What is Subgame Perfect Equilibrium?

In game theory, a subgame perfect equilibrium (SPE) is a refinement of the Nash equilibrium concept used to predict outcomes in sequential games where players take turns making decisions. This article explains the definition of a subgame perfect equilibrium, why it is crucial for eliminating unrealistic outcomes based on non-credible threats, and how to find it using the method of backward induction.

Understanding Subgames and Sequential Games

To understand subgame perfect equilibrium, you must first understand sequential games and subgames. Unlike simultaneous games where players move at the same time, sequential games are played in stages. One player moves, and the next player observes that move before making their own choice. These games are visually represented using a game tree (extensive form).

A subgame is a smaller part of the larger game that meets three conditions: 1. It begins at a single decision node. 2. It includes all the subsequent nodes (and ultimate outcomes) that follow that node. 3. It does not break up any information sets (meaning players have perfect information about how they reached that point).

The entire game itself is always considered a subgame.

Definition of Subgame Perfect Equilibrium

A strategy profile is a subgame perfect equilibrium if it constitutes a Nash equilibrium in every single subgame of the original game.

In a standard Nash equilibrium, players choose strategies that are best responses to each other. However, in sequential games, standard Nash equilibria can sometimes rely on “non-credible threats”—promises to take actions that would actually harm the player making the threat if they were forced to make the choice. Subgame perfect equilibrium rules out these unrealistic threats by requiring that players act rationally at every potential stage of the game, even in parts of the game tree that are never actually reached during play.

Eliminating Non-Credible Threats: An Example

Imagine a market entry game with an Incumbent and an Entrant. 1. The Entrant decides whether to Enter or Stay Out. 2. If the Entrant enters, the Incumbent decides whether to Fight (wage a price war, hurting both) or Accommodate (share the market).

There are two Nash equilibria in this game: 1. (Enter, Accommodate) 2. (Stay Out, Fight)

In the second equilibrium, the Incumbent threatens to “Fight” if the Entrant enters, which scares the Entrant into staying out. However, this threat is non-credible. If the Entrant actually enters, the Incumbent’s best response is to “Accommodate” because fighting would result in a lower payout for the Incumbent.

By applying subgame perfection, we analyze the subgame that starts after the Entrant has entered. In this subgame, the Incumbent’s only rational choice is to Accommodate. Knowing this, the Entrant will choose to Enter. Therefore, the only subgame perfect equilibrium is (Enter, Accommodate).

How to Find SPE: Backward Induction

The standard method for finding a subgame perfect equilibrium in a finite game is backward induction. This process involves working backward from the end of the game tree to the beginning:

1. Start at the terminal decision nodes: Look at the very last decisions to be made in the game. Identify the optimal choice (the one with the highest payoff) for the player moving at each of these final nodes.
2. Prune the tree: Assume that these optimal choices will be made, and substitute the payoffs of these choices into the decision nodes.
3. Move one step backward: Go to the next-to-last decision nodes and determine the optimal choices for the players moving at those stages, given the rational choices that will follow.
4. Repeat: Continue this process backward until you reach the initial node of the game.

The path of decisions generated by this process represents the subgame perfect equilibrium.