Game Dynamics In Populations Research Articles

Consensus-Based Multipopulation Game Dynamics for Distributed Nash Equilibria Seeking and Optimization

In this article, a consensus-based multipopulation game dynamics approach is proposed for distributed Nash equilibria seeking and optimization. The approach is fundamentally different from existing population dynamics from that: 1) the underlying communication network underlying the population game dynamics could be arbitrary undirected connected graph and more importantly 2) the proposed approach works in a distributed manner even when the objective functions of players are strongly coupled with each other. The proposed approach greatly extends the applicability of population game dynamics in distributed optimization and learning problems. A distributed constrained optimization problem and a traffic routing problem are utilized to illustrate the feasibility of the proposed multipopulation game dynamics approach.

Stability of strict equilibria in best experienced payoff dynamics: Simple formulas and applications

We consider a family of population game dynamics known as Best Experienced Payoff Dynamics. Under these dynamics, when agents are given the opportunity to revise their strategy, they test some of their possible strategies a fixed number of times. Crucially, each strategy is tested against a new randomly drawn set of opponents. The revising agent then chooses the strategy whose total payoff was highest in the test, breaking ties according to a given tie-breaking rule. Strict Nash equilibria are rest points of these dynamics, but need not be stable. We provide some simple formulas and algorithms to determine the stability or instability of strict Nash equilibria.

Distributed Population Dynamics for Searching Generalized Nash Equilibria of Population Games With Graphical Strategy Interactions

Evolutionary games and population dynamics are finding increasing applications in design learning and control protocols for a variety of resource allocation problems. The implicit requirement for full communication has been the main limitation of the evolutionary game dynamic approach in engineering tasks with various information constraints. This article intends to build population games and dynamics with both static and dynamical graphical communication structures. To this end, we formulate a population game model with graphical strategy interactions and derive its corresponding population dynamics. In particular, we first introduce the concept of generalized Nash equilibria for population games with graphical strategy interactions, and establish the equivalence between the set of generalized Nash equilibria and the set of rest points of its distributed population dynamics. Furthermore, the conditions for convergence to generalized Nash equilibrium and particularly to Nash equilibrium are obtained for the distributed population dynamics with both static and dynamical graphical structures. These results provide a new approach to design distributed Nash equilibrium seeking algorithms for population games with both static and dynamical communication networks, and hence, expand the applicability of the population game dynamics in the design of learning and control protocols under distributed circumstances.

Symmetric replicator dynamics with depletable resources.

The replicator equation is a standard model of evolutionary population game dynamics. In this article, we consider a modification of replicator dynamics, in which playing a particular strategy depletes an associated resource, and the payoff for that strategy is a function of the availability of the resource. Resources are assumed to replenish themselves, given time. Overuse of a resource causes it to crash. If the depletion rate is low enough, most trajectories converge to a stable equilibrium at which all initially present strategies are equally popular. As the depletion rate increases, these fixed points vanish in bifurcations. The phase space is periodic in each of the resource variables, and it is possible for trajectories to whirl around different numbers of times in these variables before converging to the stable equilibrium, resulting in a wide variety of topological types of orbits. Numerical solutions in a low-dimensional case show that in a cross section of the phase space, the topological types are separated by intricately folded separatrices. Once the depletion rate is high enough that the stable equilibrium in the interior of the phase space vanishes, the dynamics immediately become chaotic, without going through a period-doubling cascade; a numerical method reveals horseshoes in a Poincaré map. It appears that the multitude of topological types of orbits present before this final bifurcation generate this chaotic behavior. A periodic orbit of saddle type can be found using the symmetries of the dynamics, and its stable and unstable manifolds may generate a homoclinic tangle.

Stability for best experienced payoff dynamics

We study a family of population game dynamics under which each revising agent randomly selects a set of strategies according to a given test-set rule; tests each strategy in this set a fixed number of times, with each play of each strategy being against a newly drawn opponent; and chooses the strategy whose total payoff was highest, breaking ties according to a given tie-breaking rule. These dynamics need not respect dominance and related properties except as the number of trials become large. Strict Nash equilibria are rest points but need not be stable. We provide a variety of sufficient conditions for stability and for instability, and illustrate their use through a range of applications from the literature.

Best experienced payoff dynamics and cooperation in the centipede game

We study population game dynamics under which each revising agent tests each of his strategies a fixed number of times, with each play of each strategy being against a newly drawn opponent, and chooses the strategy whose total payoff was highest. In the centipede game, thesebest experienced payoff dynamicslead to cooperative play. When strategies are tested once, play at the almost globally stable state is concentrated on the last few nodes of the game, with the proportions of agents playing each strategy being largely independent of the length of the game. Testing strategies many times leads to cyclical play.

Public goods games in populations with fluctuating size

Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright–Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation–selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a trait’s long-term success.

Deterministic Equilibrium Selection Under Payoff-Perturbed Dynamics

This paper studies models of population game dynamics where players make stochastic choices because of payoff perturbations. The goal is to obtain deterministic equilibrium selection, where the action distribution in the population globally converges to Nash equilibria with probability 1 in a finite time, under a moderate level of payoff perturbations. Exploiting the tractable nature of bounded perturbations, we characterize almost-sure global stability in a finite population setting by generalizing risk-dominance in many action games. We also characterize conditions for (approximate) global asymptotic stability under a continuum population setting allowing unbounded perturbations; this leads to a stronger conclusion that waiting times are uniformly bounded in the population size.

