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

Evolutionary stability in quantum games

In evolutionary game theory an Evolutionarily Stable Strategy (ESS) is a refinement of the Nash equilibrium concept that is sometimes also recognized as evolutionary stability. It is a game-theoretic model, well known to mathematical biologists, that was found quite useful in the understanding of evolutionary dynamics of a population. This chapter presents an analysis of evolutionary stability in the emerging field of quantum games.Comment: 38 pages, 2 figures, contributed chapter to the book "Quantum Aspects of Life" edited by D. Abbott, P. Davies and A. Pat

A Survey of the Current Status of Research on Quantum Games

Quantum games have gained considerable interest from researchers. In this paper, on the basis of the Web of Science database, through the use of the social network analysis methods, the literature on quantum games is analyzed from three aspects: the keywords co-occurrence, co-authorship, and co-citation. In the process of analysis, the main quantum game models are reviewed with graphical illustrations. Our paper provides a survey and outline of the current Status of research in this field, and identify directions for future work

Quantum Strategies Win in a Defector-Dominated Population

Quantum strategies are introduced into evolutionary games. The agents using quantum strategies are regarded as invaders whose fraction generally is 1% of a population in contrast to the 50% defectors. In this paper, the evolution of strategies on networks is investigated in a defector-dominated population, when three networks (Regular Lattice, Newman-Watts small world network, scale-free network) are constructed and three games (Prisoners' Dilemma, Snowdrift, Stag-Hunt) are employed. As far as these three games are concerned, the results show that quantum strategies can always invade the population successfully. Comparing the three networks, we find that the regular lattice is most easily invaded by agents that adopt quantum strategies. However, for a scale-free network it can be invaded by agents adopting quantum strategies only if a hub is occupied by an agent with a quantum strategy or if the fraction of agents with quantum strategies in the population is significant.Comment: 8 pages, 7figure

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.Comment: Review, final version, 133 pages, 65 figure

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