Slightly Altruistic Equilibria

We introduce a refinement concept for Nash equilibria (slightly altruistic equilibrium) defined by a limit process and which captures the idea of reciprocal altruism as presented in Binmore (Proceedings of the XV Italian Meeting on Game Theory and Applications, [2003]). Existence is guaranteed for every finite game and for a large class of games with a continuum of strategies. Results and examples emphasize the (lack of) connections with classical refinement concepts. Finally, it is shown that, under a pseudomonotonicity assumption on a particular operator associated to the game, it is possible, by selecting slightly altruistic equilibria, to eliminate those equilibria in which a player can switch to a strategy that is better for the others without leaving the set of equilibria. Part of the results in this paper have been presented at: First Spain, Italy, Netherlands Meeting on Game Theory, Maastricht, 2005; Fifth International ISDG Workshop, Segovia, 2005; GATE, Université Lumière Lyon 2, 2005; XXX AMASES Workshop, Trieste 2006; CSEF, Università di Salerno, 2006.     

Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games

In this paper, we discuss nonzero-sum linear-quadratic differential games. For this kind of games, the Nash equilibria for different kinds of information structures were first studied by Starr and Ho. Most of the literature on the topic of nonzero-sum linear-quadratic differential games is concerned with games of fixed, finite duration; i.e., games are studied over a finite time horizon t _f. In this paper, we study the behavior of feedback Nash equilibria for t _f.In the case of memoryless perfect-state information, we study the so-called feedback Nash equilibrium. Contrary to the open-loop case, we note that the coupled Riccati equations for the feedback Nash equilibrium are inherently nonlinear. Therefore, we limit the dynamic analysis to the scalar case. For the special case that all parameters are scalar, a detailed dynamical analysis is given for the quadratic system of coupled Riccati equations. We show that the asymptotic behavior of the solutions of the Riccati equations depends strongly on the specified terminal values. Finally, we show that, although the feedback Nash equilibrium over any fixed finite horizon is generically unique, there can exist several different feedback Nash equilibria in stationary strategies for the infinite-horizon problem, even when we restrict our attention to Nash equilibria that are stable in the dynamical sense.     

On differential games with a feedback nash equilibrium

This note provides a lemma on differential games which possess a feedback Nash equilibrium (FNE). In particular, it shows that (i) a class of games with a degenerate FNE can be constructucted from every game which has a nondegenerate FNE and (ii) a class of games with a nondegenerate FNE can be constructed from every game which has a degenerate FNE.The author would like to thank an anonymous referee for invaluable comments and suggestions.     

Feedback Solution of a Class of Differential Games with Endogenous Horizons

In this paper, we consider a class of differential games in which the game ends when a subset of its state variables reaches a certain target at the terminal time. A special feature of the game is that its horizon is not fixed at the outset, but is determined endogenously by the actions of the players; conditions characterizing a feedback Nash equilibrium (FNE) solution of the game are derived for the first time. Extensions and illustrations of the derivation of FNE solutions of the game are provided.     

A Discrete-Time Dynamic Game of Seasonal Water Allocation

We present a method for the derivation of feedback Nash equi- libria in discrete-time finite-horizon nonstationary dynamic games. A partic- ular motivation for such games stems from environmental economics, where problems of seasonal competition for water levels occur frequently among heterogeneous economic agents. These agents are coupled through a state variable, which is the water level. Actions are strategically chosen to max- imize the agents individual season-dependent utility functions. We observe that, although a feedback Nash equilibrium exists, it does not satisfy the (exogenous) environmental watchdog expectations. We devise an incentive scheme to help meeting those expectations and calculate a feedback Nash equilibrium for the new game that uses the scheme. This solution is more environmentally friendly than the previous one. The water allocation game solutions help us to draw some conclusions regarding the agents behavior and also about the existence of feedback Nash equilibria in dynamic games. The paper draws from Refs.1–2. Its earlier version was presented at the Victoria International Conference 2004, Victoria University of Wellington, Wellington, New Zealand, February 9–13, 2004. We thank the anonymous referee and Christophe Deissenberg for insightful comments, which have helped us to clarify its message. We also thank our colleagues Sophie Thoyer, Robert Lifran, Odile Pourtalier, and Vladimir Petkov for helpful discussions on the model and techniques used in this Paper. Gratitude is expressed to the Kyoto Institute for Economic Research, Kyoto University, for this author's support in the final stages of the paper preparation     

Identification of classes of differential games for which the open loop is a degenerate feedback Nash equilibrium

In general, it is clear that open-loop Nash equilibrium and feedback Nash equilibrium do not coincide. In this paper, we study the structure of differential games and develop a technique using which we can identify classes of games for which the open-loop Nash equilibrium is a degenerate feedback equilibrium. This technique clarifies the relationship between the assumptions made on the structure of the game and the resultant equilibrium.The author would like to thank E. Dockner, A. Mehlmann, and an anonymous referee for helpful comments.     

