Flexibility in a Stackelberg leadership with differentiated goods

We study the effects of product differentiation in a Stackelberg model with demand uncertainty for the first mover. We do an ex-ante and ex-post analysis of the profits of the leader and of the follower firms in terms of product differentiation and of the demand uncertainty. We show that even with small uncertainty about the demand, the follower firm can achieve greater profits than the leader, if their products are sufficiently differentiated. We also compute the probability of the second firm having higher profit than the leading firm, subsequently showing the advantages and disadvantages of being either the leader or the follower firm.     

Duality for generalized equilibrium problem

We introduce a generalized equilibrium problem (GEP) that allow us to develop a robust dual scheme for this problem, based on the theory of conjugate functions. We obtain a unified dual analysis for interesting problems. Indeed, the Lagrangian duality for convex optimization is a particular case of our dual problem. We establish necessary and sufficient optimality conditions for GEP that become a well-known theorem given by Mosco and the dual results obtained by Morgan and Romaniello, which extend those introduced by Auslender and Teboulle for a variational inequality problem.     

A best-response approach for equilibrium selection in two-player generalized Nash equilibrium problems

ABSTRACT

In this paper, we propose a best-response approach to select an equilibrium in a two-player generalized Nash equilibrium problem. In our model we solve, at each of a finite number of time steps, two independent optimization problems. We prove that convergence of our Jacobi-type method, for the number of time steps going to infinity, implies the selection of the same equilibrium as in a recently introduced continuous equilibrium selection theory. Thus the presented approach is a different motivation for the existing equilibrium selection theory, and it can also be seen as a numerical method. We show convergence of our numerical scheme for some special cases of generalized Nash equilibrium problems with linear constraints and linear or quadratic cost functions.     

Existence and generic stability of solutions for multiclass multicriteria traffic equilibrium problems with capacity constraints of arcs

ABSTRACT

In this paper, we introduce the multiclass multicriteria traffic equilibrium problem with capacity constraints of arcs and its equilibrium principle. Using Fan–Browder's fixed points theorem and Fort's lemma to prove the existence and generic stability results of multiclass multicriteria traffic equilibrium flows with capacity constraints of arcs.     

Split equilibrium problems for related games and applications to economic theory

ABSTRACT

In this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan-KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As an application of this existence to economics, an example is provided that studies the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual (playing twice) extended Bertrand duopoly model of price competition.     

Smoothing method in multi-criteria transportation network equilibrium problem

ABSTRACT

This paper is aimed to study a single-product multi-criteria transportation network with capacity constraints. We use a vector version of the Heaviside Step function to construct an optimization problem, the solutions of which form the set of equilibria of our model. We propose two methods to solve this problem. The first one is based on a modified Frank-Wolfe gradient algorithm, and the second one is based on smoothing the objective function, the optimal solutions of which can be obtained by optimization tools. Numerical examples are also given to illustrate our approaches.     

Some kind of Pareto stationarity for multiobjective problems with equilibrium constraints

ABSTRACT

In this paper, we derive some optimality and stationarity conditions for a multiobjective problem with equilibrium constraints (MOPEC). In particular, under a generalized Guignard constraint qualification, we show that any locally Pareto optimal solution of MOPEC must satisfy the strong Pareto Kuhn-Tucker optimality conditions. We also prove that the generalized Guignard constraint qualification is the weakest constraint qualification for the strong Pareto Kuhn-Tucker optimality. Furthermore, under certain convexity or generalized convexity assumptions, we show that the strong Pareto Kuhn-Tucker optimality conditions are also sufficient for several popular locally Pareto-type optimality conditions for MOPEC.     

带平衡约束的离散网络平衡设计问题的遗传算法

本文研究了带平衡约束的离散网络设计问题及其求解算法.模型中上层是一个离散网络设计的数学规划模型,采用遗传算法来求解.下层是采用变分不等式描述的用户平衡配流问题,利用对角化方法直接求解.通过实例对算法进行验证,结果表明该算法是有效的.     

Erratum to “Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey” [Appl. Math. Modell. (2011) 366-381]

The aim of this paper is to correct two mistakes in [Appl. Math. Modell. (2011) 366-381], which are: the function defining the time rescaling given and the inclusion of a parameter outside of model.For a modified Leslie-Gower type predator-prey model considering the Allee effect on prey, a change of variables and a new time rescaling generating a diffeomorphism is proved; a topologically equivalent system to the original one is obtained, which is the same studied in the mentioned paper; we claim that the results and conclusions obtained are correct and the errors have not further implications.     

Stochastically perturbed vector-borne disease models with direct transmission

In this paper we study a stochastic epidemic model of vector-borne diseases with direct mode of transmission and its delay modification. More precisely, we extend the deterministic epidemic models by introducing random perturbations around the endemic equilibrium state. By using suitable Lyapunov functions and functionals, we obtain stability conditions for the considered models and study the effect of the delay on the stability of the endemic equilibrium. Finally, numerical simulations for the stochastic model of malaria disease transmission are presented to illustrate our mathematical findings.