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 behaviour, but hurt worth -2 from punishment. Thus, a child who behaves well and is not punished gets a 0; 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.
(d) How do your answers to parts (b) and (c) differ? Explain the reason for the difference
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 behaviour, but hurt worth -2 from punishment. Thus, a child who behaves well and is not punished gets a 0; 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.
(d) How do your answers to parts (b) and (c) differ? Explain the reason for the difference
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Consider the following game, which comes from James Andreoni and Hal Varian at the University of Michigan. A neutral referee runs the game.There are two players, Row and Column. The referee gives two cards to each:
2 and 7 to Row and 4 and 8 to Column. This is common knowledge. Then, playing simultaneously and independently, each player is · asked to hand over to the referee either his high card or his low card. The referee hands out payoffs- which come from a central kitty, not from the players' pockets- which come from a central kitty, not from the players' pockets-that are measured in dollars and depend on the cards that he collects. If row chooses his low card, 2, then row gets $2; if he choses his High card, 7 then Column gets $7. If column chooses his low card, 4, then column gets $4; if he chooses his high card, 8, then row gets $8.
(a) Show that the complete payoff table is as follows:
| Column | Column | ||
| low | high | ||
| row | Low | 2,4 | 10,0 |
| row | High | 0,11 | 8,7 |
(b) What is the Nash equilibrium? Verify that this game is a prisoners' dilemma.
Now suppose the game has the following stages. The referee hands
out cards as before; who gets what cards is common knowledge. At stage
I, each player, out of his own pocket, can hand . over a sum of money,
which the referee is to hold in an escrow account. This amount can be
zero 'out cannot 'oe negative. When both have made then Stage l choices,
these are publicly disclosed. Then at stage II, the two make their choices
of cards, again simultaneously and independently. The referee hands
out payoffs from the central kitty in the same way as in the single-stage
game before. In addition, he disposes of the escrow account as follows.
If Column chooses his high card, the referee hands over to Column the
sum that Row put into the account; if Column chooses his low card,
Row's sum reverts back to him. The disposition of the sum that Column
deposited depends similarly on Row's card choice. All these rules are
common knowledge.
(c) Find the rollback (subgame-perfect) equilibrium of this two-stage game.
Does it resolve the prisoners' dilemma? What is the role of the escrow account?