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.     

Equilibria in load balancing games

A Nash equilibrium （NE） in a multi-agent game is a strategy profile that is resilient to unilateral deviations. A strong Nash equilibrium （SE） is one that is stable against coordinated deviations of any coalition. We show that, in the load balancing games, NEs approximate SEs in the sense that the benefit of each member of any coalition from coordinated deviations is well limited. Furthermore, we show that an easily recognizable special subset of NEs exhibit even better approximation of SEs.     

A differential game of industrial pollution management

This paper explores a differential game between a policy maker and a profit maximizing entrepreneur in which production generates pollution. The government levies a pollution tax on output and uses the tax received for pollution abatement. The entrepreneur determines the level of output. A feedback Nash equilibrium is derived. Using more specific functional forms, the game is extended to cover the multiple firm case.     

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.     

Infinite horizon noncooperative differential games

For a noncooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we consider a class of infinite horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value functions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability.     

Existence and Uniqueness of Open-Loop Stackelberg Equilibria in Linear-Quadratic Differential Games

We present existence and uniqueness results for a hierarchical or Stackelberg equilibrium in a two-player differential game with open-loop information structure. There is a known convexity condition ensuring the existence of a Stackelberg equilibrium, which was derived by Simaan and Cruz (Ref. 1). This condition applies to games with a rather nonconflicting structure of their cost criteria. By another approach, we obtain here new sufficient existence conditions for an open-loop equilibrium in terms of the solvability of a terminal-value problem of two symmetric Riccati differential equations and a coupled system of Riccati matrix differential equations. The latter coupled system appears also in the necessary conditions, but contrary to the above as a boundary-value problem. In case that the convexity condition holds, both symmetric equations are of standard type and admit globally a positive-semidefinite solution. But the conditions apply also to more conflicting situations. Then, the corresponding Riccati differential equations may be of H-type. We obtain also different uniqueness conditions using a Lyapunov-type approach. The case of time-invariant parameters is discussed in more detail and we present a numerical example.     

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.     

An algorithm for computing the stable coalition structures in tree-graph communication games

We construct an algorithm which provides in finite steps the stable coalition structure(s) of tree-graph communication games and an allocation of the core: the restricted marginal contribution allocation. This paper has been presented at the St. Petersburg Institute for Economics and Mathematics (Russian Academy of Sciences), University of Santiago de Compostela (International Workshop on Game Theory), Universidad Autónoma de Barcelona, and Universidad de Sevilla. This research has been supported partially by: DGICYT PB94-1372 and UPV 035.321-HB146/96     

Necessary and Sufficient Conditions for Feedback Nash Equilibria for the Affine-Quadratic Differential Game

In this note, we consider the non-cooperative linear feedback Nash quadratic differential game with an infinite planning horizon. The performance function is assumed to be indefinite and the underlying system affine. We derive both necessary and sufficient conditions under which this game has a Nash equilibrium. As a special case, we derive existence conditions for the multi-player zero-sum game.     

Discontinuous differential equations,II

The concepts and results of the first part of this paper are applied in three situations: differential games, specifically feedback controls in a linear game with hyperplane target; an example of Nash equilibrium with feedback strategies; uniqueness properties, theorems, and stability with respect to measurement; control theory, specifically time-optimal feedback.     

Feedback Nash equilibria in a problem of optimal fishery management

The paper deals with a problem of optimal management of a common-property fishery, modelled as a two-player differential game. Under nonclassical assumptions on harvest rates and utilities, a feedback Nash equilibrium is determined, using a bionomic equilibrium concept. Later on, this assumption is relaxed and a feedback Nash equilibrium is established under minimal hypotheses.     

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.     

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.     

Nash Equilibria of Risk-Sensitive Nonlinear Stochastic Differential Games

This paper considers a class of risk-sensitive stochastic nonzero-sum differential games with parametrized nonlinear dynamics and parametrized cost functions. The parametrization is such that, if all or some of the parameters are set equal to some nominal values, then the differential game either becomes equivalent to a risk-sensitive stochastic control (RSSC) problem or decouples into several independent RSSC problems, which in turn are equivalent to a class of stochastic zero-sum differential games. This framework allows us to study the sensitivity of the Nash equilibrium (NE) of the original stochastic game to changes in the values of these parameters, and to relate the NE (generally difficult to compute and to establish existence and uniqueness, at least directly) to solutions of RSSC problems, which are relatively easier to obtain. It also allows us to quantify the sensitivity of solutions to RSSC problems (and thereby nonlinear H-control problems) to unmodeled subsystem dynamics controlled by multiple players.