A note on Berge equilibrium

This work is a contribution on the problem of the existence of Berge equilibrium. Abalo and Kostreva give an existence theorem for this equilibrium that appears in the papers [K.Y. Abalo, M.M. Kostreva, Berge equilibrium: Some recent results from fixed-point theorems, Appl. Math. Comput. 169 (2005) 624–638; K.Y. Abalo, M.M. Kostreva, Some existence theorems of Nash and Berge equilibria, Appl. Math. Lett. 17 (2004) 569–573]. We found that the assumptions of these theorems are not sufficient for the existence of Berge equilibrium. Indeed, we construct a game that verifies Abalo and Kostreva’s assumptions, but has no Berge equilibrium. Then we provide a condition that overcomes the problem in these theorems. Our conclusion is also valid for Radjef’s theorem, which is the basic reference for [K.Y. Abalo, M.M. Kostreva, Berge equilibrium: Some recent results from fixed-point theorems, Appl. Math. Comput. 169 (2005) 624–638; K.Y. Abalo, M.M. Kostreva, Some existence theorems of Nash and Berge equilibria, Appl. Math. Lett. 17 (2004) 569–573; K.Y. Abalo, M.M. Kostreva, Fixed points, Nash games and their organizations, Topol. Methods Nonlinear Anal. 8 (1996) 205–215; K.Y. Abalo, M.M. Kostreva, Equi-well-posed games, J. Optim. Theory Appl. 89 (1996) 89–99].     

A note on vector network equilibrium principles

An example is given to show that the necessary conditions of Theorem 4.5 [in Chen et al. Math Methods Oper Res 49:239–253, 1999] and Theorem 2.1 (i) [in Goh and Yang Eur J Oper Res 116:615–628, 1999] for (weak) vector equilibrium flows may not hold. New ξ-equilibrium and parametric equilibrium flows are introduced. As a result, necessary and sufficient conditions between a weak vector equilibrium flow and an ξ-equilibrium flow and between a vector equilibrium flow and a parametric equilibrium flow are established.     

A note on equilibrium points of Green's function

We answer a question raised by Ahmet Sebbar and Thérèse Falliero (2007) by showing that for every finitely connected planar domain there exists a compact subset , independent of , containing all critical points of Green's function of with pole at .

     

A note on economic equilibrium and financial networks

Nagurney （1999） used variational inequalities to study economic equilibrium and financial networks and applied the modified projection method to solve the problem. In this paper, we formulate the problem as a nonlinear complementarity problem. The complementarity model is just the KKT condition for the model of Nagurney （1999）. It is a simpler model than that of Nagurney （1999）. We also establish sufficient conditions for existence and uniqueness of the equilibrium pattern, which are weaker than those in Nagurney （]999）. Finally, we apply a smoothing Newton-type algorithm to solve the problem and report some numerical results.     

A note on duality of generalized equilibrium problem

The main aim of this note is to extend the dual (in the form of an optimization problem) given for equilibrium problem by Martinez-Legaz and Sosa (in J Glob Optim 35:311–319, 2006) for a generalized equilibrium problem in finite dimensional setting and to establish its equivalence with the dual derived by Bigi et al. (in J Math Anal Appl 342:17–26, 2008) (in the form of inclusion conditions) under a mild condition.     

A note on stability for parametric equilibrium problems

In this paper we consider equilibrium problems in vector metric spaces where the function f and the set K are perturbed by the parameters ε,η. We study the stability of the solutions, providing some results in the peculiar framework of generalized monotone functions, first in the particular case where K is fixed, then under both data perturbation.     

A generalization of correlated equilibrium: A new protocol

A new correlation scheme (leading to a special equilibrium called “soft” correlated equilibrium) is introduced for finite games. After randomization over the outcome space, players have the choice either to follow the recommendation of an umpire blindly or freely choose some other action except the one suggested. This scheme can lead to Pareto-better outcomes than the simple extension introduced by [Moulin, H., Vial, J.-P., 1978. Strategically zero-sum games: the class of games whose completely mixed equilibria cannot be improved upon. International Journal of Game Theory 7, 201–221]. The informational and interpretational aspects of soft correlated equilibria are also discussed in detail. The power of the generalization is illustrated in the prisoners’s dilemma and a congestion game.     

