@ARTICLE{26583204_150390809_2015, 
	author = {Vladimir Gurvich}, 
	keywords = {, positional game, stochastic chess-like game, perfect information, move of chance, Nash equilibrium, terminal positiondirected cycle},
	title = {<p class="text">A four-person chess-like game without Nash equilibria in pure stationary strategies</p>},
	journal = {},
	year = {2015},
	number = {1 (31)},
	pages = {31-40},
	url = {https://bijournal.hse.ru/en/2015--1 (31)/150390809.html},
	publisher = {},
	abstract = {Vladimir A. Gurvich -&nbsp;Professor of Applied Mathematics and Computer Science,&nbsp;Rutgers Center for Operations Research,&nbsp;Business School, Rutgers, the State University of New Jersey, USA.Address: 100, Rockafeller Road, Piscataway, NJ, 08854, USA. &nbsp;E-mail: gurvich@rutcor.rutgers.edu, vladimir.gurvich@gmail.com&nbsp; &nbsp; &nbsp; In this paper we give an example of a finite positional game with perfect information and without moves of chance (a chess-like game) that has no Nash equilibria in pure stationary strategies. In this example the number n of players is 4, the number p of terminals is 5; furthermore, there is only one directed cycle. On the other hand, it is known that a chess-like game has a Nash equilibrium (NE) in pure stationary strategies if (A) n &pound; 2, or (B) p &pound; 3 and (C) any infinite play is worse than each terminal for every player. It remains open whether a NE-free chess-like game exists for n = 3, or when 2 &pound; p &pound; 4, or can such a game satisfy (C) for some n and p.&nbsp;},
	annote = {Vladimir A. Gurvich -&nbsp;Professor of Applied Mathematics and Computer Science,&nbsp;Rutgers Center for Operations Research,&nbsp;Business School, Rutgers, the State University of New Jersey, USA.Address: 100, Rockafeller Road, Piscataway, NJ, 08854, USA. &nbsp;E-mail: gurvich@rutcor.rutgers.edu, vladimir.gurvich@gmail.com&nbsp; &nbsp; &nbsp; In this paper we give an example of a finite positional game with perfect information and without moves of chance (a chess-like game) that has no Nash equilibria in pure stationary strategies. In this example the number n of players is 4, the number p of terminals is 5; furthermore, there is only one directed cycle. On the other hand, it is known that a chess-like game has a Nash equilibrium (NE) in pure stationary strategies if (A) n &pound; 2, or (B) p &pound; 3 and (C) any infinite play is worse than each terminal for every player. It remains open whether a NE-free chess-like game exists for n = 3, or when 2 &pound; p &pound; 4, or can such a game satisfy (C) for some n and p.&nbsp;}
}