Nash Equilibria of Two-Player Matrix Games Repeated Until Collision

Authors Aniket Murhekar , Eklavya Sharma

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Author Details

Aniket Murhekar
  • University of Illinois, Urbana-Champaign, IL, USA
Eklavya Sharma
  • University of Illinois, Urbana-Champaign, IL, USA


We thank Prof. Jugal Garg and Prof. Ruta Mehta for their helpful comments.

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Aniket Murhekar and Eklavya Sharma. Nash Equilibria of Two-Player Matrix Games Repeated Until Collision. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We introduce and initiate the study of a natural class of repeated two-player matrix games, called Repeated-Until-Collision (RUC) games. In each round, both players simultaneously pick an action from a common action set {1, 2, … , n}. Depending on their chosen actions, they derive payoffs given by n × n matrices A and B, respectively. If their actions collide (i.e., they pick the same action), the game ends, otherwise, it proceeds to the next round. Both players want to maximize their total payoff until the game ends. RUC games can be interpreted as pursuit-evasion games or repeated hide-and-seek games. They also generalize hand cricket, a popular game among children in India. We show that under mild assumptions on the payoff matrices, every RUC game admits a Nash equilibrium (NE). Moreover, we show the existence of a stationary NE, where each player chooses their action according to a probability distribution over the action set that does not change across rounds. Remarkably, we show that all NE are effectively the same as the stationary NE, thus showing that RUC games admit an almost unique NE. Lastly, we also show how to compute (approximate) NE for RUC games.

Subject Classification

ACM Subject Classification
  • Applied computing → Economics
  • Two player games
  • Nash equilibrium
  • Repeated games


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  1. Franklin Allen and Stephen Morris. Finance applications of game theory. Center for financial institutions working papers, Wharton School Center for Financial Institutions, University of Pennsylvania, 1998. URL:
  2. Noga Alon, Troy Lee, Adi Shraibman, and Santosh Vempala. The approximate rank of a matrix and its algorithmic applications: Approximate rank. In Symposium on Theory of Computing (STOC), pages 675-684. ACM, 2013. URL:
  3. Shubham Arya. Hand cricket (iOS app). URL:
  4. Robert J. Aumann and Lloyd S. Shapley. Long-Term Competition - A Game-Theoretic Analysis, pages 1-15. Springer New York, 1994. URL:
  5. Siddharth Barman. Approximating nash equilibria and dense bipartite subgraphs via an approximate version of caratheodory’s theorem. In Symposium on Theory of Computing (STOC), pages 361-369. ACM, 2015. URL:
  6. Jean-Pierre Benoit and Vijay Krishna. Finitely repeated games. Econometrica, 53(4):905-922, 1985. URL:
  7. Richard Borie, Craig Tovey, and Sven Koenig. Algorithms and complexity results for pursuit-evasion problems. In International Joint Conference on Artificial Intelligence (IJCAI), pages 59-66, 2009. URL:
  8. Dimitris E. Charilas and Athanasios D. Panagopoulos. A survey on game theory applications in wireless networks. Computer Networks, 54(18):3421-3430, 2010. URL:
  9. Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Computing nash equilibria: Approximation and smoothed complexity, 2006. URL:
  10. Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Sparse games are hard. In Workshop on Internet and Network Economics (WINE), pages 262-273. Springer, 2006. Google Scholar
  11. Xi Chen and Shang-hua Teng. The approximation complexity of win-lose games. In Symposium on Discrete Algorithms (SODA), pages 159-168. SIAM, 2007. Google Scholar
  12. Konstantinos Daskalakis and Christos Papadimitriou. Three-player games are hard. Electronic Colloquium on Computational Complexity (ECCC), 2005. Google Scholar
  13. dissknight. How to play hand cricket. URL:
  14. James W. Friedman. A non-cooperative equilibrium for supergames. The Review of Economic Studies, 38(1):1-12, 1971. URL:
  15. Drew Fudenberg and Eric Maskin. The folk theorem in repeated games with discounting or with incomplete information. Econometrica, 54(3):533-554, 1986. URL:
  16. Ravi Kannan and Thorsten Theobald. Games of fixed rank: A hierarchy of bimatrix games. In Symposium on Discrete Algorithms (SODA), pages 1124-1132. SIAM, 2007. URL:
  17. Mohammad Emtiyaz Khan. Game theory models for pursuit evasion games, 2007. URL:
  18. S.M. LaValle and S. Hutchinson. Game theory as a unifying structure for a variety of robot tasks. In International Symposium on Intelligent Control, pages 429-434. IEEE, 1993. URL:
  19. Ruta Mehta. Constant rank bimatrix games are ppad-hard. In Symposium on Theory of Computing (STOC), pages 545-554. ACM, 2014. URL:
  20. Jean-François Mertens, Sylvain Sorin, and Shmuel Zamir. Repeated Games. Cambridge University Press, 2015. URL:
  21. Carl D. Meyer and Ian Stewart. Matrix Analysis and Applied Linear Algebra, Second Edition. SIAM, 2023. URL:
  22. R. V. Mises and H. Pollaczek-Geiringer. Praktische verfahren der gleichungsauflösung. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 9(2):152-164, 1929. URL:
  23. Aniket Murhekar and Eklavya Sharma. Nash equilibria of two-player matrix games repeated until collision, 2023. URL:
  24. John Nash. Non-cooperative games. Annals of mathematics, pages 286-295, 1951. URL:
  25. J. von Neumann. Zur theorie der gesellschaftsspiele. Mathematische Annalen, 100:295-320, 1928. URL:
  26. Noam Nisan, Tim Roughgarden, Éva Tardos, and Vijay V. Vazirani, editors. Algorithmic Game Theory. Cambridge University Press, 2007. URL:
  27. Martin J. Osborne and Ariel Rubinstein. A Course in Game Theory. The MIT Press, 1994. URL:
  28. Christos H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci., 48(3):498-532, 1994. URL:
  29. A. Rubinstein. Strong perfect equilibrium in supergames. International Journal of Game Theory, 9(1):1-12, 1980. URL:
  30. Ariel Rubinstein and A. Tversky. Naive strategies in zero-sum games. Working papers, Tel Aviv - The Sackler Institute of Economic Studies, 1993. URL:
  31. Ariel Rubinstein, Amos Tversky, and Dana Heller. Naive Strategies in Competitive Games, pages 394-402. Springer Berlin Heidelberg, 1997. URL:
  32. L. S. Shapley. Stochastic games. Proceedings of the National Academy of Sciences, 39(10):1095-1100, 1953. URL:
  33. Sylvain Sorin. A First Course on Zero-Sum Repeated Games. Springer, 2003. Google Scholar
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