Equilibrium Existence Theorem Research Articles

Cooperative equilibria of strategy-form games with both nontransferable and transferable utilities

In this paper, we consider a class of strategy-form games with both nontransferable and transferable utilities. Inspired by NTU and TU α-core concepts, we first introduce the notion of cooperative equilibria, and prove the existence theorem in this model with finite dimensional strategy spaces. Furthermore, we extend the cooperative equilibrium existence theorem to strategy-form games with infinite dimensional strategy spaces.

Existence and generic stability of open-loop Nash equilibria for noncooperative fuzzy differential games

Differential game is a critical area of study not only in theory but also in application. Based on the concept of the fuzzy process introduced by Liu, a fuzzy differential game model described using fuzzy differential equations is studied. Firstly, the existence theorem of open-loop Nash equilibrium for fuzzy differential games is given using Ky Fan inequality, and an example shows the applicability of the theorem. Secondly, the fuzzy differential game space Γ1 is constructed, and the stability of open-loop Nash equilibria of the fuzzy differential game γ∈Γ1 is studied. The conclusion shows that the fuzzy differential games whose all the open-loop Nash equilibria are stable form a dense residual set, i.e., most of the fuzzy differential games are stable in the sense of Baire classification.

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>

Equilibrium non-existence in generalized games

A generalized game is a strategic situation in which agents' behavior restricts their opponents' available action choices, giving rise to interdependencies in terms of what strategy profiles remain mutually feasible. This paper proposes a novel example of a simple jointly convex generalized game in which the well-known convexity, compactness, continuity, and concavity assumptions are satisfied, but no Nash equilibrium exists. The essence of this contribution lies in answering a question left open by Banks and Duggan (2004): whether the supplemental condition of lower hemicontinuity of feasibility correspondences can be dropped from these authors' equilibrium-existence theorem. It cannot.

A social coalitional weak equilibrium existence theorem with a continuum of agents and applications

In this paper, we establish a social coalitional weak equilibrium existence theorem with a continuum of agents, which is an infinite-player extension of the fully cooperative version of Ichiishi (1981). As a unified approach, we obtain the weak α-core existence theorems of normal-form games and exchange economies with a continuum of agents by the social coalitional weak equilibrium existence theorem.

On the approximate purification of mixed strategies in games with infinite action sets

We consider a game in which the action set of each player is uncountable, and show that, from weak assumptions on the common prior, any mixed strategy has an approximately equivalent pure strategy. The assumption of this result can be further weakened if we consider the purification of a Nash equilibrium. Combined with the existence theorem for a Nash equilibrium, we derive an existence theorem for a pure strategy approximated Nash equilibrium under sufficiently weak assumptions. All of the pure strategies we derive in this paper can take a finite number of possible actions.

Well-posedness and existence for the weak multicriteria Nash equilibrium of multicriteria games

In this paper, we mainly study a class of multicriteria games. We establish an existence theorem for the weak multicriteria Nash equilibrium of multicriteria games in which a weak form of upper semicontinuity and quasiconvexity on payoff functions is allowed. Moreover, we introduce the concept of -convergence for the sequence of payoff mappings and obtain the sufficient conditions for generalized Hadamard well-posedness of multicriteria games.

Теоремы существования равновесия Нэша и равновесия в безопасных стратегиях

The paper proposes a method for constructing existence theorems for equilibrium in secure strategies (EinSS) on the basis of the well-known existence theorems for Nash equilibrium. The system of defi nitions of the EinSS is interpreted as the development of the logic of rational behavior in the defi nition of Nash equilibrium. It is proved that under the condition of «strong threats» for the existence of EinSS, it is suffi cient to fulfi ll the conditions of the known existence theorem for the Nash equilibrium only on the sets of secure strategies. This statement is formulated both globally and locally, and is an effective tool for application to practical problems. It opens up the possibility of constructing various particular existence theorems for EinSS. As a demonstration of the proposed approach, from the Debreu’s theorem of the existence of social equilibrium, the corresponding theorem of the existence of the EinSS is obtained.

