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arXiv:quant-ph/0104091 · Physical Review A

Quantum mechanics gives stability to a Nash equilibrium

Azhar Iqbal, A. H. Toor

Published Jan 4, 2002• 86 citations
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Abstract

We consider a slightly modified version of the rock-scissors-paper (RSP) game from the point of view of evolutionary stability. In its classical version the game has a mixed Nash equilibrium (NE) not stable against mutants appearing in small numbers. We find a quantized version of the RSP game for which the classical mixed NE becomes stable.

DOI
10.1103/physreva.65.022306

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We consider a slightly modified version of the rock-scissors-paper (RSP) game from the point of view of evolutionary stability. In its classical version the game has a mixed Nash equilibrium (NE) not stable against mutants appearing in small numbers. We find a quantized version of the RSP game for which the classical mixed NE becomes stable.
With respect to probabilistic mixtures of the strategies in non-cooperative games, quantum game theory provides guarantee of fixed-point stability, the so-called Nash equilibrium. This permits players to choose mixed quantum strategies that prepare mixed quantum states optimally under constraints. We show here that fixed-point stability of Nash equilibrium can also be guaranteed for pure quantum strategies via an application of the Nash embedding theorem, permitting players to prepare pure quantum states optimally under constraints.
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. I.
A version of the Monty Hall problem is presented where the players are permitted to select quantum strategies. If the initial state involves no entanglement the Nash equilibrium in the quantum game offers the players nothing more than that obtained with a classical mixed strategy. However, if the initial state involves entanglement of the qutrits of the two players, it is advantageous for one player to have access to a quantum strategy while the other does not. Where both players have access to quantum strategies there is no Nash equilibrium in pure strategies, however, there is a Nash equilibrium in quantum mixed strategies that gives the same average payoff as the classical game.
Non-transitivity can arise in games with three or more strategies A,B,CA,B,CA,B,C, when AAA beats BBB, BBB beats CCC, and CCC beats AAA, (A>B>C>AA>B>C>AA>B>C>A). An example is the children's game \textquotedblleft rock, scissors, paper" (R,S,PR,S,PR,S,P) where R>S>P>RR>S>P>RR>S>P>R. We discuss the conditions under which quantum versions of R,S,PR,S,PR,S,P retain the non-transitive characteristics of the corresponding classical game. Some physical implications of non-transitivity in quantum game theory are also considered.