Aggregation in Game Theoretical Situations

The thesis deals with the class of Aggregative Games, namely strategic form games where each payoff function depends on the corresponding player's strategy and on some aggregation among strategies of all involved players. The first part of the thesis is devoted to the multi-leader multi-follower equilibrium concept for the class of aggregative games: the considered game presents aymmetry between two groups of players, acting noncooperatively within the group and one group is the leader in a leader-follower hierarchical model. Moreover, as it happens in concrete situations, the model is affected by uncertainty and the game is considered in a stochastic context. Assuming an exogenous uncertainty affecting the aggregator, the multi-leader multi-follower equilibrium model is presented and existence results for the stochastic resulting game are obtained in the smooth case of nice aggregative games, where payoff functions are continuous and concave in own strategies, as well as in the general case of aggregative games with strategic substitutes. These results apply to the global emission game and the teamwork project game. Then, an investment in Common-Pool Resources is studied: the situation of many agents interested in a common-pool resource, like water resource, is modeled as an aggregative game and existence results of Nash equilibria are obtained with or without convexity-like assumptions. In the special case of quadratic return functions, the game is also considered under uncertainty i.e. when the possibility of a natural disaster with a given probability may occur. In the second part of the thesis, in line with the literature on additively separable aggregative games, a class of non cooperative games, called Social Purpose Games, is introduced. In this class of games the payoff of each player depends separately on his own strategy and on a function of the strategy profile, the aggregation function, which is the same for all players, weighted by an individual benefit parameter which enlightens the asymmetry between agents toward the social part of the benefit. The two parts of the payoff function represent respectively the individual and the social benefits. For the class of social purpose games it has been showed that they have a potential, providing also a comparison between the Nash equilibrium strategies and the social optimum strategies, namely when all the players agree in maximizing the aggregate profit. For social purpose games we study the existence of the so called coalition leadership equilibrium: it is a multi-leader multi-follower model where a cooperative behaviour is assumed between players of the leading group and they decide to maximize the aggregation of their payoffs. The rest of the players act noncooperatively. This kind of equilibrium presents a mixture of cooperative and noncooperative behaviour, situation that often occurs in many applicative examples. The weights affecting the aggregation function allow to derive explicit conditions under which the leading coalition is stable. An application to a water resource game is illustrated.