Game theory III: Subgames

In game theory, a subgame is a subset of any game that includes an initial node (which has to be independent from any information set) and all its successor nodes. It’s quite easy to understand how subgames work using the extensive form when describing the game. In the following game tree there are six separate subgames other than the game itself, two of them containing two subgames each.

Subgames

A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. Consider the following game: player 1 has to decide between going up or down (U/D), while player 2 has to decide between going left or right (L/R). In this case, we can represent this game using the strategic form by laying down all the possible strategies for player 2:

-go Right if player 1 goes Up, go Left otherwise;

-go Left if player 1 goes Up, go Right otherwise;

-go Right no matter what;

-go Left no matter what.

We can see how this game is described using the extensive form (game tree on the left) and using the strategic form (game matrix on the left). Since this is a sequential game, we must describe all possible outcomes depending on player 2 decisions, as seen in the game matrix. 

Perfect subgame equilibrium

If we look for the equilibrium of this game, considered as a whole, we find that Up-Left is a Nash equilibrium (red). However, it’s not a perfect equilibrium. In order to find the subgame-perfect equilibrium, we must do a backwards induction, starting at the last move of the game, then proceed to the second to last move, and so on. In this particular case, we know that player 2 will choose Left if player 1 goes Up, and Right if player 1 goes Down, since these are the moves that maximise his utility. Because there is complete information (and therefore each player’s payoffs are known), player 1 knows these choices in advance, and will therefore choose to go Down, because the payoff will be greater. Therefore, Down-Right is the perfect subgame equilibrium (green).

 

Now, let’s see what the Folk theorem used in game theory tells us. We’ll need it to understand how stable collusion agreements can be.