Subgame perfect equilibrium

In game theory, a subgame perfect equilibrium (SPE), or subgame perfect Nash equilibrium (SPNE), is a refinement of the Nash equilibrium concept, specifically designed for dynamic games where players make sequential decisions. A strategy profile is an SPE if it represents a Nash equilibrium in every possible subgame of the original game. Informally, this means that at any point in the game, the players' behavior from that point onward should represent a Nash equilibrium of the continuation game (i.e. of the subgame), no matter what happened before. This ensures that strategies are credible and rational throughout the entire game, eliminating non-credible threats.

Every finite extensive game with perfect recall (each player remembers all their previous actions and knowledge throughout the game) has a subgame perfect equilibrium.^[1] A common method for finding SPE in finite games is backward induction, where one starts by analyzing the last actions the final mover should take to maximize his/her utility and works backward. While backward induction is a common method for finding SPE in finite games, it is not always applicable to games with infinite horizons, or those with imperfect or incomplete information. In infinite horizon games, other techniques, like the one-shot deviation principle, are often used to verify SPE.

Subgame perfect equilibrium necessarily satisfies the one-shot deviation principle and is always a subset of the Nash equilibria for a given game. The ultimatum game is a classic example of a game with fewer subgame perfect equilibria than Nash equilibria.