by open ascending bid ("English") auction Auctioneer raises asking price until all but one bidder drops out

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1 Auctions. Auction off a single item (a) () (c) (d) y open ascending id ("English") auction Auctioneer raises asking price until all ut one idder drops out y Dutch auction (descending asking price) Auctioneer starts a clock with high price and lets the clock tick down until a uyer cries out stop y sealed first-price (high-id) auction. Sealed ids. High idder pays his id. If a tie, the winner is assigned at random. By sealed second-price auction Sealed ids. This time high idder pays the second highest id. Tie rule as aove. What does game theory have to say aout the equilirium outcome? In the ascending price and sealed second-price auctions, there is a dominant strategy equilirium. In the former, stay in up to your valuation. In the latter, suppose idder i ids lower than his valuation ** *** * i v i * If the highest of the other ids is, uyer i loses if he ids either v i or ** < v. If the highest of the other ids is ** i i uyer i wins and ays *** The only time it matters is if the highest of the other ids is etween i and v i. In this case uyer i wishes he had id higher since he could have *** profited y v i..

2 Simple example: Auction off a $0 ill Sealed high-id auction with 0 cent raises Can you see why there can e no pure strategy equilirium id less than $9.90? If other idders id $9.90, your est response is to id the same since you may then win the coin toss. If other idders id $0.00, you can do no etter than id the same. (Of course you could also id zero. However it is not an equilirium for all ut one to id zero. Why is this? With two equiliria, which is more likely? Apply the Pareto criterion. All uyers are etter off in the first equilirium so it seems reasonale that they would all choose to id $9.90. Dutch auction with a 0 cent tick. See if you can explain why idding $9.90 must again e the an equilirium.. Auctions with private information Two idders. Each idder knows his own valuation ut neither knows the valuation of his opponent. Each elieves his opponents valuation is equally likely to e anywhere etween 0 and 00. What is a strategy in a private information game? v Valuations Bids A strategy is a plan indicating what to do for every possile information that a player may have. In this case, every possile valuation.

3 Can theory guide idders? Sealed high id auction Suppose you thought your opponent was going to id his full valuation (overly aggressive.) How aggressive should you e if your valuation is 00? If you id 0 you win with proaility 0/00 0 with proaility 0/00 60 with proaility 60/00 with proaility /00 Expected gain = Pr{opponent ids less than } profit u( v, ) = ( v ) = ( v ) Differentiating y, u = ( v ) 00 = 0at= v. 3

4 Extending this argument, suppose that your opponent adopts a linear strategy: = a v. What should you do? = a v / a v If you id you win with proaility Pr{ < } = Pr{ a v < } = Pr{ v < } a But values are evenly distriuted. Thus v < =. Pr{ v v} 00 Then uyer s win proaility is 00a His expected gain is therefore uv (, ) = ( v ) ( v ) 00a = 00a. Arguing as efore, uyer s est response is = v. 4

5 Appealing to the symmetry of the game, each therefore has a est response = v. i i Comparison with the open ascending id auction. In the open auction, if you win and your valuation is 60, what do you expect to pay? 40, what do you expect to pay? v, what do you expect to pay? v/ In the sealed high id auction, if you win when your valuation is v you pay = v/. Thus the expected payments are the same in the two auctions. It follows immediately that the expected receipts of the seller (the revenue) is the same as well. Revenue equivalence theorem If (a) valuations are independent draws from the same distriution and () idders are risk neutral Then equilirium idding results in the same expected winning id and hence the same expected revenue. Sealed high-id Auctions with more than idders. Revenue equivalence theorem continues to hold 5

6 Reserve prices? (Revenue equivalence still holds) "Auction" with idder. Reserve price r Pr{sale} Expected revenue Another way to answer the question. Consider the sealed second price auction. Let the possile valuations e 0,,,.,v. Let f t e the proaility that the valuation is t. Suppose that the reserve price is raised from t to t+. A particular uyer only affects the outcome if he is the only one with a valuation of t or higher. [Why is this?]. Conditional upon this eing the case, with proaility f t the item is not sold. The seller thus gets to keep the item rather than sell it at a price of t. With proaility Ft = ft+ + ft fv the price is sold for $ more. The expected profit is therefore ftt+ Gt In the uniform case with values on [0,00] t Then ft t G t = t+ (00 t ) = 99 t > 0 f = and G = 00 ( t+ ) + if and only if t > 49. Thus the profit maximizing reserve price is 50. t 6

7 Class exercises:. What if valuations are uniformly distriuted on [00,00]?. What if the sellers valuation is s? Risk aversion Consider the Dutch auction. How does it fare in comparison with the sealed high-id and open ascending id auctions? Modelling Utility gain U(x) x Monetary gain 7

8 Special case: U( x) = x α, 0< α < Consider the uniform case with idders. Suppose your opponent s strategy is linear. = θ v. If you id you win if = θv (this is exact only if you win all ties.) That is if v < θ In the uniform case the proaility of this event, Pr{ v } = θ 00 θ Buyer s expected utility is therefore α α e( ) = Pr{ v } U( v ) = ( v ) = [ ( v ) ] θ 00θ 00θ It is left as an exercise to show that this is maximized if = v. + α Auctions with more than item Example: Bidder s valuations: 80, 60 Bidder s valuations: 50, 0 Bidder s valuations: 30, 5 Consider the open ascending id auction 8