Pure strategy Nash equilibria and the probabilistic prospects of Stackelberg players

We consider the set of all m×n bimatrix games with ordinal payoffs. We show that on the subset E of such games possessing at least one pure strategy Nash equilibrium, both players prefer the role of leader to that of follower in the corresponding Stackelberg games. This preference is in the sense of first-degree stochastic dominance by leader payoffs of follower payoffs. It follows easily that on the complement of E, the follower’s role is preferred in the same sense. Thus we see a tendency for leadership preference to obtain in the presence of multiple pure strategy Nash equilibria in the underlying game.     

Inequalities and Nash equilibria

This paper concentrates on the problem of the existence of equilibrium points for non-cooperative generalized N-person games, N-person games of normal form and their related inequalities. We utilize the K-K-M lemma to obtain a theorem and then use it to obtain a new Fan-type inequality and minimax theorems. Various new equilibrium point theorems are derived, with the necessary and sufficient conditions and with strategy spaces with no fixed point property. Examples are given to demonstrate that these existence theorems cover areas where other existence theorems break down.     

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.     

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.     

On stationary equilibria of a single-controller stochastic game

We consider a two-person, general-sum, rational-data, undiscounted stochastic game in which one player (player II) controls the transition probabilities. We show that the set of stationary equilibrium points is the union of a finite number of sets such that, every element of each of these sets can be constructed from a finite number of extreme equilibrium strategies for player I and from a finite number of pseudo-extreme equilibrium strategies for player II. These extreme and pseudo-extreme strategies can themselves be constructed by finite (but inefficient) algorithms. Analogous results can also be established in the more straightforward case of discounted single-controller games.     

On existence of undominated pure strategy Nash equilibria in anonymous nonatomic games: a generalization

In this paper, we generalize the exitence result for pure strategy Nash equilibria in anonymous nonatomic games. By working directly on integrals of pure strategies, we also generalize, for the same class of games, the existence result for undominated pure strategy Nash equilibria even though, in general, the set of pure strategy Nash equilibria may fail to be weakly compact. Received August 2001     

A two-population game in observable double-ended queuing systems

In this study, the existing game theoretical framework is extended to strategic queuing in search of solutions for a two-population game in observable double-ended queuing systems with zero matching times. We show that multiple Nash equilibria and one unique subgame perfect Nash equilibrium exist in this game.     

Definition and properties of cooperative equilibria in a two-player game of infinite duration

A two-player multistage game, with an infinite number of stages is considered. The concepts of overtaking and weakly overtaking payoff sequences are introduced. The class of strategies considered consists of memory strategies, which are based on the past history of the control and the initial state from where the game has been played. Weak equilibria are defined in this class of strategies. It is then shown how such equilibria can be constructed by composing into a trigger strategy a nominal cooperative control sequence and two threat strategies representing the announced retaliation by each player in the case where the other player does not play according to the nominal control. When the threats consists of a feedback equilibrium pair, the resulting cooperative equilibrium is perfect. Another result shows that, if each player can use a most effective threat based on a saddle-point feedback strategy, then any weak equilibrium in the class of memory strategies is in some sense related to this particular kind of equilibrium in the class of trigger strategies.Dedicated to G. LeitmannThis research was supported by SSHRC Grant No. 410-81-0722 and FCAC Grant No. EQ-428 to the first author. This research has also been made possible by a financial support from the University of Puerto Rico.     

A differential game model of a market of substitutable products

Modern industrial organization often classifies groups of differentiated products that are fairly good substitutes to belong to the same market. This paper develops a differential game model of a market of substitutable products. To avoid the problem of time-inconsistency, we solve a feedback Nash equilibrium solution for the game. A set of state-dependent equilibrium strategies is derived. Extensions of the model to a stochastic formulation and to an infinite time horizon specification are also provided.     

一类博弈排序问题的纳什均衡存在性证明

研究机器带有激活费用的博弈排序问题. 机器集由两类组成: 一类是速度为1、 激活费用为B的k_1台同型机; 另一类是速度为a(>1)、激活费用为aB的k_2台同型机, 其中k_1与k_2是任意正整数. 工件作为``局中人", 其目的是极小化自身的费用, 工件的费用是由其所在机器的负载和其所承担的激活费用组成, 其中工件承担的激活费用与工件的加工时间成正比. 针对不同的情况, 设计不同的算法, 并证明各算法得到的排序都是纳什均衡.     

Nash equilibrium in a pay-as-bid electricity market Part 2 - best response of a producer

We consider a multi-leader-common-follower model of a pay-as-bid electricity market in which the producers provide the regulator with either linear or quadratic bids. We prove that for a given producer only linear bids can maximize his profit. Such linear bids are referred as the ‘best response’ of the given producer. They are obtained assuming the demand is known and some estimate of the bids of the other producers is available. Nevertheless we also show that whenever no best response exists, the optimal profit can be asymptotically attained by a sequence of quadratic bids converging to the so-called ‘limiting best response’. An explicit formula for such a sequence is provided.     

A model of a 2-player stopping game with priority and asynchronous observation

Various models of 2-player stopping games have been considered which assume that players simultaneously observe a sequence of objects. Nash equilibria for such games can be found by first solving the optimal stopping problems arising when one player remains and then defining by recursion the normal form of the game played at each stage when both players are still searching (a 2 × 2 matrix game). The model considered here assumes that Player 1 always observes an object before Player 2. If Player 1 accepts the object, then Player 2 does not see that object. If Player 1 rejects an object, then Player 2 observes it and may choose to accept or reject it. It is shown that such a game can be solved using recursion by solving appropriately defined subgames, which are played at each moment when both players are still searching. In these subgames Player 1 chooses a threshold, such that an object is accepted iff its value is above this threshold. The strategy of Player 2 in this subgame is a stopping rule to be used when Player 1 accepts this object, together with a threshold to be used when Player 1 rejects the object. Whenever the payoff of Player 1 does not depend on the value of the object taken by Player 2, such a game can be treated as two optimisation problems. Two examples are given to illustrate these approaches.     

