Entanglement and dynamic stability of Nash equilibria in a symmetric quantum game

We study the evolutionary stability of Nash equilibria (NE) in a symmetric quantum game played by the recently proposed scheme of applying ‘identity’ and ‘Pauli spin-flip’ operators on an initial state with classical probabilities. We show that in this symmetric game dynamic stability of a NE can be changed when the game changes its form, for example, from classical to quantum. It happens even when the NE remains intact in both forms.     

Modeling and analyzing of botnet interactions

The dynamics of interacting botnets and the effects of the strategies selected by interacting botnet owners on the spread of botnets remain unclear. As a result, in this paper, we present a botnet interaction model, obtained by coupling a fast evolutionary game dynamics to a slow population dynamics model, in which two botnet types are considered. We analyze the fast evolutionary game model and obtain two stable equilibria. Additionally, we substitute them into the complete model and get two reduced models. Such models allow us to study the effects of strategies selected by botnet owners. Analysis of the models shows that when all owners adopt the cooperative strategy both types of botnets can survive with much lower contact rates. However, while they choose the competitive strategy one type of botnet will become extinct and the other will persist with a lower infection rate. The equilibrium conditions of the evolutionary game model, which can guide us in designing effective counter-botnet methods, are also obtained.     

Evolutionary games in a generalized Moran process with arbitrary selection strength and mutation

By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2×2 games in a well-mixed population with a finite size is investigated. In the model, the individuals' payoff accumulating from games is mapped into fitness using an exponent function. Both selection strength β and mutation rate ε are considered. The process is an ergodic birth-death process. Based on the limit distribution of the process, we give the analysis results for which strategy will be favoured when ε is small enough. The results depend on not only the payoff matrix of the game, but also on the population size. Especially, we prove that natural selection favours the strategy which is risk-dominant when the population size is large enough. For arbitrary β and ε values, the 'Hawk--Dove' game and the 'Coordinate' game are used to illustrate our model. We give the evolutionary stable strategy (ESS) of the games and compare the results with those of the replicator dynamics in the infinite population. The results are determined by simulation experiments.     

Some Analytical Properties of the Model for Stochastic Evolutionary Games in Finite Populations with Non-uniform Interaction Rate

Traditional evolutionary games assume uniform interaction rate, which means that the rate at which individuals meet and interact is independent of their strategies. But in some systems, especially biological systems, the players interact with each other discriminately. Taylor and Nowak (2006) were the first to establish the corresponding non-uniform interaction rate model by allowing the interaction rates to depend on strategies. Their model is based on replicator dynamics which assumes an infinite size population. But in reality, the number of individuals in the population is always finite, and there will be some random interference in the individuals' strategy selection process. Therefore, it is more practical to establish the corresponding stochastic evolutionary model in finite populations. In fact, the analysis of evolutionary games in a finite size population is more difficult. Just as Taylor and Nowak said in the outlook section of their paper, "The analysis of non-uniform interaction rates should be extended to stochastic game dynamics of finite populations." In this paper, we are exactly doing this work. We extend Taylor and Nowak's model from infinite to finite case, especially focusing on the influence of non-uniform connection characteristics on the evolutionary stable state of the system. We model the strategy evolutionary process of the population by a continuous ergodic Markov process. Based on the limit distribution of the process, we can give the evolutionary stable state of the system. We make a complete classification of the symmetric 2×2 games. For each case game, the corresponding limit distribution of the Markov-based process is given when noise intensity is small enough. In contrast with most literatures in evolutionary games using the simulation method, all our results obtained are analytical. Especially, in the dominant-case game, coexistence of the two strategies may become evolutionary stable states in our model. This result can be used to explain the emergence of cooperation in the Prisoner is Dilemma Games to some extent. Some specific examples are given to illustrate our results.     

