Exp 32 2 nd price sealed bid auction

Exp 32: 2 nd price sealed bid auction § All submit bids, highest bidder wins + pays 2 nd highest bid a) Derive the equilibrium bidding strategy of a bidder with a value of V. What changes if you raise/lower your bid? Should you submit a bid equal to your value? Why, or why not? Does your strategy depend on the strategies and values of other players? b) If all bidders bid this way, who will be the auction winner? Which price will she pay? © WU IMS 1

Exp 32: 2 nd price sealed bid auction § All submit bids, highest bidder wins + pays 2 nd highest bid § What should I bid? • This is equivalent to the English auction: if you are the highest bidder, it is not you who determines the price, but the second highest bidder. • You only determine what you would maximally pay, and thereby the probability that you win. • Bidding your value is a weakly dominant strategy. © WU IMS 2

Exp 32: 2 nd price sealed bid auction § To show that bidding your value is a weakly dominant strategy, we compare this strategy with bidding higher or lower (both conditional on where your value and eventual bid is) You Win You Lose § In the following slides: Others’ bids higher Your bid B Your value V © WU IMS 3

Exp 32: 2 nd price sealed bid auction § If you bid higher than your value: V not highest B not highest No difference © WU IMS V highest B highest V not highest B highest No difference Lose money 4

Exp 32: 2 nd price sealed bid auction § If you bid lower than your value: V not highest B not highest No difference © WU IMS V highest B highest V highest B not highest No difference Lose money 5

Exp 32: 2 nd price sealed bid auction § All submit bids, highest bidder wins + pays 2 nd highest bid § What should I bid? • This is equivalent to the English auction: if you are highest bidder, it is not you who determines the price, but the second highest bidder. • You only determine what you would maximally pay, and thereby the probability that you win. • Bidding your value is a weakly dominant strategy. § Who wins? • Thus, the bidder with the highest value. § What is the price to be paid by the winner? • The 2 nd highest value. § Also called Vickrey Auction after William Vickrey, Nobel laureate in Economics © WU IMS 6

Exp 32: 2 nd price sealed bid auction § Experiment data c) Analyze the data set of the experiment. How do participants bid? © WU IMS 7

Exp 32: 2 nd price sealed bid auction © WU IMS 8

All experiment auctions § Realized prices in the 5 auctions § Ex-ante expected price (before knowing values) = expected 2 nd highest value = 10 + 40 * 3/5 = 34 © WU IMS 9

All experiment auctions § Realized prices vs. prices we should have observed if all bidders submit equilibrium bids © WU IMS 10

Revenue equivalence Auction format Optimal Bid Auction winner Japanese Stay in until the price reaches your value Bidder with exp. 2 nd highest value English Bid in small increments up to your Bidder with exp. 2 nd value highest value Dutch Bid at expected 2 nd highest value, conditional on being highest Bidder with exp. 2 nd highest value 1 st priced sealed Bid at expected 2 nd highest value, conditional on being highest Bidder with exp. 2 nd highest value 2 nd price sealed Bidder with exp. 2 nd highest value Bid your value Expected price § Revenue equivalence between different auction formats: seller’s expected revenue the are same, buyers’ expected profits are the same © WU IMS 11

Revenue equivalence § Conditions for revenue equivalence between 2 auctions: 1. Bidders play equilibrium strategies in both auctions 2. Bidders have private values that are not correlated with one another 3. Both auctions lead (in equilibrium) to the same allocation of the prize 4. Bidders are risk neutral 5. A bidder with the lowest possible value gets zero surplus in both auctions § Note: revenue equivalence means that expected profits are the same, but does not imply strategic equivalence: There is more uncertainty in Dutch and 1 st price auctions © WU IMS 12

Strategic properties of auction § Does your bid determine the auction price, or just your probability of winning? • If just probability: bid truthfully. • If price: Think about what your best bid would have been in case you win. § But: this assumes pure private value auctions, that means others’ bids do not influence your valuation (i. e the bidders’ values are not correlated). © WU IMS 13

e. Bay § What kind of auction is e. Bay? • English auction? Really? § Who determines the current price on e. Bay? • The bidder who is overbid, as the bidding agent places a bid just one increment above the previous highest bid. The bidder who is overbid is, by definition, the second highest bidder. § What’s the final price? • The final price equals the second highest bid in the auction, plus an increment. § Thus, e. Bay’s auctions are in essence second price sealed bid auctions. © WU IMS 15

Revenue equivalence in reality § Perspective of seller § If RET conditions are met, for given bidders auction format does not matter just focus on attracting more bidders! § Risk Aversion • Does not influence bids in 2 nd price auctions • But risk averse bidders bid more aggressively in first price auctions (don’t like the risk to lose, and reduce risk by increasing bid, giving up some expected profit) • Risk aversion 1 st price or Dutch auctions better for S § Non-familiarity with auctions • More overbidding in second-price auctions • More overbidding in sealed-bid auctions • Inexperience 2 nd price sealed bid auction better for S © WU IMS 16

More bidders lead to higher prices § Example: • Second price auction (rational bidders bid their value), each bidder has a valuation of either $20 or $40, each with equal probability What is the expected revenue? § Two bidders: • Values are either 20, 20 or 20, 40 or 40, 20 or 40, 40 • expected price= ( 20 + 40 )/4= 25 § Three bidders: • Values are either 20, 20 or 20, 40, 20 or 20, 40 or 40, 20 or 40, 40 • expected price=(20+20+20+40+40+49)/8=30 © WU IMS 17

The revenue-optimal auction § If the seller is concerned with efficiency (the bidder with the highest valuation should win), any of the discussed auctions is sufficient. § If he is interested in revenues, however, he can increase the expected price at the cost of expected efficiency by setting a positive reserve price. § If the reservation price is between the highest and second highest bid, this increases revenues. § However, if it is higher than the largest bid, the revenue will be zero and the outcome will be inefficient. § Optimization § But: reserve prices might scare bidders away! © WU IMS 18