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...

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 22 from punishment. Thus, a child who behaves well and is not pun‑ ished gets a 0; one who behaves badly and is punished gets 1 2 2 5 21; and so on. The parent gets 22 from the child’s bad behavior and 21 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.

(d) How do your answers to parts (b) and (c) differ? Explain the reason for the difference.