A two stage stochastic equilibrium model for electricity markets with two way contracts

This paper investigates generators’ strategic behaviors in contract signing in the forward market and power transaction in the electricity spot market. A stochastic equilibrium program with equilibrium constraints (SEPEC) model is proposed to characterize the interaction of generators’ competition in the two markets. The model is an extension of a similar model proposed by Gans et al. (Aust J Manage 23:83–96, 1998) for a duopoly market to an oligopoly market. The main results of the paper concern the structure of a Nash–Cournot equilibrium in the forward-spot market: first, we develop a result on the existence and uniqueness of the equilibrium in the spot market for every demand scenario. Then, we show the monotonicity and convexity of each generator’s dispatch quantity in the spot equilibrium by taking it as a function of the forward contracts. Finally, we establish some sufficient conditions for the existence of a local and global Nash equilibrium in the forward-spot markets. Numerical experiments are carried out to illustrate how the proposed SEPEC model can be used to analyze interactions of the markets.     

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.     

Logit equilibrium as an approximation of Nash equilibrium

We prove that the graph of the logit equilibrium correspondence is a smooth manifold, which is homeomorphic to the space of payoff functions and uniformly approximates the graph of the Nash equilibrium manifold.     

Cournot–Walras equilibrium as a subgame perfect equilibrium

In this paper, we investigate the problem of the strategic foundation of the Cournot–Walras equilibrium approach. To this end, we respecify à la Cournot–Walras the mixed version of a model of simultaneous, noncooperative exchange, originally proposed by Lloyd S. Shapley. We show, through an example, that the set of the Cournot–Walras equilibrium allocations of this respecification does not coincide with the set of the Cournot–Nash equilibrium allocations of the mixed version of the original Shapley’s model. As the nonequivalence, in a one-stage setting, can be explained by the intrinsic two-stage nature of the Cournot–Walras equilibrium concept, we are led to consider a further reformulation of the Shapley’s model as a two-stage game, where the atoms move in the first stage and the atomless sector moves in the second stage. Our main result shows that the set of the Cournot–Walras equilibrium allocations coincides with a specific set of subgame perfect equilibrium allocations of this two-stage game, which we call the set of the Pseudo–Markov perfect equilibrium allocations. We would like to thank Pierpaolo Battigalli, Marcellino Gaudenzi, and an anonymous referee for their comments and suggestions.     

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.     

不确定条件下n人非合作博弈均衡点集的通有稳定性

基于经典博弈模型的Nash均衡点集的通有稳定性和具有不确定参数的n人非合作博弈均衡点的概念,探讨了具有不确定参数博弈的均衡点集的通有稳定性.参照Nash均衡点集稳定性的统一模式,构造了不确定博弈的问题空间和解空间,并证明了问题空间是一个完备度量空间,解映射是上半连续的,且解集是紧集（即usco（upper semicontinuous and compact-valued）映射）,得到不确定参数博弈模型的解集通有稳定性的相关结论.     

Imitation,coordination and the emergence of Nash equilibrium

We consider a learning dynamic in which players imitate and better reply. Sufficient conditions are provided for Nash equilibrium play to emerge over time. The role of imitation in the learning dynamic is discussed through a series of examples. Most interestingly we demonstrate how imitation can ‘help’ the emergence of Nash equilibrium where ‘more rational’ methods do not.     

Multiclass multicriteria mixed equilibrium on networks and uniform link tolls for system optimum

In this paper, we consider a unified framework of multiclass multicriteria mixed equilibrium, and the existence of uniform link tolls supporting such a mixed equilibrium as a system optimum. The network users are divided into different classes, and each class of traveler perceives his/her disutility associated with a route as a combination of two criteria given, respectively, by the travel time disutility and the time-irrelevant travel disutility. And users in a common class follow either user equilibrium (UE) principle or Cournot–Nash (CN) principle. A variational inequality model characterizing the multiclass multicriteria UE–CN mixed equilibrium behavior is developed. By utilizing the dual theory, we establish the existence of uniform link tolls supporting such mixed equilibrium as a system optimum.     

