Strategic Stability

Strategic stability is a solution concept for finite games introduced by Elon Kohlberg and Jean-Francois Mertens in 1986. Rather than refining the set of equilibria through restrictions on beliefs — as the intuitive criterion, divinity, and universal divinity do — strategic stability defines a set of equilibria that remain robust to all small perturbations of the game. An equilibrium is strategically stable if it is the limit of equilibria in all nearby games, where "nearby" includes perturbations of payoffs, strategies, and the game tree itself.

Kohlberg and Mertens proposed several axioms that any satisfactory solution concept should satisfy, including: existence (every game has at least one stable equilibrium), admissibility (no equilibrium should assign positive probability to weakly dominated strategies), and backward induction (stable equilibria should respect the logic of subgame perfection). They proved that a set-valued solution concept satisfying these axioms exists, and that it selects a subset of sequential equilibria with strong predictive power.

Strategic stability shifts the focus of refinement theory from "what do players believe after surprises?" to "what predictions survive if the game is slightly misspecified?" This is a methodological revolution: instead of restricting beliefs, it restricts the analyst's confidence. A prediction is valid not because it follows from a particular belief assumption, but because it persists across all nearby specifications.

Strategic stability is the refinement program's final form — not because it solved the multiplicity problem, but because it changed the question. Instead of asking "which equilibrium is most reasonable?" it asks "which predictions are robust?" The shift is subtle but profound. Reasonableness is a psychological question; robustness is a mathematical one. And while psychology has never been game theory's strong suit, mathematics is its native tongue. The irony is that by abandoning the belief-based approach, strategic stability returned game theory to what it does best: proving theorems about abstract structures, even if those structures have less and less to do with the games people actually play.