1. 1 Competitive Auctions Authors: A. V. Goldberg, J. D. Hartline, A. Wright, A. R. Karlin and M. Saks Presented By: Arik Friedman and Itai Sharon

2. 2 Motivation – Current Trends Negligible cost of duplicating digital goods Emergence of the internet the problem: profit optimization for seller in an auction Possible uses: PPV-TV, audio files

3. 3 Traditional Solution – Bayesian Auction Example: VCG selling mechanism However: –Accurate prior distribution unavailable or expensive –Might be infeasible or unacceptable to consumers Required: dynamic selling mechanism, for any market condition

4. 4 Settings Single-round, sealed-bid, truthful auction mechanism Performance of algorithms gauged in terms of optimal algorithm –Worst-case analysis –Success  competitive algorithm Unlimited supply –Can be extended for limited supply

5. 5 Agenda Optimal Auctions Bid-independent auction –Equivalent to truthful auction No symmetric, truthful, deterministic auction is competitive Two competitive randomized auctions: –DSOT –SCS Justyfing optimality of F (m)

6. 6 Optimal Auctions A gauge for measuring auction performance

7. 7 Single Round Sealed Bid Auctions n bidders b – vector of bids –Maximum amount each bidder will pay Auctioneer computes: (Randomized?) –Allocation x = (x 1,x 2,…,x n ) –Prices p = (p 1,p 2,…,p n ) For winning bidders (x i =1): 0≤p i ≤b i For losing bidders (x i =0): p i =0 Profit: R(b) = Σ i p i

8. 8 Assumptions Each bidder i has private utility value u i Bidders want to maximize profit, u i x i -p i Bidders have full knowledge of auctioneer’s strategy Bidders do not collude

9. 9 Some More Definitions Symmetric auctions: Values of x and p are independent of order of bids Deterministic Truthful auction: Bidder i’s profit is maximized by bidding u i Randomized Truthful auction: May be described as a probability distribution over deterministic truthful auctions

10. 10 Optimal Auction – First Try The Optimal multiple-price omniscient auction: But: not truthful…  As we will see – not a good bound…

11. 11 Optimal Auction – Second Try The Optimal single-price omniscient auction: –v i is the i th largest bid in b –All bidders with b i ≥v k win at price v k However – impossible to compete with... –As will be shown later

12. 12 Theorem (T(b) vs. F(b)) For all bid vectors b F (b) ≥ T(b)/ln n There exist bid vectors b for which F(b) = Θ(T(b)/ln n)

13. 13 Optimal Auction – Final Try The m-optimal single-price omniscient auction: –v i is the i th largest bid in b –Determines k such that k≥m and k  v k is maximized –All bidders with b i ≥v k win at price v k

14. 14 Competitive Auctions – Definitions A – truthful auction β - the competitive ratio of A. A is β-competitive against F (m) if for all bid vectors b: E[A(b)] ≥ F (m) (b) / β A is competitive against F (m) if it is β-competitive against F (m) for constant β For m=2: A is [β-]competitive

15. 15 Bid-Independent Auctions And Other Definitions …

16. 16 Bid-Independent Auctions: Definitions 1≤i≤n f i : bid vectors  prices The deterministic bid-independent auction defined by the functions f i. For each bidder i: –t i = f i (b -i ), b -i = (b 1,…,b i-1,b i+1,…,b n ) –if b i ≥t i, bidder i wins at price t i –Otherwise, bidder i is rejected Bid-Independent = Truthful

17. 17 Bid Independent  Truthful u i ≥ t i – bid at least t i and pay t i –specifically, bid u i u i < t i – can’t win without losing… –so bid u i and lose  u i maximizes bidder i’s profit.

18. 18 Truthful  Bid Independent A – any truthful deterministic auction –We want to find f such that A f is identical to A. b i x =(b 1,…,b i-1,x,b i+1,…,b n ) If  x * such that in A(b i x* ) i wins and pays p then: f(b -i )=p otherwise f(b -i )=∞ Given p, We can show for A(b i x ) that: –If bidder i wins, he pays p. –Bidder i wins by bidding any x≥p.

19. 19 Which Implies… A deterministic auction is truthful if and only if it is equivalent to a bid-independent auction Definition: a randomized bid-independent auction is a probability distribution over bid- independent auctions. Corollary: a randomized auction is truthful if and only if it is equivalent to a bid- independent auction

20. 20 Measuring Performance

21. 21 Theorem (can’t compete F (1) (b)) For any truthful auction A f and constant β≥1, there is a bid vector b such that E[R(b)] <F(b)/β

22. 22 Proof Consider a bid-independent randomized auction on two bids, 1 and x≥1. let h be the smallest value greater or equal to 1 such that Pr[f(1)≥h] ≤ 1/2β. Then the profit on input vector b = (1,H) with H = 4βh is at most  For any constant β≥1, no auction is β-competitive against F =F (1)

23. 23 Theorem (deterministic auctions are not competitive) Let A f be any symmetric, truthful, deterministic auction defined by a function f. Then A f is not competitive: for any 1≤m≤n, there exists a bid vector b of length n, such that R[b] ≤ F(m)(b) / ( ) nmnm

24. 24 Proof Consider bid vectors whose bids are all n or 1. –For 0≤j≤n-1: f(j) is the price the auction assigns to a vector with j bids at n and n-1-j bids at 1. We can assume f(0)≤1. Let k be the largest integer in {0,…,n-1} such that f(k)≤1. Let b be the bid vector with k+1 bids at n, and n-k-1 bids at 1. –Profit of A f on b is (k+1)f(k) ≤ k+1 –If k≤m-1 then F (m) has profit n. –If k≥m then F (m) has profit at least (k+1)n. –Either case the profit is at most F (m) (b) / ( ). nmnm