Berge's maximum theorem ensures the continuity of the value function and the upper semicontinuity of the solution mapping in parametric optimization problems. This theorem plays a central role in optimization theory, game theory, and dynamic programming. Motivated by the inherent inaccuracies in optimization data, this paper investigates the stability of such problems under sequential perturbations of both the objective function and the feasible mapping. The analysis focuses on the convergence of sequences of value functions and solution mappings via variational approximations of the data. To this end, we employ lower and upper continuous, epi- and hypo-convergence notions for functions, together with lower and upper continuous and graphical convergence notions for multifunctions. In addition, we study some relationships among these types of convergence and provide examples and counterexamples associated with the corresponding notions. Our results extend and complement existing stability results in the literature. We provide applications to generalized Nash equilibrium problems, where stability is obtained via a direct approach, as well as to finite-horizon dynamic programming models under novel perturbation assumptions.

Lower semicontinuity is a key continuity concept in set-valued analysis, but it is also known to be somehow difficult to establish when one wants to tackle specific set-valued maps like constraint maps of quasi-variational inequalities or generalized Nash equilibrium problems or quasi-optimization problems. A classical sufficient condition to obtain the lower semicontinuity is the openness of the graph of the map or of its fibers. Our aim in this work is to propose a weaker concept, called densely locally constant, ensuring the lower semicontinuity and, at the same time, with powerful properties like local intersection property or stability over convexification. Moreover, this new concept allows us to prove alternative selection results, fixed-point theorems and the maximum principle.

In this paper, we study a recently introduced vector-valued version of the Nikaidô–Isoda bifunction that allows Nash games to be reformulated as strong vector equilibrium problems. We show that classical assumptions in Nash equilibrium theory, such as the quasiconvexity of the players’ payoff functions, naturally translate into generalized convexity properties of the vector Nikaidô–Isoda bifunction. This correspondence does not generally hold for the scalar Nikaidô–Isoda bifunction, since it is defined through summation and quasiconvexity is not preserved under sums. Based on this reformulation, we investigate the existence of solutions to strong vector equilibrium problems by means of the finite intersection property and transfer lower continuity, thereby extending and unifying several results in the literature. We further analyze the main properties of the vector Nikaidô–Isoda bifunction and identify classes of games for which this bifunction satisfies generalized quasiconvexity and generalized monotonicity assumptions. Finally, we establish a connection between the classical notion of weak transfer lower continuity in games and the transfer lower continuity of the vector Nikaidô–Isoda bifunction.

This paper addresses the problem of stochastic optimization with decision-dependent uncertainty, a class of problems where the probability distribution of the uncertain parameters is influenced by the decision-maker's actions. While recent literature primarily focuses on solving or analyzing these problems by directly imposing hypotheses on the distribution mapping, we explore in this work some of these properties for a specific construction by means of the moving support and a density function. The construction is motivated by the Bayesian approach to bilevel programming, where the response of a follower is modeled as the uncertainty, drawn from the moving set of optimal responses, which depends on the leader's decision. Our main contribution is to establish sufficient conditions for the Lipschitz continuity of the expected value function. We show that Lipschitz continuity can be achieved when the moving support is a Lipschitz continuous set-valued map with full-dimensional, convex, compact values, or when it is the solution set of a fully linear parametric problem. We also provide an example showing that the sole Lipschitz assumption on the moving set itself is not sufficient and that additional conditions are necessary.

In this paper, we focus on the concept of projected solutions in generalized Nash games and examine their relationship with generalized Nash equilibria. We also establish the existence of projected solutions through an elegant reformulation that allows the use of a result by Arrow and Debreu. Finally, abstract economy is considered as an application.

Quasi-equilibrium problems represent a general framework covering, in many situations, quasi- variational inequalities, complementarity problems and generalized Nash equilibrium problems. In this work, we provide necessary and sufficient conditions guaranteeing the existence of a new kind of solution for quasi-equilibrium problems defined on Hausdorff topological spaces with non-self constraint maps. The main machineries for proving our results are a Tian's fixed point theorem for the convex case, and the finite intersection property for the non-convex case. Furthermore, quasi-variational inequality problems and generalized Nash equilibrium problems are considered as applications.

We study the generalized Nash game proposed by Rosen, which involves strategy sets coupled across players through a shared constraint. We demonstrate a reduction to a classical game, allowing Rosen’s result to be deduced from the work of Arrow and Debreu. Additionally, we establish an existence result under the quasi-convexity assumption in Banach spaces. New existence results are also provided for the non-compact case under coerciveness conditions. Finally, an abstract economy is considered as an application.

