Dutch Books and Combinatorial Games

TL;DR

This paper explores the integration of combinatorial game theory with social game theory, interpreting payoffs as combinatorial games and extending Dutch Book arguments to game-valued payoffs, revealing new insights into decision consistency.

Contribution

It introduces a novel framework combining combinatorial and social game theories, extending Dutch Book arguments to game-valued payoffs and analyzing implications for decision consistency.

Findings

Number valued payoffs can be extended to game-valued payoffs.

Non-negative mean payoff does not guarantee game loss due to infinitesimal games.

Discussion of Ramsay/de Finetti theorem on exchangeable sequences.

Abstract

The theory of combinatorial game (like board games) and the theory of social games (where one looks for Nash equilibria) are normally considered as two separate theories. Here we shall see what comes out of combining the ideas. The central idea is Conway's observation that real numbers can be interpreted as special types of combinatorial games. Therefore the payoff function of a social game is a combinatorial game. Probability theory should be considered as a safety net that prevents inconsistent decisions via the Dutch Book Argument. This result can be extended to situations where the payoff function is a more general game than a real number. The main difference between number valued payoff and game valued payoff is that a probability distribution that gives non-negative mean payoff does not ensure that the game will be lost due to the existence of infinitisimal games. Also the Ramsay/de Finetti theorem on exchangable sequences is discussed.