COS 444 Internet Auctions: Theory and Practice

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

2. FP equilibrium for general v distribution The set up: • Baseline IPV model, values iid with cdf F(v) • E[surplus of 1] = pr{1 wins} (v1 – b(v1 )) • Bidders 2,…,n bid b (v) • What is best b( v1 ) ?

3. In bid space… We need to express the prob. that 1 wins, which is For this, we assume for now that βis monotonically increasing, and hence invertible. We thus need to check our answer!

4. Because the v ’s are independent, we can now write and the equilbrium condition is then

5. One more thing: When we differentiate this, we’ll need the derivative of β-1 . If you rotate the picture it’s (almost) obvious that

6. Use chain rule, set β=b, use v instead Of v1 , so β-1(b(v1)) = v. Leads to a linear, first-order differential equation for b(v): This is of the form:

7. Use the ancient trick of multiplying by the integrating factor

8. In this case, C=(n-1)f/F; D=vC; and Solution: (check monotonicity assumption) …optimal shade Integrate by parts, use b(0)=0 to determine γ=0.

11. To check monotonicity assumption: From the differential equation: 

12. eBay observed • Assignment 2 provides a tool for visualizing behavior… • We’ll look at some examples, but first…

13. eBay’s algorithm • Open vs. secret reserve • Increments • Raising your own (highest) bid when you're less than a bidding increment above the posted second price • Raising your own (highest) bid when you’re below secret reserve • Buy-it-Now with and without offer invitation, and with and without bidding opportunity at lower reserve, when bids will remove buy-it-now. Question: is it rational to bid above buy-it-now?

14. Simplest case:Open reserve (minimum bid) Assume for simplicity that bidding increment = tick = $1; open reserve = $10 •  First bid: $20 • Posted: $10 • Minimum next bid is this + tick = $11 • New bid and bidder: $15 Posted: $16 (“proxy bid”) • Minimum next bid is this + tick = $17 • New bid: $19.90 Posted: $ 20.00 (“proxy” can’t exceed high bid) Minimum next bid is this + tick = $21.00 *In all cases the posted price is the one paid

15.  First bid: $20 • Posted: $10 • Minimum next bid is this + tick = $11 • New bid and bidder: $15 Posted: $16 • Minimum next bid is this + tick = $17 • New bid: $20.10 Posted: $20.10  would pay Minimum next bid is this + tick = $21.10 • New bid by high bidder: $24.00 • Posted: $21.00  now pays(basis of law suit!) • Minimum next bid is this + tick = $22.00 If instead:

16. With a secret reserve, say $100  First bid: $20 Posted: $20 and “reserve not met” Minimum next bid is this + tick = $21 • Further bids: treated as usual if secret reserve not met, with the warning “reserve not met”. (High bidder does not bid against herself.) If and when secret reserve is met, highest bidder’s bid is hidden and (formerly secret) reserve is posted, plus “reserve met”.

17. The inside of the algorithm After (open or secret) reserve is met: H = max { H, acceptable bid by new bidder } … tie keeps the same high bidder L = min { H, acceptable bid by new bidder } posted price = min {L + tick, H }

18. Early bidding vs. Sniping • But early bidding affects behavior WAR

20. Dangers of early bidding, con’t As bait

22. Dangers of early bidding, con’t Curiosity

24. A (likely) shill Reserve = $95 ______ • Bidder 3 bids $94 when the reserve is $95 and the high bid is below that. She has feedback of 1. A likely shill.

26. First-price equil. derivation in value space A slicker way to do business, the way the pros do it: If the assumed monotonically increasing bidding function is b(v) , then bid as if your value is z. The equil. condition is then where now

27. The rest is now much easier Differentiating wrt z : Leads to the same differential equation.