Michèle Breton

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 Başar 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.

A differential game of joint implementation of environmental projects

This paper proposes a two-player, finite-horizon differential game model to analyze joint implementation in environmental projects, one of the flexible mechanisms considered in the Kyoto Protocol. Our results show that allowing for foreign investments could in some cases increase the welfares of both parties involved in the project. Further, imposing an environmental target constraint does not necessarily deteriorate the payoffs of both players. Finally, a leakage effect does occur when foreign investments are possible.

A game-theoretic formulation of joint implementation of environmental projects

Abstract The aim of this paper is to provide a game-theoretic interpretation of joint implementation in environmental projects and to assess the merit of such a strategy. More specifically, we consider a two-player game and solve it under three different cases. In the first case, countries play a non-cooperative game and optimize their welfare under an environmental constraint without having access to joint implementation. In the second case, we assume countries do have access to JI, which allows us to assess its merits by comparing the players’ welfare levels achieved with and without JI. In the last case, the players jointly optimize their welfare under a collective environmental constraint. Comparing welfare levels in this case to those in the second case allows us to assess the merits of cooperation.

A Dynamic Programming Procedure for Pricing American-Style Asian Options

Pricing European-style Asian options based on the arithmetic average, under the Black and Scholes model, involves estimating an integral (a mathematical expectation) for which no easily computable analytical solution is available. Pricing their American-style counterparts, which provide early exercise opportunities, poses the additional difficulty of solving a dynamic optimization problem to determine the optimal exercise strategy. A procedure for pricing American-style Asian options of the Bermudan flavor, based on dynamic programming combined with finite-element piecewise-polynomial approximation of the value function, is developed here. A convergence proof is provided. Numerical experiments illustrate the consistency and efficiency of the procedure. Theoretical properties of the value function and of the optimal exercise strategy are also established.

A dynamic programming approach to price installment options

Installment options are Bermudan-style options where the holder periodically decides whether to exercise or not and then to keep the option alive or not (by paying the installment). We develop a dynamic programming procedure to price installment options. We study in particular the geometric Brownian motion case and derive some theoretical properties of the IO contract within this framework. We also characterize the range of installments within which the installment option is not redundant with the European contract. Numerical experiments show the method yields monotonically converging prices, and satisfactory trade-offs between accuracy and computational time. Our approach is finally applied to installment warrants, which are actively traded on the Australian Stock Exchange. Numerical investigation shows the various capital dilution effects resulting from different installment warrant designs.

A Note on Feedback Sequential Equilibria in a Lanchester Model with Empirical Application

We study in this paper dynamic equilibrium advertising strategies in a duopoly with asymmetric information structure and sequential play. The advertising model of Lanchester is used in a game where the relevant solution concept is feedback Stackelberg equilibrium, which is subgame perfect. An algorithm is devised for the computation of this equilibrium, and numerical results are reported and discussed. Using a data set from the cola market, we obtain the resulting advertising strategies and provide a comparison with closed-loop and open-loop Nash equilibria.

Algorithms for Stochastic Games

In this paper, we present algorithms for the solution of finite discounted stochastic games, without special structure. Three equilibrium concepts are considered: saddle points in two-person zero-sum games, Nash equilibrium points in N-person non-cooperative games and finally Stackelberg equilibrium in two-person games.

Dynamic Model of R&D, Spillovers, and Efficiency of Bertrand and Cournot Equilibria

Using an infinite-horizon two-player differential game, we derive and compare Bertrand and Cournot equilibria for a differentiated duopoly engaging in the process of R&D competition. The main findings of this study are as follows. First, Bertrand competition is more efficient if either R&D productivity is low or products are very different. Second, Cournot competition is more efficient provided that R&D productivity is high, products are close substitutes, and spillovers are not close to zero. This last result is different from what has been obtained in the literature. Hence, this shows that considering a dynamic model and more general investment costs does have an impact on the efficiency results.

Incentive equilibrium in an overlapping-generations environmental game☆

We consider two overlapping generations that want to coordinate their strategies of working, consuming and controlling pollution. Since the cooperative solution is not an equilibrium, and hence is not a self-enforcing contract, a mechanism is required to sustain it. We show how incentive strategies, and the resulting incentive equilibrium, could provide such a mechanism. We also derive the conditions that ensure the credibility of these strategies.

Equilibria in an asymmetric duopoly facing a security constraint

Abstract We consider an asymmetric duopoly producing a homogeneous commodity and facing a competitive demand. Our model incorporates two asymmetries. The first one is relative to the performance indices. Indeed, we assume that one of the player maximizes its profit and the second one, being eager to earn hard currencies, maximizes its revenue. The second asymmetry involves a security (or diversification) constraint which states that the second player is not allowed to sell more than a certain proportion of the quantity sold by its rival. This game is an abstraction of the European natural gas market during the eighties. Cournot and Stackelberg equilibria are characterized and compared. An assessment of the impact on consumers and producers of the security constraint is also made.