Consider this "all-pay" auction in which the highest bidder wins and both bidders must pay an amount equal to their bid regardless of whether they win or lose. It is common knowledge that Bidder 1 has a valuation of 2 and Bidder 2 has a valuation of 3. The bids must be whole numbers and Bidder 2 wins in the event of tie. Which of the following are Nash equilibria? Bidder 2 1 2 4 0, 3 0, 2 0, 1 0,0 0, -1 1 1, 0 -1, 2 -1, 1 -1, O -1, -1 Bidder 1 2 0, 0 0, -1 -2, 1 -2, 0 -2, -1 3 -1, O -1, -1 -1, -2 -3, 0 -3, -1 4 -2, 0 -2, -1 -2, -2 -2, -3 -4, -1 Note: The notation [0], [1], etc. denotes the pure strategies in which the bidder bids 0,1, etc. Mixed strategies are represented by weighted averages of the pure strategies, e.g the mixed strategy in which a bidder bids 1 with a probability of 1/4 and 4 with a probability of 3/4 is written as 1/4 [1] + 3/4 [4]. O (5/12 [0] + 1/3 [1] + 1/4 [2], 1/2 [0] + 1/2 [1]) O ([0), [0]) O (2/3 [0] + 1/3 [1], 1/2 [0] + 1/2 [1]) O ([0], [3]) O ([0], [1]) O None of these options O ((1), [0])
Consider this "all-pay" auction in which the highest bidder wins and both bidders must pay an amount equal to their bid regardless of whether they win or lose. It is common knowledge that Bidder 1 has a valuation of 2 and Bidder 2 has a valuation of 3. The bids must be whole numbers and Bidder 2 wins in the event of tie. Which of the following are Nash equilibria? Bidder 2 1 2 4 0, 3 0, 2 0, 1 0,0 0, -1 1 1, 0 -1, 2 -1, 1 -1, O -1, -1 Bidder 1 2 0, 0 0, -1 -2, 1 -2, 0 -2, -1 3 -1, O -1, -1 -1, -2 -3, 0 -3, -1 4 -2, 0 -2, -1 -2, -2 -2, -3 -4, -1 Note: The notation [0], [1], etc. denotes the pure strategies in which the bidder bids 0,1, etc. Mixed strategies are represented by weighted averages of the pure strategies, e.g the mixed strategy in which a bidder bids 1 with a probability of 1/4 and 4 with a probability of 3/4 is written as 1/4 [1] + 3/4 [4]. O (5/12 [0] + 1/3 [1] + 1/4 [2], 1/2 [0] + 1/2 [1]) O ([0), [0]) O (2/3 [0] + 1/3 [1], 1/2 [0] + 1/2 [1]) O ([0], [3]) O ([0], [1]) O None of these options O ((1), [0])
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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5.
![Consider this "all-pay" auction in which the highest bidder wins and both bidders must pay an amount equal to their
bid regardless of whether they win or lose. It is common knowledge that Bidder 1 has a valuation of 2 and Bidder 2 has
a valuation of 3. The bids must be whole numbers and Bidder 2 wins in the event of tie. Which of the following are
Nash equilibria?
Bidder 2
1
2
3
4
0, 3
0, 2
0, 1
0, 0
0, -1
1
1, 0
-1, 2
-1, 1
-1, O
-1, -1
Bidder 1
2
0, 0
0, -1
-2, 1
-2, 0
-2, -1
3
-1, 0
-1, -1
-1, -2
-3, 0
-3, -1
4
-2,0
-2, -1
-2, -2
-2, -3
-4, -1
Note: The notation [0], [1], etc. denotes the pure strategies in which the bidder bids 0,1, etc. Mixed strategies are
represented by weighted averages of the pure strategies, e.g the mixed strategy in which a bidder bids 1 with a
probability of 1/4 and 4 with a probability of 3/4 is written as 1/4 [1] + 3/4 [4].
(5/12 [이 + 1/3 [1] + 1/4 [2], 1/2 [이 + 1/2 [1])
O ([0], [0])
O (2/3 [0] + 1/3 [1], 1/2 [0] + 1/2 [1])
O ([0), [3])
O ([0], [1])
O None of these options
O ((1), [0])](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d6b41d0-2c02-4c1f-addb-01bcbdab0813%2F462ded37-02bc-4f38-9f74-65f69d39d594%2Fk0znl7q_processed.png&w=3840&q=75)
Transcribed Image Text:Consider this "all-pay" auction in which the highest bidder wins and both bidders must pay an amount equal to their
bid regardless of whether they win or lose. It is common knowledge that Bidder 1 has a valuation of 2 and Bidder 2 has
a valuation of 3. The bids must be whole numbers and Bidder 2 wins in the event of tie. Which of the following are
Nash equilibria?
Bidder 2
1
2
3
4
0, 3
0, 2
0, 1
0, 0
0, -1
1
1, 0
-1, 2
-1, 1
-1, O
-1, -1
Bidder 1
2
0, 0
0, -1
-2, 1
-2, 0
-2, -1
3
-1, 0
-1, -1
-1, -2
-3, 0
-3, -1
4
-2,0
-2, -1
-2, -2
-2, -3
-4, -1
Note: The notation [0], [1], etc. denotes the pure strategies in which the bidder bids 0,1, etc. Mixed strategies are
represented by weighted averages of the pure strategies, e.g the mixed strategy in which a bidder bids 1 with a
probability of 1/4 and 4 with a probability of 3/4 is written as 1/4 [1] + 3/4 [4].
(5/12 [이 + 1/3 [1] + 1/4 [2], 1/2 [이 + 1/2 [1])
O ([0], [0])
O (2/3 [0] + 1/3 [1], 1/2 [0] + 1/2 [1])
O ([0), [3])
O ([0], [1])
O None of these options
O ((1), [0])
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