Correction: The Emergence of Chaos in Population Game Dynamics Induced by Comparisons

Precise description of population game dynamics introduced by revision protocols—an economic model describing the agent’s propensity to switch to a better-performing strategy—is of importance in economics and social sciences in general. In this setting imitation of others and innovation are forces which drive the evolution of the economic system. As the continuous-time game dynamics is relatively well understood, the same cannot be said about revision driven dynamics in the discrete time. We investigate the behavior of agents using revision protocols in a $$2\times 2$$ anti-coordination game with symmetric random matching and a unique mixed Nash equilibrium. We show that in discrete time one can construct a revision protocol (either innovative or imitative) such that if a large enough fraction of agents revise their choices the game dynamics becomes Li-Yorke chaotic, inducing complex and unpredictable behavior of the system, precluding stable predictions of equilibrium. This is in stark contrast to the continuous case. Moreover, we reveal that this unpredictability is encoded in any imitative revision protocol. Furthermore, we show that for any such game there exists a perturbed pairwise proportional imitation protocol introducing chaotic behavior of agents when a sufficiently large part of the population reconsiders their strategies.

Public goods games in populations with fluctuating size

Replicator equation-a paradigm equation in evolutionary game dynamics-mathematizes the frequency dependent selection of competing strategies vying to enhance their fitness (quantified by the average payoffs) with respect to the average fitnesses of the evolving population under consideration. In this paper, we deal with two discrete versions of the replicator equation employed to study evolution in a population where any two players' interaction is modelled by a two-strategy symmetric normal-form game. There are twelve distinct classes of such games, each typified by a particular ordinal relationship among the elements of the corresponding payoff matrix. Here, we find the sufficient conditions for the existence of asymptotic solutions of the replicator equations such that the solutions-fixed points, periodic orbits, and chaotic trajectories-are all strictly physical, meaning that the frequency of any strategy lies inside the closed interval zero to one at all times. Thus, we elaborate on which of the twelve types of games are capable of showing meaningful physical solutions and for which of the two types of replicator equation. Subsequently, we introduce the concept of the weight of fitness deviation that is the scaling factor in a positive affine transformation connecting two payoff matrices such that the corresponding one-shot games have exactly same Nash equilibria and evolutionary stable states. The weight also quantifies how much the excess of fitness of a strategy over the average fitness of the population affects the per capita change in the frequency of the strategy. Intriguingly, the weight's variation is capable of making the Nash equilibria and the evolutionary stable states, useless by introducing strict physical chaos in the replicator dynamics based on the normal-form game.

Two examples demonstrate the possibility of extremely complicated non-convergent behavior in evolutionary game dynamics. For the Taylor-Jonker flow, the stable orbits for three strategies were investigated by Zeeman. Chaos does not occur with three strategies. This papers presents numerical evidence that chaotic dynamics on a “strange attractor” does occur with four strategies. Thus phenomenon is closely related to known examples of complicated behavior in Lotka-Volterra ecological models.

Recent studies have raised concerns on the inevitability of chaos in congestion games with large learning rates. We further investigate this phenomenon by exploring the learning dynamics in simple two-resource congestion games, where a continuum of agents learns according to a simplified experience-weighted attraction algorithm. The model is characterized by three key parameters: a population intensity of choice (learning rate), a discount factor (recency bias or exploration parameter), and the cost function asymmetry. The intensity of choice captures agents' economic rationality in their tendency to approximately best respond to the other agent's behavior. The discount factor captures a type of memory loss of agents, where past outcomes matter exponentially less than the recent ones. Our main findings reveal that while increasing the intensity of choice destabilizes the system for any discount factor, whether the resulting dynamics remains predictable or becomes unpredictable and chaotic depends on both the memory loss and the cost asymmetry. As memory loss increases, the chaotic regime gives place to a periodic orbit of period 2 that is globally attracting except for a countable set of points that lead to the equilibrium. Therefore, memory loss can suppress chaotic behaviors. The results highlight the crucial role of memory loss in mitigating chaos and promoting predictable outcomes in congestion games, providing insights into designing control strategies in resource allocation systems susceptible to chaotic behaviors.

