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Decision theory

In game theory, a **subgame perfect equilibrium** is a refinement of a Nash equilibrium used in dynamic games. A strategy set is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. More informally, this means that if (1) the players played any smaller game that consisted of only one part of the larger game and (2) their behavior represents a Nash equilibrium of that smaller game, then their behavior is a subgame perfect equilibrium of the larger game

A common method for determining subgame perfect equilibria is backward induction. Here one first considers the last actions of the game and determine which actions the final mover should take in each possible circumstance to maximize 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 all subgame perfect equilibria.

The Ultimatum game provides an intuitive example of a game with fewer subgame perfect equilibria than Nash equilibria.

## Option pricing example[]

A subgame perfect Nash equilibrium is a set of strategies for all players optimised to take into account the *order* of each player's moves. A strategy is a subgame perfect Nash equilibrium if it leads to the optimal outcome for every player based on a given strategy for all other players at each "sub-game" position. These positions in the extensive form of the game are called "nodes". A common example of the use of nodes and subgame perfect Nash equilibrium is in the estimate of optimal American option exercise through a decision lattice such as a binomial tree.

## Finding subgame perfect equilibria[]

Reinhard Selten proved that any game which can be broken into "sub-games" containing a sub-set of all the available choices in the main game will have a subgame perfect Nash Equilibrium strategy (possibly as a mixed strategy giving non-deterministic sub-game decisions).

The subgame perfect Nash equilibrium is normally deduced by "backward induction" from the various ultimate outcomes of the game, eliminating branches which would involve any player making a move that is not credible (optimal) from that node. An example game of this type is tic-tac-toe, but in theory go has such an optimum strategy for all players. Again, the widest common application of the backward induction technique is in numerical approximations of early-exercise options in finance.

The interesting aspect of the word "credible" in the preceding paragraph is that taken as a whole (disregarding the irreversibility of reaching sub-games) superior strategies exist to subgame perfect strategies, but they are not credible in the sense that a threat to carry them out will harm the player making the threat and prevent that combination of strategies. For instance in the game of "chicken" if one player has the option of ripping the steering wheel from their car they should always take it because it leads to a "sub game" in which their rational opponent is precluded from doing the same thing (and killing them both). The wheel-ripper will always win the game (making his opponent swerve away), and the opponent's threat to suicidally follow suit is not credible. In fact, having seen the first player discard any means of steering his car, the second player's rational options are reduced from "<Discard wheel>, <Keep wheel>" to "<Keep wheel>", leading to a subgame perfect Nash equilibrium.

## See also[]

- Chess
- Dynamic inconsistency
- John Nash
- Minimax theorem
- Solution concept
- Glossary of Game theory

## External links[]

- Example of Extensive Form Games with imperfect information
- Java applet to find a subgame perfect nash equilibrium solution for an extensive form game from gametheory.net, although this has a problem with firewalls.

Simple example: http://www.uni-graz.at/~baigent/pdfs/WS06-07/SPNE_example.pdf

- de:Teilspielperfektes Gleichgewicht

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