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On the relationship between uniqueness and stability in sum-aggregative, symmetric and general differentiable games

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  • Hefti, Andreas

Abstract

This article explores the relationship between uniqueness and stability in differentiable regular games, with a major focus on the important classes of sum-aggregative, two-player and symmetric games. We consider three types of popular dynamics, continuous-time gradient dynamics as well as continuous- and discrete-time best-reply dynamics, and include aggregate-taking behavior as a non-strategic behavioral variant. We show that while in general games stability conditions are only sufficient for uniqueness, they are likely to be necessary as well in models with sum-aggregative or symmetric payoff functions. In particular, a unique equilibrium always verifies the stability conditions of all dynamics if strategies are equilibrium complements, and this also holds for both continuous-time dynamics if strategies are equilibrium substitutes with bounded slopes. These findings extend to the case of aggregate-taking equilibria. We further analyze the stability relations between the various dynamics, and demonstrate that the restrictive nature of the discrete dynamics originates from simultaneity of adjustments. Asynchronous decisions or heterogeneous forward thinking may stabilize the adjustment process.

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  • Hefti, Andreas, 2016. "On the relationship between uniqueness and stability in sum-aggregative, symmetric and general differentiable games," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 83-96.
  • Handle: RePEc:eee:matsoc:v:80:y:2016:i:c:p:83-96
    DOI: 10.1016/j.mathsocsci.2016.02.008
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    1. repec:eee:proeco:v:193:y:2017:i:c:p:726-736 is not listed on IDEAS
    2. Christensen, Finn & Cornwell, Christopher R., 2018. "A strong correspondence principle for smooth, monotone environments," Journal of Mathematical Economics, Elsevier, vol. 77(C), pages 15-24.
    3. Andreas Hefti, 2016. "Distributional comparative statics with heterogeneous agents," ECON - Working Papers 237, Department of Economics - University of Zurich.
    4. Christensen, Finn, 2017. "A necessary and sufficient condition for a unique maximum with an application to potential games," Economics Letters, Elsevier, vol. 161(C), pages 120-123.

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