An equilibrium model for urban transit assignment based on game theory

The urban public transport system is portrayed as a special commodity market where passenger is consumer, transit operator is producer and the special goods is the service for passenger’s trip. The generalized Nash equilibrium game is applied to describe how passengers adjust their route choices and trip modes. We present a market equilibrium model for urban public transport system as a series of mathematical programmings and equations, which is to describe both the competitions among different transit operators and the interactive influences among passengers. The proposed model can simultaneously predict how passengers choose their optimal routes and trip modes. An algorithm is designed to obtain the equilibrium solution. Finally, a simple numerical example is given and some conclusions are drawn.     

不动点与平衡点

介绍Brouwer不动点定理、Kakutani不动点定理与数理经济学中平衡点和博弈论中Nash平衡点存在性定理的等价性结果.     

Discriminatory power and pure strategy Nash equilibrium in the lottery Blotto game

This paper highlights the role of discriminatory power in showing the existence of pure strategy Nash equilibrium in a two-player lottery Blotto game. Precisely, when the discriminatory power of the contest success function in every battle is less than or equal to one, a pure strategy Nash equilibrium always exists for any prizes and budgets. However, we construct examples in which pure strategy Nash equilibrium fails to exist as long as the discriminatory power of only one battle exceeds one.     

Constrained equilibrium in a bidding problem

There are a number of situations where firms vie for some constrained total quantity by bidding individual quantities but where the allocation is determined on a ‘pro-rated’ basis. One such example is a licenses-on-demand method of import quota allocation. Bidders are constrained to bid at most the total quantity, and if the sum of all bids exceeds this total, the awards are granted proportional to the fractional amounts that the bidders request. In this note, we identify the (Nash) equilibrium of the model, present some examples, and do a sensitivity analysis of the bids as the total amount available changes. We also look into changes of the bids when a new participant enters the bidding process. In an economic context, such allocation methods are shown to be biased in favor of the least efficient firms.     

Generalized Nash equilibrium problem, variational inequality and quasiconvexity

It is well known that the generalized Nash equilibrium problem, a model for multi-leader–follower games, can be reformulated as a quasivariational inequality. We show that, in fact, a reformulation in terms of a variational inequality can be obtained in the general setting of quasiconvex nondifferentiable decision functions. An existence result is deduced.     

A Nash equilibrium filter

The Wonham filter, which estimates a Markov chain observed in Brownian noise, is considered. However, the parameters of the observation process are not known. Maximizing the un-normalized probabilities of the Zakai equation over the parameters leads to a Nash equilibrium whose solution is discussed using the stochastic control results of Peng and Yong and Zhou.     

On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables

ABSTRACT

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branch-and-bound methods based on merit functions for the mixed-integer game, and branch-and-prune methods that exploit the concept of dominance to make effective cuts. We show that under mild assumptions the equilibrium set of the game is finite and we define an enumerative method to compute the whole of it. We show that our branch-and-prune method can be suitably modified in order to make a general equilibrium selection over the solution set of the mixed-integer game. We define an application in economics that can be modelled as a Nash game with linear coupling constraints and mixed-integer variables, and we adapt the branch-and-prune method to efficiently solve it.     

Fall back equilibrium

Fall back equilibrium is a refinement of the Nash equilibrium concept. In the underlying thought experiment each player faces the possibility that, after all players decided on their action, his chosen action turns out to be blocked. Therefore, each player has to decide beforehand on a back-up action, which he plays in case he is unable to play his primary action.     

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.     

Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game

Existence and uniqueness of a Nash equilibrium feedback is established for a simple class nonzero-sum differential games on the line.     

Searching for equilibrium positions in a game of political competition with restrictions

This paper considers a problem of political economy in which a Nash equilibrium study is performed in a proposed game with restrictions where the two major parties in a country vary their position within a politically flexible framework to increase their number of voters. The model as presented fits the reality of many countries. Moreover, it avoids the uniqueness of equilibrium positions. The problem is stated and solved from a geometric point of view.     

Maximal element with applications to Nash equilibrium problems in Hadamard manifolds

ABSTRACT

The purpose of this paper is to study the existence of maximal elements with applications to Nash equilibrium problems for generalized games in Hadamard manifolds. By employing a KKM lemma, we establish a new maximal element theorem in Hadamard manifolds. As applications, some existence results of Nash equilibria for generalized games are derived. The results in this paper unify, improve and extend some known results from the literature.