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.     

Dynamic programming approach to discrete time dynamic feedback Stackelberg games with independent and dependent followers

Stackelberg games play an extremely important role in such fields as economics, management, politics and behavioral sciences. Stackelberg game can be modelled as a bilevel optimization problem. There exists extensive literature about static bilevel optimization problems. However, the studies on dynamic bilevel optimization problems are relatively 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. Dynamic programming algorithms are presented for the solution of discrete time dynamic feedback Stackelberg games with multiple players both for independent followers and for dependent followers. When the followers act dependently, the game in this paper is a combination of Stackelberg game and Nash game.     

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 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.     

Phenomena in Inverse Stackelberg Games,Part 1: Static Problems

Games are considered in which the role of the players is a hierarchical one. Some players behave as leaders, others as followers. Such games are named after Stackelberg. In the current paper, a special type of these games is considered, known in the literature as inverse Stackelberg games. In such games, the leader (or: leaders) announces his strategy as a mapping from the follower (or: followers) decision space into his own decision space. Arguments for studying such problems are given. The routine way of analysis, leading to a study of composed functions, is not very fruitful. Other approaches are given, mainly by studying specific examples. Phenomena in problems with more than one leader and/or follower are studied within the context of the inverse Stackelberg concept. As a side issue, expressions like “two captains on a ship” and “divide and conquer” are given a mathematical foundation.     

Phenomena in Inverse Stackelberg Games,Part 2: Dynamic Problems

Games are considered in which the role of the players is a hierarchical one. Some players behave as leaders, others as followers. Such games are named after Stackelberg. In the current paper, a special type of these games is considered, known in the literature as inverse Stackelberg games. In such games, the leader (or: leaders) announces his strategy as a mapping from the follower (or: followers) decision space into his own decision space. Arguments for studying such problems are given. The routine way of analysis, leading to a study of composed functions, is not very fruitful. Other approaches are given, mainly by studying specific examples. Phenomena in problems with more than one leader and/or follower are studied within the context of the inverse Stackelberg concept. As a side issue, expressions like “two captains on a ship” and “divide and conquer” are given a mathematical foundation.     

A note on dynamic Stackelberg games with leaders in turn

Recently, a model of dynamic Stackelberg games with leaders in turn has been proposed, and dynamic Stackelberg games with leaders in turn have been exploited under a feedback information structure. This paper characterizes dynamic Stackelberg games with leaders in turn under other information structures, both closed-loop and open-loop information structures. Explicit solutions are given for linear-quadratic systems under an open-loop information structure for dynamic Stackelberg games with leaders in turn.     

Closed-loop Stackelberg solution to a multistage linear-quadratic game

A new solution of a two-person, nonzero-sum Stackelberg game, with linear dynamics, quadratic performance criteion, and closed-loop information available to both players, is presented. This solution is applicable to all problems where the leader is able to influence the objective function of the follower, and this function is strictly convex with respect to the control variable handled by the follower. The resulting equilibrium strategies adapt to the possible nonoptimal behavior of players at some stages of the game. The strategy of the leader has a simple interpretation of a threat formulated by the leader toward the follower and, if necessary, carried out one stage after the follower has played inconsistently with the leader's wishes.     

Performance versus informativeness in linear-quadratic Gaussian noncooperative games

In this paper, we study the impact of informativeness on the performance of linear quadratic Gaussian Nash and Stackelberg games. We first show that, in two-person static Nash games, if one of the players acquires more information, then this extra information is beneficial to him, provided that it is orthogonal to both players' information. A special case is that when one of the players is informationally stronger than the other, then any new information is beneficial to him. We then show that a similar result holds for dynamic Nash games. In the dynamic games, the players use strategies that are linear functions of the current estimates of the state, generated by two Kalman filters. The same properties are proved to hold in static and feedback Stackelberg games as well.This work was partially supported by the US Air Force Office of Scientific Research under Grant No. AFOSR-82-0174.     

Variational Inequality Formulation of a Class of Multi-Leader-Follower Games

The multi-leader-follower game can be looked on as a generalization of the Nash equilibrium problem and the Stackelberg game, which contains several leaders and a number of followers. Recently, the multi-leader-follower game has been drawing more and more attention, for example, in electricity power markets. However, when we formulate a general multi-leader-follower game as a single-level game, it will give rise to a lot of problems, such as the lack of convexity and the failure of constraint qualifications. In this paper, to get rid of these difficulties, we focus on a class of multi-leader-follower games that satisfy some particular, but still reasonable assumptions, and show that these games can be formulated as ordinary Nash equilibrium problems, and then as variational inequalities. We establish some results on the existence and uniqueness of a leader-follower Nash equilibrium. We also present illustrative numerical examples from an electricity power market model.     

Bundle Trust-Region Algorithm for Bilinear Bilevel Programming

The bilevel programming problem (BLPP) is equivalent to a two-person Stackelberg game in which the leader and follower pursue individual objectives. Play is sequential and the choices of one affect the choices and attainable payoffs of the other. The purpose of this paper is to investigate an extension of the linear BLPP where the objective functions of both players are bilinear. To overcome certain discontinuities in the master problem, a regularized term is added to the follower objective function. Using ideas from parametric programming, the generalized Jacobian and the pseudodifferential of the regularized follower solution function are computed. This allows us to develop a bundle trust-region algorithm. Convergence analysis of the proposed methodology is given.     

