Symmetric versus asymmetric equilibria in symmetric supermodular games

This paper is an attempt to develop a unified approach to endogenous heterogeneity and symmetry-breaking by constructing two general classes of two-player symmetric games that always possess only asymmetric purestrategy Nash equilibria. These classes of games are characterized in some abstract sense by two general properties: payoff non-concavities and some form of strategic substitutability. Our framework relies on easily verified assumptions on the primitives of the game, which are satisfied in a number of two-stage models with an investment decision preceding product market competition. To illustrate the generality and wide scope for application of our approach, we present some existing models dealing with R&D and capacity expansion, which motivated this study.

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“…The role of A4 is simply to rule out PSNEs at (0, 0) or (c, c). 4 We say that a function f : R → R is increasing (strictly increasing) if…”

Section: The Resultsmentioning

confidence: 99%

Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities

2006

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“…Indeed, if a supermodular game is symmetric, then a greatest and a least equilibrium exist and they are symmetric. Amir et al (2008) show that monotonicity induces the greatest and the least equilibrium converge to the highest and the lowest symmetric steady states, respectively. A game is symmetric supermodular if on top of the usual conditions for supermodularity, the agents' strategy spaces are identical and…”

Section: Dynamic Properties Of Open-loop Nash Equilibrium and Steady mentioning

confidence: 93%

Strategic interaction and dynamics under endogenous time preference

Camacho

Sağlam

Turan

2013

Journal of Mathematical Economics

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a b s t r a c tThis paper presents a strategic growth model with endogenous time preference. Due to the potential lack of concavity and the differentiability of the value functions associated with each agent's problem, we employ the theory of monotone comparative statics and supermodular games based on order and monotonicity properties on lattices. In particular, we provide the sufficient conditions of supermodularity for dynamic games with open-loop strategies based on two fundamental elements: the ability to order elements in the strategy space of the agents and the strategic complementarity which implies upward sloping best responses. The supermodular game structure of the model lets us provide the existence and the monotonicity results on the greatest and the least equilibria. We sharpen these results by showing the differentiability of the value function and the uniqueness of the best response correspondences almost everywhere and show that the stationary state Nash equilibria tend to be symmetric. Finally, we numerically analyze to what extent the strategic complementarity inherent in agents' strategies can alter the convergence results that could have emerged under a single agent optimal growth model. In particular, we show that the initially rich can pull the poor out of the poverty trap even when sustaining a higher level of steady state capital stock for itself.

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“…That two-dimensional symmetric supermodular games may have asymmetric equilibria has been shown to hold by means of (nondifferentiable) examples (e.g. Amir et al (2008), p. 311). Let N = 2.…”

Section: Super-and Submodular Gamesmentioning

confidence: 99%

On Uniqueness and Stability of Symmetric Equilibria in Differentiable Symmetric Games

Hefti

2011

SSRN Journal

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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.

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Symmetric versus asymmetric equilibria in symmetric supermodular games

Cited by 28 publications

References 19 publications

Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities

Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities

Strategic interaction and dynamics under endogenous time preference

On Uniqueness and Stability of Symmetric Equilibria in Differentiable Symmetric Games

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