# Subgame > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Subgame > Markdown URL: https://mediated.wiki/source/Subgame.md > Source: https://en.wikipedia.org/wiki/Subgame > Source revision: 1182359078 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Subset of a game; used in game theory This article is about subgames in game theory. For subgame as a short video game contained in another, see [minigame](/source/Minigame). In [game theory](/source/Game_theory), a **subgame** is any part (a subset) of a game that meets the following criteria (the following terms allude to a game described in [extensive form](/source/Extensive_form_game)):[1] 1. It has a single initial node that is the only member of that node's [information set](/source/Information_set_(game_theory)) (i.e. the initial node is in a [singleton](/source/Singleton_(mathematics)) information set). 1. If a node is contained in the subgame then so are all of its successors. 1. If a node in a particular [information set](/source/Information_set_(game_theory)) is in the subgame then all members of that information set belong to the subgame. It is a notion used in the [solution concept](/source/Solution_concept) of [subgame perfect Nash equilibrium](/source/Subgame_perfect_equilibrium), a refinement of the [Nash equilibrium](/source/Nash_equilibrium) that eliminates [non-credible threats](/source/Non-credible_threat). The key feature of a subgame is that it, when seen in isolation, constitutes a game in its own right. When the initial node of a subgame is reached in a larger game, players can concentrate only on that subgame; they can ignore the history of the rest of the game (provided they know [what subgame they are playing](/source/Bayesian_game)). This is the intuition behind the definition given above of a subgame. It must contain an initial node that is a singleton information set since this is a requirement of a game. Otherwise, it would be unclear where the player with first move should start at the beginning of a game (but see [nature's choice](/source/Bayesian_game)). Even if it is clear in the context of the larger game which node of a non-singleton information set has been reached, players could not ignore the history of the larger game once they reached the initial node of a subgame if subgames cut across information sets. Furthermore, a subgame can be treated as a game in its own right, but it must reflect the strategies available to players in the larger game of which it is a subset. This is the reasoning behind 2 and 3 of the definition. All the strategies (or subsets of strategies) available to a player at a node in a game must be available to that player in the subgame the initial node of which is that node. ## Subgame perfection One of the principal uses of the notion of a subgame is in the [solution concept](/source/Solution_concept) subgame perfection, which stipulates that an equilibrium strategy profile be a [Nash equilibrium](/source/Nash_equilibrium) in *every subgame*. In a Nash equilibrium, there is some sense in which the outcome is optimal - every player is playing a best response to the other players. However, in some dynamic games this can yield implausible equilibria. Consider a two-player game in which player 1 has a strategy S to which player 2 can play B as a best response. Suppose also that S is a best response to B. Hence, {S,B} is a Nash equilibrium. Let there be another Nash equilibrium {S',B'}, the outcome of which player 1 prefers and B' is the only best response to S'. In a dynamic game, the first Nash equilibrium is implausible (if player 1 moves first) because player 1 will play S', forcing the response (say) B' from player 2 and thereby attaining the second equilibrium (regardless of the preferences of player 2 over the equilibria). The first equilibrium is subgame imperfect because B does not constitute a best response to S' once S' has been played, i.e. in the subgame reached by player 1 playing S', B is not optimal for player 2. If not all strategies at a particular node were available in a subgame containing that node, it would be unhelpful in subgame perfection. One could trivially call an equilibrium subgame perfect by ignoring playable strategies to which a strategy was not a best response. Furthermore, if subgames cut across information sets, then a Nash equilibrium in a subgame might suppose a player had information in that subgame, he did not have in the larger game. ## References 1. **[^](#cite_ref-1)** ["Table of Contents for Morrow, J.D.: Game Theory for Political Scientists"](http://press.princeton.edu/TOCs/c5590.html). press.princeton.edu. Retrieved 2008-03-26. v t e Game theory Glossary Game theorists Games 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 Win–win game Zero-sum game 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 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. 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