Dutch Books and Combinatorial Games

Abstract

The theory of combinatorial games (like board games) and the theory of social games (where one looks for Nash equilibria) are normally considered two separate theories. Here we shall see what comes out of combining the ideas. J. Conway observed that there is a one-to-one correspondence between the real numbers and a special type of combinatorial games. Therefore the payoffs of a social games are combinatorial games. Probability theory should be considered a safety net that prevents inconsistent decisions via the Dutch Book Argument. This result can be extended to situations where the payoff function yields a more general game than a real number. The main difference between number-valued payoff and game-valued payoff is that the existence of a probability distribution that gives non-negative mean payoff does not ensure that the game will not be lost.