Consider a game between a parent and a child. The child can choose to be good (G) or bad (B); the…

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.

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(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.

 

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