Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that player's 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. Economics (deriving from the Greek words οίκω [okos], house, and νέμω [nemo], rules hence household management) is the social science that studies the allocation of scarce resources to satisfy unlimited wants. ... Game theory is a branch of applied mathematics that uses models to study interactions with formalised incentive structures (games). It has applications in a variety of fields, including economics, international relations, evolutionary biology, political science, and military strategy. ... For the 2002 science fiction movie see Equilibrium (2002 movie) Equilibrium or balance is any of a number of related phenomena in the natural and social sciences. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only ones own strategy. ... A minigame is a (usually short) segment of a video game that uses a different style of gameplay than the rest of the game. ... An extensive form game is a specification of a game in game theory. ...

Option pricing example

A subgame perfect Nash equilibrium is a set of strategies for all players optimized 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. A strategy is a long term plan of action designed to achieve a particular goal. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only ones own strategy. ... The term node can refer to: Node, a spatial locus along a standing wave where the wave has minimal amplitude. ... The style or family of a financial option is a general term denoting the class into which the option falls, usually defined by the manner in which the option may be exercised. ... In colloquial usage, a lattice is a structure of crossed laths with open spaces left between them. ...

Finding subgame perfect equilibia

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). Reinhard Selten (born October 5, 1930) is a German economist. ... A mixed strategy is used in game theory economics to describe a strategy compromising of possible moves and a probability distribution which corresponds to how frequently each move is chosen. ...

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. Credibility is the believability of a statement, action, or source, and the ability of the observer to believe the above. ... The term node can refer to: Node, a spatial locus along a standing wave where the wave has minimal amplitude. ... Tic-tac-toe, also called noughts and crosses and many other names, is a paper and pencil game between two players, O and X, who alternate in marking the spaces in a 3×3 board. ... Go is a strategic, two-player board game originating in ancient China between 2000 BC and 200 BC. Go is a popular game in East Asia. ... In finance, an option is a contract whereby the contract buyer has a right to exercise a feature of the contract (the option) on or before a future date (the exercise date). ... Finance studies and addresses the ways in which individuals, businesses and organizations raise, allocate and use monetary resources over time, taking into account the risks entailed in their projects. ...

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. Infact, having seen the first player discard any means of steering his car, the first player's rational options are reduced from "<Discard wheel>, <Keep wheel>" to "<Keep wheel>", leading to a subgame perfect Nash equilibrium. The game of chicken (also referred to as playing chicken) is a game in which two players each drive a vehicle of some sort towards each other, and the first to swerve loses and is humiliated as the chicken. In practice, this sort of game, if played at all, is... Rational may be: the adjective for the state of rationality acting according to the philosophical principles of rationalism a mathematical term for certain numbers; the rational numbers the software company Rational Software; now owned by IBM, and formerly Rational Software Corporation This is a disambiguation page — a navigational aid which...

See also

• Chess
• Dynamic inconsistency
• John Nash
• Minimax theorem
• Solution concept

• Glossary of Game theory

Chess is a board game and mental sport for two players. ... Dynamic inconsistency, or time inconsistency is a term used in game theory economics to describe a situation in a dynamic game, where a players best plan for some future period will not be optimal when that future period arrives; The plan is dynamically inconsistent. ... John Nash may refer to: John Nash (architect) (1752-1835), British architect John Forbes Nash (born 1928), mathematician and recipient of the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel. ... Minimax is a method in decision theory for minimizing the expected maximum loss. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. ...

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.