COS 444 Internet Auctions: Theory and Practice

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

2. Field experiment “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. 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. The experiment V* = $4 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* = $8

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

14. Back to optimal IPV auctions Introduce v0= valueof 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. Optimal reserve b0 We now ask the the question: how should the seller choose the reserve (opening bid) b0 optimally---that is, to max exp. rev.? b0 determines v*, so we differentiate wrt v* :

16. Optimal reserve b0 Notice: v* does does not depend on the number of bidders, nor on the particular form of the auction! In the uniform case with v0=0, eg, F(x) = x, and v* = ½ , for any auction in Ars.

17. Optimal reserve b0 Lemma: In a first- or second-price auction in Ars, v* = b0 . Proof: In either FP or SP there is no incentive to bid if your value ≤ b0 . Therefore v*≥b0 . On the other hand, as soon as our value reaches b0+ εwe can realize a positive expected surplus. The point at which we are indifferent to bidding is therefore v* = b0 . □

19. Optimal reserve b0 Notice that in FP and SP auctions in the class Ars the seller’s optimal reserve is … above the seller’s value! Intuition?

20. FP equilibrium with reserve b0 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. FP equilibrium with reserve b0 Therefore, Simple example: v* = b0 = ½, F(v) = v, n = 2. Then

23. FP equilibrium with reserve b0 Checking revenue… use (with v0 = 0 ) v* = 0 : Revenue = 4/12 v* = ½ : Revenue = 5/12 > 4/12 P 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. SP equilibrium with reserve b0 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 b0 being reduced to b0 rather than the second-highest bid when it’s below b0 . Notice that this requires much less in the way of strategic thinking on the part of the bidders.

25. Reserves: testing the benchmark theory “Field Experiments on the Effects of Reserve Prices in Auctions: More Magic on the Internet” Lucking-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)

26. 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. 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. Reserves: testing the benchmark theory R&S 81 IPV Theory predicts that higher (open) reserves b0 : • 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. open reserve

30. “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.”