Equilibrium Existence Theorem Research Articles

Maximal Element Theorems for W K -Majorized Mappings in FC-spaces and the Application to Equilibrium of Abstract Economies

In this paper,maximal elements theorems for Wk-majorized mappings are established in FC-spaces.As application,existence theorems of equilibrium for qualitative games and abstract economies are obtained in FC-spaces.

AN APPLICATION OF A DISCRETE FIXED POINT THEOREM TO A GAME IN EXPANSIVE FORM

In this paper, we first present a discrete fixed point theorem for contraction mappings from the product set of integer intervals into itself, which is an extension of Robert's discrete fixed point theorem. Next, we derive an existence theorem of a pure-strategy Nash equilibrium for a noncooperative n-person game from our fixed point theorem. Finally, we show that Kuhn's theorem for a game in expansive form can be explained by our existence theorem.

Potential Game for Dynamic Spectrum Management in Cognitive Radio Networks

Dynamic spectrum management is one of the key technologies in cognitive radio networks, and more and more researchers have been paying much attention to algorithms and models for Spectrum sharing. In this paper, we present a dynamic spectrum management application model based on potential game. Firstly, the mapping relationship is constructed for sections in cognitive radio network and game theory model. Game participants include primary network users and cognitive network users. Game process is defined as participants want to gain personal best interests and use different strategies in game. Secondly, we present a model for dynamic spectrum management in cognitive radio networks. In accordance with the definition and existence theorem of Nash equilibrium, we describe the users’ behaviors in cognitive radio networks. In accordance with the definition and existence theorem of Nash equilibrium, we describe the users’ behaviors in cognitive radio networks. Lastly, we simulate the realistic scenarios with the presented model and the results show that it can solve the problem for dynamic spectrum management with lower efficiency and non-flexibility.

NONEMPTY INTERSECTION THEOREMS AND SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS IN FC-SPACES

By using some existence theorems of maximal elements for a family of set-valued mappings involving a better admissible set-valued mapping under noncompact setting of FC-spaces, we present some non-empty intersection theorems for a family in product FC-spaces. Then, as applications, some new existence theorems of equilibrium for a system of generalized vector equilibrium problems are proved in product FC-spaces. Our results improve and generalize some recent results.

A Collectively Fixed Point Theorem in Abstract Convex Spaces and Its Applications

The main purpose of this paper is to establish a new collectively fixed point theorem in noncompact abstract convex spaces. As applications of this theorem, we obtain some new existence theorems of equilibria for generalized abstract economies in noncompact abstract convex spaces.

On Abstract Economies and Their Applications

We establish a new equilibrium existence theorem of generalized abstract economies with general preference correspondences. As an application, we derive an existence theorem of generalized quasi-variational inequalities in the general setting of -spaces without any linear structure.

A Note on the Equilibrium Existence Problem in Discontinuous Games

In this note, we prove an equilibrium existence theorem for games with discontinuous payoffs and convex and compact strategy spaces. It generalizes the classical result of Reny (1999), as well as the recent paper of McLennan, Monteiro, and Tourky (2011). Our conditions are simple and easy to verify. Importantly, examples of spatial location models show that our conditions allow for economically meaningful payoff discontinuities, that are not covered by other conditions in the literature.

Representing equilibrium aggregates in aggregate games with applications to common agency

An aggregate game is a normal-form game with the property that each playerʼs payoff is a function of only his own strategy and an aggregate of the strategy profile of all players. Such games possess properties that can often yield simple characterizations of equilibrium aggregates without requiring that one solves for the equilibrium strategy profile. When payoffs have a quasi-linear structure and a degree of symmetry, we construct a self-generating maximization program over the space of aggregates with the property that the solution set corresponds to the set of equilibrium aggregates of the original n-player game. We illustrate the value of this approach in common-agency games where the playersʼ strategy space is an infinite-dimensional space of nonlinear contracts. We derive equilibrium existence and characterization theorems for both the adverse selection and moral hazard versions of these games.

SOCIAL EQUILIBRIUM IN A GENERALIZED NASH GAME WITH THE AFFINE CONDITION

In this paper, we will prove a social equilibrium existence theorem of a generalized Nash game with affine constraint correspondences which is comparable with Nash's equilibrium existence theorem in several aspects.

Generalized Browder-type fixed point theorem with strongly geodesic convexity on Hadamard manifolds with applications

In this paper, a generalized Browder-type fixed point theorem on Hadamard manifolds is introduced, which can be regarded as a generalization of the Browder-type fixed point theorem for the set-valued mapping on an Euclidean space to a Hadamard manifold. As applications, a maximal element theorem, a section theorem, a Ky Fan-type Minimax Inequality and an existence theorem of Nash equilibrium for non-cooperative games on Hadamard manifolds are established.

The single deviation property in games with discontinuous payoffs

We study equilibrium existence in normal form games in which it is possible to associate with each nonequilibrium point an open neighborhood, a set of players, and a collection of deviation strategies, such that at any nonequilibrium point of the neighborhood, a player from the set can increase her payoff by switching to the deviation strategy designated for her. An equilibrium existence theorem for compact, quasiconcave games with two players is established as an application of a general equilibrium existence result for qualitative games. A new form of the better-reply security condition, called the strong single deviation property, is proposed.

KKM Theorems in GFC-Spaces and the Application to the System of General Quasiequilibrium Problems

In this paper,new KKM theorems and a Browder type fixed point theorem are established for GFC-spaces.As applications,equilibrium existence theorems for the system of general quasiequilibrium problems and the system of quasiequilibrium problems are obtained in GFC-spaces.

