Rooted Potential and Static Stability in Games
49 Pages Posted: 27 Jul 2008 Last revised: 12 Apr 2023
Date Written: July 25, 2008
Abstract
Rooted potential is a generalization of (exact) potential that exists in every game. In a potential game, it coincided with the potential. In general games, it varies according to the choice of root, which is any strategy (in symmetric and population games) or strategy profile (in asymmetric games). Roots that are local maxima of the rooted potential have interesting stability features. Depending on the class of games considered, the set of equilibrium roots with this property may coincide with the evolutionarily stable strategies, ESS (in symmetric 𝑛×𝑛 games); with the continuously stable strategies, CSS (in symmetric games with a unidimensional strategy space); with the risk dominant equilibria (in finite symmetric 2×2 games); or with the strict equilibria (in bimatrix games). Local maximization of the rooted potential may thus be viewed as a general, unifying notion of static stability. Like the earlier notions of static stability it generalizes, it is formulated solely in terms of the players’ incentives to change their strategies close to the point in question. This differs from dynamic stability, which is specific to an extraneous law of motion that dictates the players’ reactions to the incentives. Depending on the choice of the latter, dynamic stability may be weaker than static stability or the two may be incomparable. Static stability is also connected with comparative statics of altruism. In general, internalization of the aggregate payoff or some other measure of social payoff by all players may paradoxically result in a decrease of that payoff. But this is never so if the equilibria involved satisfy static stability.
Keywords: Static stability, evolutionarily stable strategy, continuously stable strategy, risk dominance, potential games, comparative statics, altruism
JEL Classification: C72
Suggested Citation: Suggested Citation