Economics 4360 – Two players play the following prisoners

Subject: Economics / General Economics
Question
Game Theory homework. Only #3 and #4

Spring Semester 2017
Economics 4360 Game Theory
Homework Assignment # 6

1. Dixit, Skeath and Reiley, chapter 11, question U2.
2. Dixit, Skeath and Reiley, chapter 11, question U7.

3. Two players play the following prisoners’ dilemma game repeatedly. Suppose that L = 1, D = 4,
C = 5, and H = 8. The one period interest rate is r = 5%. At the beginning of each period, the
two play the game with probability p. (a) Describe the tit-for-tat and grim strategy for a player in this game.
(b) Suppose that p = 0.5 would (tit-for-tat, tit-for-tat) be an Nash equilibrium of the repeated
game? What about p = 0.8?

4. A neutral referee runs the following game. There are two players, Row and Column. The referee
gives two cards to each: 3 and 7 to Row and 5 and 8 to Column. This is common knowledge. Then
each player is asked to hand over to the referee either his high card or his low card. The referee
hands out payoffs —not from the players pockets. If Row chooses his Low card, 3, then Row gets
$3; if he chooses his high card, 7, then Column gets $7. If Column chooses his Low card, 5, then
Column gets $5; if he chooses his High card, 8, then Row gets $8.
(a) Write down the game table for the game.
(b) Now suppose that at stage 1, 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 must be nonnegative.
When both have made their Stage 1 choices, these are publicly disclosed. At stage 2, the two
make their choices of cards 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, the
referee disposes the escrow account as follows. If Row chooses his high card, the referee hands
over to Row the sum that Column put into the account; if Row chooses his low card, Columns
sum reverts back to him and Row gets nothing. Similar for Columns card choice. All these
rules are common knowledge.
i. Let r be the amount Row puts into the escrow account and c be Column’s amount. Rewrite
the game table for the new situation.
ii. Choose a pair of (r, c) such that High Card is the dominant strategy for both players.
5. A firm has two managers. If both managers expend “high effort”, each earns $150k a year. If both
expend “low effort”, each earns $100k. But if one manager shows high effort and the other shows
low effort, the high effort manager is paid $150k plus a $50k bonus, but the other manager gets
$80k. The cost of expending low effort is zero.
(a) Suppose expending high effort costs an equivalent of $60k a year. Write down the game table
for the game. Is it a prisoners dilemma game? (b) Suppose expending high effort costs $110k a year. Draw the game table again. Find the Nash
equilibrium.
6. A worker has two types, hard-working (H) and shirking (L). A manager must decide whether to
hire the worker, only knowing that the worker is type H with probability p. The payoffs for all
possible outcomes are shown in the following game tree. For what values of p will Manager hire
Worker?

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