Evolutionary game dynamics in populations with heterogenous structures.

Evolutionary graph theory is a well established framework for modelling the evolution of social behaviours in structured populations. An emerging consensus in this field is that graphs that exhibit heterogeneity in the number of connections between individuals are more conducive to the spread of cooperative behaviours. In this article we show that such a conclusion largely depends on the individual-level interactions that take place. In particular, averaging payoffs garnered through game interactions rather than accumulating the payoffs can altogether remove the cooperative advantage of heterogeneous graphs while such a difference does not affect the outcome on homogeneous structures. In addition, the rate at which game interactions occur can alter the evolutionary outcome. Less interactions allow heterogeneous graphs to support more cooperation than homogeneous graphs, while higher rates of interactions make homogeneous and heterogeneous graphs virtually indistinguishable in their ability to support cooperation. Most importantly, we show that common measures of evolutionary advantage used in homogeneous populations, such as a comparison of the fixation probability of a rare mutant to that of the resident type, are no longer valid in heterogeneous populations. Heterogeneity causes a bias in where mutations occur in the population which affects the mutant's fixation probability. We derive the appropriate measures for heterogeneous populations that account for this bias.

Evolutionary game dynamics in populations with different learners

We study evolutionary game theory in a setting where individuals learn from each other. We extend the traditional approach by assuming that a population contains individuals with different learning abilities. In particular, we explore the situation where individuals have different search spaces, when attempting to learn the strategies of others. The search space of an individual specifies the set of strategies learnable by that individual. The search space is genetically given and does not change under social evolutionary dynamics. We introduce a general framework and study a specific example in the context of direct reciprocity. For this example, we obtain the counter intuitive result that cooperation can only evolve for intermediate benefit-to-cost ratios, while small and large benefit-to-cost ratios favor defection. Our paper is a step toward making a connection between computational learning theory and evolutionary game dynamics.

Graph-based quadratic optimization: A fast evolutionary approach

Quadratic optimization lies at the very heart of many structural pattern recognition and computer vision problems, such as graph matching, object recognition, image segmentation, etc., and it is therefore of crucial importance to devise algorithmic solutions that are both efficient and effective. As it turns out, a large class of quadratic optimization problems can be formulated in terms of so-called “standard quadratic programs” (StQPs), which ask for finding the extrema of a quadratic polynomial over the standard simplex. Computationally, the standard approach for attacking this class of problems is to use replicator dynamics, a well-known family of algorithms from evolutionary game theory inspired by Darwinian selection processes. Despite their effectiveness in finding good solutions in a variety of applications, however, replicator dynamics suffer from being computationally expensive, as they require a number of operations per step which grows quadratically with the dimensionality of the problem being solved. In order to avoid this drawback, in this paper we propose a new population game dynamics (I nI mD yn) which is motivated by the analogy with infection and immunization processes within a population of “players.” We prove that the evolution of our dynamics is governed by a quadratic Lyapunov function, representing the average population payoff, which strictly increases along non-constant trajectories and that local solutions of StQPs are asymptotically stable (i.e., attractive) points. Each step of I nI mD yn is shown to have a linear time/space complexity, thereby allowing us to use it as a more efficient alternative to standard approaches for solving StQPs and related optimization problems. Indeed, we demonstrate experimentally that I nI mD yn is orders of magnitude faster than, and as accurate as, replicator dynamics on various applications ranging from tree matching to image registration, matching and segmentation.

Game Dynamics with Learning and Evolution of Universal Grammar

We investigate a model of language evolution, based on population game dynamics with learning. First, we examine the case of two genetic variants of universal grammar (UG), the heart of the human language faculty, assuming each admits two possible grammars. The dynamics are driven by a communication game. We prove using dynamical systems techniques that if the payoff matrix obeys certain constraints, then the two UGs are stable against invasion by each other, that is, they are evolutionarily stable. Then, we prove a similar theorem for an arbitrary number of disjoint UGs. In both theorems, the constraints are independent of the learning process. Intuitively, if a mutation in UG results in grammars that are incompatible with the established languages, then the mutation will die out because mutants will be unable to communicate and therefore unable to realize any potential benefit of the mutation. An example for which these theorems do not apply shows that compatible mutations may or may not be able to invade, depending on the population's history and the learning process. These results suggest that the genetic history of language is constrained by the need for compatibility and that mutations in the language faculty may have died out or taken over due more to historical accident than to any straightforward notion of relative fitness.

Stochastic payoff evaluation increases the temperature of selection

We study stochastic evolutionary game dynamics in populations of finite size. Moreover, each individual has a randomly distributed number of interactions with other individuals. Therefore, the payoff of two individuals using the same strategy can be different. The resulting “payoff stochasticity” reduces the intensity of selection and therefore increases the temperature of selection. A simple mean-field approximation is derived that captures the average effect of the payoff stochasticity. Correction terms to the mean-field theory are computed and discussed.