Solutions of system of generalized vector quasi-equilibrium problems in locally FC-uniform spaces

In this paper, we establish a collectively fixed point theorem and an equilibrium existence theorem for generalized games in product locally FC-uniform spaces. As applications, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are derived in product locally FC-uniform spaces. These theorems are new and generalize some known results in the literature.

Global stability for a new predator\u2013prey model with cross-dispersal among patches based on graph theory

In this paper, cross-dispersal is considered in a predator–prey model with a patchy environment. A new predator–prey model with cross-dispersal among patches is constructed. A new cross-dispersal matrix is established by the coupling relationship between vertices. First, an existence theorem of the positive equilibrium for the new model is obtained. Secondly, based on the idea of constructing Lyapunov functions and a graph-theoretical approach for coupled systems, sufficient conditions that the positive equilibrium of the new model is globally asymptotically stable in R^{2n}_{+} are derived on a network with strongly connected graphs. Thirdly, based on the theory of asymptotically autonomous systems, Lyapunov functions method and graph theory, a stability theorem for the positive equilibrium of the new model is established on a complex network without strongly connected graphs. Finally, two examples are given to illustrate main results.

Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences, without free disposal, and without compact consumption sets

We extend a result on existence of Walrasian equilibria in He and Yannelis (Econ Theory 61:497–513, 2016) by replacing the compactness assumption on consumption sets made there by the standard assumption that these sets are closed and bounded from below. This provides a positive answer to a question explicitly raised in He and Yannelis (Econ Theory 61:497–513, 2016). Our new equilibrium existence theorem generalizes many results in the literature as we do not require any transitivity or completeness or continuity assumption on preferences, initial endowments need not be in the interior of the consumption sets, preferences may be interdependent and price-dependent, and no monotonicity or local non satiation is needed for any of the agents.

Existence of equilibrium in an abstract economy without SS-convexity

In this paper, we introduce the concept of Δ-convexity in an abstract economy with all choice sets being general topological spaces. We give a new generalization of the classical Yannelis-Prabhakar equilibrium existence theorem in abstract economies by relaxing SS-convexity to Δ-convexity.

Browder type fixed point theorems and Nash equilibria in generalized games

In this paper, we present two generalizations of the well-known Browder fixed point theorem, one of which is equivalent to the well-known Fan–Knaster–Kuratowski–Mazurkiewicz theorem. As applications, we apply these fixed point theorems to derive existence theorems for Nash equilibria in generalized games which generalize some existing existence theorems in the literature, including the well-known equilibrium existence theorem by Arrow and Debreu (Econometrica 22:265–290, 1954) and the existence theorem by Cubiotti (Int J Game Theory 26:267–273, 1997).

A Stackelberg-population competition model via variational inequalities and fixed points

In this paper, we introduce and study a new Stackelberg-population competition model which captures the desired features of both population games and Stackelberg competition model within the same framework. We obtain some characterization results for the Stackelberg-population equilibrium response set and the Stackelberg-population equilibrium leader set by using the variational inequality technique and Brouwer’s fixed point theorem. We also show an existence theorem of Nash equilibrium for Stackelbergpopulation competition model under some mild conditions. Finally, we give an example to illustrate our main results.

Coalitional Equilibria in Coalitional Abstract Economies with Nonordered Preferences

This paper investigates the existence of coalitional equilibria for coalitional abstract economies with nonordered preferences. Accordingly, the authors give a fuzzy extension of the coalitional equilibrium existence theorem. The main result is illustrated with applications to coalitional abstract economies with payoff functions, strong Nash equilibria and absolute optimal solutions.

A coalitional extension of generalized fuzzy games

In this paper, we give a coalitional extension of generalized fuzzy games and introduce the notion of coalitional fuzzy equilibria. We first prove the existence of coalitional fuzzy equilibria in our model. As an application, we further show the existence theorem of strong fuzzy equilibria for fuzzy games.