We show that projected solutions of a generalized game correspond to classical ones of an auxiliary generalized game obtained by doubling the number of players. Based on this reformulation and using known results for the existence of classical solutions, we deduce some new existence results for projected solutions of generalized Nash equilibrium problems and quasivariational inequalities.

In this work we extend a maximum theorem proposed by Morgan and Scalzo. We also show some results of minimax inequalities which are equivalent to the famous Ky Fan minimax inequality. Additionally, we prove that the existence result of Nash equilibria proposed by Morgan and Scalzo is actually equivalent to a classical result in the literature.

This note qualifies some results in Cotrina and Flores-Bazán (2024) on the ‘Finite intersection property’. We remind the essential role of this property for proving our results. In this note, we recall the statement of this property. We then specify the restrictive conditions that should be provided. An example proves the interest of this note revising some points in Cotrina and Flores-Bazán. Furthermore, it is showed that a class of generalized Nash equilibrium problems can be viewed as a particular case of a vector optimization problem, whose vector-valued function always possesses the Finite intersection property on its natural domain. Thus, the existence of a generalized Nash equilibrium will be a consequence of our Berge-type theorem and the Kakutani fixed point theorem, being it more general than those existing in recent literature, as shown by our examples.

In this work, we deal with a vector extension of the generalized the Weierstrass and Berge maximum theorems, notably in mathematical economics. This is carried out by introducing the notions of transfer continuity and pseudo-continuity for those functions. Here the preference relation is given via a closed convex cone, having possibly empty interior. As a consequence, we present an existence of strong-Nash equilibria for generalized multiobjective games, and the Rosen model is revisited.

It is known that the generalized Nash equilibrium problem can be reformulated as a quasivariational inequality. Our aim in this work is to introduce a variational approach to study the existence of solutions for generalized ordinal Nash games, that is, generalized games where the player preferences are binary relations that do not necessarily admit a utility representation.

In this note we are interested in a relevant generalized Nash equilibrium problem, which was proposed by Rosen in 1965. An existence result is established in the general setting of quasiconvexity, by relaxing the continuity hypothesis of the objective functions, which is independent from the one given by Aussel and Dutta in 2008.

In this work, we reformulate the problem of existence of maximal elements for preference relations as a variational inequality problem in the sense of Stampacchia. Similarly, we establish the uniqueness of maximal elements using a variational inequality problem in the sense of Minty. In both of these approaches, we use the normal cone operator to find existence and uniqueness results, under mild assumptions. In addition, we provide an algorithm for finding such maximal elements, which is inspired by the steepest descent method for minimization. Under certain conditions, we prove that the sequence generated by this algorithm converges to a maximal element.

This work is devoted to inverse maximum theorems and is motivated by the ones given by Komiya and Yamauchi. Also, we show that the results by Arrow and Debreu, on the existence of solutions for generalized Nash games, and the Kakutani–Fan–Glicksberg fixed point theorem can be derived from the Debreu–Fan–Glicksberg theorem, for classical Nash games.

In this work, we focus on the concept of projected solutions for generalized Nash equilibrium problems. We present new existence results by considering sets of strategies that are not necessarily compact. The relationship between projected solutions and Nash equilibria is studied for the generalized Nash game proposed by Rosen. Finally, we demonstrate that every projected solution of a game is associated with a Nash equilibrium, but in a different game.

We deal with the generalized Nash game proposed by Rosen, which is a game with strategy sets that are coupled across players through a shared constraint. A reduction to a classical game is shown, and as a consequence, Rosen's result can be deduced from the one given by Arrow and Debreu. We also establish necessary and sufficient conditions for a point to be a generalized Nash equilibrium employing the variational inequality approach. Finally, some existence results are given in the non-compact case under coerciveness conditions.

A quasi-equilibrium problem is an equilibrium problem where the constraint set does depend on the reference point. It generalizes important problems such as quasi-variational inequalities and generalized Nash equilibrium problems. We study the existence of equilibria on unbounded sets under a coerciveness condition. We discuss the relation of our results with others from the literature.

In this note we are interested in a relevant generalized Nash equilibrium problem, which was proposed by Rosen in 1965. An existence result is established in the general setting of quasiconvexity, which is independent from the one given by Aussel and Dutta in 2008.

We deal with inverse maximum theorems, which are inspired by the ones given by Aoyama, Komiya, Li et al., Park and Komiya, and Yamauchi. As a consequence of our results, we state and prove an inverse maximum Nash theorem and show that any generalized Nash game can be reduced to a classical Nash game, under suitable assumptions. Additionally, we show that a result by Arrow and Debreu, on the existence of solutions for generalized Nash games, is actually equivalent to the one given by Debreu-Fan-Glicksberg for classical Nash games, which in turn is equivalent to Kakutani-Fan-Glisckberg's fixed point theorem.