University of Seville , July 1 - 3 , 2002 13 4 Approximate solutions and Well - posedness in Multicriteria Games

For long time, real valued functions have played a central role in game theory. More recently, motivated by applications to real-world situations, much attention has been attracted to multicriteria games, i.e. games with vector payoffs. Different concepts of solutions have been introduced and existence of such solutions has been investigated by various authors, mostly in the setting of finite dimensional spaces (cf. In the first part of the lecture I will present, together with illustrative examples, some existence results for different concepts of solutions under sufficient conditions of minimal character, also in the case of pseudo-games where the players can affect the feasible sets of the other players. The results obtained in reflexive Banach spaces will allow to consider differential non-zero sum multicriteria games. On the contrary, no attention has been paid to well-posedness of multicriteria games, an interesting question from theoretical and numerical point of view. In Scalar Optimization, the two standard notions of well-posedness are based either on the behaviour of minimizing (or maximizing) sequences (Tychonov well-posedness) or on the dependence of the optimal solutions from the data of the optimization problem (Hadamard well-posedness). These notions for Scalar Optimization have been extensively investigated in many papers (cf. for example, [4] in which a good list of references can be found). but, differently from what happens in Scalar Optimization, the notion of approximate solution, which plays a central role for Tychonov well-posedness, is far from being univoquely determined. So, different definitions of Tychonov well-posedness for the same problem can be considered. Clearly, the situation is the same for the more complex problems defined by multicriteria games so, in the second part of the lecture, various notions of approximate solutions for multicriteria will be presented together with suitable concepts of well-posedness which can be useful for numerical purposes. More precisely, I will introduce notions of approximating sequences for multicriteria