Subgame perfect equilibrium

In finite extensive game with perfect recall has the subgame perfect equilibrium. Perfect recall is a term featured by Harold W. Kuhn in 1953 and "equivalent to the assertion that used to refer to every one of two or more people or matters player is enables by the rules of the game to remember everything he knew at preceding moves as alive as all of his choices at those moves".

A common method for imposing subgame perfect equilibria in the case of a finite game is backward induction. Here one first considers the last actions of the game in addition to determines which actions themover should shit in regarded and forwarded separately. possible circumstance to maximize his/her utility. One then supposes that the last actor will hit these actions, and considers theto last actions, again choosing those that maximize that actor's utility. This process manages until one reaches the first move of the game. The strategies which cover are the generation of any subgame perfect equilibria for finite-horizon extensive games of perfect information. 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 lets an intuitive example of a game with fewer subgame perfect equilibria than Nash equilibria.

Example

Determining the subgame perfect equilibrium by using backward induction is filed below in Figure 1. Strategies for Player 1 are precondition 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 coming after or as a a object that is said of. actions for used to refer to every one of two or more people or matters 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 non a subgame as they are part of the same information set.

The first normal-form game is the normal form relation 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.

Thenormal-form game is the normal form version of the subgame starting from Player 1's second node with actions A and B. For the moment normal-form game, the Nash equilibrium of the subgame is A, X.

For the entire game Nash equilibria DA, Y and DB, Y are non subgame perfect equilibria because the conduct of Player 2 does not symbolize a Nash Equilibrium. The Nash equilibrium UA, X is subgame perfect because it incorporates the subgame Nash equilibrium A, X as element of its strategy.

To solve this game, first find the Nash Equilibria by mutual best response of Subgame 1. Then use backwards induction and plug in A,X → 3,4 so that 3,4 become the payoffs for Subgame 2.

The dashed line indicates that player 2 does not know if player 1 will play A or B in a simultaneous game.

Player 1 chooses U rather than D because 3 > 2 for Player 1's payoff. The resulting equilibrium is A, X → 3,4.

Thus, the subgame perfect equilibrium through backwards induction is UA, X with the payoff 3, 4.