Evolutionarily stable strategies in quantum games

Abstract

Evolutionarily Stable Strategy (ESS) in classical game theory is a refinement of Nash equilibrium concept. We investigate the consequences when a small group of mutants using quantum strategies try to invade a classical ESS in a population engaged in symmetric bimatrix game of Prisoner's Dilemma.Secondly we show that in an asymmetric quantum game between two players an ESS pair can be made to appear or disappear by resorting to entangled or unentangled initial states used to play the game even when the strategy pair remains a Nash equilibrium in both forms of the game.when |b| 2 = 0 i.e. when the initial state becomes unentangled. NE inequalities are thenThe parameters of the initial entangled state a and b may decide some of the possible NE. Three symmetric NE are 1.The first two NE are independent of the parameters a and b of the initial state. However, the third NE depends on these. We now ask which of these NE can be ESS's assuming that a particular NE exists with reference to a particular set of initial states |ψ in for which it can be found. The payoff to a player using I with probability p when the opponent uses I with probability q is9For the first case ⋆ p = ⋆ q = 0. The payoff P (0, 0) > P (p, 0) when 3 |b| 2 < 1 and P (0, 0) = P (p, 0) imply 3 |b| 2 = 1. Also P (q, q) = −q 2 + 5 3 (q + 1) and P (0, q) = 5 3 (q + 1). Now P (0, q) > P (q, q) when q = 0.Therefore, ⋆ p = ⋆ q = 0 is an ESS when 3 |b| 2 ≤ 1.Consider ⋆ p = ⋆ q = 1 now. P (1, 1) > P (p, 1) means 3 |b| 2 > 2 if p = 1. And P (1, 1) = P (p, 1) means for p = 1 we have 3 |b| 2 = 2. In such case P (q, q) = −q 2 + 1 3 (q + 7) and P (1, q) = 5 3 (2 − q). Now P (1, q) > P (q, q) because (1 − q) 2 > 0 for q = 1. Therefore ⋆ p = ⋆ q = 1 is an ESS when 3 |b| 2 ≥ 2.The third case ⋆ p = ⋆ q = 3 |b| 2 − 1. Here P (3 |b| 2 − 1, 3 |b| 2 − 1) = −36 |b| 6 + 36 |b| 4 − 5 |b| 2 + 6.Also we find P (p, 3 |b| 2 − 1) = −21 |b| 4 + 21 |b| 2 − 3. Therefore, the condition P (3 |b| 2 − 1, 3 |b| 2 − 1) > P (p, 3 |b| 2 − 1) holds and ⋆ p = ⋆ q = 3 |b| 2 − 1 is an ESS too for 1 < 3 |b| 2 < 2.All three possible symmetric NE definable for different ranges of |b| 2 turn out ESS's. Each of the three sets of initial states |ψ in give a unique NE that is an ESS too. Switching from one to the other sets of initial states also changes the NE and ESS accordingly. A question rises here: is it possible that a particular NE switches over between 'ESS' and 'not ESS' when the initial state changes between certain possible choices?. The transition between classical and quantum game is also controlled by a change in the initial state. For example classical payoffs can be obtained when the initial state is unentangled. It implies that it may be possible to switch over between 'ESS' and 'not ESS' by a change between 'classical' and 'quantum' forms of a game i.e. when the initial state is unentangled and entangled respectively. This possibility makes ESS interesting for the quantum game theory as well.Because PD does not allow such a possibility we now investigate asymmetric games to look for ...

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