Heifetz, Aviad and Peña, Jorge
(2025)
Nash equilibria are extremely unstable in most games under the utility-taking gradient dynamics.
TSE Working Paper, n. 25-1627, Toulouse
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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.
Item Type: | Monograph (Working Paper) |
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Language: | English |
Date: | March 2025 |
Place of Publication: | Toulouse |
Subjects: | B- ECONOMIE ET FINANCE |
Divisions: | TSE-R (Toulouse) |
Institution: | Université Toulouse Capitole |
Site: | UT1 |
Date Deposited: | 17 Mar 2025 09:38 |
Last Modified: | 17 Mar 2025 09:49 |
OAI Identifier: | oai:tse-fr.eu:130433 |
URI: | https://publications.ut-capitole.fr/id/eprint/50660 |