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

Dynamics in atomic signaling games

We study an atomic signaling game under stochastic evolutionary dynamics. There is a finite number of players who repeatedly update from a finite number of available languages/signaling strategies. Players imitate the most fit agents with high probability or mutate with low probability. We analyze the long-run distribution of states and show that, for sufficiently small mutation probability, its support is limited to efficient communication systems. We find that this behavior is insensitive to the particular choice of evolutionary dynamic, a property that is due to the game having a potential structure with a potential function corresponding to average fitness. Consequently, the model supports conclusions similar to those found in the literature on language competition. That is, we show that efficient languages eventually predominate the society while reproducing the empirical phenomenon of linguistic drift. The emergence of efficiency in the atomic case can be contrasted with results for non-atomic signaling games that establish the non-negligible possibility of convergence, under replicator dynamics, to states of unbounded efficiency loss

Social Dilemmas and Cooperation in Complex Networks

In this paper we extend the investigation of cooperation in some classical evolutionary games on populations were the network of interactions among individuals is of the scale-free type. We show that the update rule, the payoff computation and, to some extent the timing of the operations, have a marked influence on the transient dynamics and on the amount of cooperation that can be established at equilibrium. We also study the dynamical behavior of the populations and their evolutionary stability.Comment: 12 pages, 7 figures. to appea

Rational Samaritans, Strategic Moves, and Rule-Governed Behavior: Some Remarks on James Buchanan's "Samaritan's Dilemma"

Using James Buchanan's "Samaritan's Dilemma" as a basic example, this paper analyses the problems that have to be solved if strategic behavior is necessary to escape from dilemma situations by changing the opponents' incentives. These problems are addressed within one-shot games as well as repeated games. Furthermore, the implications resulting from the assumption of common knowledge of rationality are analyzed. By using the concept of finite automata to model strategies for supergames, the possible relations with evolutionary game theory are spelled out. -- Dieser Beitrag untersucht am Beispiel von James Buchanans "Samariter-Dilemma" die Probleme, die es zu lösen gilt, wenn "strategisches Verhalten" im Sinne der Veränderung von Handlungsanreizen durch glaubhafte Drohungen und Versprechen notwendig ist, um Dilemmasituationen zu vermeiden. Dabei werden neben den grundelgenden Strukturen "Einmalspiel" und "Superspiel" auch die Besonderheiten der Annahme von "common knowledge" analysiert, und mit dem Konzept der als finite Automaten abbildbaren Superspielstrategien Brücken zur evolutorischen Spieltheorie geschlagen.rationality,altruism,evolutionary stability,strategic moves

On the robustness of learning in games with stochastically perturbed payoff observations

Motivated by the scarcity of accurate payoff feedback in practical applications of game theory, we examine a class of learning dynamics where players adjust their choices based on past payoff observations that are subject to noise and random disturbances. First, in the single-player case (corresponding to an agent trying to adapt to an arbitrarily changing environment), we show that the stochastic dynamics under study lead to no regret almost surely, irrespective of the noise level in the player's observations. In the multi-player case, we find that dominated strategies become extinct and we show that strict Nash equilibria are stochastically stable and attracting; conversely, if a state is stable or attracting with positive probability, then it is a Nash equilibrium. Finally, we provide an averaging principle for 2-player games, and we show that in zero-sum games with an interior equilibrium, time averages converge to Nash equilibrium for any noise level.Comment: 36 pages, 4 figure

On evolutionarily stable strategies and replicator dynamics in asymmetric two-population games

We analyze the main dynamical properties of the evolutionarily stable strategy ESS for asymmetric two-population games of finite size in its corresponding replicator dynamics. We introduce a defnition of ESS for two-population asymmetric games and a method of symmetrizing such an asymmetric game. Then, we show that every strategy profile of the asymmetric game corresponds to a strategy in the symmetric game, and that every Nash equilibrium (NE) of the asymmetric game corresponds to a (symmetric) NE of the symmetric version game. So, we study (standard) replicator dynamics for the asymmetric game and define corresponding (non-standard) dynamics of the symmetric game.Asymmetric game; Evolutionary games; ESS; Replicator dynamics.

Evolution via imitation among like-minded individuals

In social situations with which evolutionary game is concerned, individuals are considered to be heterogeneous in various aspects. In particular, they may differently perceive the same outcome of the game owing to heterogeneity in idiosyncratic preferences, fighting abilities, and positions in a social network. In such a population, an individual may imitate successful and similar others, where similarity refers to that in the idiosyncratic fitness function. I propose an evolutionary game model with two subpopulations on the basis of multipopulation replicator dynamics to describe such a situation. In the proposed model, pairs of players are involved in a two-person game as a well-mixed population, and imitation occurs within subpopulations in each of which players have the same payoff matrix. It is shown that the model does not allow any internal equilibrium such that the dynamics differs from that of other related models such as the bimatrix game. In particular, even a slight difference in the payoff matrix in the two subpopulations can make the opposite strategies to be stably selected in the two subpopulations in the snowdrift and coordination games.Comment: 3 figure