Quantization of the stag hunt game and the Nash equilibrilum

TL;DR

This paper quantizes the stag hunt game using Marinatto and Weber's framework, analyzes Nash equilibria in entangled and non-entangled states, and explores the impact of initial quantum states on optimal strategies and payoffs.

Contribution

It introduces a comprehensive quantum formulation for the four-parameter stag hunt game and compares its Nash equilibria structure with related symmetric games like Chicken and Prisoner's Dilemma.

Findings

Nash equilibria depend on initial quantum states.

The structure of equilibria is richer due to four parameters.

Optimal strategies vary with entanglement and payoff parameters.

Abstract

In this paper I quantize the stag hunt game in the framework proposed by Marinatto and Weber which, is introduced to quantize the Battle of the Sexes game and gives a general quntization scheme of various game theories. Then I discuss the Nash equibilium solution in the cases of which starting strategies are taken in both non entangled state and entangled state and uncover the structure of Nash Equilibrium solutions and compare the case of the Battle of the Sexes game. Since the game has 4 parameters in the payoff matrix has rather rich structure than the Battle of the Sexes game with 3-parameters in the payoff matrix, the relations of the magnitude of these payoff values in Nash Equilibriums are much involuved. This structure is uncovered completly and it is found that the best strategy which give the maximal sum of the payoffs of both players strongly depends on the initial quntum state. As the bonus of the formulation the stag hunt game with four parameters we can discuss various types of symmetric games played by two players by using the latter formulation, i.e. Chicken game. As result some common properties are found between them and the stag hunt game. Lastly a little remark is made on Prisoner's Dillemma.