Existence of equilibria in economies with externalities and non-convexities in an infinite-dimensional commodity space

We prove an equilibrium existence theorem for economies with externalities, general types of non-convexities in the production sector, and infinitely many commodities. The consumption sets, the preferences of the consumers, and the production possibilities are represented by set-valued mappings to take into account the external effects. The firms set their prices according to general pricing rules which are supposed to have bounded losses and may depend upon the actions of the other economic agents. The commodity space is L ∞ ( M , M , μ ) , the space of all μ -essentially bounded M -measurable functions on M . As for our existence result, we consider the framework of Bewley (1972). However, there are four major problems in using this technique. To overcome two of these difficulties, we impose strong lower hemi-continuity assumptions upon the economies. The remaining problems are removed when the finite economies are large enough. Our model encompasses previous works on the existence of general equilibria when there are externalities and non-convexities but the commodity space is finite dimensional and those on general equilibria in non-convex economies with infinitely many commodities when no external effect is taken into account.

Consistent Conjectural Variations Equilibrium in a Mixed Duopoly

In this paper, we consider a model of mixed duopoly with Conjectured Variations Equilibrium (CVE). The agents’ conjectures concern the price variations depending on the increase or decrease in their production outputs. We establish existence and uniqueness results for the conjectured variations equilibrium (called an exterior equilibrium) for any set of feasible conjectures. To introduce the notion of an interior equilibrium, we develop a consistency criterion for the conjectures (referred to as influence coefficients) and prove the existence theorem for the interior equilibrium (understood as a CVE with consistent conjectures). To prepare the base for the extension of our results to the case of non-differentiable demand functions, we also investigate the behavior of the consistent conjectures in dependence upon a parameter representing the demand function derivative with respect to the market price.

Domain L-Majorization and Equilibrium Existence in Discontinuous Games

We study the equilibrium existence problem in normal form and qualitative games in which it is possible to associate with each nonequilibrium point an open neighborhood and a collection of deviation strategies such that, at any nonequilibrium point of the neighborhood, a player can increase her payoff by switching to the deviation strategy designated for her. An equilibrium existence theorem for compact, quasiconcave games with two players is established. We propose a new form of the better-reply security condition, called the strong single deviation property, that covers games whose set of Nash equilibria is not necessarily closed. We introduce domain L-majorized correspondences and use them to study equilibrium existence in qualitative games.

Equilibrium existence theorems for multi-leader-follower generalized multiobjective games in FC-spaces

In this paper, we introduce and study a class of multi-leader-follower generalized multiobjective games in FC-spaces where the number of leaders and followers may be finite or infinite and the objective functions of leaders and followers get their values in infinite-dimensional spaces. By using a Pareto equilibrium existence theorem of generalized constrained multiobjective games in FC-spaces due to author, some equilibrium existence theorems for the multi-leader-follower generalized multiobjective games are established in noncompact FC-spaces. These results improve and generalize some corresponding results in recent literatures.

Bayesian abstract economies

We have a Bayesian approach for an equilibrium problem in abstract economies of the Yannelis–Prabhakar type. We consider an economy with a countable or uncountable set of agents, with private information defined by subalgebras as per Yannelis [Yannelis, A Bayesian equilibrium existence theorem, Adv. Math. Econ. 4 (2002), pp. 61–72] and the preferences defined by correspondences. We prove an existence equilibrium results which extends the known results, as Yannelis and Prabhakar theorise in [Yannelis and Prabhakar, Existence of maximal elements and equilibrium in linear topological spaces, J. Math. Econom. 12 (1983), pp. 233–245].

Mixed oligopoly with consistent conjectures

In this paper, we consider a model of mixed oligopoly with conjectural variations equilibrium (CVE). The agents’ conjectures concern the price variations depending upon their production output’s increase or decrease. We establish existence and uniqueness results for the conjectural variations equilibrium (called an exterior equilibrium) for any set of feasible conjectures. To introduce the notion of an interior equilibrium, we develop a consistency criterion for the conjectures (referred to as influence coefficients) and prove the existence theorem for the interior equilibrium (understood as a CVE with consistent conjectures). To prepare the base for the extension of our results to the case of non-differentiable demand functions, we also investigate the behavior of the consistent conjectures in dependence upon a parameter representing the demand function’s derivative with respect to the market price.

Fixed Point Theorems on Product Topological Semilattice Spaces, Generalized Abstract Economies and Systems of Generalized Vector Quasi-equilibrium Problems

In this paper, we establish fixed point theorems for a family of multivalued maps defined on the product space of topological semilattice spaces. By using our fixed point theorems, we derive a result on the nonempty intersection of sets without convex structure and equilibrium existence theorems for generalized abstract economies with two constraint correspondences. We present some special cases of our results which generalize several known results in the literature. We consider systems of generalized vector quasi-equilibrium problems and their special cases. As an application of our equilibrium existence theorems, we establish some existence results for solutions of systems of generalized vector quasi-equilibrium problems and their special cases. The results of this paper improve and extend several results in the literature.

Games With Discontinuous Payoffs: A Strengthening of Reny's Existence Theorem

We provide a pure Nash equilibrium existence theorem for games with discontinuous payoffs whose hypotheses are in a number of ways weaker than those of the theorem of Reny (1999). In comparison with Reny's argument, our proof is brief. Our result subsumes a prior existence result of Reny (1999) that is not covered by his theorem. We use the main result to prove the existence of pure Nash equilibrium in a class of finite games in which agents' pure strategies are subsets of a given set, and in turn use this to prove the existence of stable configurations for games, similar to those used by Schelling (1971, 1972) to study residential segregation, in which agents choose locations.