Chess-Like Games May Have No Uniform Nash Equilibria Even in Mixed Strategies
Figure 3
The normal form of the positional game structures from Figure 1. There are two players controlling positions each. Again, in every position there are only two options: to terminate () or proceed (). Hence, in , each player has strategies, which are naturally coded by the -letter words in the alphabet . Respectively, is represented by the table, each entry of which contains terminals corresponding to the (nonterminal) potential initial positions of . Again, players and control the rows and columns, respectively. The corresponding game has no uniform NE whenever a utility function satisfies the constraints specified under the table. Indeed, a (unique) uniformly best response of the player (resp. ) to each strategy of (resp. ) is shown by the white discs (resp. black squares). Since the obtained two sets are disjoint, no uniform NE exists in .
