One-shot deviation principle

In game theory, the one-shot deviation principle is a principle used to determine whether a strategy in a sequential game constitutes a subgame perfect equilibrium. An SPE is a Nash equilibrium where no player has an incentive to deviate in any subgame. It is closely related to the principle of optimality in dynamic programming.
The one-shot deviation principle states that a strategy profile of a finite multi-stage extensive-form game with observed actions is an SPE if and only if there exist no profitable single deviation for each subgame and every player. In simpler terms, if no player can profit by deviating from their original strategy via a single action, then the strategy profile is an SPE.
The one-shot deviation principle is very important for infinite horizon games, in which the backward induction method typically doesn't work to find SPE. In an infinite horizon game where the discount factor is less than 1, a strategy profile is a subgame perfect equilibrium if and only if it satisfies the one-shot deviation principle.

Definitions

The following is the paraphrased definition from Watson.
To check whether strategy s is a subgame perfect Nash equilibrium, we have to ask every player i and every subgame, if considering s, there is a strategy s’ that yields a strictly higher payoff for player i than does s in the subgame. In a finite multi-stage game with observed actions, this analysis is equivalent to looking at single deviations from s, meaning s’ differs from s at only one information set. Note that the choices associated with s and s’ are the same at all nodes that are successors of nodes in the information set where s and s’ prescribe different actions.

Example

Consider a symmetric game with two players in which each player makes binary choice decisions, A or B, in each of three stages. In each stage, the players observe the choices made in the previous stages. Note that each player has 21 information sets, one in the first stage, four in the second stage, and 16 in the third stage. The single-deviation condition requires checking each of these information sets, asking in each case whether the expected payoff of the player on the move would strictly increase by deviating at only this information set.