimization: Chaos and Butterfly Effects of Online Learning in Zero-Sum Games [article]

Yun Kuen Cheung, Georgios Piliouras
2019 arXiv   pre-print
the day-to-day behavior of online learning dynamics in zero-sum games.  ...  Contradicting nearly a century of economic thought that treats zero-sum games nearly axiomatically as the exemplar symbol of economic stability, we prove that no meaningful prediction can be made about  ...  Non-Zero-Sum Games: 2 × 2 Bimatrix Games We consider general 2×2 bimatrix game here.  ... 
arXiv:1905.08396v2 fatcat:pur4e5o7azdxnopb5hghgqjpzy

Constrained Pure Nash Equilibria in Polymatrix Games [article]

Sunil Simon, Dominik Wojtczak
2016 arXiv   pre-print
We study the problem of checking for the existence of constrained pure Nash equilibria in a subclass of polymatrix games defined on weighted directed graphs.  ...  The payoff of a player is defined as the sum of nonnegative rational weights on incoming edges from players who picked the same strategy augmented by a fixed integer bonus for picking a given strategy.  ...  Polymatrix games are well-studied in the literature and include game classes with good computational properties like the two-player zero-sum games.  ... 
arXiv:1611.09515v1 fatcat:xahlyudshne4zb6tmxidc4bjga

Orderfield property of mixtures of stochastic games

Nagarajan Krishnamurthy, T. Parthasarathy, G. Ravindran
2010 Sankhya A  
For 2-player zero-sum and non-zero sum stochastic games, we prove that if we mix a set of states S1 where the transitions are controlled by one player with a set of states S2 constituting a sub-game having  ...  In the zero-sum case, S1 can be a mixture of SC and ARAT as well.  ...  The first author would like to thank the National Board for Higher Mathematics (NBHM) for providing travel grant to present a part of this work at the International Conference on Game Theory held at the  ... 
doi:10.1007/s13171-010-0012-7 fatcat:qrbs22pprbb7podljia2wijb3a

Computational Results for Extensive-Form Adversarial Team Games [article]

Andrea Celli, Nicola Gatti
2017 arXiv   pre-print
These games are sequential, zero-sum games in which a team of players, sharing the same utility function, faces an adversary.  ...  We provide, to the best of our knowledge, the first computational study of extensive-form adversarial team games.  ...  Von Stengel and Koller (1997) analyze zero-sum normal-form games where a single team plays against an adversary. We extend this game model to the scenario of extensive-form games.  ... 
arXiv:1711.06930v1 fatcat:bbevuxkpkfh4pip5vkipcgtonu

Interdependent Defense Games with Applications to Internet Security at the Level of Autonomous Systems

Hau Chan, Michael Ceyko, Luis Ortiz
2017 Games  
A Double Oracle Algorithm for Zero-Sum Security Games on Graphs. In Proceedings of the AAMAS, Taipei, Taiwan, 2–6 May 2011. 13.  ...  On Minmax Theorems for Multiplayer Games.  ... 
doi:10.3390/g8010013 fatcat:si2w33luendyrlxy3mcgnivo7q

Tight last-iterate convergence rates for no-regret learning in multi-player games [article]

Noah Golowich, Sarath Pattathil, Constantinos Daskalakis
2020 arXiv   pre-print
We show that the optimistic gradient (OG) algorithm with a constant step-size, which is no-regret, achieves a last-iterate rate of O(1/√(T)) with respect to the gap function in smooth monotone games.  ...  We study the question of obtaining last-iterate convergence rates for no-regret learning algorithms in multi-player games.  ...  Acknowledgements We thank Yossi Arjevani for a helpful conversation. References [Ahl79] L.V. Ahlfors. Complex Analysis. McGraw-Hill, 1979.  ... 
arXiv:2010.13724v1 fatcat:ybwuvqiwi5cu3nmiebh7arh75i
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