Consider the second price auction of a single object and with a reserve price R°
ID: 3326005 • 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, 3, or 4. For each bidder i, the probabilities of Vi-1,Vi = 2, v' = 3 and vi = 4 are each I. If there is a tie at a bid of 2 R for the highest bid, the winner is selected at random from among the highest bidders and the price is r. Assume that bidder i can choose a discrete amount of bid as follows: b, E (1,2,3,4) (a) If ui = 3, how much should bidder J bid? (b) What is the optimal amount of reserve price R for the seller? (c) Which of the following auctions (is/are) strategically equivalent to the second price auction10? - First-Price -Dutch - English 9R can take only discrete values: 0, 1, 2, 3, or 4 1°From the bidders' point of view.Explanation / Answer
a)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
b) 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
c) dutch
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