Two-strategy Games Research Articles

Existence of a pure strategy equilibrium in finite symmetric games where payoff functions are integrally concave

In this paper we show that a finite symmetric game has a pure strategy equilibrium if the payoff functions of players are integrally concave due to Favati and Tardella (1990). Since the payoff functions of any two-strategy game are integrally concave, this generalizes the result of Cheng et al. (2004). A simple algorithm to find a pure strategy equilibrium is also provided.

Extrapolating Weak Selection in Evolutionary Games

In evolutionary games, reproductive success is determined by payoffs. Weak selection means that even large differences in game outcomes translate into small fitness differences. Many results have been derived using weak selection approximations, in which perturbation analysis facilitates the derivation of analytical results. Here, we ask whether results derived under weak selection are also qualitatively valid for intermediate and strong selection. By “qualitatively valid” we mean that the ranking of strategies induced by an evolutionary process does not change when the intensity of selection increases. For two-strategy games, we show that the ranking obtained under weak selection cannot be carried over to higher selection intensity if the number of players exceeds two. For games with three (or more) strategies, previous examples for multiplayer games have shown that the ranking of strategies can change with the intensity of selection. In particular, rank changes imply that the most abundant strategy at one intensity of selection can become the least abundant for another. We show that this applies already to pairwise interactions for a broad class of evolutionary processes. Even when both weak and strong selection limits lead to consistent predictions, rank changes can occur for intermediate intensities of selection. To analyze how common such games are, we show numerically that for randomly drawn two-player games with three or more strategies, rank changes frequently occur and their likelihood increases rapidly with the number of strategies . In particular, rank changes are almost certain for , which jeopardizes the predictive power of results derived for weak selection.

The Explanatory Relevance of Nash Equilibrium: One-Dimensional Chaos in Boundedly Rational Learning

Game theory is often used to explain behavior. Such explanations often proceed by demonstrating that the behavior in question is a Nash equilibrium. Agents are in Nash equilibrium if each agent’s strategy maximizes her payoff given her opponents’ strategies. Nash equilibriums are fundamentally static, but it is usually assumed that equilibriums will be the outcome of a dynamic process of learning or evolution. This article demonstrates that, even in the most simple setting, this need not be true. In two-strategy games with just a single equilibrium, a family of imitative learning dynamics does not lead to equilibrium.

Fence-sitters protect cooperation in complex networks

Evolutionary game theory is one of the key paradigms behind many scientific disciplines from science to engineering. In complex networks, because of the difficulty of formulating the replicator dynamics, most of the previous studies are confined to a numerical level. In this paper, we introduce a vectorial formulation to derive three classes of individuals' payoff analytically. The three classes are pure cooperators, pure defectors, and fence-sitters. Here, fence-sitters are the individuals who change their strategies at least once in the strategy evolutionary process. As a general approach, our vectorial formalization can be applied to all the two-strategy games. To clarify the function of the fence-sitters, we define a parameter, payoff memory, as the number of rounds that the individuals' payoffs are aggregated. We observe that the payoff memory can control the fence-sitters' effects and the level of cooperation efficiently. Our results indicate that the fence-sitters' role is nontrivial in the complex topologies, which protects cooperation in an indirect way. Our results may provide a better understanding of the composition of cooperators in a circumstance where the temptation to defect is larger.

USING PARTICLE SWARM OPTIMIZATION TO EVOLVE COOPERATION IN MULTIPLE CHOICES ITERATED PRISONER'S DILEMMA GAME

Mechanisms of promoting the evolution of cooperation in two-player, two-strategy evolutionary games have been discussed in great detail over the past decades. Understanding the effects of repeated interactions in n-player with n-choice is a formidable challenge. This paper presents and investigates the application of co-evolutionary training techniques based on particle swarm optimization (PSO) to evolve cooperation for the iterated prisoner's dilemma (IPD) game with multiple choices. Several issues will be addressed, which include the evolution of cooperation and the evolutionary stability in the presence of multiple choices and noise. First is using PSO approach to evolve cooperation. The second is the consideration of real-dilemma between social cohesion and individual profit. Experimental results show that the PSO approach evolves the cooperation. Agents with stronger social cognition choose higher levels of cooperation. Finally the impact of noise on the evolution of cooperation is examined. Experiments show the noise has a negative impact on the evolution of cooperation.

