Trembling-Hand Perfection

An equilibrium robust to small probabilities of mistakes.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

$\sigma^*$ is trembling-hand perfect if it is the limit of a sequence of totally mixed $\varepsilon$ -equilibria as $\varepsilon\to 0$ .

Equivalently, best-responding against small trembles ( $\sigma_{-i}^\varepsilon$ ) still prescribes the original strategy.

Players may occasionally make mistakes, and equilibrium should remain optimal under this perturbation.

It rules out weakly dominated strategies that are fragile to tiny errors.

Selten (1975) motivated the concept using a simple game where weakly dominated strategies survive Nash but not trembling-hand perfection.

In 2-player normal form games, trembling-hand perfection coincides with playing undominated best responses.

Every trembling-hand perfect equilibrium is a Nash equilibrium that uses no weakly dominated strategies.

In the extensive form, the analog is subgame-perfect trembling-hand perfection, strengthened further by sequential equilibrium.

Motivates robust mechanism design and evolutionary robustness.

Commonly invoked to discard weakly dominated equilibria.

Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.