Second-price auction.
There are 10 bidders. Each bidder i values the object at vi > 0. The indices are chosen in a way so that v1 >v2 >···v10 >0.
Each bidder I can submit a bid bi ¥ 0. The bidder whose bid is the highest wins the object. If there are multiple highest bids, then the winner is the bidder whose valuation is the highest (or whose index is the smallest) among the highest bidders. (For example, if bidder 3 and bidder 9 have the highest bid, then bidder 3 is the winner.)
The winner, says bidder I, gets a payoff vi p, where p is the highest bid made by other bidders. Losing players all receive zero payoffs.
(a) Use the definition of Nash equilibrium to explain why (b1 = v1 + 1, b2 = v2,b3 = v3,...,b10 = v10), is a Nash equilibrium,
(b) Is the profile (bi = v6, for i does NOT = 5 and b5 = v1 + 1) a Nash equilibrium? Explain.
(c) Can you find and verify all Nash equilibria in which player 5 wins the object?
Second-price auction.
There are 10 bidders. Each bidder i values the object at vi > 0. The indices are chosen in a way so that v1 >v2 >···v10 >0.
Each bidder I can submit a bid bi ¥ 0. The bidder whose bid is the highest wins the object. If there are multiple highest bids, then the winner is the bidder whose valuation is the highest (or whose index is the smallest) among the highest bidders. (For example, if bidder 3 and bidder 9 have the highest bid, then bidder 3 is the winner.)
The winner, says bidder I, gets a payoff vi p, where p is the highest bid made by other bidders. Losing players all receive zero payoffs.
(a) Use the definition of Nash equilibrium to explain why (b1 = v1 + 1, b2 = v2,b3 = v3,...,b10 = v10), is a Nash equilibrium,
(b) Is the profile (bi = v6, for i does NOT = 5 and b5 = v1 + 1) a Nash equilibrium? Explain.
(c) Can you find and verify all Nash equilibria in which player 5 wins the object?
Related textbook solutions
Related questions
(Please show as much work as possible)
1. You are bidding in a second-price auction for a painting that you value at $800. You estimate that other bidders are most likely to value the painting at between $200 and $600. Which of these is likely to be your best bid?
a. $1,000
b. $800
c. $600
d. $400
2. Which of the following is true about different ways of conducting a private-value auction?
a. A first-price auction is strategically equivalent to a second-price auction.
b. A first-price auction is strategically equivalent to an English auction.
c. A second-price auction is strategically equivalent to an English auction.
d. None of the above
3. Suppose that five bidders with values of $500, $400, $300, $200, and $100 attend an oral auction. Which of these is closest to the winning price?
a. $500
b. $400
c. $300
d. $200
4. In the above auction, if the bidders with the first- and third-highest values ($500 and
$300) collude, which of these is closest to the winning price?
a. $500
b. $400
c. $300
d. $200
5. If a seller is concerned about collusion among bidders, which of the following changes to the auction, should the seller make?
a. Hold frequent, small auctions instead of infrequent large auctions.
b. Conceal the amount of winning bids.
c. Publically announce the name of each auction's winner.
d. Hold a second-price instead of a first-price auction.
6. You're holding an auction to license a new technology that your company has developed. One of your assistants raises a concern that bidders' fear of the winner's curse may encourage them to shade their bids. How might you address this concern?
a. Release your analyst's positive scenario for the technology's future profitability.
b. Release your analyst's negative scenario for the technology's future profitability.
c. Use an oral auction.
d. All of the above
7. In a first-price auction, you bid ________ your value, and in a second-price auction you bid _________ your value.
a. at; above
b. below; above
c. below; at
d. below; below
8. You hold an auction among three bidders. You estimate that each bidder has a value of either $16 or $20 for the item, and you attach probabilities to each value of 50%. What is the expected price? If two of the three bidders collude, what is the price?
9. In Sweden, firms that fail to meet their debt obligations are immediately auctioned off to the highest bidder. (There is no reorganization through Chapter 11 bankruptcy.) The current managers are often high bidders for the company. Why?
10. When a famous painting becomes available for sale, it is often known which museum or collector will be the likely winner. Yet, representatives of other museums that have no chance of winning are actively wooed by the auctioneer to attend anyway. Why?
11. The deities Mars and Venus often do battle to create the weather conditions on Earth. Venus prefers extreme temperatures (especially heat), while Mars prefers temperate conditions. The payoffs (expressed in Points of Wrath) are given below.
|
|
|
Venus |
|
|
|
|
Warm |
Chill |
|
Mars |
Warm |
20 , 0 |
0 , 10 |
|
Chill |
0 , 90 |
20 , 0 |
What is the unique mixed-strategy equilibrium of the above game?
(Let p be the probability of "Warm" for Mars, and q the probability of "Warm" for Venus.)
a) p=9/10, q=1/2
b) p=1/2, q=1/10
c) p=1/2, q=1/2
d) p=1/10, q=1/10
Player 2
|
|
|
H |
D |
|
Player 1 |
H |
0 , 0 |
4 , 1 |
|
D |
1 , 4 |
2 , 2 |
12. The above game is the title of the hawk-dove game and used by evolutionary biologists to describe evolutionary processes. It is also used to model how a business should grow. In the above game, what is the Nash equilibrium in pure strategies and mixed strategies.?
Assume the cost of producing the goods is zero and that each consumer will purchase each good as long as the price is less than or equal to value. Consumer values are the entries in the table.
|
|
Good 1 |
Good 2 |
|
Consumer A |
$2,300 |
$1,700 |
|
Consumer B |
$2,800 |
$1,200 |
13. Suppose the monopolist only sold the goods separately. What price will the monopolist charge for good 1 to maximize revenues for good 1?
a. $2,300
b. $2,800
c. $1,200
d. $1,700
14. What is the total profit to the monopolist from selling the goods separately?
a. $4,500
b. $6,300
c. $7,000
d. $6,000
15. What is a better pricing strategy for the monopolist? At this price, what are the total profits to the monopolist?
a. Bundle the goods at $2,800; Profits = $5,600
b. Bundle the goods at $4,000; Profits = $8,000
c. Charge $2,800 for good 1 and charge $1,700 for good 2; Profits = $4,500
d. Charging the lowest price for each good individually is the best pricing strategy; Profits = $7,000
16. The prisoners' dilemma is an example of
a. a sequential game.
b. a simultaneous game.
c.a shirking game.
d. a dating game
17. Nash equilibrium
a. is where one player maximizes his payoff, and the other doesn't.
b. is where each player maximizes his own payoff given the action of the other player.
c.is where both players are maximizing their total payoff.
d. is a unique prediction of the likely outcome of a game.
