Subgame

In 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):^[1]

1. It has a single initial node that is the only member of that node's information set (i.e. the initial node is in a singleton information set).
2. It contains all the nodes that are successors of the initial node.
3. It contains all the nodes that are successors of any node it contains.
4. If a node in a particular information set is in the subgame then all members of that information set belong to the subgame.

It is a notion used in the solution concept of subgame perfect Nash equilibrium, a refinement of the Nash equilibrium that eliminates non-credible threats.

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). 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). 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 subgame perfection, which stipulates that an equilibrium strategy profile be a 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.

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