On the Stackelberg strategy in nonzero-sum games

The properties of the Stackelberg solution in static and dynamic nonzero-sum two-player games are investigated, and necessary and sufficient conditions for its existence are derived. Several game problems, such as games where one of the two players does not know the other's performance criterion or games with different speeds in computing the strategies, are best modeled and solved within this solution concept. In the case of dynamic games, linear-quadratic problems are formulated and solved in a Hilbert space setting. As a special case, nonzero-sum linear-quadratic differential games are treated in detail, and the open-loop Stackelberg solution is obtained in terms of Riccati-like matrix differential equations. The results are applied to a simple nonzero-sum pursuit-evasion problem.This work was supported in part by the US Air Force under Grant No. AFOSR-68-1579D, in part by NSF under Grant No. GK-36276, and in part by the Joint Services Electronics Program under Contract No. DAAB-07-72-C-0259 with the Coordinated Science Laboratory, University of Illinois, Urbana, Illinois.     

Further properties of nonzero-sum differential games

The general nonzero-sum differential game hasN players, each controlling a different set of inputs to a single nonlinear dynamic system and each trying to minimize a different performance criterion. These general games have several interesting features which are absent in the two bestknown special cases (the optimal control problem and the two-person, zero-sum differential game). This paper considers some of the difficulties which arise in attempting to generalize ideas which are well known in optimal control theory, such as theprinciple of optimality and the relation betweenopen-loop andclosed-loop controls. Two types of solutions are discussed: theNash equilibrium and thenoninferior set. Some simple multistage discrete games are used to illustrate phenomena which also arise in the continuous formulation.This research was supported by Joint Services Electronics Contracts Nos. N00014-67-A-0298-0006, 0005, 0008 and by NASA Grant No. NGR 22-007-068.     

A linearization method inN-person nonzero-sum differential games

In the present treatment, a nonlinear system of anN-person nonzero-sum differential game is linearized with respect to the controls. It is shown that the optimal trajectory and the optimal costs of the linearized system lead, under certain conditions, to an approximation of the optimal trajectory and the optimal costs of the original nonlinear system.     

Extra-proximal methods for solving two-person nonzero-sum games

We consider two-person nonzero-sum game, both in the classical form and in the form of a game with coupled variables. An extra-proximal approach for finding the game’s solutions is suggested and justified. We provide our algorithm with an analysis of its convergence.      

Constrained nonzero-sum games with partially controllable strategies

The present paper deals with a class of nonzero-sum, two-person games with finite strategies when there are constraints on the strategies selected by the players. The constraints arise due to the subjective difficulty that each player often has in assigning to the states probabilities with which he is completely satisfied, and the model specifies how much each player must perturb his initial probability estimate in order to change his maximum utility alternative from the alternative originally best under the initial estimate. It is shown that the Nash-equilibrium solution of this class of nonzero-sum games can be characterized by an equivalent nonlinear program which leads in some cases to a pair of complementary eigenvalue problems. Applications to normal or approximate solutions of linear programming problems are also indicated.     

Sequential Stackelberg equilibria in two-person games

The concept of sequential Stackelberg equilibrium is introduced in the general framework of dynamic, two-person games defined in the Denardo contracting operator formalism. A relationship between this solution concept and the sequential Nash equilibrium for an associated extended game is established. This correspondence result, which can be related to previous results obtained by Baar and Haurie (1984), is then used for studying the existence of such solutions in a class of sequential games. For the zero-sum case, the sequential Stackelberg equilibrium corresponds to a sequential maxmin equilibrium. An algorithm is proposed for the computation of this particular case of equilibrium.This research was supported by SSHRC Grant No. 410-83-1012, NSERC Grant No. A4952, and FCAR Grants Nos. 86-CE-130 and EQ-0428.The authors thank T. R. Bielecki and J. A. Filar, who pointed out some mistakes and helped improving the paper.At the time of this research, this author was with GERMA, Ecole Mohammedia d'Ingénieurs, Rabat, Morocco.     

Stackelberg equilibria in managerial delegation games

Managerial compensation packages do not only influence managers’ behavior, but also have an impact on competing firms. In a managerial delegation game investigating the latter aspect, it is shown that the inherent prisoner’s dilemma situation can be resolved (without changing the normally studied setup or timing). In the first stage, owners choose an incentive function for their managers, in the second stage they choose the weights assigned to that function besides profits and in the third stage managers play a Cournot game. Solving this continuous optimization problem with the implicit function theorem shows that choosing an incentive from the set of “multiplicative incentives”, i.e. any generalized affine transformation of the product of both firms’ quantities, which includes e.g. relative profit, ensures that the Stackelberg outcome is among the set of equilibrium outcomes. Furthermore, it is the unique outcome if the rival owner opts for one of the well-known incentives like sales, revenue or market share. The general approach used allows demonstrating that with no other linear incentive a Stackelberg outcome results and that incentives like profit-to-cost ratio should be avoided. Selecting a multiplicative incentive is a dominant strategy of the game.     

