ENDOGENOUS NETWORKS IN RANDOM POPULATION GAMES

Endogenous Networks in Random Population Games

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.

Because of nonstorability of electric energy, temporal effects are important in electricity markets. Demand fluctuation and generator outages cause stochastic behaviors in the market. Therefore, viewing the generation competition as a dynamic and stochastic game, rather than the traditional static deterministic game commonly considered in economics seems more appropriate. This paper studies the generator bidding using stochastic optimal control formulations. A stochastic and dynamic feedback system is proposed to model generation competition and then a stochastic optimal control problem is formulated to investigate individual's optimization behavior. With the proposed approach, two interesting issues are addressed: 1) what is the effect of generators' strategic behaviors on individual's payoffs, and 2) how market efficiency evolves with generators' strategic behavior. The result shows that that generators' strategic behaviors have positive effects on market efficiency and demonstrates that the evolution of individual's payoffs w.r.t. generators' strategic behavior resembles a game of "prisoner dilemma."

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

Public goods games in populations with fluctuating size

Evolutionary dynamics of finite populations in games with polymorphic fitness equilibria

In this article, we report on new sufficient conditions for the stability of evolutionary dynamics in population games. A large number of agents interact noncooperatively in a population game by selecting strategies based on their payoffs. Each agent is allowed to revise its strategy repeatedly with an average frequency referred to as the revision rate. We are interested in the case where an agent's current strategy directly influences the revision rate. Existing stability results for this case assume that a memoryless potential game generates the strategies' payoffs. This article extends these results to allow for payoff mechanisms that can be either dynamic or memoryless games that do not have to be potential. To make our analysis concrete, we assume that the agents' revision preferences follow a so-called pairwise comparison protocol. These protocols are ubiquitous because they operate fully decentralized and with minimal information requirements (they need to access only the payoff values, not the mechanism). We use a well-motivated example to illustrate an application of our framework.

Gradient dynamics in population games: Some basic results

In population games, one of the main interests is the evolution of the dynamics, i.e., how the distribution of individuals change along time. This is an abstract but elegant way to model population of drivers selecting routes. In this paper, a three-population asymmetric game is used to investigate the co-evolution of drivers' strategies. It is shown that the convergence to one of the Nash equilibria is achieved when the three populations co-evolve, under different rates of mutants in these populations.

We model a wireless sensors’ connectivity scenario mathematically and analyze it using capacity provision mechanisms, with the objective of maximizing the profits of a network operator. The scenario has several sensors’ clusters with each one having one sink node, which uploads the sensing data gathered in the cluster through the wireless connectivity of a network operator. The scenario is analyzed both as a static game and as a dynamic game, each one with two stages, using game theory. The sinks’ behavior is characterized with a utility function related to the mean service time and the price paid to the operator for the service. The objective of the operator is to maximize its profits by optimizing the network capacity. In the static game, the sinks’ subscription decision is modeled using a population game. In the dynamic game, the sinks’ behavior is modeled using an evolutionary game and the replicator dynamic, while the operator optimal capacity is obtained solving an optimal control problem. The scenario is shown feasible from an economic point of view. In addition, the dynamic capacity provision optimization is shown as a valid mechanism for maximizing the operator profits, as well as a useful tool to analyze evolving scenarios. Finally, the dynamic analysis opens the possibility to study more complex scenarios using the differential game extension.

EvoBank: An evolutionary game solution for Bankruptcy problem

Although learning has found wide application in multi-agent systems, its effects on the temporal evolution of a system are far from understood. This paper focuses on the dynamics of Q-learning in large-scale multi-agent systems modeled as population games. We revisit the replicator equation model for Q-learning dynamics and observe that this model is inappropriate for our concerned setting. Motivated by this, we develop a new formal model, which bears a formal connection with the continuity equation in physics. We show that our model always accurately describes the Q-learning dynamics in population games across different initial settings of MASs and game configurations. We also show that our model can be applied to different exploration mechanisms, describe the mean dynamics, and be extended to Q-learning in 2-player and n-player games. Last but not least, we show that our model can provide insights into algorithm parameters and facilitate parameter tuning.