Game Theoretical Interactions of Moving Agents

Game theory has been one of the most successful quantitative concepts to describe social interactions, their strategical aspects, and outcomes. Among the payoff matrix quantifying the result of a social interaction, the interaction conditions have been varied, such as the number of repeated interactions, the number of interaction partners, the possibility to punish defective behavior etc. While an extension to spatial interactions has been considered early on such as in the "game of life", recent studies have focussed on effects of the structure of social interaction networks. However, the possibility of individuals to move and, thereby, evade areas with a high level of defection, and to seek areas with a high level of cooperation, has not been fully explored so far. This contribution presents a model combining game theoretical interactions with success-driven motion in space, and studies the consequences that this may have for the degree of cooperation and the spatio-temporal dynamics in the population. It is demonstrated that the combination of game theoretical interactions with motion gives rise to many self-organized behavioral patterns on an aggregate level, which can explain a variety of empirically observed social behaviors

Relative Best Response Dynamics in Finite and Convex Network Games

Motivated by theoretical and experimental economics, we propose novel evolutionary dynamics for games on networks, called the h-Relative Best Response (h–RBR) dynamics, that mixes the relative performance considerations of imitation dynamics with the rationality of best responses. Under such a class of dynamics, the players optimize their payoffs over the set of strategies employed by a time–varying subset of their neighbors. As such, the h-RBR dynamics share the defining non–innovative characteristic of imitation based dynamics and can lead to equilibria that differ from classic Nash equilibria. We study the asymptotic behavior of the h–RBR dynamics for both finite and convex games in which the strategy spaces are discrete and compact, respectively, and provide preliminary sufficient conditions for finite–time convergence to a generalized Nash equilibrium

Time-varying partitioning for predictive control design: density-games approach

The design of distributed optimization-based controllers for large-scale systems (LSSs) implies every time new challenges. The fact that LSSs are generally located throughout large geographical areas makes dicult the recollection of measurements and their transmission. In this regard, the communication network that is required for a centralized control approach might have high associated economic costs. Furthermore, the computation of a large amount of data implies a high computational burden to manage, process and use them in order to make decisions over the system operation. A plausible solution to mitigate the aforementioned issues associated with the control of LSSs consists in dividing this type of systems into smaller sub-systems able to be handled by independent local controllers. This paper studies two fundamental components of the design of distributed optimization-based controllers for LSSs, i.e., the system partitioning and distributed optimization algorithms. The design of distributed model predictive control (DMPC) strategies with a system partitioning and by using density-dependent population games (DDPG) is presented.Peer ReviewedPostprint (author's final draft

On imitation dynamics in population games with Markov switching

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 lower convergence tendency of imitators compared to best responders

Imitation is widely observed in nature and often used to model populations of decision-making agents, but it is not yet known under what conditions a network of imitators will reach a state where they are satisfied with their decisions. We show that every network in which agents imitate the best performing strategy in their neighborhood will reach an equilibrium in finite time, provided that all agents are opponent coordinating, i.e., earn a higher payoff if their opponent plays the same strategy as they do. It follows that any non-convergence observed in imitative networks is not necessarily a result of population heterogeneity nor special network topology, but rather must be caused by other factors such as the presence of non-opponent-coordinating agents. To strengthen this result, we show that large classes of imitative networks containing non-opponent-coordinating agents never equilibrate even when the population is homogeneous. Comparing to best-response dynamics where equilibration is guaranteed for every network of homogeneous agents playing 2 × 2 matrix games, our results imply that networks of imitators have a lower equilibration tendency

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.Comment: Review, final version, 133 pages, 65 figure