Evolutionary games with facilitators: When does selection favor cooperation?

We study the combined influence of selection and random fluctuations on the evolutionary dynamics of two-strategy (“cooperation” and “defection”) games in populations comprising cooperation facilitators. The latter are individuals that support cooperation by enhancing the reproductive potential of cooperators relative to the fitness of defectors. By computing the fixation probability of a single cooperator in finite and well-mixed populations that include a fixed number of facilitators, and by using mean field analysis, we determine when selection promotes cooperation in the important classes of prisoner’s dilemma, snowdrift and stag-hunt games. In particular, we identify the circumstances under which selection favors the replacement and invasion of defection by cooperation. Our findings, corroborated by stochastic simulations, show that the spread of cooperation can be promoted through various scenarios when the density of facilitators exceeds a critical value whose dependence on the population size and selection strength is analyzed. We also determine under which conditions cooperation is more likely to replace defection than vice versa.

Spatial pattern dynamics due to the fitness gradient flux in evolutionary games

We introduce a nondiffusive spatial coupling term into the replicator equation of evolutionary game theory. The spatial flux is based on motion due to local gradients in the relative fitness of each strategy, providing a game-dependent alternative to diffusive coupling. We study numerically the development of patterns in one dimension (1D) for two-strategy games including the coordination game and the prisoner's dilemma, and in two dimensions (2D) for the rock-paper-scissors game. In 1D we observe modified traveling wave solutions in the presence of diffusion, and asymptotic attracting states under a frozen-strategy assumption without diffusion. In 2D we observe spiral formation and breakup in the frozen-strategy rock-paper-scissors game without diffusion. A change of variables appropriate to replicator dynamics is shown to correctly capture the 1D asymptotic steady state via a nonlinear diffusion equation.

Density games

The basic idea of evolutionary game theory is that payoff determines reproductive rate. Successful individuals have a higher payoff and produce more offspring. But in evolutionary and ecological situations there is not only reproductive rate but also carrying capacity. Individuals may differ in their exposure to density limiting effects. Here we explore an alternative approach to evolutionary game theory by assuming that the payoff from the game determines the carrying capacity of individual phenotypes. Successful strategies are less affected by density limitation (crowding) and reach higher equilibrium abundance. We demonstrate similarities and differences between our framework and the standard replicator equation. Our equation is defined on the positive orthant, instead of the simplex, but has the same equilibrium points as the replicator equation. Linear stability analysis produces the classical conditions for asymptotic stability of pure strategies, but the stability properties of internal equilibria can differ in the two frameworks. For example, in a two-strategy game with an internal equilibrium that is always stable under the replicator equation, the corresponding equilibrium can be unstable in the new framework resulting in a limit cycle.

Complex Coevolutionary Dynamics—Structural Stability and Finite Population Effects

Unlike evolutionary dynamics, coevolutionary dynamics can exhibit a wide variety of complex regimes. This has been confirmed by numerical studies, e.g., in the context of evolutionary game theory (EGT) and population dynamics of simple two-strategy games with various types of replication and selection mechanisms. Using the framework of shadowing lemma, we study to what degree can such infinite population dynamics: 1) be reliably simulated on finite precision computers; and 2) be trusted to represent coevolutionary dynamics of possibly very large, but finite, populations. In a simple EGT setting of two-player symmetric games with two pure strategies and a polymorphic equilibrium, we prove that for (μ,λ), truncation, sequential tournament, best-of-group tournament, and linear ranking selections, the coevolutionary dynamics do not possess the shadowing property. In other words, infinite population simulations cannot be guaranteed to represent real trajectories or to be representative of coevolutionary dynamics of potentially very large, but finite, populations.

PSO Algorithm for IPD Game

Mechanisms promoting the evolution of cooperation in two-player, two-strategy evolutionary games have been discussed in great detail over the past decades. Understanding the effects of repeated interactions in multi-player with multi-choice is a formidable challenge. This paper presents and investigates the application of co-evolutionary training techniques based on particle swarm optimization (PSO) to evolve cooperation for the iterated prisoner’s dilemma (IPD) game with multiple choices in noisy environment. Several issues will be addressed, which include the evolution of cooperation and the evolutionary stability in the presence of multiple choices and noise. First is using PSO approach to evolve cooperation. The second is the impact of noise on the evolution of cooperation is examined.

