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 com- bining the ideas. J. Conway observed that there is a one-to-one correspondence between the real numbers and a special type of combinatorial games. There- fore the payo§s 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 payo§ function yields a more general game than a real number. The main di§er- ence between number-valued payooff and game-valued payo§ is that the existence of a probability distrib- ution that gives non-negative mean payo§ does not ensure that the game will not be lost.
Keywords. Combinatorial game, Dutch book theorem, exchangable sequences, game theory, surreal number.
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