Consistent Conjectural Variations Equilibrium in a Semi-Mixed Duopoly

This paper considers conjectural variations equilibrium (CVE) in the one item market with a mixed duopoly of competitors. The duopoly is calledsemi-mixedbecause one (semi-public) company’s objective is to maximize a convex combination of her net profit and domestic social surplus (DSS). The two agents make conjectures about fluctuations of the equilibrium price occurring after their supplies having been varied. Based on the concepts of theexteriorandinterior equilibrium, as well as the existence theorem for the interior equilibrium (a.k.a. the consistent CVE, or the exterior equilibrium withconsistent conjectures) demonstrated in the authors’ previous papers, we analyze the behavior of the interior equilibrium as a function of the semi-public firm’s level of socialization. When this parameter reflected by the convex combination coefficient tends to 1, thus transforming the semi-public company into a completely public one, and the considered model into the classical mixed duopoly, two trends are apparent. First, for the private company, the equilibrium with consistent conjectures (CCVE) becomes more attractive (lucrative) than the Cournot-Nash equilibrium. Second, there exists a (unique in the case of an affine demand function) value of the convex combination coefficient such that the private agent’s profit is the same in both of the above-mentioned equilibrium types, thus making no subsidy to the producer or to the consumers necessary. Numerical experiments with various mixed duopoly models confirm the robustness of the proposed algorithm for finding the optimal value of the above-mentioned combination coefficient (a.k.a. the semi-public company’s socialization level).

General Equilibrium Model for an Asymmetric Information Economy Without Delivery Upper Bounds

AbstractIn this paper, we introduce production into the standard general equilibrium model with asymmetric information, which was proposed by Dubey et al. (Cowles Foundation Discussion Paper 2000; Econometrica 2005). In such an economy, there is no rational explanation for producers’ delivery upper bounds while the endowments naturally limit consumers’ deliveries. However, we show that the typical equilibrium allocation of the asymmetric information economy necessarily and substantially depends on such exogenous upper bounds (Example 1 and Theorem 1). In other words, an equilibrium existence theorem without such upper bounds, even if such exists, will typically fail to treat the asymmetric information problem, e.g., the adverse selection problem. Hence, to treat the equilibrium existence problem under the informational asymmetry appropriately, we have to extend the standard model so that the delivery upper bounds need not to be specified explicitly. For this purpose, we propose a quite natural and realistic assumption with respect to the technological condition related to the market delivery, i.e., the existence of some small standardization, commoditization, and/or transaction costs of market deliveries is shown to be sufficient (Theorem 3).

An Existence Theorem of Nash Equilibrium in Coq and Isabelle

Nash equilibrium (NE) is a central concept in game theory. Here we prove formally a published theorem on existence of an NE in two proof assistants, Coq and Isabelle: starting from a game with finitely many outcomes, one may derive a game by rewriting each of these outcomes with either of two basic outcomes, namely that Player 1 wins or that Player 2 wins. If all ways of deriving such a win/lose game lead to a game where one player has a winning strategy, the original game also has a Nash equilibrium. This article makes three other contributions: first, while the original proof invoked linear extension of strict partial orders, here we avoid it by generalizing the relevant definition. Second, we notice that the theorem also implies the existence of a secure equilibrium, a stronger version of NE that was introduced for model checking. Third, we also notice that the constructive proof of the theorem computes secure equilibria for non-zero-sum priority games (generalizing parity games) in quasi-polynomial time.

Existence of competitive equilibrium in coalition production economies with a continuum of agents

Aumann (Econometrica 34: 1–17, 1966) established the existence of competitive equilibria in exchange economies with a continuum of traders. This result has been extended to special production economies under the condition that the production sets are countably additive or superadditive in the literature. In this paper, we extend Aumann’s result to coalition production economies with a continuum of traders in which each coalition can have a different production set and the production sets need not to be additive or superadditive. Our new existence theorem for competitive equilibrium implies the classical result by Hildenbrand (Econometrica 38: 608–623, 1970), and implies those by Sondermann (J Econ Theory 8: 259–291, 1974) and Greenberg et al. (J Math Econ 6: 31–41, 1979) under the assumption that the production sets are convex.