Characterizing Stability Properties in Games with Strategic Substitutes

In games with strategic substitutes (GSS), convergence of the best response dynamic starting from the inf (or sup) of the strategy space is equivalent to global stability (convergence of every adaptive dynamic to the same pure strategy Nash equilibrium). Consequently, in GSS, global stability can be analyzed using a single best response dynamic. Moreover, in GSS, global stability is equivalent to dominance solvability, showing that in this class of games, two different foundations for robustness of predicted outcomes are equivalent, and both can be checked using a single best response dynamic. These equivalences are useful to study stability of equilibria in a variety of applications. Furthermore, in parameterized GSS, under natural conditions, dynamically stable equilibrium selections can be viewed in terms of monotone selections of equilibria. Several examples are provided.Global stability, adaptive dynamics, strategic substitutes, dominance solvable, learning, monotone comparative statics, Correspondence principle

Endougenous Timing in a Mixed Duopoly

This paper applies the framework of endogenous timing in games to mixed quantity duopoly, wherein a private – domestic or foreign – firm competes with a public, welfare maximizing firm. We show that simultaneous play never emerges as a subgame-perfect equilibrium of the extended game, in sharp contrast to private duopoly games. We provide sufficient conditions for the emergence of public and/or private leadership equilibrium. In all cases, private profits and social welfare are higher than under the corresponding Cournot equilibrium. From a methodological viewpoint we make extensive use of the basic results from the theory of supermodular games in order to avoid common extraneous assumptions such as concavity, existence and uniqueness of the different equilibria, whenever possible. Some policy implications are drawn, in particular those relating to the merits of privatization.Mixed markets, endogenous timing, Cournot equilibrium, Stackelberg equilibrium, privatization.

Comparative Statics by Adaptive Dynamics and the Correspondence Principle

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Finding all equilibria in games of strategic complements

I present a simple and fast algorithm that finds all the pure-strategy Nash equilibria in games with strategic complementarities. This is the first non-trivial algorithm for finding all pure-strategy Nash equilibria

The complementarity foundations of industrial organization

In this paper we review the state of the art of Games with Strategic Complementarities (GSC), which are fundamental tools in modern Industrial Organization. The originality of the paper lies in the way the material is presented. Indeed, the mathematical aspects of GSC are complex and scattered in a literature which spans a long time period and a variety of research fields such as economics, applied mathematics and operations research. We organize a large amount of material in a unified and self-contained way, and concentrate on the intuitions and conceptual points that lie in the background of the mathematical modeling, with special emphasis on the modeling of complementarity. On the technical side, we investigate in details the choice and content of the assumptions. The scope of the paper is to allow the applied researcher to understand the theory, so that she may rapidly develop her own ability to deal with concrete problems.strategic complementarity, oligopoly theory, supermodularity, Nash equilibria, lattices

A non-robustness in the order structure of the equilibrium set in lattice games

The order and lattice structure of the equilibrium set in games with strategic complements do not survive a minimal introduction of strategic substitutes: in a lattice game in which all-but-one players exhibit strategic complements (with one player exhibiting strict strategic complements), and the remaining player exhibits strict strategic substitutes, no two equilibria are comparable. More generally, in a lattice game, if either (1) just one player has strict strategic complements and another player has strict strategic substitutes, or (2) just one player has strict strategic substitutes and has singleton-valued best-responses, then without any restrictions on the strategic interaction among the other players, no two equilibria are comparable. In such cases, the equilibrium set is a non-empty, complete lattice, if, and only if, there is a unique equilibrium. Moreover, in such cases, with linearly ordered strategy spaces, the game has at most one symmetric equilibrium. Several examples are presented.Lattice games, strategic complements, strategic substitutes, equilibrium set

The prisoner’s dilemma in Cournot models: when endogenizing the level of competition leads to competitive behaviors.

In resource based economies, regulating the production and export activities have always been an important challenge. Examples in oil and gas show that different behaviors have been adopted ranging from the export monopoly to the complete opening of the export market. This paper tries to explain this multitude of solutions via strategic interactions. When modeling imperfect competition, players are separated in two categories: those who exert market power and those who are competitive and propose the good at their marginal supply cost. Letting a player freely choose whether it wants to exert market power or not when it optimizes its utility is not discussed in the literature. This paper addresses this issue by letting the players choose the level of competition they want to exert in the market. To do so, we analyze the behavior of two countries competing to supply a market with a homogeneous good in an imperfect competition setting. Each country decides the number of firms it authorizes to sell in the market. The interaction between the firms is of a Nash-Cournot type, where each one exerts market power and is in competition with all other firms allowed to sell, whether they belong to the same country or not. Each country optimizes its utility, that is the sum of the profits of its firms. We have studied four kinds of interaction between the countries. The first calculates the closed loop Nash equilibrium of the game between the countries. The second setup analyzes the cartel when the countries collude. The third focuses on the open loop Nash equilibrium and the fourth models a bi-level Stackelberg interaction where one country plays before the other. We demonstrate that in the closed loop Nash equilibrium, our setting leads to the prisoner’s dilemma: the equilibrium occurs when both countries authorize all their firms to sell in the market. In other words, countries willingly chose not to exert market power. This result is at first sight similar to the Allaz & Vila (1993) result but is driven by a completely different economic reasoning. In the Stackelberg and coordinated solutions, the market is on the contrary very concentrated and the countries strongly reduce the number of firms that enter the market in order to fully exert market power and increase the price. The open loop result lies in between: the countries let all their firms sell but market power remains strong. These results suggest that the prisoner’s dilemma outcome is due to the conjectural inconsistency of the Nash equilibrium. Finally, in the Stackelberg setting, we give countries the choice of being leader or follower and demonstrate that the counter-intuitive competitive outcome is very unlikely to occur in the market