You have decided to hold a second-price auction with no reserve to sell a painting. There are currently 3 bidders who would participate in the auction. You can also hire a marketing firm, who can bring up to 20 additional bidders to the auction at a cost of $2 per bidder. The bidder's valuations (including the 3 initial bidders) are drawn independently and uniformly at random over the interval [0,$100]. How many additional bidders will you ask the marketing firm for? Answer this question assuming that your objective is ? maximizing profit ? maximizing the winner's surplus (i.e., her valuation for the good minus her payment)

you need to consider the two cases (two bullets) separately. Try to get exact answers (a single number for each bullet). If you cannot do that, then describe how you can compute such an answer to get partial credit. *if that helps* you may use analytical tools to compute/plot expressions.

Ascending auction revenue (i) e What revenue can we expect from the ascending auction? (Assume two bidders with values U[0,100]). e Revenue will be equal to the second highest value. e Expected revenue is equal to the expected second highest value. e |f we repeatedly draw two values from U[0,100], e On average, the highest draw will be 66.66. e On average, the second highest will be 33.33. e What is the revenue for n bidders whose valuations are drawn from U[0,100]? Ascending auction revenue (vi) e S0, the average revenue from the auction with n bidders is " %100 e Because that is the expectation of the bid of the second highest bidder Competition e How is the \"value\" created in the auction divided between the seller and the winning bidder? e Two bidders with values U[0,100]. e Expected value of winner: 66.66 e Expected (sale) price: 33.33 e If seller has zero value, her revenue equals the bidder's surplus e |.e., the seller and the winner split the generated value equally e With n bidders o ..7 e With higher demand, the expected value created is ... and the seller is able to capture a ... share of generated value. Competition e How is the \"value\" created in the auction divided between the seller and the winning bidder? Two bidders with values U[0,100]. e Expected value of winner: 66.66 e Expected (sale) price: 33.33 e If seller has zero value, equal split of price, consumer surplus. With n bidders e Expected value of winner: 100 and Expected price: 100 2= n+1 n+1 With higher demand, the expected created value is higher (100 nn? but winner's surplus is lower, nil) and seller captures a higher share of generated value (a surplus of 100 n+1 Second price auction e Bidders submit sealed bids. e Seller opens the bids. e Bidder who submitted the highest bid wins. e Winner pays the second highest bid. e How should you bid? Second price auction Claim. An optimal strategy in the second price auction (regardless of what opponents do) is to bid your value. Proof e Compare bidding v to bidding some b>v. e Only way the higher bid matters is if opponent bid is above v but less than b. Then you lose if you bid v and win if you bid b, but you pay above your value. So better to have bid v. e Compare bidding v to bidding some bv 0 If opponent bid is here, v/ If bid is b If opponent 100 win and make money here, win bid is here, but lose lose auction money! Second price auction Bid value v 0 If opponent bid is here, If opponent bid is here, 100 win and make money lose the auction Bid b