Nash Equilibria And Partition Functions Of Games With Many Dependent Players

TL;DR

This paper models a complex game with dependent agents, revealing how interdependencies influence Nash equilibria and can induce phase transitions to cooperative, Pareto-efficient outcomes, with applications in stock market shareholder dynamics.

Contribution

It introduces a novel model linking game theory with the Ising model, showing how dependencies among players lead to phase transitions in equilibrium states.

Findings

High dependency causes a phase transition to Pareto-efficient equilibrium.

The model connects game theory with statistical physics concepts.

Interdependencies promote cooperation among decision-makers.

Abstract

We discuss and solve a model for a game with many players, where a subset of truely deciding players is embedded into a hierarchy of dependent agents. These interdependencies modify the game matrix and the Nash equilibria for the deciding players. In a concrete example, we recognize the partition function of the Ising model and for high dependency we observe a phase transition to a new Nash equilibrium, which is the Pareto-efficient outcome. An example we have in mind is the game theory for major shareholders in a stock market, where intermediate companies decide according to a majority vote of their owners and compete for the final profit. In our model, these interdependency eventually forces cooperation.