Equilibrium uniqueness in aggregative games: very practical conditions

Various Nash equilibrium results for a broad class of aggregative games are presented. The main ones concern equilibrium uniqueness. The setting presupposes that each player has $$\mathbb {R}_+$$ R + as strategy set, makes smoothness assumptions but allows for a discontinuity of stand-alone payoff functions at 0; this possibility is especially important for various contest and oligopolistic games. Conditions are completely in terms of marginal reductions which may be considered as primitives of the game. For many games in the literature they can easily be checked. They automatically imply that conditional payoff functions are strictly quasi-concave. The results are proved by means of the Szidarovszky variant of the Selten–Szidarovszky technique. Their power is illustrated by reproducing quickly and improving upon various results for economic games.

Mean Field Equilibrium: Uniqueness, Existence, and Comparative Statics

The Uniqueness of a Mean Field Equilibrium

On the Equilibrium Uniqueness in Cournot Competition with Demand Uncertainty

We revisit the linear Cournot model with uncertain demand that is studied in Lagerlöf (2006. “Equilibrium Uniqueness in a Cournot Model with Demand Uncertainty.” The B.E. Journal of Theoretical Economics 6, no. 1. (Topics), Article 19: 1–6.) and provide sufficient conditions for equilibrium uniqueness that complement the existing results. We show that if the distribution of the demand intercept has the decreasing mean residual demand (DMRD) or the increasing generalized failure rate (IGFR) property, then uniqueness of equilibrium is guaranteed. The DMRD condition implies log-concavity of the expected profits per unit of output without additional assumptions on the existence or the shape of the density of the demand intercept and, hence, answers in the affirmative the conjecture of Lagerlöf (2006. “Equilibrium Uniqueness in a Cournot Model with Demand Uncertainty.” The B.E. Journal of Theoretical Economics 6, no. 1. (Topics), Article 19: 1–6.) that such conditions may not be necessary.

Cournot Equilibrium Uniqueness: At 0 Discontinuous Industry Revenue and Decreasing Price Flexibility

We consider the equilibrium uniqueness problem for a large class of Cournot oligopolies with convex cost functions and a proper price function [Formula: see text] with decreasing price flexibility. This class allows for (at [Formula: see text]) discontinuous industry revenue and in particular for [Formula: see text]. The paper illustrates in an exemplary way the Selten–Szidarovszky technique based on virtual backward reply functions. An algorithm for the calculation of the unique equilibrium is provided.