Multiobjective Games Research Articles

A note on the Finite intersection property in “Existence of Nash equilibria for generalized multiobjective games through the vector extension of Weierstrass and Berge maximum theorems”

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.

Approximate weak optimality conditions in multiobjective generalized Nash equilibrium problems

A multiobjective generalized Nash equilibrium problem (MGNEP) is a Nash equilibrium problem with constraints that include multiobjective games. We focus in this paper on examining the approximate Karush–Kuhn–Tucker (KKT) conditions for MGNEPs and their impact on the global convergence of algorithms. To begin, we define standard approximate weak KKT (standard-AWKKT) conditions for MGNEPs. We demonstrate that every weak efficient solution meets the standard-AWKKT condition. As these optimality conditions are strong, we propose a new set of approximate weak KKT conditions, specifically tailored for MGNEPs, and show their convergence towards weak KKT points provided a cone-continuity property is satisfied. It is important to note that while these newly introduced approximate weak KKT conditions do not serve as optimality conditions for general MGNEPs, we illustrate that they hold true for the special case of a weak variational equilibrium point in a jointly convex MGNEP. Finally, we propose an enhanced Lagrangian-type algorithm for estimating an approximate weak KKT of an MGNEP and demonstrate its global convergence.

On the existence of solutions for systems of generalized vector quasi-variational equilibrium problems in abstract convex spaces with applications

<p>In this paper, we first introduced systems of generalized vector quasi-variational equilibrium problems as well as systems of vector quasi-variational equilibrium problems as their special cases in abstract convex spaces. Next, we established some existence theorems of solutions for systems of generalized vector quasi-variational equilibrium problems and systems of vector quasi-variational equilibrium problems in non-compact abstract convex spaces. Furthermore, we applied the obtained existence theorem of solutions for systems of vector quasi-variational equilibrium problems to solve the existence problem of Nash equilibria for noncooperative games. Then, as applications of the existence result of Nash equilibria for noncooperative games, we studied the existence of weighted Nash equilibria and Pareto Nash equilibria for multi-objective games. The results derived in this paper extended and unified the primary findings presented by some authors in the literature.</p>

Existence of Nash equilibria for generalized multiobjective games through the vector extension of Weierstrass and Berge maximum theorems

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.

Existence results and equilibrium stability conditions to fuzzy [formula omitted]-player generalized multiobjective games with application to economic equilibrium models

The purpose of this article is to study existence conditions and generic stability of solution sets to the fuzzy generalized multiobjective games in infinite dimensional spaces. First, we revisit a new class of the fuzzy generalized multiobjective games. Second, we introduce the concept of the extended Ci-quasi-concavity and establish existence conditions for fuzzy generalized multiobjective games using Browder type fixed point theorem in the noncompact cases. Third, we establish some sufficient conditions for generic solution stability conditions to such n-player multiobjective games under uncertainty. Finally, as a real-world application, we consider the special case of economic equilibrium models. Existence conditions and generic stability of solution sets for these real models are also obtained. The results obtained in this paper are new and different from the main results given by some authors in the literature.

Optimal control of generalized multiobjective games with application to traffic networks modeling

AbstractThe purpose of this paper is to study some new results on the existence and convergence of the solutions to controlled systems of generalized multiobjective games, controlled systems of traffic networks, and optimal control problems (OCPs). First, we introduce the controlled systems of generalized multiobjective games and establish the existence of the solutions for these systems using Browder‐type fixed point theorem in the noncompact case and the ‐quasi‐concavity. Results on the convergence of controlled systems of the solutions for such problems using the auxiliary solution sets and the extended ‐convexity of the objective functions are studied. Second, we investigate OCPs governed by generalized multiobjective games. The existence and convergence of the solutions to these problems are also obtained. Finally, as a real‐world application, we consider the special case of controlled systems of traffic networks. Many examples are given for the illustration of our results.

Analysis of the n-person noncooperative supermodular multiobjective games

This paper investigates a class of noncooperative multiobjective games in the supermodular form played by a finite number of players. We prove the existence of two different equilibria of those games, namely, the Pareto equilibrium and the weighted Nash equilibrium. Using Python programming, we apply a genetic algorithm to obtain the Pareto equilibrium and a modified payoff matrix to obtain the weighted Nash equilibrium. We demonstrate the application of our result in an inventory game problem.

