1. For the following 2-player nonzero-sum game, simplify the game as much as you can by eliminating dominated strategies.
|
|
L
|
C
|
R
|
|
U
|
1, 3
|
4, 5
|
9, 0
|
|
M
|
2, 7
|
2, 4
|
4, 0
|
|
D
|
8, 0
|
3, 5
|
1, 2
|
2. List the Pareto efficient outcomes of the game in Question 1 (e.g., (U,L),. . .).
3. Consider the following general 2 × 2 zero-sum game, with the payoff numbers a, b, c, d all distinct.
a) Show that if (U, L) are the maxmin strategies then b > a > c.
b) Show that if (U, L) are the maxmin strategies then at least one player has a dominated strategy.
4. For each of the following two nonzero-sum games, is the game strategically equivalent to a ZSG? If your answer is "YES," give a strategically equivalent ZSG and show why it is strategically equivalent. If your answer is "NO," explain why not.
a)
|
U
|
L
4, -2
|
R
0, 0
|
|
D -2, 1
(2 points)
|
6, -3
|
|
|
L
|
R
|
|
U
|
-1, 2
|
4, 4
|
|
D
|
-2, 0
|
3, 2
|
|
|
b
5. Consider the following sequential ZSG. First, nature chooses heads or tails, each with probability one-half. Player 1 then sees nature's choice, and chooses heads or tails. Player 2 then sees player 1's choice but not nature's choice, and chooses heads or tails, which ends the game. If player 2's choice matches nature's choice, player 2 wins a dollar from player 1. If player 2's choice does not match nature's choice, player 1 wins a dollar from player 2. Draw an extensive form for this game (Hint: look at the game tree on page 393 in the text).