Subgame perfect equilibrium is an economics term used in game theory to describe 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. In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... In game theory, the Nash equilibrium (named after John Nash, who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only his or her own strategy. ... In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ... Reinhard Selten (born October 5, 1930) is a German economist. ... It has been suggested that Game tree be merged into this article or section. ... The Ultimatum game is an experimental economics game in which two parties interact anonymously and only once, so reciprocation is not an issue. ... Buyers bargain for good prices while sellers put forth their best front in Chichicastenango Market, Guatemala. ... Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ... Look up equilibrium in Wiktionary, the free dictionary. ... In game theory, the Nash equilibrium (named after John Nash, who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only his or her 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. ... In game theory, backward induction is one of dynamic programming algorithms used to compute subgame perfect equilibria in sequential games. ... It has been suggested that Game tree be merged into this article or section. ... A non-credible threat is a term used in game theory economics to describe a threat by a player in a sequential game that they would not carry out as it would not be in their best interest to do so. ...

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. In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ... In game theory, the Nash equilibrium (named after John Nash, who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only his or her 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. ... The ordinary meaning of lattice is the basis for several technical usages A cherry lattice pastry A mathematical lattice that is a type of partially ordered set. ... In computer science, a binomial heap is a data structure similar to binary heap but also supporting the operation of merging two heaps quickly. ...

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). Reinhard Selten (born October 5, 1930) is a German economist. ... A mixed strategy is used in game theory economics to describe a strategy comprising 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. ... Node may mean: Node (botany), the place on a plant stem where a leaf is attached Node (physics), a spatial locus along a standing wave where the wave has minimal amplitude Node (networking), a device connected to a network, such as a computer or router Node (computer science), a basic... 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, also known as Wéiqí in Mandarin Chinese (Traditional Chinese: 圍棋; Simplified Chinese: 围棋), is a strategic, deterministic Chinese board game played by alternately placing stones on a grid. ... In finance, an option is a contract whereby one party (the holder or buyer) has the right but not the obligation to exercise a feature of the contract (the option) on or before a future date (the exercise date or expiry). ... 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. 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. 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 an abstract strategy board game for two players. ... In economics, dynamic inconsistency, or time inconsistency, describes a situation where a decision-makers preferences change over time, such that what is preferred at one point in time is inconsistent with what is preferred at another point in time. ... 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.