Lucia Pusillo

Vector Optimization Problems via Improvement Sets

Motivated by applications to the real world, various optimality criteria (also approximate ones) are developed for situations in vector optimization. We propose a new type of solution based on upper comprehensive sets and we discuss the existence of optimal points in multicriteria situations.

On the Tikhonov well-posedness of concave games and Cournot oligopoly games

The purpose of this paper is to investigate whether theorems known to guarantee the existence and uniqueness of Nash equilibria, provide also sufficient conditions for the Tikhonov well-posedness (T-wp). We consider several hypotheses that ensure the existence and uniqueness of a Nash equilibrium (NE), such as strong positivity of the Jacobian of the utility function derivatives (Ref. 1), pseudoconcavity, and strict diagonal dominance of the Jacobian of the best reply functions in implicit form (Ref. 2). The aforesaid assumptions imply the existence and uniqueness of NE. We show that the hypotheses in Ref. 2 guarantee also the T-wp property of the Nash equilibrium.As far as the hypotheses in Ref. 1 are concerned, the result is true for quadratic games and zero-sum games. A standard way to prove the T-wp property is to show that the sets of ∈-equilibria are compact. This last approach is used to demonstrate directly the T-wp property for the Cournot oligopoly model given in Ref. 3. The compactness of ∈-equilibria is related also to the condition that the best reply surfaces do not approach each other near infinity.

E-Equilibria for Multicriteria Games

In this paper we propose an equilibrium definition for non-cooperative multicriteria games based on improvement sets. Our new definition generalizes the idea of exact and approximate equilibria. We obtain existence theorems for some multicriteria games.

Interactive decisions and potential games

The aim of this contribution is an overview on Potential Games. This class of games is special, in fact we can investigate their properties by a unique function: the potential function. We consider several types of potential games: exact, ordinal, bayesian and hierarchical. Some results are generalized to multicriteria decisions.

(∊,k) EQUILIBRIA AND WELL-POSEDNESS

The aim of this paper is to discuss a new concept of well-posedness for non cooperative games. Starting from the definition of (∊,k) equilibrium as the point where every player either guarantees at least k or he (she) does not lose more than ∊, we introduce an original definition of well-posedness. We study characterizations of this well-posedness and its relations with the more known Tikhonov well-posedness. We prove that this well-posedness is an ordinal property if the payoff functions are bounded from below.

Potential games and well-posedness properties

The aim of this article is to study potential games which are a special class of games, in fact their properties are dictated by a single function called the potential function. We consider Tikhonov well-posedness and other well-posedness properties introduced by the authors in Margiocco et al. (Margiocco, M., Patrone, F. and Pusillo Chicco, L., 1997, A new approach to Tikhonov well–posedness for Nash equilibria. Optimization, 40, 385–400) Margiocco and Pusillo (Margiocco, M. and Pusillo, L., Value bounded approximations for Nash equilibria, Preprint, Submitted). We relate these properties with the Tikhonov well posedness of the potential function as maximum problem.

Share Opportunity Sets and Cooperative Games

In many share problems there is a priori given a natural set of possible divisions to solve the sharing problem.Cooperative games related to such share sets are introduced, which may be helpful in solving share problems.Relations between properties of share sets and properties of games are investigated.The average lexicographic value for share sets and for cooperative games is studied.

StackelbergWell-Posedness and Hierarchical Potential Games

We consider the Stackelberg well-posedness for hierarchical potential games and relate it to the Tikhonov well-posedness of the potential function as a maximum problem. We also make some considerations about the strong and weak Stackelberg approximate equilibria.

Multiobjective Optimization Problems

In this paper I give a survey on some results about optimality criteria (also approximate) in vector optimization. A special attention is dedicated to a new approach to approximate solutions for multicriteria situations as recently given in L. Pusillo and S. Tijs 2009.

Approximate Equilibria for Bayesian Multi-Criteria Games

In this paper the existence of Bayesian multi-criteria equilibria in mixed strategies is established for finite Bayesian multi-criteria games.Sufficient conditions are given under which approximate equilibria exist for non-finite Bayesian multi-criteria games.