subgames

Subgames are components of an extensive-form game that themselves meet the formal criteria of a game. In an extensive-form representation, a subgame begins at a single decision node that is not part of any information set containing nodes outside the subgame. Equivalently, the initial node must be a singleton information set. The subgame includes that node and all of its successors, together with the actions, payoffs, and information sets that lie within it, forming a complete extensive-form game.

Subgames can be nested; the entire game is itself a subgame of itself. Subgames are central to the concept of subgame perfect equilibrium (SPE), a refinement of Nash equilibrium that requires the strategy profile to constitute a Nash equilibrium in every subgame. The concept of subgame perfection was introduced by Reinhard Selten. In finite games with perfect information, SPE can be found by backward induction; in games with imperfect information, the concept generalizes to SPNE.

The notion is used to evaluate the credibility of contingent threats and to enable dynamic analysis of strategic interactions. When a game has a subgame following a particular history, players’ strategies must specify actions for that subgame as part of the overall equilibrium. Subgames ensure sequential rationality, forcing consistency of choices at every stage of the game.

Limitations include that not all portions of a game form subgames; the initial node must be compatible with the information structure. Subgames rely on the formal framework of extensive-form games and are a standard tool in economic theory and related fields.