Conservation Law of Utility and Equilibria in Non-Zero Sum Games

TL;DR

This paper introduces a method to transform non-zero sum games into zero-sum games by adding a passive player, ensuring the existence of equilibrium strategies through a conservation law of utility.

Contribution

It proposes a novel transformation technique converting non-zero sum games into zero-sum games, guaranteeing equilibrium existence via a passive player and utility conservation.

Findings

Transformation ensures equilibrium existence in non-zero sum games.

The approach applies to classic examples like Prisoner's Dilemma.

The transformed game may differ in strategies and value from the original.

Abstract

This short note demonstrates how one can define a transformation of a non-zero sum game into a zero sum, so that the optimal mixed strategy achieving equilibrium always exists. The transformation is equivalent to introduction of a passive player into a game (a player with a singleton set of pure strategies), whose payoff depends on the actions of the active players, and it is justified by the law of conservation of utility in a game. In a transformed game, each participant plays against all other players, including the passive player. The advantage of this approach is that the transformed game is zero-sum and has an equilibrium solution. The optimal strategy and the value of the new game, however, can be different from strategies that are rational in the original game. We demonstrate the principle using the Prisoner's Dilemma example.