Proper Equilibrium

A refinement of trembling-hand perfection where more costly mistakes are infinitely less likely.

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

Definition

$\sigma^*$ is proper if there exists a sequence $\sigma^n\to\sigma^*$ of fully mixed strategies with $\sigma^n_i(s_i)\le \varepsilon^n \sigma^n_i(s_i')$ whenever $s_i$ is strictly worse than $s_i'$ against $\sigma^n_{-i}$ , and $\varepsilon^n\to 0$ .

Proper equilibria are always trembling-hand perfect and are invariant under elimination of strictly dominated strategies.

Intuition

Players are infinitely more careful about costly errors: a mistake that hurts more is made with vanishingly smaller probability.

Myerson (1978) introduced the concept to address the limitations of trembling-hand perfection.

Worked example

In any normal-form game, proper equilibrium refines trembling-hand perfect by ruling out equilibria that rely on certain domination relationships.

A proper equilibrium always exists in every finite normal-form game (Myerson).

The math

Every proper equilibrium is trembling-hand perfect, and every trembling-hand perfect equilibrium is proper in generic games.

Van Damme (1984): proper equilibria are characterized by a lexicographic hierarchy of caution.

Where it is used

Refined equilibrium selection in signaling and bargaining games.

Used in mechanism design to eliminate fragile Nash equilibria.

More in Game theory

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