On Uniqueness and Stability of Symmetric Equilibria in Differentiable Symmetric Games

Abstract

Higher-dimensional symmetric games become of more and more importance for applied micro-and macroeconomic research. Standard approaches to uniqueness of equilibria have the drawback that they are restrictive or not easy to evaluate analytically. In this paper I provide some general but comparably simple tools to verify whether a symmetric game has a unique symmetric equilibrium or not. I distinguish between the possibility of multiple symmetric equilibria and asymmetric equilibria which may be economically interesting and is useful to gain further insights into the causes of asymmetric equilibria in symmetric games with higher-dimensional strategy spaces. Moreover, symmetric games may be used to derive some properties of the equilibrium set of certain asymmetric versions of the symmetric game.I further use my approach to discuss the relationship between stability and (in)existence of multiple symmetric equilibria. While there is an equivalence between stability, inexistence of multiple symmetric equilibria and the unimportance of strategic effects for the comparative statics, this relationship breaks down in higher dimensions. Stability under symmetric adjustments is a minimum requirement of a symmetric equilibrium for reasonable comparative statics of symmetric changes. Finally, I present an alternative condition for a symmetric equilibrium to be a local contraction which is more general than the conventional approach of diagonal dominance and yet simpler to evaluate than the eigenvalue condition of continuous adjustment processes.

Local Contraction-Stability and Uniqueness

“…Symmetric games have been of considerable importance in applied work (e.g. Salop (1979), Grossman and Shapiro (1984), Hefti (2011)), mainly because symmetry simplifies the analysis. In this section I apply previous results to symmetric games.…”

“…Any equilibrium in a symmetric games must either be symmetric or asymmetric, and if x * is an asymmetric equilibrium, there must be at least N − 1 further asymmetric equilibria. Under fairly general conditions, a symmetric game possesses at least one symmetric equilibrium (Hefti (2011)). Hence uniqueness may fail to hold in symmetric games because there are asymmetric equilibria or multiple symmetric equilibria.…”

“…I call a symmetric game, where ∇ 1Π 1 points into S(k) at boundary points 9 and Det(H(x 1 )) = 0 on Cr s a symmetric index game. For such an index game we can use the index theorem to count the multiplicity of symmetric equilibria (their number must always be odd, see Hefti (2011) for details). Moreover, a symmetric index games has exactly one symmetric equilibrium if and only if Det(−H(x 1 )) > 0 on Cr s .…”

“…Further, these properties are independent of whether or not there are asymmetric equilibria. In Hefti (2011) it is shown that…”

“…Dixit (1986), Vives (1999), Dastidar (2000) or Hefti (2011) for applications). s > 0 is an arbitrary speed of adjustment.…”