Subgame perfect equilibrium
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In game theory, Subgame perfect equilibrium is an equilibrium such that players' strategies constitute a Nash equilibrium in every subgame of the original game. It may be found by backward induction, an iterative process for solving finite extensive form or sequential games. First, one determines the optimal strategy of the player who makes the last move of the game. Then, the optimal action of the next to last moving player is determined assuming the last player's action as given. The process continues until all player's actions have been determined. Subgame perfect equilibria eliminate non-credible threats.
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 "
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