Isomorphic Strategy Spaces in Game Theory

TL;DR

This paper introduces the use of probability space isomorphic mappings in game theory to align it more closely with probability theory, potentially resolving paradoxes and producing outcomes consistent with human behavior.

Contribution

It proposes a novel approach of applying strong isomorphic mappings to game strategy spaces, altering rational outcomes and addressing discrepancies between probability theory and traditional game theory.

Findings

Altered rational outcomes in simple games

Resolution of some classical game theory paradoxes

Alignment of game outcomes with observed human behavior

Abstract

This book summarizes ongoing research introducing probability space isomorphic mappings into the strategy spaces of game theory. This approach is motivated by discrepancies between probability theory and game theory when applied to the same strategic situation. In particular, probability theory and game theory can disagree on calculated values of the Fisher information, the log likelihood function, entropy gradients, the rank and Jacobian of variable transforms, and even the dimensionality and volume of the underlying probability parameter spaces. These differences arise as probability theory employs structure preserving isomorphic mappings when constructing strategy spaces to analyze games. In contrast, game theory uses weaker mappings which change some of the properties of the underlying probability distributions within the mixed strategy space. Here, we explore how using strong isomorphic mappings to define game strategy spaces can alter rational outcomes in simple games . Specific example games considered are the chain store paradox, the trust game, the ultimatum game, the public goods game, the centipede game, and the iterated prisoner's dilemma. In general, our approach provides rational outcomes which are consistent with observed human play and might thereby resolve some of the paradoxes of game theory.