Sequential game

In game theory, a sequential game is a game where one player chooses their action before the others choose theirs.^[1] 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.

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.^[2]

Repeated games 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 and revenge, when each player makes a decision at every stage game based on how the game has been played out so far.^[2]

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, where each player draws at the same time not knowing whether their opponent will choose rock, paper, or scissors. Extensive form 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, infinite chess, backgammon, tic-tac-toe and Go are examples of sequential games. The size of the decision trees can vary according to game complexity, ranging from the small game tree of tic-tac-toe, to an immensely complex game tree of chess so large that even computers cannot map it completely.^[3]

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.^[4]

In sequential games with perfect information, a subgame perfect equilibrium can be found by backward induction.^[5]

References

^ Brocas; Carrillo; Sachdeva (2018). "The Path to Equilibrium in Sequential and Simultaneous Games". Journal of Economic Theory. 178: 246–274. doi:10.1016/j.jet.2018.09.011. S2CID 12989080.
^ ^a ^b Aumann, R. J. Game Theory.^{[full citation needed]}
^ Claude Shannon (1950). "Programming a Computer for Playing Chess" (PDF). Philosophical Magazine. 41 (314).
^ Aumann, R.J. (2008), Palgrave Macmillan (ed.), "Game Theory", The New Palgrave Dictionary of Economics, London: Palgrave Macmillan UK, pp. 1–40, doi:10.1057/978-1-349-95121-5_942-2, ISBN 978-1-349-95121-5, retrieved 2021-12-08
^ Aliprantis, Charalambos D. (August 1999). "On the backward induction method". Economics Letters. 64 (2): 125–131. doi:10.1016/s0165-1765(99)00068-3.

v t e Topics of game theory
Definitions	Congestion game Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game Mechanism design
Equilibrium concepts	Bayes correlated equilibrium Bayesian Nash equilibrium Berge equilibrium Core Correlated equilibrium Coalition-proof Nash equilibrium Epsilon-equilibrium Evolutionarily stable strategy Gibbs equilibrium Mertens-stable equilibrium Markov perfect equilibrium Nash equilibrium Pareto efficiency Perfect Bayesian equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Risk dominance Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfection Trembling hand equilibrium
Strategies	Appeasement Backward induction Bid shading Collusion Cheap talk De-escalation Deterrence Escalation Forward induction Grim trigger Markov strategy Pairing strategy Dominant strategies Pure strategy Mixed strategy Strategy-stealing argument Tit for tat
Classes of games	Auction Bargaining problem Global game Intransitive game Mean-field game n-player game Perfect information Large Poisson game Potential game Repeated game Screening game Signaling game Strictly determined game Stochastic game Symmetric game Zero-sum game
Games	Go Chess Infinite chess Checkers All-pay auction Prisoner's dilemma Gift-exchange game Optional prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Lewis signaling game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Electronic mail game Rock paper scissors Pirate game Dictator game Public goods game Blotto game War of attrition El Farol Bar problem Fair division Fair cake-cutting Bertrand competition Cournot competition Stackelberg competition Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Induction puzzles Trust game Princess and monster game Rendezvous problem
Theorems	Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Antoine Augustin Cournot Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin John Conway Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
Search optimizations	Alpha–beta pruning Aspiration window Principal variation search max^n algorithm Paranoid algorithm Lazy SMP
Miscellaneous	Bounded rationality Combinatorial game theory Confrontation analysis Coopetition Evolutionary game theory Glossary of game theory List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons

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