Stochastic imitative game dynamics with committed agents

We consider models of stochastic evolution in two-strategy games in which agents employ imitative decision rules. We introduce committed agents: for each strategy, we suppose that there is at least one agent who plays that strategy without fail. We show that unlike the standard imitative model, the model with committed agents generates unambiguous infinite horizon predictions: the asymptotics of the stationary distribution do not depend on the order in which the mutation rate and population size are taken to their limits.

Spatiotemporal dynamics of cooperation and spite behavior by conformist transmission

In this paper, we propose a model describing the dynamics of human society.Then we show that classic theorems on traveling wave solutions in a reaction diffusion equationcan be readily applied and we obtain some mathematical result.Our model considers a human society where people play a two-strategy multi-player game.The key concepts are 1) payoff-dependent update rule of strategy and2) social learning with conformism.Because of conformism, the system can be bistable and our primary concern is whetherone global majority appears or not when multiple societies that initially havedifferent local majorities are spatially connected.Applying the result of this general framework to a public goods game example,we show that cooperation is less likely maintained by conformism and that the spread ofirrational spite behavior can occur.

Critical transition induced by neighbourhood size in evolutionary spatial games

We consider an evolutionary two-strategy two-player game where individual strategies evolve by imitation of players with large payoffs, and affected by noise. In a well-mixed population, the system approaches a mixed state with a part of the population in each strategy. On the other hand, when the population is distributed along a one-dimensional array and interactions are limited to a small neighbourhood of each player, the system falls in an absorbing state with a pure dominant strategy. We characterize the transition between these two regimes as a function of the neighbourhood size, providing evidence that it belongs to the universality class of directed percolation. Critical exponents are numerically evaluated, and the dependence of the critical point on the payoff and noise parameters is analyzed.

The Diffusion Approximation of Stochastic Evolutionary Game Dynamics: Mean Effective Fixation Time and the Significance of the One-Third Law

The one-third law introduced by Nowak et al. (Nature 428:646–650, 2004) for the Moran stochastic process has proven to be a robust criterion to predict when weak selection will favor a strategy invading a finite population. In this paper, we investigate fixation probability, mean effective fixation time, and average and expected fitnesses in the diffusion approximation of the stochastic evolutionary game. Our main results show that in two-strategy games with strict Nash equilibria A and B: (i) the one-third law means that, if selection favors strategy A when a single individual is using it initially, then one-third of the opponents one meets before fixation are A-individuals; and (ii) the average fitness of strategy A about the mean effective fixation time is larger than that of strategy B. The analysis reinforces the universal nature of the one-third law as of fundamental importance in models of selection. We also connect risk dominance of strategy A to its larger expected fitness with respect to the stationary distribution of the diffusion approximation that includes a small mutation rate between the two strategies.

Chaos and Unpredictability in Evolutionary Dynamics in Discrete Time

A discrete-time version of the replicator equation for two-strategy games is studied. The stationary properties differ from those of continuous time for sufficiently large values of the parameters, where periodic and chaotic behavior replace the usual fixed-point population solutions. We observe the familiar period-doubling and chaotic-band-splitting attractor cascades of unimodal maps but in some cases more elaborate variations appear due to bimodality. Also unphysical stationary solutions can have unusual physical implications, such as the uncertainty of the final population caused by sensitivity to initial conditions and fractality of attractor preimage manifolds.

Selfishness, fraternity, and other-regarding preference in spatial evolutionary games

Spatial evolutionary games are studied with myopic players whose payoff interest, as a personal character, is tuned from selfishness to other-regarding preference via fraternity. The players are located on a square lattice and collect income from symmetric two-person two-strategy (called cooperation and defection) games with their nearest neighbors. During the elementary steps of evolution a randomly chosen player modifies her strategy in order to maximize stochastically her utility function composed from her own and the co-players' income with weight factors 1−Q and Q. These models are studied within a wide range of payoff parameters using Monte Carlo simulations for noisy strategy updates and by spatial stability analysis in the low noise limit. For fraternal players (Q=1/2) the system evolves into ordered arrangements of strategies in the low noise limit in a way providing optimum payoff for the whole society. Dominance of defectors, representing the “tragedy of the commons”, is found within the regions of prisoner's dilemma and stag hunt game for selfish players (Q=0). Due to the symmetry in the effective utility function the system exhibits similar behavior even for Q=1 that can be interpreted as the “lovers' dilemma”.

