Consider the following game. Player 1 has an infinite set of pure strategies, which is the interval [0, 1]. Player 2 has only two pure strategies, L and R. When Player 1 chooses his pure strategy x â [0, 1], payoffs to Player 1 are u1(x, L) = 2 â 2x, u1(x, R) = x; and payoffs to Player 2 are u2(x, L) = x, u2(x, R) = 1 â x.
(i) Show that the game has no Nash equilibrium in pure strategies.
(ii) Find all Nash equilibria of the game when Player 2 is allowed to use a mixed strategy. Explain carefully why they are Nash equilibria. Give the payoffs to the two players.
Consider the following game. Player 1 has an infinite set of pure strategies, which is the interval [0, 1]. Player 2 has only two pure strategies, L and R. When Player 1 chooses his pure strategy x â [0, 1], payoffs to Player 1 are u1(x, L) = 2 â 2x, u1(x, R) = x; and payoffs to Player 2 are u2(x, L) = x, u2(x, R) = 1 â x.
(i) Show that the game has no Nash equilibrium in pure strategies.
(ii) Find all Nash equilibria of the game when Player 2 is allowed to use a mixed strategy. Explain carefully why they are Nash equilibria. Give the payoffs to the two players.
Related textbook solutions
Related questions
Problem 1
In each of the three games shown below, let p be the probability that player 1 plays cooperates (and 1- p the probability that player 1 defects), and let q be the probability that Player 2 plays cooperates (and 1- q the probability that player 2 defects).
Prisonerââ¬â¢s Dilemma
| Player 2 | |||
| Player 1 | cooperate | defect | |
| cooperate | 70,70 | 10,80 | |
| defect | 80,10 | 40,40 | |
Stag Hunt
| Player 2 | |||
| Player 1 | cooperate | defect | |
| cooperate | 70,70 | 5,40 | |
| defect | 40,5 | 40,40 | |
Chicken
| Player 2 | |||
| Player 1 | cooperate | defect | |
| cooperate | 70,70 | 50,80 | |
| defect | 80,50 | 40,40 | |
1. For each game, draw a graph with player 1ââ¬â¢s best response function (choice of p as a function of q), and player 2ââ¬â¢s best response function (choice of q as a function of p), with p on the horizontal axis and q on the vertical axis.
2. Using this graphs, find all the Nash equilibriums for the game, both pure and mixed strategy Nash equilibriums (if any). Label these equilibriums on the corresponding graph.
3. In those games that have multiple pure strategy Nash equilibriums, how do the expected payoffs from playing the mixed strategy Nash equilibrium compare with the payoffs from playing the pure strategy Nash equilibriums? Which type of strategy (mixed or pure) would players prefer to play in these games?
Problem 2
Two people are involved in a dispute. Player 1 does not know whether player 2 is strong or weak; she assigns probability ñ to player 2 being strong. Player 2 is fully informed. Each player can either fight or yield. Each player obtains a payoff of 0 is she yields (regardless of the other personââ¬â¢s action) and a payoff of 1 if she fights and her opponent yields. If both players fight, then their payoffs are (-1; 1) if player 2 is strong and (1;-1) if player 2 is weak. The Bayesian game is the following, depending on the type of player 2:
| Y | F | Y | F | ||||||
| Y | 0, 0 | 0, 1 | Y | 0, 0 | 0, 1 | ||||
| F | 1, 0 | -1, 1 | F | 1, 0 | 1, -1 | ||||
| Player 2 is strong (ñ) | Player 2 is weak (1-ñ) | Player 2 is strong (ñ) | |||||||
After writing all the strategies and payoffs in the same matrix, find the Bayesian Nash equilibriums, depending on the value of ñ (ñ ââ°Â¤ 1/2 or ñ ââ°Â¥1/2).
7.
|
1 \ 2 |
L |
R |
|
U |
4, 4 |
0, 3 |
|
D |
5, 0 |
1, 1 |
If this game becomes a perfect information game and is played sequentially, two things can happen: (i) case 1: player 1 moves first and player 2 moves second, and (ii) case 2: player 2 moves first and player 1 moves later. In case 2, the perfect equilibrium strategies and corresponding payoffs are __________ and in case 1, the perfect equilibrium strategies and corresponding payoffs are ____________. This suggests that players will prefer to play ________ rather than __________.
|
A. |
(D,R) (1,1); (U,L) (4,4) ; sequentially under case 1; simultaneously |
|
|
B. |
(D,R) (1,1); (U,L) (4,4) ; sequentially under case 2; sequentially under case 1 |
|
|
C. |
(D,L) (5,0); (U,L) (4,4) ; game under case 2; game under case 1 |
|
|
D. |
(U,L) (4,4); (D,R) (1,1) ; sequentially under case 2; simultaneously |
