2nd Price Auctions
Assumptions
1. There is a single unit of the good, a Louis 14^{th} escritoire, to be sold.
2. There are two buyers: Kristy and Sutheby.
a. Kristy values the good at , although Sutheby thinks her valuation of the good is with probability 1/2 and with probability 1/2.
b. If Kristy wins the desk then her payoff is - price. If she doesn't win the desk then her payoff is 0.
c. Sutheby has the same attributes as Kristy, but he values the desk at. Kristy doesn't know Sutheby's
valuation and thinks that
and are equally likely.
If he wins the desk then his payoff is -price. If he loses then his payoff is 0.
3. >
4. The seller of the desk wants to maximize the expected selling price.
Rules
Kristy and Sutheby don't know each other's valuation of the desk and they are not allowed to exchange this information.
Each player submits a sealed bid. The sealed bids are opened and announced by the auctioneer.
The higher bidder gets the object but pays the lower bid as the price. If there is a tie then the winner is decided by the toss of a fair coin.
A Dominant Strategy
Kristy knows that she values the desk at . If Kristy bids and Sutheby turns out to value the desk at then she wins the desk. When bidding truthfully she can compute her probability of winning as follows: Given that Sutheby values the good at , Kristy's probability of winning is 1 {in the notation of probability P(win|)}. Before the start of the auction Kristy believes that Sutheby values the desk at with probability 1/2 {in the notation of probability P()}.
Alternatively, if Sutheby values the good at , which she believes occurs with probability 1/2, then Kristy can win the good with the toss of a coin. She wins the good 50% of the time when there is a tie valuation/bid.
Now put the pieces together to get Kristy's ex ante probability of winning the auction: 1(1/2)+(1/2)(1/2) = 3/4.
Kristy's expected total gain is (3/4)()+(1/4)(0).
But don't forget that she has to pay for the desk. Since she values the desk at and she thinks that she pays a price for the desk 50% of the time. She pays with probability 1/2. The other 50% of the time her opposite is also of type and the auction is settled by a coin toss, so she pays with probabity (1/2)(1/2).
Kristy's net gain will be (3/4) - /2 - /4 = (-)/2.
Can Kristy increase her expected payoff by bidding less than her valuation? Remember she doesn't know Sutheby's valuation.
Can you use the same line of reasoning to demonstrate that Sutheby will bid and that his net surplus is 0?
The seller, Louis 14^{th} , doesn't know whether either or both bidders are of type or type . He thinks that they are equally likely. We can calculate his expected revenue from the auction from Kristy's expected payment, /2 + /4 (one quarter of the time she wins against a type player and pays her bid of and half the time she bids truthfully and wins against a -type player).
We can analyze this auction using the machinery that we adopted for static games of incomplete information. We'll do it from Kristy's perspective. Begin by setting up the two possible games that she might be playing.
Sutheby |
||||||
- type |
- type |
|||||
Kristy |
0, 0 | , 0 | 0, | , 0 | ||
0, | , | 0, 0 | , 0 |
Kristy believes that Sutheby is a - type bidder with probability 1/2. Furthermore, when we convert the game to one of imperfect information, Sutheby has four strategic plans. Therefore, the normal form of the game, with expected payoffs, can be written as:
Sutheby |
|||||
, | , | , | , | ||
Kristy |
0, | , (0, 0) | , (0, 0) | ||
0, (, 0) | , |
The game is dominance solvable. Truthful bidding by Sutheby dominates his other three bidding strategies. Kristy sees that her dominant strategy is to bid truthfully as well.
We can also do the 2nd price auction from Sutheby's perspective. The payoffs are given in the following table.
Kristy | ||||||
Sutheby | , 0 | 0, 0 | , | 0, 0 | ||
0, | 0, | 0, 0 | 0, 0 |
The normal form of the game is then:
Kristy | |||||
Sutheby | , | , (0, 0) | , | 0, (0,0) | |
0, | 0, | 0, | 0, |
Again we see that Kristy and Sutheby will bid truthfully.