1. week 71 COS 444 Internet Auctions: Theory and Practice Spring 2009 Ken Steiglitz ken@cs.princeton.edu

2. week 72 Field experiment “A Test of the Revenue Equivalence Theorem using Field Experiments on eBay” “A Test of the Revenue Equivalence Theorem using Field Experiments on eBay” T. Hossain, J. Morgan, 2004 We have just seen that Riley & Samuelson 1981 predicts that the revenue for a wide class of auctions depends only on the entry value (v * ), also called the “effective reserve”. This paper uses eBay to field-test this prediction.

3. week 73 The experiment 80 auctions in all, 40 for CDs, 40 for Xbox games. Four copies each of 10 CDs, four copies each of 10 Xboxes. Private values is a good assumption. Auctions were held for v * = $4 (low effective reserve) and v * = $8 (high effective reserve). For each of these cases, the opening bid was varied and the shipping charges adjusted to achieve the desired v *.

4. week 74 The experiment Treat. A: Opening bid = $4.00 ship = $0.00 Treat. B: Opening bid = $0.01 ship = $3.99 Treat. C: Opening bid = $6.00 ship = $2.00 Treat. D: Opening bid = $2.00 ship = $6.00 All other experimental variables held as close to fixed as possible, order randomized V * = $4 V * = $8

5. week 75 Revenue results A: low v *, high opening bid B: low v *, low opening bid C: high v *, high opening bid D: high v *, low opening bid Explain…? Higher revenue in B rev. eq. for CDs, Higher revenue in D for Xbox games

6. week 76 Revenue results A: low v *, high opening bid B: low v *, low opening bid C: high v *, high opening bid D: high v *, low opening bid Explain…? Higher revenue in B rev. eq. for CDs, V * > 50% retail, people notice! Higher revenue in D for Xbox games

7. week 77 Explanations of revenue results Mental accounting (Kahneman & Tversky 84; Thaler 85). Modeled in Hossain & Morgan. Salience Bidders suspicious of free shipping Love of winning Costly search (usual searches ignore shipping) Sequential auctions

8. week 78 Hypothesis testing We often want to test the statistical significance of observations (as in Hossein-Morgan 04) Many common tests use normal distributions and their derivatives The one-tailed binomial test is the simplest Such tests can easily be abused, and are often blindly applied

9. week 79 Using the one-sided binomial test in Hossein & Morgan 04 Consider Treatment A (v * = $4, high opening bid) vs. Treatment B (v * = $4, low opening bid) Null hypothesis: A and B are rev. equiv. One-sided alternative: rev. in B > A Data: B>A 9/10 for CDs, 7/10 for Xboxes  16/20

10. week 710 One-sided binomial test Bernoulli trials: n independent coin flips, say in this case with a coin that comes up heads with prob. p So we ask what the probability is that we get 16 or more heads out of 20 flips if the coin is fair (one-sided test of null hypothesis) Add these for k = 16, …, 20

11. week 711 table of cumulative binomial distribution Weight of tail up and including k=4, for n=20 = 0.0059 Hossein & Morgan 04, p. 11: “The p-value of the one-sided binomial test is 0.005, which implies that we can reject the null hypothesis implied by the revenue equivalence theorem at the 99.5% level.

12. week 712 Warning: the Normal approximation When n is “reasonably” large, the binomial distribution is well approximated by the normal distribution… usually that means n > ~ 30. If you use normal tables for this problem you get a one-sided p value of 0.00368 --- not very close to the true value.

13. week 713 Warning: inference and priors This test tells us Prob (DATA|NULL). We might worry more about Prob (NULL|DATA) Bayes’ Rule tells us But do we know the priors: P(NULL)? P(DATA)?

14. week 714 Back to optimal IPV auctions Introduce v 0 = value of the item to the seller, which we’ve taken to be 0 till now, and which we will often do in the future. Then the total expected revenue is The first term is due to the possibility that all values are below v * and the seller retains the item.

