Consider the second price auction of a single object and with a reserve price R.
ID: 1124848 • Letter: C
Question
Consider the second price auction of a single object and with a reserve price R. There are two bidders who have independent private values vi , which are either 1, 2, or 3. For each bidder i, the probabilities of vi = 1, vi = 2 and vi = 3 are each 1/3 . If there is a tie at a bid of x R for the highest bid, the winner is selected at random from among the highest bidders and the price is x.
(a) Calculate seller’s revenue when R = 3 and everyone plays the dominant strategy.
(b) Calculate seller’s revenue when R = 2 and everyone plays the dominant strategy.
Explanation / Answer
The implicit assumption is that the reserve price R is secret and bidders don’t know it.
We get the following payoff matrix for the final price x, where x1 and x2 are the individual bids
2, 1
Both reach their max individual payoffs by bidding max, given that both follow a dominant strategy.
x1 = x2 = 3 and the probability of both conditions being true is 1/9
Thus, x = 3 is the winning bid and the revenue is 3 * (1/9) = 1/3
Since everyone plays the dominant strategy, R becomes immaterial.
Both players will bid max and the revenue is 1/3.
x2 = 1 x2 = 2 x2 = 3 x1 = 1 1, 1 1, 2 1, 3 x1 = 22, 1
2, 2 2, 3 x1 = 3 3, 1 3, 2 3, 3