On a class of nonzero-sum,linear-quadratic differential games

A class of two-player, nonzero-sum, linear-quadratic differential games is investigated for Nash equilibrium solutions when both players use closed-loop control and when one or both of the players are required to use open-loop control. For three formulations of the game, necessary and sufficient conditions are obtained for a particular strategy set to be a Nash equilibrium strategy set. For a fourth formulation of the game, where both players use open-loop control, necessary and sufficient conditions for the existence of a Nash equilibrium strategy set are developed. Several examples are presented in order to illustrate the differences between this class of differential games and its zero-sum analog.This research was supported by the National Science Foundation under Grant No. GK-3341.     

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.     

Cheap talk and cooperation in Stackelberg games

Central European Journal of Operations Research - Previous literature on cheap talk suggests that it is used to increase cooperation. We study cheap talk and the effect of the leader’s...     

An associated maximization problem for two-person nonzero-sum separable games

It has been shown that two-person zero-sum separable games are associated with a maximization problem involving the cones and dual cones of the generalized moments of the players' mixed strategy sets. In this note it is observed that those results extend almost immediately to two-person nonzero-sum separable games, but that then-person extension does not follow except in a special case.     

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).     

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

In this paper we deal with the problem of existence of a smooth solution of the Hamilton–Jacobi–Bellman–Isaacs (HJBI for short) system of equations associated with nonzero-sum stochastic differential games. We consider the problem in unbounded domains either in the case of continuous generators or for discontinuous ones. In each case we show the existence of a smooth solution of the system. As a consequence, we show that the game has smooth Nash payoffs which are given by means of the solution of the HJBI system and the stochastic process which governs the dynamic of the controlled system.     

On nash equilibrium solutions in nonzero-sum stochastic games with complete information

Two-person nonzero-sum stochastic games with complete information are considered. It is shown that it is sufficient to search the equilibrium solutions in a class of deterministic strategy pairs — the so-calledintimidation strategy pairs. Furthermore, properties of the set of all equilibrium losses of such strategy pairs are proved.     

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.     

Discrete time dynamic Stackelberg games with the leaders in turn

In many dynamic Stackelberg games, the leader changes at each stage. A new type of dynamic Stackelberg game is initially put forward in this paper and is called dynamic Stackelberg games with the leaders in turn, in which players act as the leaders in turn. There exist extremely comprehensive applications for dynamic Stackelberg games with the leaders in turn. On the one hand, in this work we aim to establish models for a new type of game, dynamic Stackelberg games of multiple players with leaders in turn, which are induced from some economic and political phenomena, which play exceedingly important roles in many fields. On the other hand, we hope to extend dynamic programming algorithms to the new model under feedback information structure.     

Dynamic feedback Stackelberg games with alternating leaders

In all past researches on dynamic Stackelberg games, the leader(s) and the followers are always assumed to be fixed. In practice, the roles of the players in a game may change from time to time. Some player in contract bridge, for example, acts as a leader at some stage but as a follower at the subsequent stage, which motivates the Stackelberg games with unfixed leaders. We aim to analyze the dynamic Stackelberg games with two players under such circumstances and call them dynamic Stackelberg games with alternating leaders. There are two goals in this paper. One goal is to establish models for a new type of games, dynamic Stackelberg games of alternating leaders with two players. The other goal is to extend dynamic programming algorithms to discrete time dynamic Stackelberg games with alternating leaders under feedback information structure.     

Closed-form solution for discrete-time linear-quadratic Stackelberg games

A method is proposed for solving the two-point boundary-value problem occurring in discrete-time linear-quadratic Stackelberg games. It is shown that, for open-loop information structure, the necessary conditions can be ordered to form a symplectic matrix. The solution is then obtained by exploiting the properties of such a matrix.     

Dynamic feedback Stackelberg games with non-unique solutions

Stackelberg games, which was originally introduced by Stackelberg, are widely applied in such fields as economics, management, politics and behavioral sciences. Stackelberg games can be modelled as a bi-level optimization problem. There exists an extensive literature about static bi-level optimization problems. However, the studies on dynamic bi-level optimization problems are fairly scarce in spite of the importance in explaining and predicting some phenomena rationally. In this paper, we consider discrete time dynamic Stackelberg games with feedback information. In general, the lower-level strategies are non-unique in practice. For a unique solution, dynamic programming algorithms have been presented with multiple players. We revisit dynamic programming for feedback information dynamic Stackelberg games with non-unique lower-level solution. First, we define some kind of solutions related to the decisions styles. Then, we analyze them, respectively. Moreover, dynamic programming algorithm is successful in solving solve feedback information dynamic Stackelberg games with non-unique lower-level solutions.