零行列式策略在雪堆博弈中的演化

零行列式策略不仅可以单方面设置对手收益,而且可以对双方的收益施加一个线性关系,从而达到敲诈对手的目的.本文针对零行列式策略博弈前期与稳态期的收益存在偏差,基于Markov链理论给出零行列式策略与全合作策略博弈的瞬态分布、瞬态收益及达到稳态所需时间.发现在小的敲诈因子下,敲诈者前期收益高于稳态期收益,敲诈因子较大时,情况截然相反,并且敲诈因子越大,越不利于双方合作,达到稳态也越慢.这为现实生活中频繁更新策略的博弈提供了一种计算实时收益的方法.此外针对敲诈策略与进化人的博弈,论证了双方均背叛状态下,进化人下次博弈时一定进化为全合作策略.通过对所有状态下策略更新过程仿真,发现进化人在四种情况下的进化速度有显著差异,并最终演化为全合作策略,表明零行列式策略是合作产生的催化剂.     

Coexistence of cooperators and defectors in well mixed populations mediated by limiting resources

Traditionally, resource limitation in evolutionary game theory is assumed just to impose a constant population size. Here we show that resource limitations may generate dynamical payoffs able to alter an original prisoner's dilemma, and to allow for the stable coexistence between unconditional cooperators and defectors in well-mixed populations. This is a consequence of a self-organizing process that turns the interaction payoff matrix into evolutionary neutral, and represents a resource-based control mechanism preventing the spread of defectors. To our knowledge, this is the first example of coexistence in well-mixed populations with a game structure different from a snowdrift game.     

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 a consequence of the financial crisis, 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 well-known hawk-dove game by a quantum approach, we can show that dependent on entanglement, evolutionary stable strategies also can emerge, which are not predicted by the classical evolutionary game theory and where the total economic population uses a non-aggressive quantum strategy.     

Cycle frequency in standard Rock–Paper–Scissors games: Evidence from experimental economics

The Rock–Paper–Scissors (RPS) game is a widely used model system in game theory. Evolutionary game theory predicts the existence of persistent cycles in the evolutionary trajectories of the RPS game, but experimental evidence has remained to be rather weak. In this work, we performed laboratory experiments on the RPS game and analyzed the social-state evolutionary trajectories of twelve populations of N=6$N=6$ players. We found strong evidence supporting the existence of persistent cycles. The mean cycling frequency was measured to be 0.029±0.009$0.029\pm 0.009$ period per experimental round. Our experimental observations can be quantitatively explained by a simple non-equilibrium model, namely the discrete-time logit dynamical process with a noise parameter. Our work therefore favors the evolutionary game theory over the classical game theory for describing the dynamical behavior of the RPS game.     

Mean-field approximations of fixation time distributions of evolutionary game dynamics on graphs

The mean fixation time is often not accurate for describing the timescales of fixation probabilities of evolutionary games taking place on complex networks. We simulate the game dynamics on top of complex network topologies and approximate the fixation time distributions using a mean-field approach. We assume that there are two absorbing states. Numerically, we show that the mean fixation time is sufficient in characterizing the evolutionary timescales when network structures are close to the well-mixing condition. In contrast, the mean fixation time shows large inaccuracies when networks become sparse. The approximation accuracy is determined by the network structure, and hence by the suitability of the mean-field approach. The numerical results show good agreement with the theoretical predictions.     

Evolution of Cooperation in Evolutionary Games for Heterogeneous Interactions

When a population structure is modelled as a square lattice,the cooperation may be improved for an evolutionary prisoner dilemma game or be inhibited for an evolutionary snowdrift game.In this work,we investigate cooperation in a population on a square lattice where the interaction among players contains both prisoner dilemma game and snowdrift game.The heterogeneity in interaction is introduced to the population in two different ways:the heterogenous character of interaction assigned to every player(HCP) or the heterogenous character of interaction assigned to every link between any two players(HCL).The resonant enhancement of cooperation in the case of HCP is observed while the resonant inhibition of cooperation in the case of HCL is prominent.The explanations on the enhancement or inhibition of cooperation are presented for these two cases.     