A note on equilibrium of eventually strongly monotone skew-product semiflows

In this paper, we show that for an eventually strongly monotone skew-product semiflow τ, the strict ordering on Ec （the set consisting of continuous equilibria of τ） implies the strong one.     

A note on existence of weak efficient solutions for vector equilibrium problems

This work concerns with vector equilibrium problems where the image space of the bifunction is not endowed with any topology. To be precise, a kind of “semi-algebraic” upper semicontinuity notion is introduced and, by means of a recent algebraic version of the so-called Gerstewitz’s functional, a new existence result of weak efficient solutions is obtained that significantly improves some previous ones stated in the topological setting since it requires weaker assumptions.     

A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems

By using a new assumption, the lower semicontinuity of the efficient solution mappings for parametric vector equilibrium problems without a monotonicity assumption is established. Our result is different from recent ones in the literature. Some examples are given to illustrate the case.     

A note on immunization under a general stochastic equilibrium model of the term structure

The paper investigates the actuarial immunization problem under recent stochastic equilibrium models for the term structure of interest rates developed in the theory of finance, thereby generalizing results of Boyle (1978, 1980).     

Selection of a correlated equilibrium in Markov stopping games

This paper deals with an extension of the concept of correlated strategies to Markov stopping games. The Nash equilibrium approach to solving nonzero-sum stopping games may give multiple solutions. An arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players’ decisions according to some optimality criterion. This is a form of equilibrium selection. Examples of correlated equilibria in nonzero-sum games related to the best choice problem are given. Several concepts of criteria for selecting a correlated equilibrium are used.     

A detailed note on the finite-buffer queueing system with correlated batch-arrivals and batch-size-/phase-dependent bulk-service

4OR - This paper analyzes a finite-buffer queueing system, where customers arrive in batches and the accepted customers are served in batches by a single server. The service is assumed to be...     

Evolutionary dynamics may eliminate all strategies used in correlated equilibrium

We show on a 4×4 example that many dynamics may eliminate all strategies used in correlated equilibria, and this for an open set of games. This holds for the best-response dynamics, the Brown–von Neumann–Nash dynamics and any monotonic or weakly sign-preserving dynamics satisfying some standard regularity conditions. For the replicator dynamics and the best-response dynamics, elimination of all strategies used in correlated equilibrium is shown to be robust to the addition of mixed strategies as new pure strategies.     

Implementability of correlated and communication equilibrium outcomes in incomplete information games

In a correlated equilibrium, the players’ choice of actions is directed by correlated random messages received from an outside source, or mechanism. These messages allow for more equilibrium outcomes than without any messages (pure-strategy equilibrium) or with statistically independent ones (mixed-strategy equilibrium). In an incomplete information game, the messages may also reflect the types of the players, either because they are affected by extraneous factors that also affect the types (correlated equilibrium) or because the players themselves report their types to the mechanism (communication equilibrium). Mechanisms may be further differentiated by the connections between the messages that the players receive and their own and the other players’ types, by whether the messages are statistically dependent or independent, and by whether they are random or deterministic. Consequently, whereas for complete information games there are only three classes of equilibrium outcomes, with incomplete information the corresponding number is 14 or 15 for correlated equilibria and even larger—15, 16 or 17—for communication equilibria. For both solution concepts, the implication relations between the different kinds of equilibria form a two-dimensional lattice, which is considerably more intricate than the single-dimensional one of the complete information case.     

A note on pseudocompactifications

There exists a countable spaceV _ω such that:

1. V _ω has a single non-isolated point,
2. V _ω has no pseudocompactificationX witht(X)=ω.

     

A note on rank-additivity

A new criterion is given for rank additivity of a collection of m × n complex matrices.     

A note on P-convexity

In this paper we study and characterize the structure of the linear transformations which preserve P-convex sets.