Research Projects

A game model consists of three objects: (1) players, (2) Strategy sets of players, and (3) payoff functions. Game theory is widely used in two ways. First, several conflict can be modeled using game theory, and Nash equilibria can be found in each game to predict the behavior and optimal income of players. Second way is the design of the game to ensure that the players behave in a desired manner. Thus the stability of the set of Nash equilibria is crucial.The requirement of infinitely rational players is not realistic in the classical definition of Nash equilibrium. An alternate interpretation and prediction in game theory are found in evolutionary game theory. Here the players are randomly chosen individuals from large populations, and they need not be fully rational, instead they are programmed to play certain way and can switch their strategies according to success of players’ strategies. At any time, the state of these populations may be represented by a vector each of whose component represents the state of the population corresponding to a fixed player position. That is, the population state at any given time may be identified with a mixed strategy profile. The evolution of population states are modeled using various types of differential equations. Then the large time behavior of the population dynamics is studied and is related to the set of Nash equilibrium profiles of the game. Thus a fully rational prediction of the model is given a population dynamics interpretation where the individuals need not be rational. This idea of reducing social behavior to individual actions yields the sought after interpretation. This methodology is reasonably well explored in games where players have finitely many pure strategies. However, in several games related to Economics and Biology, players have a pure strategy set which is a continuum, and the aforementioned population dynamics approach is not much developed for these continuous strategy games.The proposed project aims to investigate the stability of sets of Nash equilibria of games with continuous strategy spaces with respect to a class of regular selection dynamics. This is challenging because the associated population dynamics now lives in the (infinite-dimensional) space of probability measures on the Cartesian product of pure strategy sets of all players.Most of the existing results in the stability of profiles related to continuous games are with respect to the replicator dynamics. However there are no theorems in the literature concerning the stability of sets of strategy profiles in asymmetric continuous games, w.r.t. dynamics other than replicator dynamics. We expect that the relevant stable sets of equilibria are expressible in terms of the payoff functions and the associated game dynamics vector field. Under appropriate regularity assumptions, one can thus obtain characterizations of stable sets of profiles in terms of payoff functions and vector field.