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 complete information (all players know the complete state of the game) and perfect recall (each player remembers all their previous actions and knowledge throughout the game) has a subgame perfect equilibrium. 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.

Determining the subgame perfect equilibrium by using backward induction is shown below in Figure 1. Strategies for Player 1 are given by {Up, Uq, Dp, Dq}, whereas Player 2 has the strategies among {TL, TR, BL, BR}. There are 4 subgames in this example, with 3 proper subgames.

Using the backward induction, the players will take the following actions for each subgame:

Thus, the subgame perfect equilibrium is {Dp, TL} with the payoff (3, 3).

An extensive-form game with incomplete information is presented below in Figure 2. Note that the node for Player 1 with actions A and B, and all succeeding actions is a subgame. Player 2's nodes are not a subgame as they are part of the same information set.

The first normal-form game is the normal form representation of the whole extensive-form game. Based on the provided information, (UA, X), (DA, Y), and (DB, Y) are all Nash equilibria for the entire game.