Using strong isomorphisms to construct game strategy spaces

TL;DR

This paper investigates how using strong isomorphisms to define game strategy spaces can alter rational outcomes and potentially resolve paradoxes in game theory by preserving structural properties of probability distributions.

Contribution

It introduces the use of strong isomorphic mappings for constructing game strategy spaces, contrasting with traditional weaker mappings in game theory.

Findings

Strong isomorphisms can change rational outcomes in simple games

Using isomorphisms may resolve certain paradoxes of game theory

Structural preservation affects analysis of game strategy spaces

Abstract

When applied to the same game, 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. In this paper, we explore how using strong isomorphic mappings to define game strategy spaces can alter rational outcomes in simple games, and might resolve some of the paradoxes of game theory.