Orders of limits for stationary distributions, stochastic dominance, and stochastic stability

A population of agents recurrently plays a two-strategy population game. When an agent receives a revision opportunity, he chooses a new strategy using a noisy best response rule that satisfies mild regularity conditions; best response with mutations, logit choice, and probit choice are all permitted. We study the long run behavior of the resulting Markov process when the noise level η is small and the population size N is large. We obtain a precise characterization of the asymptotics of the stationary distributions μ N�η as η approaches zero and N approaches infinity, and we establish that these asymptotics are the same for either order of limits and for all simultaneous limits. In general, different noisy best response rules can generate different stochastically stable states. To obtain a robust selection result, we introduce a refinement of risk dominance called stochastic dominance, and we prove that coordination on a given strategy is stochastically stable under every noisy best response rule if and only if that strategy is stochastically dominant.

Evolution of cooperation by phenotypic similarity.

The emergence of cooperation in populations of selfish individuals is a fascinating topic that has inspired much work in theoretical biology. Here, we study the evolution of cooperation in a model where individuals are characterized by phenotypic properties that are visible to others. The population is well mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies can depend on distance in phenotype space. We study the interaction of cooperators and defectors. In our model, cooperators cooperate with those who are similar and defect otherwise. Defectors always defect. Individuals mutate to nearby phenotypes, which generates a random walk of the population in phenotype space. Our analysis brings together ideas from coalescence theory and evolutionary game dynamics. We obtain a precise condition for natural selection to favor cooperators over defectors. Cooperation is favored when the phenotypic mutation rate is large and the strategy mutation rate is small. In the optimal case for cooperators, in a one-dimensional phenotype space and for large population size, the critical benefit-to-cost ratio is given by b/c = 1 + 2/square root(3). We also derive the fundamental condition for any two-strategy symmetric game and consider high-dimensional phenotype spaces.

Fixation probabilities in evolutionary game dynamics with a two-strategy game in finite diploid populations

Fixation processes in evolutionary game dynamics in finite diploid populations are investigated. Traditionally, frequency dependent evolutionary dynamics is modeled as deterministic replicator dynamics. This implies that the infinite size of the population is assumed implicitly. In nature, however, population sizes are finite. Recently, stochastic processes in finite populations have been introduced in order to study finite size effects in evolutionary game dynamics. One of the most significant studies on evolutionary dynamics in finite populations was carried out by Nowak et al. which describes “one-third law” [Nowak, et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646–650]. It states that under weak selection, if the fitness of strategy α is greater than that of strategy β when α has a frequency 1 3 , strategy α fixates in a β -population with selective advantage. In their study, it is assumed that the inheritance of strategies is asexual, i.e. the population is haploid. In this study, we apply their framework to a diploid population that plays a two-strategy game with two ESSs (a bistable game). The fixation probability of a mutant allele in this diploid population is derived. A “three-tenth law” for a completely recessive mutant allele and a “two-fifth law” for a completely dominant mutant allele are found; other cases are also discussed.

Evolutionary prisoner’s dilemma game on Newman-Watts networks

Maintenance of cooperation was studied for a two-strategy evolutionary prisoner's dilemma game where the players are located on a one-dimensional chain and their payoff comes from games with the nearest- and next-nearest-neighbor interactions. The applied host geometry makes it possible to study the impacts of two conflicting topological features. The evolutionary rule involves some noise affecting the strategy adoptions between the interacting players. Using Monte Carlo simulations and the extended versions of dynamical mean-field theory we determined the phase diagram as a function of noise level and a payoff parameter. The peculiar feature of the diagram is changed significantly when the connectivity structure is extended by extra links as suggested by Newman and Watts.