Existence and stability of fuzzy Pareto-Nash equilibria for fuzzy constrained multi-objective games with fuzzy payoffs

<abstract><p>A new class of fuzzy constrained multi-objective games with fuzzy payoffs (FCMGFPs) is considered in this paper. First, Berge's maximum theorem for the fuzzy-vector-valued function is obtained. Based on this theorem and the Fan-Glicksberg fixed point theorem, the existence theorem of the fuzzy Pareto-Nash equilibrium for the FCMGFP is established. Second, the abstract rationality function for the FCMGFP is given by using a nonlinear scalarization function of interval vectors. Finally, a series of results, such as structural stability ($ (\gamma, \epsilon) $-stability) and robustness to $ \epsilon $-equilibrium ($ (\gamma, \epsilon) $-robustness), are obtained.</p></abstract>

Production Planning Games in Uncertain Environment

In this paper, a production planning game with single and multiobjectives in uncertain environment is considered. In this problem, the parameters are assumed to be fuzzy variables. By the notions of the expected value and the critical value, the concept of the solutions in the game is introduced. It is shown that the game is balanced and the core of game is nonempty. The duality theory in single-objective and the weighted sum method in multiobjective games are proposed to obtain the payoffs of the players. Finally, the validity and applicability of the method are illustrated by a practical example.

Existence and stability of weakly Pareto-Nash equilibria for discontinuous multiobjective games

We introduce two new concepts of the vector-valued generalized single deviation property and the vector-valued positive generalized single deviation property for discontinuous multiobjective games. Then, some new existence theorems of weakly Pareto-Nash equilibria for discontinuous multiobjective games are established. Moreover, we prove that most of the discontinuous multiobjective games are essential with perturbed payoffs in the sense of Baire category.

Generalized well-posedness for parametric fuzzy generalized multiobjective games

The purpose of this article is to study well-posedness for parametric fuzzy generalized multiobjective games. First, we introduce the concepts of the well-posedness and well-posedness in the generalized sense for parametric fuzzy generalized multiobjective games. Then, under suitable conditions, we establish some sufficient conditions for the well-posedness and the well-posedness in the generalized sense for such problem. Further, results on metric characterization of the well-posedness and the well-posedness in the generalized sense for parametric fuzzy generalized multiobjective games in terms of the behavior of the approximate solution sets are also obtained. Finally, we state results on the well-posedness in the generalized sense for the parametric fuzzy generalized multiobjective games by using the Kuratowski measure of noncompactness. Examples are given to illustrate our results. The results presented in the paper are new.

Analysis of a Single Manufacturer Multi-Retailer Inventory Competition Using Supermodular Multi-Objective Games

In this research, the optimum decision of a multi-objective inventory game problem has been investigated under strategic complementarities in a multiplayer supply chain. The supply chain comprises a single retailer and multi-retailers under the synchronization process, the wholesale contract, and the buy-back contract policy. We apply some results in supermodular multi-objective game theory to solve these problems. The optimal decision for all players is formed in two equilibria, namely the Pareto equilibrium and the weighted Nash equilibrium. For numerical results, A genetic algorithm and dominance principle elements of the payoff matrix are used to obtain the weighted Nash equilibrium and the weighted Nash equilibrium, respectively. The analytical and numerical results using the supermodular multi-objective games concept can be significant results to solve the supply chain competition problems in the industrial engineering.

Essential stability of fuzzy equilibria for generalized multiobjective games with fuzzy constraint mappings

Under the assumption that the constraint mappings of players are fuzzy, we study the stability of fuzzy equilibria for a group of generalized multiobjective games. First, we prove that most of generalized multiobjective games with fuzzy constraint mappings are essential on the meaning of Baire category by Fort theorem. Furthermore, an example is constructed to reveal that not all generalized multiobjective games with fuzzy constraint mappings come into possession of essential points. Finally, we obtain the existence of essential components for the games.

