N-Player impartial combinatorial games with random players

Abstract

In this paper we develop a method of analyzing n-player impartial combinatorial games where n−1 players behave optimally while one of the players plays randomly. More specifically, we develop a function gr that maps impartial combinatorial games to n-tuples called game values. A game value of some impartial combinatorial game G describes the probability that any player can win assuming player r plays randomly (i.e. player r chooses games at random without strategy).After devising our game value function gr, we apply it to the analysis of three-player Chomp whereby the second player plays randomly. Our analysis gives game values for single rowed Chomp positions and two rowed Chomp positions where the bottom row has 1 or 2 elements.We next discuss three alternative methods of analysis. First we extend our game value function to support more than just a single random player. Our second extension allows the random player to choose from an arbitrary distribution over games (as opposed to uniform which is assumed in our analysis of Chomp). This second extension also allows us to model a player who makes correct decisions for “simple” games (those that are, for instance, close to an end game) while playing randomly for more complex games. Finally, the third method considers the case where the random player only makes a mistake with probability ϵ and otherwise plays strategically.We conclude with many open problems and some interesting observations on the behavior of games containing a random player.