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In game theory and related fields, a game form, game frame, ruleset, or outcome function is the set of rules that govern a game and determine its outcome based on each player's choices. A game form differs from a game in that it does not stipulate the utilities or payoffs for each agent.^[1]

Mathematically, a game form can be defined as a mapping going from an action space^[2]^[3]—which describes all the possible moves a player can make—to an outcome space. The action space is also often called a message space when the actions consist of providing information about beliefs or preferences, in which case it is called a direct mechanism.^[3] For example, an electoral system is a game form mapping a message space consisting of ballots to a winning candidate (the outcome).^[1] Similarly, an auction is a game form that takes each bidder's price and maps them to both a winner and a set of payments by the bidders.

Often, a game form is a set of rules or institutions designed to implement some normative goal (called a social choice function), by motivating agents to act in a particular way through an appropriate choice of incentives. Then, the game form is called an implementation or mechanism. This approach is widely used in the study of auctions and electoral systems.^[4]

The social choice function represents the desired outcome or goal of the game, such as maximizing social welfare or achieving a fair allocation of resources. The mechanism designer's task is to design the game form in such a way that when each player plays their best response (i.e. behaves strategically), the resulting equilibrium implements the desired social choice function.

References

^ ^a ^b Gibbard, Allan (1978). "Straightforwardness of Game Forms with Lotteries as Outcomes". Econometrica. 46 (3): 595–614. doi:10.2307/1914235. hdl:10419/220562. ISSN 0012-9682. JSTOR 1914235.
^ Roth, Benjamin N.; Shorrer, Ran I. (March 2015). "Mechanism Design in the Presence of a Pre-Existing Game". Working Paper.
^ ^a ^b Blumrosen, Liad; Feldman, Michal (2013-11-01). "Mechanism design with a restricted action space". Games and Economic Behavior. 82: 424–443. doi:10.1016/j.geb.2013.03.005. ISSN 0899-8256.
^ Ozdaglar, Asu. "Game Theory with Engineering Applications" (PDF). Archived (PDF) from the original on 2024-06-29.

Game theory

Traditional game theory

Definitions	Asynchrony Bayesian regret Best response Bounded rationality Cheap talk Coalition Complete contract Complete information Complete mixing 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 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 Cursed 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 Confrontation analysis Contract theory Drama theory Graphical game theory Heresthetic Mean-field game theory Negotiation theory Quantum game theory Social software
Key people	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 people	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 people	John Maynard Smith Robert Axelrod

Mechanism design

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

Other topics
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete

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