Equilibria in symmetric games: Theory and applications

Abstract

This article presents a new approach to analyze the equilibrium set of symmetric, differentiable games by separating multiple symmetric equilibria and asymmetric equilibria. This separation allows the investigation of, for example, how various parameter constellations affect the scope for multiple symmetric or asymmetric equilibria, or how the equilibrium set depends on the nature of the strategies. The approach is particularly helpful in applications because (i) it allows the complexity of the uniqueness problem to be reduced to a two‐player game, (ii) boundary conditions are less critical compared to standard procedures, and (iii) best replies need not be everywhere differentiable. The usefulness of the separation approach is illustrated with several examples, including an application to asymmetric games and to a two‐dimensional price‐information game.

Mergers and Innovation Portfolios

“…Our next result provides conditions under which a merger is a corrective device. Before presenting it, 26 Each component of the gradient evaluated at the pre-merger equilibrium is given by the expression:…”

“…x´i, the existence of equilibrium follows from the Debreu-Glicksberg-Fan theorem. Because the game is symmetric, there exists a SNE x i " x˚(see Hefti, 2017).…”

“…To check for multiple equilibria, we follow the method proposed by Hefti (2017). We use index theory approach based on Poincare-Hopf index.…”

Symmetric Stability in Symmetric Games

“…A symmetric game is (symmetrically) regular if i) ∇Π  has only regular zeroes 4 and ii) ∇Π  points inwards at the boundary of S. Any future reference to a regular game means "symmetrically" regular, which is a weaker condition than general regularity of a symmetric game (see [2]). The first theorem below reveals the general connection between symmetric stability and the existence of a single symmetric equilibrium, depending on the dimensionality of the strategy space.…”

Market Power and Information Effects in a Multi-Unit Auction

“…wind), which produces up toq at low marginal costs.18 One reason for why we chose n = 3 is that this is between competitive and collusive in oligopolistic markets(Huck et al, 2004), which is a type of tension we do not want to rule out.19 This is the generic equilibrium type in such games(Hefti, 2017). There can be asymmetric SFE as well, but any asymmetric equilibrium is subject to a "coordination dilemma" as any permutation across the players again is an asymmetric equilibrium.…”