Some Kinds of Bargaining Equilibria of Multi-objective Games

Some Kinds of Bargaining Equilibria of Multi-objective Games

Characterizations of Pareto-Nash Equilibria for Multiobjective Potential Population Games

This paper studies the characterizations of (weakly) Pareto-Nash equilibria for multiobjective population games with a vector-valued potential function called multiobjective potential population games, where agents synchronously maximize multiobjective functions with finite strategies via a partial order on the criteria-function set. In such games, multiobjective payoff functions are equal to the transpose of the Jacobi matrix of its potential function. For multiobjective potential population games, based on Kuhn-Tucker conditions of multiobjective optimization, a strongly (weakly) Kuhn-Tucker state is introduced for its vector-valued potential function and it is proven that each strongly (weakly) Kuhn-Tucker state is one (weakly) Pareto-Nash equilibrium. The converse is obtained for multiobjective potential population games with two strategies by utilizing Tucker’s Theorem of the alternative and Motzkin’s one of linear systems. Precisely, each (weakly) Pareto-Nash equilibrium is equivalent to a strongly (weakly) Kuhn-Tucker state for multiobjective potential population games with two strategies. These characterizations by a vector-valued approach are more comprehensive than an additive weighted method. Multiobjective potential population games are the extension of population potential games from a single objective to multiobjective cases. These novel results provide a theoretical basis for further computing (weakly) Pareto-Nash equilibria of multiobjective potential population games and their practical applications.

Strongly essential set of vector Ky Fan's points problem and its applications

In this paper, several existence results of strongly essential set of the solution set for Ky Fan's section problems and vector Ky Fan's point problems are obtained. Firstly, two kinds of strongly essential sets of Ky Fan's section problems are defined, and some further results on existence of the strongly essential component of solutions set of Ky Fan's section problems are proved, which generalize the conclusion in ^{[<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>]}, and further generalize the conclusions in ^{[<span class="xref"><a href="#b21" ref-type="bibr">21</a></span>,<span class="xref"><a href="#b28" ref-type="bibr">28</a></span>]}. Secondly, based on the above results, two classes of stronger perturbations of vector-valued inequality functions are proposed respectively, and several existence results of the strongly essential component of set of vector Ky Fan's points are obtained. By comparing several metrics, we give some strong and weak relationships among the various metrics involved in the text. The main results of this paper actually generalize the relevant conclusions in the current literature. Finally, as an application, we obtain an existence result of the strongly essential component of weakly Pareto-Nash equilibrium for multiobjective games.

Collectively fixed point theorems in noncompact abstract convex spaces with applications

&lt;abstract&gt;&lt;p&gt;In this paper, by using the KKM theory and the properties of $ \Gamma $-convexity and $ {\frak{RC}} $-mapping, we investigate the existence of collectively fixed points for a family with a finite number of set-valued mappings on the product space of noncompact abstract convex spaces. Consequently, as applications, some existence theorems of generalized weighted Nash equilibria and generalized Pareto Nash equilibria for constrained multiobjective games, some nonempty intersection theorems with applications to the Fan analytic alternative formulation and the existence of Nash equilibria, and some existence theorems of solutions for generalized weak implicit inclusion problems in noncompact abstract convex spaces are given. The results obtained in this paper extend and generalize many corresponding results of the existing literature.&lt;/p&gt;&lt;/abstract&gt;

Multiobjective Games for Detecting Abnormally Expressed Genes

A class of multiobjective games with applications to a medicine setting is studied. We consider the vector Shapley value and the vector Banzhaf value for a multicriteria game and we apply them to a microarray game. We give an axiomatic characterization too.

A new class of generalized multiobjective games in bounded rationality with fuzzy mappings: Structural [formula omitted]-stability and [formula omitted]-robustness to [formula omitted]-equilibria

In this paper, we introduce a new class of generalized multiobjective games with fuzzy mappings and study the solution existence for this class of games. The model of bounded rationality proposed by Anderlini and Canning (2001) and Miyazaki and Azuma (2013) is applied to a new class of generalized multiobjective games with fuzzy mappings in infinite-dimensional spaces. We introduce a related abstract rationality function for this model using the nonlinear scalarization method and we show that the structural stability (i.e., (λ,ε)-stability) of this model implies its robustness (i.e., (λ,ε)-robustness) to ε-equilibria.

An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality

In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It is also shown that the bounded rationality is an approximate way to achieve the full rationality. As a special case, we obtain some corollaries for scalar equilibrium problems. Moreover, we obtain a generic convergence theorem of the solutions of strictly-quasi-monotone vector equilibrium problems according to Baire’s theorems. As applications, we investigate vector variational inequality problems, vector optimization problems and Nash equilibrium problems of multi-objective games as special cases.