Nash equilibria are extremely unstable in most games under the utility-taking gradient dynamics

Aviad Heifetz and Jorge Peña

No 25-1627, TSE Working Papers from Toulouse School of Economics (TSE)

Abstract: In the standard continuous-time choice-taking gradient dynamics in smooth two-player games, each player implicitly assumes that their opponent momentarily main-tains their last choice. Contrastingly, in the utility-taking gradient dynamics each player implicitly assumes that their opponent momentarily maintains their utility level, by marginally adjusting their choice to that effect. Somewhat surprisingly, employing a transversality argument we find that, in an open and dense set of smooth games, this dynamics is undefined at Nash equilibria. This occurs because, at a Nash equilibrium, the opponent’s indifference curve is not locally a function of one’s own strategy, mak-ing it impossible to specify an opponent’s adjustment that would maintain their utility in response to one’s own marginal deviation from Nash behavior. Furthermore, when approaching a Nash equilibrium of such a generic game, the utility-taking gradient dy-namics either accelerates without bound towards the equilibrium or diverges away from it with unbounded speed.

Keywords: gradient; dynamics (search for similar items in EconPapers)
Date: 2025-03
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