Question

A painting is auctioned to \(n\) bidders, each with a private value for the painting that is uniformly distributed between 0 and 1 a. Compute the equilibrium bidding strategy in a firstprice sealed-bid auction. Compute the seller's expected revenue in this auction. Hint: Use the formula for the expected value of the \(k\) th-order statistic for uniform distributions in Equation 18.81 b. Compute the equilibrium bidding strategy in a second-price sealed-bid auction. Compute the seller's expected revenue in this auction using the hint from part (a). c. Do the two auction formats exhibit revenue equivalence? d. For each auction format, how do bidders' strategies and the seller's revenue change with an increase in the number of bidders?

Short answer

How do these auction formats compare in terms of revenue equivalence? What happens to the bidders' strategies and seller's revenue when the number of bidders increases? Answer: In a first-price sealed-bid auction, the equilibrium bidding strategy for each bidder is \(b_i(v_i) = \frac{n-1}{n} * v_i\), and the seller's expected revenue is \(\frac{n-1}{n+1}\). In a second-price sealed-bid auction, the equilibrium bidding strategy for each bidder is \(b_i(v_i) = v_i\), and the seller's expected revenue is \(\frac{n-2}{n+1}\). The two auction formats do not exhibit revenue equivalence. As the number of bidders increases, both auction formats yield higher expected revenues for the seller, approaching a value of 1 as the number of bidders approaches infinity.

Step-by-step solution

Determine the equilibrium bidding strategy for first-price auction.

In a first-price auction, each player submits a bid without knowing others' bids, and the highest bidder wins, paying their bid price. For a bidder with a private value v_i, their optimal bid is a function of their value and the distribution of other bidders' values. Let b_i(v_i) be bidder i's bid function. In equilibrium, bidders must submit a bid that maximizes their expected profit. The expected profit for bidder i can be written as: \(\pi_i = (v_i - b_i(v_i)) * P(win)\) Where \(P(win)\) is the probability of winning the auction. The bid function in a symmetric equilibrium for each bidder follows the form: \(b_i(v_i) = \frac{n-1}{n} * v_i\)

Compute the seller's expected revenue in the first-price auction.

To calculate the seller's expected revenue, we need to find the expected value of the winning bid. The highest bid of all bidders is the (n-1)th order statistic, which has the expected value: \(E(Y_{n-1}) = \frac{n-1}{n+1}\) Thus, the seller's expected revenue in the first-price sealed-bid auction is: \(Revenue = E(Y_{n-1}) = \frac{n-1}{n+1}\) #b. Second-price sealed-bid auction#

Determine the equilibrium bidding strategy for the second-price auction.

In a second-price auction, each player submits a bid without knowing others' bids, and the highest bidder wins but pays the second-highest bid. In this case, the dominant strategy is for bidder i to bid their true valuation, v_i. This means that the equilibrium bid function for the second-price auction is simply b_i(v_i) = v_i.

Compute the seller's expected revenue in the second-price auction.

In the second-price auction, the seller's revenue is equal to the expected value of the second-highest bid, which is the (n-2)th order statistic of a uniform distribution. Using Equation 18.81, we compute the expected value: \(E(Y_{n-2}) = \frac{n-2}{n+1}\) Thus, the seller's expected revenue in the second-price sealed-bid auction is: \(Revenue = E(Y_{n-2}) = \frac{n-2}{n+1}\) #c. Revenue equivalence#

Compare the seller's expected revenues in both auction formats.

We find that seller's expected revenues in both the first-price and second-price auctions are as follows: \(Revenue_{1st-price} = \frac{n-1}{n+1}\) \(Revenue_{2nd-price} = \frac{n-2}{n+1}\) While these two values are close, they are not equal. Thus, first-price and second-price sealed-bid auctions do not exhibit revenue equivalence. #d. Effects of an increase in the number of bidders#

Analyze the effects of an increase in n on both auction formats.

In both auction formats, an increase in n directly affects the revenue for the seller. We can see that as n approaches infinity, the expected revenues for both auction formats approach 1: \(\lim_{n\to\infty}\frac{n-1}{n+1} = 1\) \(\lim_{n\to\infty}\frac{n-2}{n+1} = 1\) Thus, the seller benefits from an increase in the number of bidders in both auction formats.