Consider a game between a parent and a child. The child can choose to be good (G) or bad (B); the parent can punish the child (P) or not (N).
The child gets enjoyment worth a 1 from bad behavior, but hurt worth -2 from punishment. Thus a child who behaves well and is not punished gets a O; one who behaves badly and is punished gets 1 - 2 = - 1;and so on.
The parent gets - 2 from the child's bad behavior and - 1 from inflicting punishment.
(a) Set up this game as a simultaneous-move game, and find the equilibrium.
(b) Next, suppose that the child chooses G or B first and that the parent chooses its P or N after having observed the child's action. Draw the game tree and find the subgame-perfect equilibrium.
(c) Now suppose that before the child acts, the parent can commit to a strategy.
For example, the threat "P if B" ("If you behave badly, I will punish you"). How many such strategies does the parent have? Write the table for this game. Find all pure-strategy Nash equilibria.