. 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 pu n-
ishment. 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 -1from inflicting pu nishment.
(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 down 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.
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