15. week 715 Optimal reserve b 0 We now ask the the question: how should the seller choose the reserve (opening bid) b 0 optimally---that is, to max exp. rev.? b 0 determines v*, so we differentiate wrt v* :

16. week 716 Optimal reserve b 0 Notice: v * does does not depend on the number of bidders, nor on the particular form of the auction! In the uniform case with v 0 =0, eg, F(x) = x, and v * = ½, for any auction in A rs.

17. week 717 Optimal reserve b 0 Lemma: Lemma: In a first- or second-price auction in A rs, v * = b 0. Proof: Proof: In either FP or SP there is no incentive to bid if your value ≤ b 0. Therefore v * ≥ b 0. On the other hand, as soon as our value reaches b 0 + ε we can realize a positive expected surplus. The point at which we are indifferent to bidding is therefore v * = b 0. □

18. week 718

19. week 719 Optimal reserve b 0 Notice that in FP and SP auctions in the class A rs the seller’s optimal reserve is … above the seller’s value! Intuition?

20. week 720 FP equilibrium with reserve b 0 The next question: What is the equilibrium when there is a positive reserve? A slick way to do this is to recall the E[pay] from the beginning of Riley & Samuelson 81: We got this when we abstracted the payment away from the particular type of auction. But in FP:

21. week 721 FP equilibrium with reserve b 0 Therefore, Simple example: v * = b 0 = ½, F(v) = v, n = 2. Then

22. week 722

23. week 723 FP equilibrium with reserve b 0 Checking revenue… use (with v 0 = 0 ) v * = 0 : Revenue = 4/12 v * = ½ : Revenue = 5/12 > 4/12  Notice that the revenue increase is a won tradeoff for seller: he rejects bids below ½, but forces increased bidding in equilibrium when bidder values are above ½.

24. week 724 SP equilibrium with reserve b 0 Not a problem: Vickrey’s argument works again: just bid truthfully, there can never be an advantage to deviating from truthful bidding. But the mechanism for increasing revenue with a reserve is completely different from that in FP. Now the increase in payments results from bids above b 0 being reduced to b 0 rather than the second-highest bid when it’s below b 0. Notice that this requires much less in the way of strategic thinking on the part of the bidders.

25. week 725 Reserves: testing the benchmark theory “Field Experiments on the Effects of Reserve Prices in Auctions: More Magic on the Internet” Lucking-Reiley, 2000Lucking-Reiley, 2000 Pre-eBay, first-price sealed-bid auctions with control over open reserve The unique window in the history of civilization when auction experiments like this were possible (recall also LR 1999)LR 1999

26. week 726 Reserves: testing the benchmark theory Design 1 (within cards): Binary variable: no-reserve vs. reserve The familiar setup with pairs of matched Magic cards: Treatment 1: 86 cards, no reserve Treatment 2: same cards, one week later, reserve Treatments 3,4: same experiments, different cards, reverse time order

27. week 727 Reserves: testing the benchmark theory Design 2 (between cards): Continuous variable: reserve level = varying percentage of Cloister price (“catalog”) Auctions 1 & 2: 99 cards, 9 at 10%; 9 at 20%; …, 9 at 110% catalog Auctions 3 & 4: equal numbers of cards at 10%; 20%; 30%; 40%; 50%; 100%; 110%; 120%; 130%; 140%; and 150% catalog

28. week 728 Reserves: testing the benchmark theory R&S 81 IPV Theory predicts that higher (open) reserves b 0 : reduces # of bidders OK decreases prob. of sale OK increases price conditional on sale OK increases total revenue NO! Bidders respond strategically to increased reserve OK

29. week 729 open reserve

30. week 730 “Optimal” reserves Lucking-Riley 2000, p. 22: “After spending months observing this market environment and after running auctions myself, it is hard for me to imagine how an auctioneer in a real-world environment could ever have enough information to choose precisely the optimal reserve price.”