On a game in manufacturing

We analyse a non-zero sum two-person game introduced by Teraoka and Yamada to model the strategic aspects of production development in manufacturing. In particular we investigate how sensitive their solution concept (Nash equilibrium) is to small variations in their assumptions. It is proved that a Nash equilibrium is unique if it exists and that a Nash equilibrium exists when the capital costs of the players are zero or when the players are equal in every respect. However, when the capital costs differ, in general a Nash equilibrium exists only when the players' capital costs are high compared to their profit rates.     

Stochastic differential game model of a common property fishery

The paper presents a stochastic differential game model of a common-property commercial fishery and determines a feedback Nash equilibrium of the game. Closed-form expressions for the value functions, the equilibrium harvesting strategies, and stationary distributions of the fish stock are derived. Sensitivity analyses with respect tot he model parameters are carried out. The paper also considers equilibrium outcomes under joint maximization and surplus maximization. In the latter case, an optimal market size (i.e., number of firms) is identified.The research of the first author was supported in part by Grant 5.20.31.02 from the Danish Research Council for the Social Sciences. The research of the second author was supported in part by the CRGC Fund of the University of Hong Kong. The authors would like to thank the editor for constructive comments.     

Bilevel optimization applied to strategic pricing in competitive electricity markets

In this paper, we present a bilevel programming formulation for the problem of strategic bidding under uncertainty in a wholesale energy market (WEM), where the economic remuneration of each generator depends on the ability of its own management to submit price and quantity bids. The leader of the bilevel problem consists of one among a group of competing generators and the follower is the electric system operator. The capability of the agent represented by the leader to affect the market price is considered by the model. We propose two solution approaches for this non-convex problem. The first one is a heuristic procedure whose efficiency is confirmed through comparisons with the optimal solutions for some instances of the problem. These optimal solutions are obtained by the second approach proposed, which consists of a mixed integer reformulation of the bilevel model. The heuristic proposed is also compared to standard solvers for nonlinearly constrained optimization problems. The application of the procedures is illustrated in case studies with configurations derived from the Brazilian power system.     

Monitoring cooperative equilibria in a stochastic differential game

This paper deals with a class of equilibria which are based on the use of memory strategies in the context of continuous-time stochastic differential games. In order to get interpretable results, we will focus the study on a stochastic differential game model of the exploitation of one species of fish by two competing fisheries. We explore the possibility of defining a so-called cooperative equilibrium, which will implement a fishing agreement. In order to obtain that equilibrium, one defines a monitoring variable and an associated retaliation scheme. Depending on the value of the monitoring variable, which provides some evidence of a deviation from the agreement, the probability increases that the mode of a game will change from a cooperative to a punitive one. Both the monitoring variable and the parameters of this jump process are design elements of the cooperative equilibrium. A cooperative equilibrium designed in this way is a solution concept for a switching diffusion game. We solve that game using the sufficient conditions for a feedback equilibrium which are given by a set of coupled HJB equations. A numerical analysis, approximating the solution of the HJB equations through an associated Markov game, enables us to show that there exist cooperative equilibria which dominate the classical feedback Nash equilibrium of the original diffusion game model.This research was supported by FNRS-Switzerland, NSERC-Canada, FCAR-Quebec.     

Finding a Nash equilibrium in noncooperativeN-person games by solving a sequence of linear stationary point problems

In this paper we present an algorithm for finding a Nash equilibrium in a noncooperative normal formN-person game. More generally, the algorithm can be applied for solving a nonlinear stationary point problem on a simplotope, being the Cartesian product of several simplices. The algorithm solves the problem by solving a sequence of linear stationary point problems. Each problem in the sequence is solved in a finite number of iterations. Although the overall convergence cannot be proved, the method performs rather well. Computational results suggest that this algorithm performs at least as good as simplicial algorithms do.For the special case of a bi-matrix game (N=2), the algorithm has an appealing game-theoretic interpretation. In that case, the problem is linear and the algorithm always finds a solution. Furthermore, the equilibrium found in a bi-matrix game is perfect whenever the algorithm starts from a strategy vector at which all actions are played with positive probability.This research is part of the VF-program Co-operation and Competition, which has been approved by the Netherlands Ministery of Education and Sciences.     

Finding all pure strategy Nash equilibria in a planar location game

In this paper, we deal with a planar location-price game where firms first select their locations and then set delivered prices in order to maximize their profits. If firms set the equilibrium prices in the second stage, the game is reduced to a location game for which pure strategy Nash equilibria are studied assuming that the marginal delivered cost is proportional to the distance between the customer and the facility from which it is served. We present characterizations of local and global Nash equilibria. Then an algorithm is shown in order to find all possible Nash equilibrium pairs of locations. The minimization of the social cost leads to a Nash equilibrium. An example shows that there may exist multiple Nash equilibria which are not minimizers of the social cost.     

On-line solution of a stochastic pursuit-evasion game

In this paper, the concepts of prior-commitment and delayed-commitment strategies for zero-sum, linear-quadratic differential games with noise-corrupted measurements are applied to a detailed example of a pursuit-evasion game. Quasilinearization is used to solve the nonlinear two-point boundary-value problem of the prior-commitment game. A closed form solution is obtained for the delayed-commitment strategies. Comparison of the payoff functionals confirms the relationships discussed in Ref. 1.