Optimal threat strategies in collective bargaining

A model of collective bargaining in differential games, similar to one considered by Leitmann (Ref. 1), is proposed. Optimal threat strategies are then found and compared with Leitmann's solution. Nash-optimal threat strategies are also found and compared with the other two solutions.The author would like to thank Professor G. Leitmann of the University of California for bringing the author's attention to this subject.     

Sufficient conditions for Stackelberg and Nash strategies with memory

Sufficiency conditions for Stackelberg strategies for a class of deterministic differential games are derived when the players have recall of the previous trajectory. Sufficient conditions for Nash strategies when the players have recall of the trajectory are also derived. The state equation is linear, and the cost functional is quadratic. The admissible strategies are restricted to be affine in the information available.This work was supported in part by the Joint Services Electronics Program under Contract No. N00014-79-C-0424, in part by the National Science Foundation under Grant No. ECS-79-19396, and in part by Department of Energy under Contract No. EX-76-C-01-2088.     

Stackelberg strategies in linear-quadratic stochastic differential games

This paper obtains the Stackelberg solution to a class of two-player stochastic differential games described by linear state dynamics and quadratic objective functionals. The information structure of the problem is such that the players make independent noisy measurements of the initial state and are permitted to utilize only this information in constructing their controls. Furthermore, by the very nature of the Stackelberg solution concept, one of the players is assumed to know, in advance, the strategy of the other player (the leader). For this class of problems, we first establish existence and uniqueness of the Stackelberg solution and then relate the derivation of the leader's Stackelberg solution to the optimal solution of a nonstandard stochastic control problem. This stochastic control problem is solved in a more general context, and its solution is utilized in constructing the Stackelberg strategy of the leader. For the special case Gaussian statistics, it is shown that this optimal strategy is affine in observation of the leader. The paper also discusses numerical aspects of the Stackelberg solution under general statistics and develops algorithms which converge to the unique Stackelberg solution.This work was performed while the second author was on sabbatical leave at the Department of Applied Mathematics, Twente University of Technology, Enschede, Holland.     

Sufficient conditions for Nash bargaining in differential games

Sufficient conditions for Nash bargaining in differential games are given. These conditions are compared with the sufficient conditions given by Liu (Ref. 1).     

A differential game with two pursuers and one evader

This paper is concerned with a coplanar pursuit-evasion problem in which a faster evaderE with constant speedw>1 must pass between two pursuersP ₁,P ₂ having unit speed, the payoff being the distance of closest approach to either one of the pursuers. The control variables are the directions of the velocities ofP ₁,P ₂, andE. The path equations are integrated, and a closed-form solution is obtained in terms of elliptic functions of the first and second kind. A closed-loop solution is given graphically in several diagrams, for different values ofw.     

Sufficient conditions for preference optimality

Preference optimality is an optimality concept in multicriteria problems, that is, in problems where several criteria are to beoptimized simultaneously. Formally, one wishes to optimizeN criteriag _i(·) or, equivalently, a criterion vectorg(·) ^N , subject to either functional constraints in programming or to side conditions which are differential equations in optimal control. Subject to these constraints, one obtains forg(·) a set of attainable values in ^N . This set is preordered by the introduction of a complete preordering ; a controlu*(·) or a decisionx*, then, is preference-optimal if it results ing(u*(·))g(u(·)) for all admissible controlsu(·) or ifg(x*)g(x) for all feasible decisionsx. The present paper concerns sufficient conditions for preference-optimal control and for preference-optimal decisions.     

A new approach toN-person,nonzero-sum,linear differential games

It is shown that there exist equilibrium strategies forn-person, nonero-sum, linear differential games if the cost to each player is convex. The approach used is believed to be novel, and is based on a theorem of Fan.This research was supported by the National Research Council of Canada under Grant No. A-7790.     

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.     

Some thoughts on saddle-point conditions and information structures in zero-sum differential games

For a very simple two-stage, linear-quadratic, zero-sum difference game with dynamic information structure, we show that (i) there exist nonlinear saddle-point strategies which require the same existence conditions as the well-known linear, closed-loop, no-memory solution and (ii) there exist both linear and nonlinear saddle-point strategies which require more stringent conditions than the unique open-loop solution. We then discuss the implication of this result with respect to the existence of saddle points in zero-sum differential games for different information patterns.     

