N-player quantum games in an EPR setting

The $N$-player quantum game is analyzed in the context of an Einstein-Podolsky-Rosen (EPR) experiment. In this setting, a player's strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players' strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for $N$-qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the mixed Nash equilibrium and the payoffs are determined. Players' payoffs are then defined with linear functions so that common two-player games can be easily extended to the $N$-player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners' Dilemma game for general $N \ge 2$. We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs.Comment: 26 pages, 2 figure

Analysis of two-player quantum games in an EPR setting using geometric algebra

The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of Clifford geometric algebra (GA). In this setting, the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, which is then obtained as proper subset of the corresponding quantum game. As examples, using GA we analyze the games of Prisoners' Dilemma and Stag Hunt when played in the EPR type setting.Comment: 20 pages, no figure, revise

Effects of adaptive degrees of trust on coevolution of quantum strategies on scale-free networks

We study the impact of adaptive degrees of trust on the evolution of cooperation in the quantum prisoner's dilemma game. In addition to the strategies, links between players are also subject to evolution. Starting with a scale-free interaction network, players adjust trust towards their neighbors based on received payoffs. The latter governs the strategy adoption process, while trust governs the rewiring of links. As soon as the degree of trust towards a neighbor drops to zero, the link is rewired to another randomly chosen player within the network. We find that for small temptations to defect cooperators always dominate, while for intermediate and strong temptations a single quantum strategy is able to outperform all other strategies. In general, reciprocal trust remains within close relationships and favors the dominance of a single strategy. Due to coevolution, the power-law degree distributions transform to Poisson distributions.Qiang Li, Minyou Chen, Matjaz Perc, Azhar Iqbal, & Derek Abbot