1. week 41 COS 444 Internet Auctions: Theory and Practice Spring 2010 Ken Steiglitz ken@cs.princeton.edu

2. week 42 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} (v 1 – b(v 1 )) Bidders 2,…,n bid  (v) What is best b( v 1 ) ?

3. week 43 In bid space… We need to express the prob. that 1 wins, which is For this, we assume for now that β is monotonically increasing, and hence invertible. We thus need to check our answer!

4. week 44 and the equilbrium condition is then Because the v ’s are independent, we can now write

5. week 45 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. week 46 Leads to a linear, first-order differential equation for b(v): Use chain rule, set β=b, use v instead Of v 1, so β -1 (b(v 1 )) = v. This is of the form:

7. week 47 Use the ancient trick of multiplying by the integrating factor

8. week 48 Solution: (check monotonicity assumption) …optimal shade In this case, C=(n-1)f/F; D=vC; and Integrate by parts, use b(0)=0 to determine γ=0.

10. n = 5 bidders

11. week 411 To check monotonicity assumption: From the differential equation: 

12. week 412 eBay observed Assignment 3 provides a tool for visualizing behavior… We’ll look at some examples, but first…

13. week 413 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. week 414 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. week 415  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. week 416 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. week 417 The inside of the algorithm After (open or secret) reserve is met: new H = max { H, acceptable bid by new bidder } new L = min { H, acceptable bid by new bidder } posted price = min {L + tick, H }

18. week 418 Early bidding vs. Sniping But early bidding affects behavior WAR

20. week 420 Dangers of early bidding, con’t As bait

22. week 422 Dangers of early bidding, con’t Curiosity

24. week 424 A (likely) shill Bidder 3 bids $94 when the reserve is $95 and the high bid is below that. She has feedback of 1. A likely shill. Reserve = $95 ______

26. week 426 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. week 427 The rest is now much easier Differentiating wrt z : Leads to the same differential equation.