game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

, a sequential game is a

game A game is a structured form of play (activity), play, usually undertaken for enjoyment, entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator s ...

where one player chooses their action before the others choose theirs. The other players must have information on the first player's choice so that the difference in time has no

strategic Strategy (from Greek στρατηγία ''stratēgia'', "art of troop leader; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the sense of the "art ...

effect. Sequential games are governed by the time axis and represented in the form of

decision tree A decision tree is a decision support tool that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains condit ...

s. Sequential games with

perfect information In economics, perfect information (sometimes referred to as "no hidden information") is a feature of perfect competition. With perfect information in a market, all consumers and producers have complete and instantaneous knowledge of all market pr ...

can be analysed mathematically using

combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players ...

. Decision trees are the extensive form of dynamic games that provide information on the possible ways that a given game can be played. They show the sequence in which players act and the number of times that they can each make a decision. Decision trees also provide information on what each player knows or does not know at the point in time they decide on an action to take. Payoffs for each player are given at the decision nodes of the tree. Extensive form representations were introduced by

Neumann Neumann is German language, German and Yiddish language, Yiddish for "new man", and one of the List of the most common surnames in Europe#Germany, 20 most common German surnames. People * Von Neumann family, a Jewish Hungarian noble family A� ...

and further developed by Kuhn in the earliest years of game theory between 1910–1930.

Repeated game In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game (called a stage game). The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a ...

s are an example of sequential games. Players perform a stage game and the results will determine how the game continues. At every new stage, both players will have complete information on how the previous stages had played out. A discount rate between the values of 0 and 1 is usually taken into account when considering the payoff of each player. Repeated games illustrate the

psychological Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries between t ...

aspect of games, such as

trust Trust often refers to: * Trust (social science), confidence in or dependence on a person or quality It may also refer to: Business and law * Trust law, a body of law under which one person holds property for the benefit of another * Trust (bus ...

and

revenge Revenge is committing a harmful action against a person or group in response to a grievance, be it real or perceived. Francis Bacon described revenge as a kind of "wild justice" that "does... offend the law ndputteth the law out of office." Pr ...

, when each player makes a decision at every stage game based on how the game has been played out so far. Unlike sequential games,

simultaneous game In game theory, a simultaneous game or static game is a game where each player chooses their action without knowledge of the actions chosen by other players. Simultaneous games contrast with sequential games, which are played by the players takin ...

s do not have a time axis so players choose their moves without being sure of the other players' decisions. Simultaneous games are usually represented in the form of payoff matrices. One example of a simultaneous game is

rock-paper-scissors Rock paper scissors (also known by other orderings of the three items, with "rock" sometimes being called "stone," or as Rochambeau, roshambo, or ro-sham-bo) is a hand game originating in China, usually played between two people, in which each p ...

, where each player draws at the same time not knowing whether their opponent will choose rock, paper, or scissors.

Extensive form An extensive-form game is a specification of a game in game theory, allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, th ...

representations are typically used for sequential games, since they explicitly illustrate the sequential aspects of a game.

Combinatorial game Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players ...

s are also usually sequential games. Games such as

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to disti ...

infinite chess Infinite chess is any variation of the game of chess played on an unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathematicians, both as a playable game and as a mod ...

backgammon Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back nearly 5,000 years to the regions of Mesopotamia and Pe ...

tic-tac-toe Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. T ...

and Go are examples of sequential games. The size of the decision trees can vary according to

game complexity Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity. Measures of game comple ...

, ranging from the small

game tree In the context of Combinatorial game theory, which typically studies sequential games with perfect information, a game tree is a graph representing all possible game states within such a game. Such games include well-known ones such as chess, check ...

of tic-tac-toe, to an immensely complex game tree of chess so large that even computers cannot map it completely. Games can be either strictly determined or determined. A strictly determined game only has one individually rational payoff profile in the 'pure' sense. For a game to be determined it can have only one individually rational payoff profile in the mixed sense. In sequential games with

, a

subgame perfect equilibrium In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every ...

can be found by

backward induction Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by examining the last point at which a decision is to be made and then identifying wha ...

References

{{DEFAULTSORT:Sequential Game Game theory Game theory game classes Mathematical and quantitative methods (economics)

See also

References