| Subgame Perfect Equilibrium | |
|---|---|
| Solution concept in game theory | |
| Relationship | |
| Subset of | Nash equilibrium |
| Intersects with | Evolutionarily stable strategy |
| Significance | |
| Proposed by | Reinhard Selten |
| Used for | Extensive form games |
| Example | Ultimatum game |
In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every 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. Every finite extensive game with perfect recall has a subgame perfect equilibrium.[1] Perfect recall is a term introduced by Harold W. Kuhn in 1953 and "equivalent to the assertion that each player is allowed by the rules of the game to remember everything he knew at previous moves and all of his choices at those moves".[2]
A common method for determining subgame perfect equilibria in the case of a finite game is backward induction. Here one first considers the last actions of the game and determines which actions the final mover should take in each possible circumstance to maximize his/her utility. One then supposes that the last actor will do these actions, and considers the second to last actions, again choosing those that maximize that actor's utility. This process continues until one reaches the first move of the game. The strategies which remain are the set of all subgame perfect equilibria for finite-horizon extensive games of perfect information.[1] However, backward induction cannot be applied to games of imperfect or incomplete information because this entails cutting through non-singleton information sets.
A subgame perfect equilibrium necessarily satisfies the one-shot deviation principle.
The set of subgame perfect equilibria for a given game is always a subset of the set of Nash equilibria for that game. In some cases the sets can be identical.
The ultimatum game provides an intuitive example of a game with fewer subgame perfect equilibria than Nash equilibria.