In game theory, a sequential game is a game 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 effect. Sequential games are governed by the time axis and represented in the form of decision trees. Sequential games with perfect information 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 player ...

. 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 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 p ...

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 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, 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 games 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 games 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 dist ...

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 mo ...

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''. ...

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 comp ...

, 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, ch ...

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 perfect information, a subgame perfect equilibrium 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