Adjusting learning motivation to promote cooperation

An evolutionary prisoner’s dilemma game with players adjusting their learning motivation is studied. At each time step, each player can adjust his/her learning motivation according to the difference between the current payoff and payoff aspiration. Greater payoff aspiration means stronger learning motivation, and vice versa. We find that the density of cooperation in a spatial prisoner’s dilemma game is enhanced when the learning motivation mechanism is considered. Meanwhile, we show that proper noise can not only induce the highest cooperation level but also can maintain the cooperation phenomenon even though there is more temptation to defect.     

Effects of expectation and noise on evolutionary games

Considering the difference between the actual and expected payoffs, we bring a stochastic learning updating rule into an evolutionary Prisoners Dilemma game and the Snowdrift game on scale-free networks, and then investigate how the expectation level A and environmental noise κ influence cooperative behavior. Interestingly, numerical results show that the mechanism of promoting cooperation exhibits a resonance-like fashion including the coaction of A, κ and the payoff parameters. High cooperator frequency is induced by some optimal parameter regions. The variation of time series has also been investigated. This work could be of particular interest in the evolutionary game dynamics of biological and social systems.     

Hidden antiunitary symmetry behind “accidental” degeneracy and its protection of degeneracy

The mean fixation time is often not accurate for describing the timescales of fixation probabilities of evolutionary games taking place on complex networks. We simulate the game dynamics on top of complex network topologies and approximate the fixation time distributions using a mean-field approach. We assume that there are two absorbing states. Numerically, we show that the mean fixation time is sufficient in characterizing the evolutionary timescales when network structures are close to the well-mixing condition. In contrast, the mean fixation time shows large inaccuracies when networks become sparse. The approximation accuracy is determined by the network structure, and hence by the suitability of the mean-field approach. The numerical results show good agreement with the theoretical predictions.

     

Stochastic evolutionary public goods game with first and second order costly punishments in finite populations

We study the stochastic evolutionary public goods game with punishment in a finite size population. Two kinds of costly punishments are considered, i.e., first-order punishment in which only the defectors are punished, and second-order punishment in which both the defectors and the cooperators who do not punish the defective behaviors are punished. We focus on the stochastic stable equilibrium of the system. In the population, the evolutionary process of strategies is described as a finite state Markov process. The evolutionary equilibrium of the system and its stochastic stability are analyzed by the limit distribution of the Markov process. By numerical experiments, our findings are as follows.(i) The first-order costly punishment can change the evolutionary dynamics and equilibrium of the public goods game, and it can promote cooperation only when both the intensity of punishment and the return on investment parameters are large enough.(ii)Under the first-order punishment, the further imposition of the second-order punishment cannot change the evolutionary dynamics of the system dramatically, but can only change the probability of the system to select the equilibrium points in the "C+P" states, which refer to the co-existence states of cooperation and punishment. The second-order punishment has limited roles in promoting cooperation, except for some critical combinations of parameters.(iii) When the system chooses"C+P" states with probability one, the increase of the punishment probability under second-order punishment will further increase the proportion of the "P" strategy in the "C+P" states.     

Dynamical spin-glass-like behavior in an evolutionary game

We study a new evolutionary game, where players are tempted to take part by the premium, but compete for being the first who take a specific move. Those, who manage to escape the bulk of players, are the winners. While for large premium the game is very similar to the Minority game studied earlier, significant new behavior, reminiscent of spin glasses is observed for premium below certain level.     

