Evolutionarily Stable Sets in Quantum Penny Flip Games

TL;DR

This paper explores the evolutionary stability of classical and quantum strategies in quantum penny flip games, revealing conditions under which quantum strategies can replace classical ones or form new stable sets.

Contribution

It introduces an evolutionary game theory model to analyze stability of strategies in quantum games, highlighting the emergence of quantum ES sets.

Findings

Quantum strategies can outperform classical strategies due to superposition interference.

Classical strategies can resist invasion by quantum strategies under certain conditions.

A new quantum ES set with zero payoff for both players can emerge when both classical strategies are invaded.

Abstract

In game theory, an Evolutionarily Stable Set (ES set) is a set of Nash Equilibrium (NE) strategies that give the same payoffs. Similar to an Evolutionarily Stable Strategy (ES strategy), an ES set is also a strict NE. This work investigates the evolutionary stability of classical and quantum strategies in the quantum penny flip games. In particular, we developed an evolutionary game theory model to conduct a series of simulations where a population of mixed classical strategies from the ES set of the game were invaded by quantum strategies. We found that when only one of the two players' mixed classical strategies were invaded, the results were different. In one case, due to the interference phenomenon of superposition, quantum strategies provided more payoff, hence successfully replaced the mixed classical strategies in the ES set. In the other case, the mixed classical strategies were able to sustain the invasion of quantum strategies and remained in the ES set. Moreover, when both players' mixed classical strategies were invaded by quantum strategies, a new quantum ES set emerged. The strategies in the quantum ES set give both players payoff 0, which is the same as the payoff of the strategies in the mixed classical ES set of this game.