On generalized Stackelberg strategies

The concept of Stackelberg strategy for a nonzero-sum two-person game is extended to allow for a nonunique rational response of the follower. This leads to the notion of a generalized Stackelberg strategy for the leader, which guarantees him a cost value that cannot be exceeded, no matter what the rational response of the follower. Then, a generalized Stackelberg strategy pair is defined. A simple example is given. The idea of a generalized Stackelberg strategy and strategy pair is then applied to the situation of one leader and many rational followers.This paper is based on research supported by the National Science Foundation.     

On a linear pursuit game with an unknown trap

A linear pursuit game with a trap, the location of which is unknown to the evader, is defined and investigated. The cases in which one of the players has complete energy dominance over his adversary are solved completely. In the general case, when no player dominates, the solution is indicated for the two-stage game.This research was supported in part by the Technion Fund for promotion of research.     

Parental care as a differential game: A dynamic extension of the Houston–Davies game

An interpretation of the conflict between male and female parents during the process of caring for their common offspring by means of Game Theory was given in Houston and Davies. [A.I. Houston, N.B. Davies, The evolution of cooperation and life history in the dunnock Prunella modularis, in: R.M. Sibly, R.H. Smith (Eds.), Behavioral Ecology, Blackwell Scientific Publications, 1985, pp. 471–487]. Mathematically, this model represents a static game with continuous strategy sets. Recently, this model was reconsidered in a dynamic discrete time framework which also included state dependencies [J.M. McNamara et al., A dynamic game-theoretic model of parental care, J. Theor. Biol. 205 (2000) 605–623]. In this article, we give an interpretation of the parental care conflict in continuous time by means of a differential game with state dependent strategies.     

Solution concepts in continuous-kernel multicriteria games

This paper considers nonzero-sum multicriteria games with continuous kernels. Solution concepts based on the notions of Pareto optimality, equilibrium, and security are extended to these games. Separate necessary and sufficient conditions and existence results are presented for equilibrium, Pareto-optimal response, and Pareto-optimal security strategies of the players.This paper is based partially on research supported by the Council of Scientific and Industrial Research, India, through a Research Associateship Grant to the first author.The authors are grateful to two anonymous referees for suggesting useful changes and pointing out some errors in a previous draft.     

Sufficient optimality conditions and duality in vector optimization with invex-convexlike functions

We prove the Kuhn-Tucker sufficient optimality condition, the Wolfe duality, and a modified Mond-Weir duality for vector optimization problems involving various types of invex-convexlike functions. The class of such functins contains many known generalized convex functions. As applications, we demonstrate that, under invex-convexlikeness assumptions, the Pontryagin maximum principle is a sufficient optimality condition for cooperative differential games. The Wolfe duality is established for these games.The author is indebted to the referees and Professor W. Stadler for valuable remarks and comments, which have been used to revise considerably the paper.     

The Leitmann-Schmitendorf advertising differential game

The paper revisits the advertising differential game suggested by Leitmann and Schmitendorf [1]. We put the model into perspective and discuss the related issues of (i) state separable (or linear-state) games and (ii) open-loop equilibria that are feedback equilibria which are constant with respect to state.     

On the uniqueness of Nash strategies for a class of analytic differential games

The uniqueness of Nash equilibria is shown for the case where the data of the problem are analytic functions and the admissible strategy spaces are restricted to analytic functions of the current state and time.This work was supported in part by the Joint Services Electronics Program (US Army, US Navy, and US Air Force) under Contract No. DAAB-07-72-C-0259, in part by the National Science Foundation under Grant No. ENG-74-20091, and in part by the Department of Energy, Electric Energy Systems Division under Contract No. US ERDA EX-76-C-01-2088.     

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.     

A note on nonzero-sum differential games with bargaining solution

The extension of Nash's bargaining solution to differential games is discussed. It is shown that a closed-loop solution verifies very stringent necessary conditions and that an open-loop solution can present serious weakness from a normative point of view.This research has been supported by the Canada Council (S73-0935) and the Ministère de l'Education du Québec (DGES).     

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.     

An extension of Jørgensen's differential game

This note extends Jørgensen's exponential differential game (Ref. 1) to a more general type of game structure. In particular, the Hamiltonians of the players are not required to be linear in the state variables and the control variables may not be in exponential form.The helpful comments of Professor C. Plourde and an anonymous referee are gratefully acknowledged.