  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.
