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.     

Multicriteria games and potentials

In this note we study how far the theory of strategic games with potentials, as reported by Monderer and Shapley (Games Econ Behav 14:124–143, 1996), can be extended to strategic games with vector payoffs, as reported by Shapley (Nav Res Logist Q 6:57–61, 1959). The problem of the existence of pure approximate Pareto equilibria for multicriteria potential games is also studied.      

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.     

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.      

Metric Characterizations of Tikhonov Well-Posedness in Value

In this paper, we discuss and give metric characterizations of Tikhonov well-posedness in value for Nash equilibria. Roughly speaking, Tikhonov well-posedness of a problem means that approximate solutions converge to the true solution when the degree of approximation goes to zero. If we add to the condition of -equilibrium that of -closeness in value to some Nash equilibrium, we obtain Tikhonov well-posedness in value, which we have defined in a previous paper. This generalization of Tikhonov well-posedness has the remarkable property of ordinality; namely, it is preserved under monotonic transformations of the payoffs. We show that a metric characterization of Tikhonov well-posedness in value is not possible unless the set of Nash equilibria is compact and nonempty.