COS 444 Internet Auctions: Theory and Practice

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

2. Conditional expectation Intuitively clear, very useful in auction theory If x is a random variable with cdf H, and A is an event, define the conditional expectation of x given A: Not defined when prob{ A } = 0.

3. Two quick examples of conditional expectation • Suppose x is uniformly distributed on [0,1]. What is the expected value of x given that it is less than a constant c ≤ 1? • Given two independent draws x1and x2 , what is the expected value of x1, given that x1≤ x2?

4. Interpretation of FP equil. Now take a look once more at the equilibrium bidding function for a first-price IPV auction: I claim this has the form of a conditional expectation. What is the event A?

5. Interpretation of FP equil. Claim: event A = you win! = {Y1,(n-1)≤ v }, where Y1,(n-1) = highest of (n-1) independent draws To check this • Let y = Y1,(n-1) , the “next highest value”. The cdf of y =Y1,(n-1) is F(v)n-1so the integral in the expected value of y given that you win is

6. Interpretation of FP equil. Therefore, That is, in equilibrium, bid the expected next-highest value conditioned on your winning. …Intuition?

7. Stronger revenue equivalence Let Psp(v) be the expected payment in equilibrium of a bidder in a SP auction (and similarly for FP). So SP and FP are revenue equivalent for each v !

8. Graphical interpretation • Once again, by parts:

9. Bidder preference revelation Theorem: Suppose there exists a symmetric Bayesian equilibrium in an IPV auction, and assume high bidder wins. Then this equilibrium bidding function is monotonically nondecreasing. *Thanks to Dilip Abreu for showing me this elegant proof.

10. Bidder preference revelation • Proof: Bid as if your value is z when it’s actually v. Let w(z) = prob. of winning as fctn. of z p(z) = exp. payment as fctn. of z For convenience, let w=w(v), w΄=w(v΄), p=p(v), p΄=p(v΄), for any v, v΄ .

11. Bidder preference revelation • From the definition of equilibrium, the expected surplus satisfies: v·w – p ≥ v·w΄ – p΄ v΄·w΄ – p΄ ≥ v΄·w – p for every v,v΄ . Add: (v – v΄ )·(w – w΄) ≥ 0 . So v > v΄ → w ≥ w΄ → b(v) ≥ b(v΄). □

12. Example of an IPV auction with no symmetric Bayesian equil.: third-price (see Krishna 02, p. 34)

13. Riley & Samuelson 1981:Optimal Auctions • Elegant, landmark paper, constructs the benchmark theory for optimal IPV auctions with reserves • Paradoxically, gets more powerful results more easily by generalizing

15. 3rd-price losers weep av. others all-pay Santa Claus last-price war of attr.

17. Riley & Samuelson’s class Ars • One seller, one indivisible object • Reserveb0(open reserve, starting bid) • n bidders, with valuations vi i=1,…,n • Values iid according to cdf F, which is strictly increasing, differentiable, with support [0,1] ( so f > 0 ) • There is a symmetric equilibrium bidding function b(v)which is strictly increasing (we know by preference revelation it must be nondecreasing) • Highest acceptable bid wins • Rules are anonymous

18. Abstracting away… • Bid as if value = z, and denote expected payment of bidder by P(z). Then the expected surplus is • For an equilibrium, this must be max at z=v, so differentiate and set to 0:

19. We need a boundary condition… • Denote by v* the value at which it becomes profitable to bid positively, called the entry value: • Now integrate d.e. from v* to our value v1:

20. Once more, integrate by parts… • And use the boundary condition: • A truly remarkable result! Why?

21. Once more, integrate by parts… • And use the boundary condition: • A truly remarkable result! Why? Where is the auction form? FP? SP? Third-price? All-pay? …

22. Revenue Equivalence Theorem Theorem: In equilibrium the expected revenue in an (optimal) Riley & Samuelson auction depends only on the entry value v* and not on the form of the auction. □

23. Marginal revenue, or virtual valuation • Let’s put some work into this expected revenue: • And integrate by parts (of course, what else?)…

24. Marginal revenue, or virtual valuation

25. Interpretation of marginal revenue Because F(v)nisthe cdf of the highest, winning value, we can interpret this as saying: The expected revenue of an (optimal) Riley & Samuelson auction is the expected marginal revenue of the winner.

26. Hazard rate Let the failure time of a device be distributed with pdf f(t) and cdf F(t). Define the “survival function” = R(t)= prob. of no failure before time t = prob. of survival till time t. Since F(t) = prob. of failure before t , R(t) = 1 – F(t).

27. Hazard rate The conditional prob. of failure in the interval (t, t+Δt ] , given survival up to time t , is The “Hazard Rate”is the limit of this divided by Δt as Δt → 0 :

28. Hazard rate Thus, the marginal revenue, which is key to finding the expected revenue in a Riley-Samuelson auction, is 1/HR is the “Inverse Hazard Rate”

29. Why call it “marginal revenue”? Consider a monopolist seller who makes a take-it-or-leave-it offer to a single seller at a price p. The buyer has value distribution F, so the prob. of her accepting the offer is 1–F(p). Think of this as the buyer’s demand curve. She buys, on the average, quantity q = 1–F(p) at price p. Or, what is the same thing, the seller offers price p(q) = F-1(1– q) to sell quantity q. after Krishna 02, BR 89

30. Why call it “marginal revenue”? The revenue function of the seller is therefore q·p(q) = q F-1(1-q) , the revenue derived from selling quantity q. The derivative of this wrt q is by definition the marginal revenue of the monopolist : F-1(1-q) = p, so this is □