Note on maximally entangled Eisert-Lewenstein-Wilkens quantum games

Maximally entangled Eisert-Lewenstein-Wilkens games are analyzed. For a general class of gate operators defined in the previous papers of the first author the general conditions are derived which allow to determine the form of gate operators leading to maximally entangled games. The construction becomes particularly simple provided one does distinguish between games differing by relabelling of strategies. Some examples are presented.Comment: 20 pages, no figures, appendix added, references added, concluding remarks extende

Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics

Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the so-called replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. While very many interesting results have been obtained with this equation in the three decades elapsed since it was first proposed, it is important to realize the limits of its applicability. One particularly relevant issue in this respect is that of non-mean-field effects, that may arise from temporal fluctuations or from spatial correlations, both neglected in the replicator equation. This review discusses these temporal and spatial effects focusing on the non-trivial modifications they induce when compared to the outcome of replicator dynamics. Alongside this question, the hypothesis of linearity and its relation to the choice of the rule for strategy update is also analyzed. The discussion is presented in terms of the emergence of cooperation, as one of the current key problems in Biology and in other disciplines.Comment: Review, 48 pages, 26 figure

Evolutionary Game Dynamics for Two Interacting Populations under Environmental Feedback

We study the evolutionary dynamics of games under environmental feedback using replicator equations for two interacting populations. One key feature is to consider jointly the co-evolution of the dynamic payoff matrices and the state of the environment: the payoff matrix varies with the changing environment and at the same time, the state of the environment is affected indirectly by the changing payoff matrix through the evolving population profiles. For such co-evolutionary dynamics, we investigate whether convergence will take place, and if so, how. In particular, we identify the scenarios where oscillation offers the best predictions of long-run behavior by using reversible system theory. The obtained results are useful to describe the evolution of multi-community societies in which individuals' payoffs and societal feedback interact.Comment: 7 pages, submitted to a conferenc

Nash Equilibria in the Response Strategy of Correlated Games

In nature and society problems arise when different interests are difficult to reconcile, which are modeled in game theory. While most applications assume uncorrelated games, a more detailed modeling is necessary to consider the correlations that influence the decisions of the players. The current theory for correlated games, however, enforces the players to obey the instructions from a third party or "correlation device" to reach equilibrium, but this cannot be achieved for all initial correlations. We extend here the existing framework of correlated games and find that there are other interesting and previously unknown Nash equilibria that make use of correlations to obtain the best payoff. This is achieved by allowing the players the freedom to follow or not to follow the suggestions of the correlation device. By assigning independent probabilities to follow every possible suggestion, the players engage in a response game that turns out to have a rich structure of Nash equilibria that goes beyond the correlated equilibrium and mixed-strategy solutions. We determine the Nash equilibria for all possible correlated Snowdrift games, which we find to be describable by Ising Models in thermal equilibrium. We believe that our approach paves the way to a study of correlations in games that uncovers the existence of interesting underlying interaction mechanisms, without compromising the independence of the players

Backwards-induction outcome in a quantum game

In economics duopoly is a market dominated by two firms large enough to influence the market price. Stackelberg presented a dynamic form of duopoly that is also called `leader-follower' model. We give a quantum perspective on Stackelberg duopoly that gives a backwards-induction outcome same as the Nash equilibrium in static form of duopoly also known as Cournot's duopoly. We find two qubit quantum pure states required for this purpose.Comment: Revised in the light of referee's comments. Latex, 16 pages, 2 figures, To appear in Phy. Rev.

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

Noisy Relativistic Quantum Games in Noninertial Frames

The influence of noise and of Unruh effect on quantum Prisoners' dilemma is investigated both for entangled and unentangled initial states. The noise is incorporated through amplitude damping channel. For unentangled initial state, the decoherence compensates for the adverse effect of acceleration of the frame and the effect of acceleration becomes irrelevant provided the game is fully decohered. It is shown that the inertial player always out scores the noninertial player by choosing defection. For maximally entangled initially state, we show that for fully decohered case every strategy profile results in either of the two possible equilibrium outcomes. Two of the four possible strategy profiles become Pareto Optimal and Nash equilibrium and no dilemma is leftover. It is shown that other equilibrium points emerge for different region of values of decoherence parameter that are either Pareto optimal or Pareto inefficient in the quantum strategic spaces. It is shown that the Eisert et al miracle move is a special move that leads always to distinguishable results compare to other moves. We show that the dilemma like situation is resolved in favor of one player or the other.Comment: 14 pages and 6 figure

Stability of mixed Nash equilibria in symmetric quantum games

In bi-matrix games the Bishop-Cannings theorem of the classical evolutionary game theory does not permit pure evolutionarily stable strategies (ESSs) when a mixed ESS exists. We find the necessary form of two-qubit initial quantum states when a switch-over to a quantum version of the game also changes the evolutionary stability of a mixed symmetric Nash equilibrium.Comment: 8 pages, no figure, to appear in Communications in Theoretical Physic

Doves and hawks in economics revisited. An evolutionary quantum game theory-based analysis of financial crises

The last financial and economic crisis demonstrated the dysfunctional long-term effects of aggressive behaviour in financial markets. Yet, evolutionary game theory predicts that under the condition of strategic dependence a certain degree of aggressive behaviour remains within a given population of agents. However, as the consequences of the financial crisis exhibit, it would be desirable to change the 'rules of the game' in a way that prevents the occurrence of any aggressive behaviour and thereby also the danger of market crashes. The paper picks up this aspect. Through the extension of the in literature well-known Hawk-Dove game by a quantum approach, we can show that dependent on entanglement, also evolutionary stable strategies can emerge, which are not predicted by classical evolutionary game theory and where the total economic population uses a non aggressive quantum strategy