# game between a parent and a childDaoud and Salma SalemLuke, Kethan, and Arjun.

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.   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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