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.

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.

Dynamics analysis and chaos control of a duopoly game with heterogeneous players and output limiter

Price competition has become a universal commercial phenomenon nowadays. This paper considers a dynamic Bertrand price game model, in which enterprises have heterogeneous expectations. By the stability theory of the dynamic behavior of the Bertrand price game model, the instability of the boundary equilibrium point and the stability condition of the internal equilibrium point are obtained. Furthermore, bifurcation diagram, basin of attraction, and critical curve are introduced to investigate the dynamic behavior of this game. Numerical analysis shows that the change of model parameters in a dynamic system has a significant impact on the stability of the system and can even lead to complex dynamic behaviors in the evolution of the entire economic system. This kind of complex dynamic behavior will cause certain damage to the stability of the whole economic system, causing the market to fall into a chaotic state, which is manifested as a kind of market disorder competition, which is very unfavorable to the stability of the economic system. Therefore, the chaotic behavior of the dynamical system is controlled by time-delay feedback control and the numerical analysis shows that the effective control of the dynamical system can be unstable behavior and the rapid recovery of the market can be stable and orderly.

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.

Chaos in evolutionary game dynamics

Chapter 13 - Population Games and Deterministic Evolutionary Dynamics

Imitation dynamics in population games are a class of evolutionary game-theoretic models, widely used to study decision-making processes in social groups. Different from other models, imitation dynamics allow players to have minimal information on the structure of the game they are playing, and are thus suitable for many applications, including traffic management, marketing, and disease control. In this work, we study a general case of imitation dynamics where the structure of the game and the imitation mechanisms change in time due to external factors, such as weather conditions or social trends. These changes are modeled using a continuous-time Markov jump process. We present tools to identify the dominant strategy that emerges from the dynamics through methodological analysis of the function parameters. Numerical simulations are provided to support our theoretical findings.

The traveler's dilemma game and the minimum-effort coordination game are social dilemmas that have received significant attention resulting from the fact that the predictions of classical game theory are inconsistent with the results found when the games are studied experimentally. Moreover, both the traveler's dilemma and the minimum-effort coordination games have potentially important applications in evolutionary biology. Interestingly, standard deterministic evolutionary game theory, as represented by the replicator dynamics in a well-mixed population, is also inadequate to account for the behavior observed in these games. Here we study the evolutionary dynamics of both these games in populations with interaction patterns described by a variety of complex network topologies. We investigate the evolutionary dynamics of these games through agent-based simulations on both model and empirical networks. In particular, we study the effects of network clustering and assortativity on the evolutionary dynamics of both games. In general, we show that the evolutionary behavior of the traveler's dilemma and minimum-effort coordination games on complex networks is in good agreement with that observed experimentally. Thus, formulating the traveler's dilemma and the minimum-effort coordination games on complex networks neatly resolves the paradoxical aspects of these games.

This is the second special issue on Population Games, following the December 2014 issue. In this issue, we continue the theme from the December issue, but focus on different aspects. An important extension of classical theory is the analysis of games with a continuous strategy set. The sex ratio game, for instance, contains a continuum of strategies, although these are in effect just probabilistic weights of pure strategies. Many relevant games, e.g. Bayesian games, lead naturally to rich strategy sets which one frequently describes by real parameters. In general, we need to be able to model cases where an arbitrarily large number of strategies are available. This theory has been extended in a number of ways and is the subject of several papers within this special issue. Ruijgrok and Ruijgrok extend the basic work of Riedel and collaborators on replicator and other dynamics in gameswith a continuum of strategies. Growing in importance over recent years since its introduction by Metz and collaborators is the area of adaptive dynamics,which is a generalisedmethodology to consider the evolution of continuous traits, where pay-offs are nonlinear. In Vutha and Golubitsky, we see further development of the general theory of adaptive dynamics and in particular classifications of singular strategies. Bernhard considers an adaptive dynamics model of the classical handicap principle from biological signalling of Amotz Zahavi. Another important departure has been the consideration of games with some underlying structure. Arieli considers a congestion game on a network where levied taxes influence