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Some equilibria are stable only because everyone is sure no one will ever slip. Allow for the smallest chance of a mistake, a trembling hand, and those equilibria fall apart, leaving the ones that would survive a real world of fumbles.

Trembling-hand perfection and sequential equilibrium

Different refinements repair different gaps. Trembling-hand perfection tests whether a Nash equilibrium survives small mistakes. Sequential equilibrium applies to extensive-form games and requires sequentially rational strategies plus beliefs obtained as limits of completely mixed play. The game below demonstrates the trembling-hand test; the formal section then separates it from sequential equilibrium.

From earlier lessons

Two quick ones before we start.

Subgame perfection requires optimal play at every...

Perfect Bayesian equilibrium adds, to strategies, a specification of...

Predict, then play

A pure coordination game: only (Top, Left) pays anything, $(1,1)$ ; every other cell pays $(0,0)$ . There are two Nash equilibria, $(T,L)$ and $(B,R)$ . Test each under a small tremble.

Why did that happen?

Take a guess, then reveal the reason

In $(B,R)$ both players choose an action that only pays off in cells that never occur, so nothing punishes them for it, and it counts as a Nash equilibrium. But the moment you admit the other side might tremble to Left, your Top action starts to earn something ( $1$ in the rare Left column) while Bottom still earns nothing. So you strictly prefer Top, and $(B,R)$ unravels. The tremble tests actions that the equilibrium path would otherwise leave untested.

The math

In the game, $(B,R)$ is a Nash equilibrium only weakly: given the other plays Right, Bottom and Top both pay $0$ , so Bottom is a best response but not a strict one, and it is weakly dominated by Top (which pays $1$ against Left and $0$ against Right). Introduce a tremble: the column player intends Right but plays Left with probability $\varepsilon$ . Then Row earns $\varepsilon\cdot 1 = \varepsilon$ from Top and $0$ from Bottom, so for any $\varepsilon > 0$ Row strictly prefers Top, and $(B,R)$ is not robust.

Definition, two distinct refinements

A trembling-hand perfect equilibrium is a limit, as trembles vanish, of equilibria of perturbed games in which every action has strictly positive probability. Sequential equilibrium requires a strategy profile and beliefs that are the limit of totally mixed behavior, so beliefs are consistent even off the path. For finite extensive-form games with perfect recall, the standard refinements nest as follows:

\text{Nash} \supseteq \text{subgame-perfect} \supseteq \text{sequential} \supseteq \text{extensive-form trembling-hand perfect}.

Proper equilibrium is a further normal-form refinement of trembling-hand perfection, but direct comparisons with sequential equilibrium depend on the chosen representation. PBE is not one single formal definition. Under the usual requirements of sequential rationality and Bayes’ rule on path, every sequential equilibrium is a PBE. Sequential equilibrium adds a particular consistency requirement on off-path beliefs, so the reverse can fail for weaker PBE definitions (Selten’s Horse is the classic separating example).

Work it out

Break the weak equilibrium with a tremble

Take $(B,R)$ . Given Right, Row is indifferent between Top and Bottom (both pay $0$ ), so Bottom is only a weak best response. Now let the column player tremble to Left with probability $\varepsilon > 0$ . Row’s payoff to Top becomes $\varepsilon\cdot 1 + (1-\varepsilon)\cdot 0 = \varepsilon$ ; to Bottom it stays $0$ . Since $\varepsilon > 0$ , Top is strictly better, so Row abandons Bottom and $(B,R)$ fails trembling-hand perfection.

Now $(T,L)$ . Given Left trembling to Right with probability $\varepsilon$ , Top pays $1-\varepsilon$ and Bottom pays $0$ , so Top is strictly best; symmetrically Left is strictly best for the column player. $(T,L)$ survives every small tremble, so it is the trembling-hand perfect (and only sensible) equilibrium.

Key idea

Refinements keep only the equilibria that survive small mistakes. An equilibrium resting on a weakly dominated action, tested only by a tremble that “never happens”, is exactly the kind these rules discard.

Check yourself

Check yourselfapply

Why does (Bottom, Right) fail once trembles are allowed?

Check yourselfanalyse

Which ordering of solution concepts is correct, from most permissive to most selective?

Check yourselfbrings back: subgame perfection

Why is subgame perfection not enough to rule out (Bottom, Right) here?

Sources

Kreps, D. M., and Wilson, R. (1982). Reputation and imperfect information. Journal of Economic Theory, 27(2), 253-279.
Selten, R. (1975). Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory, 4(1), 25-55.

Sequential equilibrium and refinements