Existence of Pure Strategy Nash Equilibrium in Bertrand-Edgeworth Oligopolies

This article is searching for necessary and sufficient conditions which are to be imposed on the demand curve to guarantee the existence of pure strategy Nash equilibrium in a Bertrand-Edgeworth game with capacity constraints.

Invariance under type morphisms: the bayesian Nash equilibrium

Ely and Peski (2006) and Friedenberg and Meier (2010) provide examples when changing the type space behind a game, taking a "bigger" type space, induces changes of Bayesian Nash Equilibria, in other words, the Bayesian Nash Equilibrium is not invariant under type morphisms. In this paper we introduce the notion of strong type morphism. Strong type morphisms are stronger than ordinary and conditional type morphisms (Ely and Peski, 2006), and we show that Bayesian Nash Equilibria are not invariant under strong type morphisms either. We present our results in a very simple, finite setting, and conclude that there is no chance to get reasonable assumptions for Bayesian Nash Equilibria to be invariant under any kind of reasonable type morphisms.

A note on the axiomatization of the Nash equilibrium correspondence

A new axiomatization of the Nash equilibrium correspondence for n-person games based on independence of irrelevant strategies is given. Using a flexible general model, it is proved that the Nash equilibrium correspondence is the only solution to satisfy the axioms of non-emptiness, weak one-person rationality, independence of irrelevant strategies and converse independence of irrelevant strategies on the class of subgames of a fixed finite n-person game which admit at least one Nash equilibrium. It is also shown that these axioms are logically independent.

A generalization of correlated equilibrium: A new protocol

A new correlation scheme (leading to a special equilibrium called “soft” correlated equilibrium) is introduced for finite games. After randomization over the outcome space, players have the choice either to follow the recommendation of an umpire blindly or freely choose some other action except the one suggested. This scheme can lead to Pareto-better outcomes than the simple extension introduced by [Moulin, H., Vial, J.-P., 1978. Strategically zero-sum games: the class of games whose completely mixed equilibria cannot be improved upon. International Journal of Game Theory 7, 201–221]. The informational and interpretational aspects of soft correlated equilibria are also discussed in detail. The power of the generalization is illustrated in the prisoners’s dilemma and a congestion game.

Generalized correlated equilibrium for two-person games in extensive form with perfect information

A correlation scheme (leading to a special equilibrium called “soft” correlated equilibrium) is applied for two-person finite games in extensive form with perfect information. Randomization by an umpire takes place over the leaves of the game tree. At every decision point players have the choice either to follow the recommendation of the umpire blindly or freely choose any other action except the one suggested. This scheme can lead to Pareto-improved outcomes of other correlated equilibria. Computational issues of maximizing a linear function over the set of soft correlated equilibria are considered and a linear-time algorithm in terms of the number of edges in the game tree is given for a special procedure called “subgame perfect optimization”.

Hirschman meets Williamson: Relationship-specific investment and loyalty

In Albert Hirschman’s theory, loyalty plays a key role in the equilibrium between exit and voice. This article extends economic (rational choice) analysis to the emergence of loyalty, which Hirschman considers an exogenous factor. This is accomplished by linking Williamson’s theory of specific investment to Hirschman’s model. Three cases are distinguished: (1) loyalty is due to specific investment; (2) loyalty is due to (intermediate) factors influenced by specific investment; and, (3) loyalty is independent of specific investment. A simple model formalizes the first case. A paradoxical dynamic of loyalty is identified: a lower degree of specificity may lead to a weakening of loyalty in the short run but astrengthening of loyalty in the long run. An application to the process of European integration is sketched.