# American Institute of Mathematical Sciences

January  2022, 9(1): 117-122. doi: 10.3934/jdg.2021028

## A note on the Nash equilibria of some multi-player reachability/safety games

 Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

Received  November 2020 Published  January 2022 Early access  November 2021

In this short note we study a class of multi-player, turn-based games with deterministic state transitions and reachability / safety objectives (this class contains as special cases "classic" two-player reachability and safety games as well as multi-player and ""stay–in-a-set" and "reach-a-set" games). Quantitative and qualitative versions of the objectives are presented and for both cases we prove the existence of a deterministic and memoryless Nash equilibrium; the proof is short and simple, using only Fink's classic result about the existence of Nash equilibria for multi-player discounted stochastic games

Citation: Athanasios Kehagias. A note on the Nash equilibria of some multi-player reachability/safety games. Journal of Dynamics and Games, 2022, 9 (1) : 117-122. doi: 10.3934/jdg.2021028
##### References:
 [1] K. Chatterjee, R. Majumdar and M. Jurdziński, On Nash Equilibria in Stochastic Games, Report No.UCB/CSD-3-1281, 2003, Computer Science Division (EECS), Univ. of California at Berkeley. [2] K. Chatterjee, R. Majumdar and M. Jurdziński, On Nash Equilibria in Stochastic Games, International Workshop on Computer Science Logic, Springer, Berlin, Heidelberg, 2004. [3] K. Chatterjee and T. A. Henzinger, A survey of stochastic $\omega$-regular games, Journal of Computer and System Sciences, 78 (2012), 394-413.  doi: 10.1016/j.jcss.2011.05.002. [4] J. Filar and K. Vrieze, Competitive Markov Decision Processes: Heory, Algorithms, and Applications, Springer-Verlag, New York, 1997. [5] A. M. Fink, Equilibrium in a stochastic $n$-person game, Journal of science of the Hiroshima University, Series Ai (Mathematics), 28 (1964), 89-93. [6] A. Maitra and W. D. Sudderth, Borel stay-in-a-set games, International Journal of Game Theory, 32 (2003), 97-108.  doi: 10.1007/s001820300148. [7] R. Mazala, Infinite games, in Automata Logics, and Infinite Games, Springer, 2500 (2002), 23–38. doi: 10.1007/3-540-36387-4_2. [8] P. Secchi and W. D. Sudderth, Stay-in-a-set games, International Journal of Game Theory, 30 (2002), 479-490.  doi: 10.1007/s001820200092. [9] M. Ummels, Stochastic Multiplayer Games: Theory and Algorithms, Amsterdam University Press, 2010.

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##### References:
 [1] K. Chatterjee, R. Majumdar and M. Jurdziński, On Nash Equilibria in Stochastic Games, Report No.UCB/CSD-3-1281, 2003, Computer Science Division (EECS), Univ. of California at Berkeley. [2] K. Chatterjee, R. Majumdar and M. Jurdziński, On Nash Equilibria in Stochastic Games, International Workshop on Computer Science Logic, Springer, Berlin, Heidelberg, 2004. [3] K. Chatterjee and T. A. Henzinger, A survey of stochastic $\omega$-regular games, Journal of Computer and System Sciences, 78 (2012), 394-413.  doi: 10.1016/j.jcss.2011.05.002. [4] J. Filar and K. Vrieze, Competitive Markov Decision Processes: Heory, Algorithms, and Applications, Springer-Verlag, New York, 1997. [5] A. M. Fink, Equilibrium in a stochastic $n$-person game, Journal of science of the Hiroshima University, Series Ai (Mathematics), 28 (1964), 89-93. [6] A. Maitra and W. D. Sudderth, Borel stay-in-a-set games, International Journal of Game Theory, 32 (2003), 97-108.  doi: 10.1007/s001820300148. [7] R. Mazala, Infinite games, in Automata Logics, and Infinite Games, Springer, 2500 (2002), 23–38. doi: 10.1007/3-540-36387-4_2. [8] P. Secchi and W. D. Sudderth, Stay-in-a-set games, International Journal of Game Theory, 30 (2002), 479-490.  doi: 10.1007/s001820200092. [9] M. Ummels, Stochastic Multiplayer Games: Theory and Algorithms, Amsterdam University Press, 2010.
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