Critical Behavior of Spatial Evolutionary Game with Altruistic to Spiteful Preferences on Two-Dimensional Lattices

Self-questioning mechanism which is similar to single spin-flip of Ising model in statistical physics is introduced into spatial evolutionary game model. We propose a game model with altruistic to spiteful preferences via weighted sums of own and opponent's payoffs. This game model can be transformed into Ising model with an external field. Both interaction between spins and the external field are determined by the elements of payoff matrix and the preference parameter. In the case of perfect rationality at zero social temperature, this game model has three different phases which are entirely cooperative phase, entirely non-cooperative phase and mixed phase. In the investigations of the game model with Monte Carlo simulation, two paths of payoff and preference parameters are taken. In one path, the system undergoes a discontinuous transition from cooperative phase to non-cooperative phase with the change of preference parameter. In another path, two continuous transitions appear one after another when system changes from cooperative phase to non-cooperative phase with the prefenrence parameter. The critical exponents ν, β, and γ of two continuous phase transitions are estimated by the finite-size scaling analysis. Both continuous phase transitions have the same critical exponents and they belong to the same universality class as the two-dimensional Ising model.     

Effects of migration on the evolutionary game dynamics in finite populations with community structures

We investigate the impacts of migration on the evolutionary game dynamics in finite populations with community structures in the framework of evolutionary game theory. In contrast to deterministic dynamics, our model incorporates stochastic factors induced by the finite population size. Based on the analysis of the stationary distribution of the evolutionary process in the limit of rare mutations, we prove that it is most likely to find the population in the community where all individuals have the lower migration rate. Furthermore, we show that reducing the difference between the migration rates of distinct communities can increase the first hitting time to the homogeneous absorbing state and can prolong the coexistence time of different species, promoting the conservation of biodiversity.     

Evolution of cooperation in the snowdrift game among mobile players with random-pairing and reinforcement learning

The evolutionary spatial game in a mobile population has attracted many researchers of biological, social and economic sciences. Considering some facts observed in the real world, this paper proposes a novel spatial evolutionary snowdrift game model with movable players. In this model, one player interacts only with the nearest neighbor in each turn, and makes decision in a reinforcement learning way. In a very large range of the parameters moving ability enhances cooperation, but under some special condition, velocity heavily depresses cooperation. Some explanations have also been given out by investigating the strategy-change behavior of players. The findings may be helpful in understanding cooperative behavior in natural and social systems consisting of mobile agents.     

Elimination mechanism promotes cooperation in coevolutionary prisoner’s dilemma games

We propose an elimination mechanism in the study of the evolutionary prisoner’s dilemma games on evolving networks. It assumes that after each round of playing, players whose payoffs are below a certain threshold will be eliminated from the game and the same number of new nodes will be added to the network to maintain the size of the network constant. Numerical results show that moderate values of elimination threshold can result in a maximum cooperation level in the evolutionary prisoner’s dilemma game. Moreover, the elimination mechanism can make the network structure evolve into a high heterogeneity in degree distribution, which is considered to be helpful in promoting cooperation in evolutionary games. The present study may provide new insight for understanding the evolution of cooperation in light of the law ‘survival of the fittest’ in nature.     

Individual strategies in complementarity games and population dynamics

We introduce and study an evolutionary complementarity game where in each round a player of population 1 is paired with a member of population 2. The game is symmetric, and each player tries to obtain an advantageous deal, but when one of them pushes too hard, no deal at all can be concluded, and they both lose. The game has many equilibria, and which of them is reached depends on the history of the interactions as the players evolve according to a fitness function that measures their gains across the interactions. We can then break the symmetry by assigning different strategy spaces to the populations, varying in particular with respect to the information available to the agents. The agents can, for example, adapt to the behavior of their opponents met in previous rounds, or they can try to copy the strategies of their successful friends. It turns out that, in general, the more restricted strategy spaces, that is, those which utilize less information, are more advantageous for a population as a whole as their adoption drives the equilibrium in a direction advantageous to that population. One reason is that a simpler strategy can be learned faster in an evolutionary setting, another is that it is good for a population to have some individuals that are unfit in the sense that they make offers that are individually unsuccessful, but have a systematic effect on the strategies of their opponents. All these effects are demonstrated through systematic simulations.