n-player game - Wikipedia

In game theory, an n-player game is a game which is well defined for any number of players. This is usually used in contrast to standard 2-player games that are only specified for two players. In defining n-player games, game theorists usually provide a definition that allow for any (finite) number of players.^[1] The limiting case of $n\to \infty$ is the subject of mean field game theory.^[2]

Changing games from 2-player games to n-player games entails some concerns. For instance, the Prisoner's dilemma is a 2-player game. One might define an n-player Prisoner's Dilemma where a single defection results everyone else getting the sucker's payoff. Alternatively, it might take certain amount of defection before the cooperators receive the sucker's payoff. (One example of an n-player Prisoner's Dilemma is the Diner's dilemma.)

Analysis

n-player games can not be solved using minimax, the theorem that is the basis of tree searching for 2-player games. Other algorithms, like maxⁿ, are required for traversing the game tree to optimize the score for a specific player.^[3]

References

^ Binmore, Ken (2007). Playing for Real : A Text on Game Theory:. Oxford University Press. p. 522. ISBN 9780198041146.
^ Fischer, Markus (2017). "On the connection between symmetric N-player games and mean field games". Annals of Applied Probability. 27 (2): 757–810. arXiv:1405.1345. doi:10.1214/16-AAP1215.
^ Luckhardt, Carol A.; Irani, Keki B. (11 August 1986). An Algorithmic Solution of N-Person Games (PDF). AAAI '86. pp. 158–162. Archived (PDF) from the original on 19 April 2024. Retrieved 20 August 2024.

Game theory

Traditional Game Theory

Definitions	Asynchrony Bayesian regret Best response Bounded rationality Cheap talk Coalition Complete contract Complete information Complete mixing Confrontation analysis Conjectural variation Contingent cooperator Coopetition Cooperative game theory Dynamic inconsistency Escalation of commitment Farsightedness Game semantics Hierarchy of beliefs Imperfect information Incomplete information Information set Move by nature Mutual knowledge Non-cooperative game theory Non-credible threat Outcome Payoff Perfect information Perfect recall Ply Preference Rationality Sequential game Simultaneous action selection Spite Strategic complements Strategic dominance Strategic form Strategic interaction Strategic move Strategy Subgame Succinct game Topological game Tragedy of the commons Uncorrelated asymmetry
Equilibrium Concepts	Backward induction Bayes correlated equilibrium Bayesian efficiency Bayesian game Bayesian Nash equilibrium Berge equilibrium Bertrand–Edgeworth model Coalition-proof Nash equilibrium Core Correlated equilibrium Edgeworth price cycle Epsilon-equilibrium Gibbs equilibrium Incomplete contracts Inequity aversion Individual rationality Iterated elimination of dominated strategies Markov perfect equilibrium Mertens-stable equilibrium Nash equilibrium Open-loop model Pareto efficiency Payoff dominance Perfect Bayesian equilibrium Price of anarchy Program equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Rational agent Rationalizability Rationalizable strategy Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfect equilibrium Trembling hand equilibrium
Strategies	Appeasement Bid shading Cheap talk Collusion Commitment device De-escalation Deterrence Escalation Fictitious play Focal point Grim trigger Hobbesian trap Markov strategy Max-dominated strategy Mixed strategy Pure strategy Tit for tat Win–stay, lose–switch
Games	All-pay auction Battle of the sexes Nash bargaining game Bertrand competition Blotto game Centipede game Coordination game Cournot competition Deadlock Dictator game Trust game Diner's dilemma Dollar auction El Farol Bar problem Electronic mail game Gift-exchange game Guess 2/3 of the average Keynesian beauty contest Kuhn poker Lewis signaling game Matching pennies Obligationes Optional prisoner's dilemma Pirate game Prisoner's dilemma Public goods game Rendezvous problem Rock paper scissors Stackelberg competition Stag hunt Traveler's dilemma Ultimatum game Volunteer's dilemma War of attrition
Theorems	Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite theorem Gibbs lemma Glicksberg's theorem Kakutani fixed-point theorem Kuhn's theorem One-shot deviation principle Prim–Read theory Rational ignorance Rational irrationality Sperner's lemma Zermelo's theorem
Subfields	Algorithmic game theory Behavioral game theory Behavioral strategy Compositional game theory Contract theory Drama theory Graphical game theory Heresthetic Mean-field game theory Negotiation theory Quantum game theory Social software
Key Figures	Albert W. Tucker Alvin E. Roth Amos Tversky Antoine Augustin Cournot Ariel Rubinstein David Gale David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Herbert Scarf Hervé Moulin Jean Tirole Jean-François Mertens Jennifer Tour Chayes Ken Binmore Kenneth Arrow Leonid Hurwicz Lloyd Shapley Martin Shubik Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert Axelrod Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey

Combinatorial Game Theory

Core Concepts	Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game tree Impartial game Misère Partisan game Solved game Sprague–Grundy theorem Strategy-stealing argument Zugzwang
Games	Chess Chomp Clobber Cram Domineering Hackenbush Nim Notakto Subtract a square Sylver coinage Toads and Frogs
Mathematical Tools	Mex Nimber On Numbers and Games Star Surreal number Winning Ways for Your Mathematical Plays
Search Algorithms	Alpha–beta pruning Expectiminimax Minimax Monte Carlo tree search Negamax Paranoid algorithm Principal variation search
Key Figures	Claude Shannon John Conway John von Neumann

Evolutionary Game Theory

Core Concepts	Bishop–Cannings theorem Evolution and the Theory of Games Evolutionarily stable set Evolutionarily stable state Evolutionarily stable strategy Replicator equation Risk dominance Stochastically stable equilibrium Weak evolutionarily stable strategy
Games	Chicken Stag hunt
Applications	Cultural group selection Fisher's principle Mobbing Terminal investment hypothesis
Key Figures	John Maynard Smith Robert Axelrod

Mechanism design

Core Concepts	Algorithmic mechanism design Bayesian-optimal mechanism Incentive compatibility Market design Monotonicity Participation constraint Revelation principle Strategyproofness Vickrey–Clarke–Groves mechanism
Theorems	Myerson–Satterthwaite theorem Revenue equivalence
Applications	Digital goods auction Knapsack auction Truthful cake-cutting