Remarks on hierarchic control for the wave equation in moving domains

We study a Stackelberg strategy subject to the evolutionary linearized Kirchhoff equation for small vibrations of a stretched elastic string when the ends are variables. We assume that we can act in the dynamic of the system by a hierarchy of controls. According to the formulation given by H. von Stackelberg (see [3]), there are local controls, called followers, and global controls, called leaders. In fact, one considers situations where there are two cost (objective) functions. One possible way is to cut the control into two parts, one being thought of as “the leader” and the other one as “the follower”. This situation is studied in the paper, with one of the cost functions being of the controllability type. Existence and uniqueness is proven. The optimality system is given in the paper.     

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.     

On Coincidence of Feedback Nash Equilibria and Stackelberg Equilibria in Economic Applications of Differential Games

The scope of the applicability of the feedback Stackelberg equilibrium concept in differential games is investigated. First, conditions for obtaining the coincidence between the stationary feedback Nash equilibrium and the stationary feedback Stackelberg equilibrium are given in terms of the instantaneous payoff functions of the players and the state equations of the game. Second, a class of differential games representing the underlying structure of a good number of economic applications of differential games is defined; for this class of differential games, it is shown that the stationary feedback Stackelberg equilibrium coincides with the stationary feedback Nash equilibrium. The conclusion is that the feedback Stackelberg solution is generally not useful to investigate leadership in the framework of a differential game, at least for a good number of economic applications This paper was presented at the 8th Viennese Workshop on Optimal Control, Dynamic Games, and Nonlinear Dynamics: Theory and Applications in Economics and OR/MS, Vienna, Austria, May 14–16, 2003, at the Seminar of the Instituto Complutense de Analisis Economico, Madrid, Spain, June 20, 2003, and at the Sevilla Workshop on Dynamic Economics and the Environment, Sevilla, Spain, July 2–3, 2003. The author is grateful to the participants in these sessions, in particular F.J. Andre and J. Ruiz, for their comments. Five referees provided particularly helpful suggestions. Financial support from the Ministerio de Ciencia y Tecnologia under Grant BEC2000-1432 is gratefully acknowledged.     

Existence theorems of equilibrium points in stackelberg

A two players static. Stackelberg game is considered with constraints on the decision variables. Existence theorems of equilibrium points are given when the reaction set of the follower to the strategy of the leader, is a singleton. Some definition are presented and discussed and a simple existence theorem is given also in the case when the follower has more than one element in his reaction set. Finally an example of dynamic Stackelbebg game- is analized to emphasize some features arising when constraints are present on control variables.     

Numerical construction of Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game

Numerical methods are proposed for constructing Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game with terminal cost functionals and geometric constraints on the players’ controls. The formalization of the players’ strategies and of the motions generated by them is based on the formalization and results from the theory of positional zero-sum differential games developed by N.N. Krasovskii and his school. It is assumed that the game is reduced to a planar game and the constraints on the players’ controls are given in the form of convex polygons. The problem of finding solutions of the game may be reduced to solving nonstandard optimal control problems. Several computational geometry algorithms are used to construct approximate trajectories in these problems, in particular, algorithms for constructing the convex hull as well as the union, intersection, and algebraic sum of polygons.     

Additional aspects of the Stackelberg strategy in nonzero-sum games

The Stackelberg strategy in nonzero-sum games is a reasonable solution concept for games where, either due to lack of information on the part of one player about the performance function of the other, or due to different speeds in computing the strategies, or due to differences in size or strength, one player dominates the entire game by imposing a solution which is favorable to himself. This paper discusses some properties of this solution concept when the players use controls that are functions of the state variables of the game in addition to time. The difficulties in determining such controls are also pointed out. A simple two-stage finite state discrete game is used to illustrate these properties.This work was supported in part by the U.S. 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.     

Should civil society organizations cooperate or compete in fighting a corrupt government?

We consider a dynamic game with a corrupt government and multiple civil society organizations as the players. We characterize feedback Stackelberg equilibria with the government as leader and two civil society organizations as the followers who can compete or cooperate when deciding their monitoring efforts. Overall, the numerical results show that a cooperation yields a higher institutional quality and output than does the competitive regime as it does for both individuals and government payoff while the players invest less efforts. In a nutshell, we found that it is in the best interest of both the government and civil society organizations that the latter coordinate their actions and efforts and cooperate in fight against corruption.     

An evolutionary game theoretic perspective on e-collaboration: The collaboration effort and media relativeness

Studies of e-collaboration from a game-theoretical perspective are practically nonexistent. This article contributes to filling this gap by focusing on the strategic interaction between players as they decide whether and how much to collaborate with each other. We use evolutionary game theory to make predictions about a two-person e-collaboration game. More specifically, we extend the traditional Prisoners’ Dilemma and Snowdrift game theory notions to discrete-strategy e-collaboration games, by explicitly including social punishments into the players’ payoff functions. We also introduce continuous-strategy e-collaboration games with both complete and incomplete information. Finally, we provide two generic dynamic programming models for e-collaboration games with media selection.