Competitive Generalized Auctions

1. Competitive Generalized Auctions Paper by Amos Fiat, Andrew Goldberg, Jason Hartine, Anna Karlin Presented by Chad R. Meiners

2. Abstract • Auction Mechanism • Truthful • Compete on profit • Auction Concepts • Generalized Auctions • Cancelable Auctions

3. Overview • Paper Motivation • Various types of auctions • Generalized Auction • Definition • How it models the motivating examples • Competitiveness • What is the competitive ratio and why • Analyze a Keystone Auction • Used to create more general auctions • Wrap Up • Short discussion about cancelability • Example auction that uses the keystone

4. Motivating Problems • Basic Auctions • K identical item single round sealed bids • Well studied • Are there truthful mechanisms that increase revenue by selling less items? • Conditional Financing • Sell fixed return junk bond iff there is sufficient (or better yet maximal) revenue generated • Can the auction be cancelable and truthful?

5. Motivating Problems • Pay-Per View Broadcast in Segmented Markets • Fixed cost segments; maximize revenue • Multicast Pricing • Select a multicast tree that maximizes the broadcasters profit and never runs a deficit

6. Motivating Problems • There is a trend in all of these problems • Profit maximization • Cancelable Auctions

7. Generalized Auction Problems • A Generalized Auction Problem A is a pair A = (S,c(•)) • S partitions n bidders into m markets • c(•) describes the cost of all possible market allocations

8. Generalized Auction Problems • S = { Si| 1 ≤ i ≤ m ≤ n} • m is the number of markets • n is the number of bidders • S partitions n bidders into m markets • c(•) maps {0,1}m to non-negative reals • Vector r = (r1,…,rm) is input • c(r) is the cost to the auctioneer to provide good to market allocation r

9. Generalized Auction Problems(Mechanism Goal) • Given a bid vector b b = (b1,…,bn) • Auctioneer Profit is the sum of the prices paid by winning bidders minus the cost of the market allocation • The goal of Generalize Auction Problems is to maximize Auctioneer Profit while remaining truthful

10. Generalized Auction Problems(Examples) • Basic Unlimited Market • m = 1 • c(r) = 0 for all r • Basis for the keystone • Multicast Pricing Problem • m = number of nodes • c(r) = cost of multicast tree for the allocation • Paper derives results from Generalized Framework and Basic Unlimited Market

11. Generalized Auction Problems(Competitive Analysis) • A Truthful mechanism M competes with an optimal mechanism • Let p(b) be the optimal profit for bid vector b • Let pM(b) be the profit for bid vector b using truthful mechanism M

12. Generalized Auction Problems(Competitive Ratio) supb (p(b) / E{pM(b)}) • The worse case ratio of optimal profit over mechanism profit • Allows for randomized mechanisms • Been shown that deterministic mechanisms perform worse than randomized • Authors weaken this notion to get results • i.e. net profit becomes gross profit

13. Competitive Ratio(Optimal Fixed-Pricing) • Given bid vector b and b[i] as the ith largest bid in b the optimal fixed price of b is F(b) = maxi i × b[i] • Finds the ith largest bids and charges the i winners b[i] • F(b) is the upper bound on the expected revenue (i.e. gross profit) for a single market

14. Competitive Ratio(Optimal Market Profit) • Given any selling mechanism for A = (S,c(•)) that uses a single price per market • The optimal net profit for A is FA(b) = maxr{0,1}m(1≤j≤m rj × F(b[sj]) – c(r)) • The maximum sum of the optimal fixed price for each market minus the allocation cost • FA(b) is the upper bound of all generalized auctions

15. Competitive Ratios • What should we use for our optimal profit for our competitive ratio? • FA(b)? • Would be nice but we can’t get a constant factor ratio • One bidder could dominate the bid and we can’t guarantee a truthful slice of this bid is within a constant factor.

16. Competitive Ratios • A single dominant bidder causes problem • Let’s give the bidder an opponent • F(2)(b) = maxi≥2i × b[i] • Furthermore, let make the optimal gross profit margin a constant factor greater the cost • For generalize auction a β competitive mechanism M satifies E{pM(b)} ≥ (1/β) × maxr{0,1}m(1≤j≤m rj × F(2)((b[sj]) – βc(r))

17. Sample Cost Sharing Auction • Randomized Mechanism • Truthful • Works for auctions with • m = 1 • c(r) = 0 • Proven 4-competitive

18. Cost Sharing • Given a bid b and a cost C the following mechanism finds a subset of bidders to share cost C CostShareC(b) : find largest k bidders such that they can share the price C/k • CostShareC(b) is truthful

19. Sample Cost Sharing Auction(Algorithm) • Given m=1 and c(r)=0 • Partition bids b into s’ and s’’ via fair coin • Compute each partitions optimal fixed price F’ = F(b[s’]) and F’’=F(b[s’’]) • Compute auction results as • CostShareF’’(b[s’]) • CostShareF’(b[s’’]) if CostShareF’’(b[s’]) does have any winners • Profit !??!

20. Sample Cost Sharing Auction(Analysis) • How good is the profit? • Authors prove a 4-competitive bound • Note that for this mechanism gross profit is net profit • So to prove the bound we must show • E{pscs(b)} ≥ (1/4) × F(2)(b) or • E{pscs(b)} / F(2)(b)≥ (1/4)

21. Sample Cost Sharing Auction(Proof) • The revenue R from the auction is the minimum of the two optimal fixed prices F’ and F’’ R = min(F’,F’’) • Suppose WLOG the F’ < F’’ • CostShareF’’(b[s’]) = 0 (rejects all bids in s’) • CostShareF’(b[s’’]) = F’

22. Sample Cost Sharing Auction(Proof) • Now let us look at F(2)(b) • F(2)(b) = k × p • k bidders is greater than 2 • These bidders are uniformly distributed though s’ and s’’ and can pay a price p • R / F(2)(b) = min(F’,F’’) / F(2)(b) • min(F’,F’’)/F(2)(b) ≥ min(p×k’,p×k’’)/(p ×k) • min(F’,F’’)/F(2)(b) ≥ min(k’,k’’)/(k) • k’ is bidders in s’ and k’’ is bidders in s’’

23. Sample Cost Sharing Auction(Proof) • Now let us look at the expected value of SCS • E{pSCS} = E{R} = 1≤i≤k-1(min(i,k-i)(ki)2-k • So with the ratio • R/F(2)(b) ≥ min(k’,k’’)/(k) • E{R}/F(2)(b) ≥ (1/k) ×1≤i≤k-1(min(i,k-i)(ki)2-k • E{R}/F(2)(b) ≥ (1/2) -(k-1└i┘)2-k • Worse cast is when k = 2, 3 when • E{R}/F(2)(b) ≥ ¼ • Ratio approaches ½ as k approaches infinity

24. Scaling up the Auctions • Authors introduce a notion of cancelable auctions • Auction mechanism is cancelable if it can be cancelled if it does not meet a revenue AND it is still truthful • This allows the analysis to dismiss no-profit situations during analysis • SCS is shown cancelable

25. Scaling up the Auctions • Authors introduce the Local Sampling Cost Sharing auction to handle the multicast problem • They build the algorithm out of SCS • Since SCS is cancelable LSCS can’t run a deficit • Author show LSCS is 4-competitive based on results from SCS • SCS becomes a keystone auction to construct mechanisms for other generalized auctions

26. Conclusions • Provide a framework for generalized auctions • Introduce a useful notion of profit competitiveness for the auction mechanisms • Provide a building block auction for generalized auctions • Questions?