Statistical Physics of Evolutionary Game Theory and its Applications

Evolutionary Game Theory represents a vibrant and interdisciplinary research field, that is attracting the interest of scientists belonging to different communities, spanning from physicists to biologists, and from mathematicians to sociologists. In few words, it represents the attempt to study the evolutionary dynamics of a population by the framework of Game Theory, taking into account the Darwinian theory of natural evolution. As result, Evolutionary Game Theory allows to model a number of scenarios, as social and biological systems, with a high level of abstraction. On one hand, the contribution of the classical Game Theory can be identified at a local level, i.e. in the interactions among the agents. For instance, when agents play games like the Prisoner’s Dilemma, according to the Nash Equilibrium, they should defect. On the other hand, in some conditions, it is possible to observe final equilibria far from the expected one. Notably, here we identify the contribution of the Darwinian theory, since the agents can change their behavior according to adaptive mechanisms. Remarkably, often populations reaching non-expected equilibria show emergent behaviors, resulting from their interaction pattern, or from specific local behaviors. For this reason, evolutionary games must be considered as complex systems. Accordingly, we believe that statistical physics constitutes one of the most suitable approaches for studying and understanding their underlying dynamics. In this scenario, one of the aims of this dissertation is to illustrate some models that let emerge a direct link between Evolutionary Game Theory and statistical physics. In addition, we show that the link between the two fields allows to envision new applications beyond the current horizon of Evolutionary Game Theory, as defining optimization strategies. So, at the beginning, we focus on a statistical physics model devised for understanding ’why’ random motion, in continuous spaces (and within a particular speed range), is able to trigger cooperation in the Prisoner’s Dilemma. Then, we study the role of the temperature in the spatial Public Goods Game, defining a link between this game and the classical Voter Model. Eventually, mapping strategies to spins, we study the spatial Public Goods Game in presence of agents susceptible to local fields, i.e. fields generated by their nearest-neighbors. It is worth to note that, from a social point of view, an agent susceptible to a local field can be considered as a conformist, since it imitates the strategy (or behavior) of majority in its neighborhood. Later, we propose three applications of Evolutionary Game Theory. In particular, the first one is a new method for solving combinatorial optimization problems. The second application is focused on the definition of a game for studying the dynamics of Poker challenges. Finally, the third application aims to represent a phenomenon of social evolution, named group formation. To conclude, we deem that the achieved results shed new light on the relation between Evolutionary Game Theory and Statistical Physics, and allow to get insights useful to devise new applications in different domains.