Abstract
The aim of the paper is to examine the notion of simple Kantian equilibrium in 2×2 symmetric games and their quantum counterparts. We focus on finding the Kantian equilibrium strategies in the general form of the games. As a result, we derive a formula that determines the reasonable strategies for any payoffs in the bimatrix game. This allowed us to compare the payoff results for classical and quantum way of playing the game. We showed that a very large part of 2×2 symmetric games, in which the arithmetic mean of the off-diagonal payoffs is greater than the other payoffs, have more beneficial Kantian equilibria when they are played with the use of quantum strategies. In that case, both players always obtain higher payoffs than when they use the classical strategies.
Highlights
Game theory was launched in 1928 by John von Neumann [1] and developed in 1944 by John von Neumann and Oskar Morgenstern [2]
Our work focuses on Kantian equilibria in 2 × 2 symmetric game and its quantum counterpart
Comparing formulae (30) and (42) for a01 + a10 − 2a00 ≤ 0 we find that simple Kantian equilibrium (SKE) implies the same payoff in both the classical and quantum game
Summary
Game theory was launched in 1928 by John von Neumann [1] and developed in 1944 by John von Neumann and Oskar Morgenstern [2]. In the case of many games, such as the Prisoner’s Dilemma, Nash equilibria may imply very low payoffs compared to other payoffs available in the game This undoubtedly has had an impact on the promotion of non-Nash equilibrium based solution concepts. The task is to maximize a payoff function with respect to a strategy of one of the players This is relevant in studying a quantum game in which a player’s unitary strategy depends evenly on three parameters. Weber [14] introduced an alternative model of playing a quantum game In their scheme for a 2 × 2 game, players’ strategies are restricted to two unitary operators (the identity and the Pauli operator X). We generalize the previous findings presented in [9] by deriving the general formula for Kantian equilibria We examine this solution concept with respect to the Eisert–Wilkens–Lewenstein quantum approach to the game.
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