Standard Evolutionary Game Theory framework is a useful tool to study large interacting systems and to understand the strategic behavior of individuals in such complex systems. Adding an individual state to model local feature of each player in this context, allows one to study a wider range of problems in various application areas as networking, biology, etc. In this paper, we introduce such an extension of evolutionary game framework and particularly, we focus on the dynamical aspects of this system. Precisely, we study the coupled dynamics of the policies and the individual states inside a population of interacting individuals. We first define a general model by coupling replicator dynamics and continuous-time Markov Decision Processes and we then consider a particular case of a two policies and two states evolutionary game. We first obtain a system of combined dynamics and we show that the rest-points of this system are equilibria profiles of our evolutionary game with individual state dynamics. Second, by assuming two different time scales between states and policies dynamics, we can compute explicitly the equilibria. Then, by transforming our evolutionary game with individual states into a standard evolutionary game, we obtain an equilibrium profile which is equivalent, in terms of occupation measures and expected fitness to the previous one. All our results are illustrated with numerical analysis.

Standard Evolutionary Game framework is a useful tool to study large interacting systems and to understand the strategic behavior of individuals in such complex systems. Adding an individual state to model local feature of each player in this context, allows one to study a wider range of problems in various application areas as networking, biology, etc. In this paper, we introduce such an extension of evolutionary game framework and particularly, we focus on the dynamical aspects of this system. Precisely, we study the coupled dynamics of the strategies and the individual states inside a population of interacting individuals. We consider here a two strategies evolutionary game. We first obtain a system of combined dynamics and we show that the rest-points of this system are equilibria of our evolutionary game with individual state. Second, by assuming two different time scales between states and strategy dynamics, we can compute explicitly the equilibria. Then, by transforming our evolutionary game with individual states into a standard evolutionary game, we obtain an equilibrium which is equivalent, in terms of occupation measure, to the previous one. All our results are illustrated with numerical results.

We study a non-zero sum game considered on the solutions of a hybrid dynamical system that evolves in continuous time and that is subjected to abrupt changes of parameters. The changes of the parameters are synchronized with (and determined by) the changes of the states/actions of two Markov decision processes, each of which is controlled by a player that aims at minimizing his or her objective function. The lengths of the time intervals between the "jumps" of the parameters are assumed to be small. We show that an asymptotic Nash equilibrium of such hybrid game can be constructed on the basis of a Nash equilibrium of a deterministic averaged dynamic game. * ilaria.brunetti@inria.fr; An essential part of this paper was written while Ilaria Brunetti was visiting the

We shall consider two competition problems between service providers with asymmetric information. The utility of each one of them depends on the demand it gets and in its price. The demand itself is also a function of the prices of the providers. In both problems there is one provider (player 1) that has more information than the other (player 2) on the demand function. The more informed provider plays first, and then the second observes the move of the first provider and chooses accordingly its own action: it determines its price per unit demand. In the first problem that we consider, the first provider does not control its price (it has a fixed price known to the other provider which does not depend on the information that is unknown to provider 2). Before player 2 takes its action it receives a signal (or a recommendation) from the more informed player, i.e. from provider 1. The pure actions of provider 1 are thus the possible choices of what signal to send. The second problem that we consider is the same as the first one except that the actions of provider 1 is to choose its price. Since player 2 observes the choice of price of player 1 before it takes its own pricing decision, we can consider the choice of price by player 1 has also a role of signalling. We reduce each one of the problem to an equivalent four by four matrix game.