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

Abstract Auction Mechanism –Truthful –Compete on profit Auction Concepts –Generalized Auctions –Cancelable Auctions

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

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?

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

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

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

Generalized Auction Problems S = { S i | 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 = (r 1,…,r m ) is input –c(r) is the cost to the auctioneer to provide good to market allocation r

Generalized Auction Problems (Mechanism Goal) Given a bid vector b b = (b 1,…,b n ) 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

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

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 p M (b) be the profit for bid vector b using truthful mechanism M

Generalized Auction Problems (Competitive Ratio) sup b (p(b) / E{p M (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

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) = max i 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

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 F A (b) = max r  {0,1} m (  1≤j≤m r j × F(b[s j ]) – c(r)) The maximum sum of the optimal fixed price for each market minus the allocation cost F A (b) is the upper bound of all generalized auctions

Competitive Ratios What should we use for our optimal profit for our competitive ratio? F A (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.

Competitive Ratios A single dominant bidder causes problem –Let’s give the bidder an opponent –F (2) (b) = max i≥2 i × 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{p M (b)} ≥ (1/β) × max r  {0,1} m (  1≤j≤m r j × F (2) ( (b[s j ]) – β c(r))

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

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

Sample Cost Sharing Auction (Algorithm) Given m=1 and c(r)=0 1.Partition bids b into s’ and s’’ via fair coin 2.Compute each partitions optimal fixed price F’ = F(b[s’]) and F’’=F(b[s’’]) 3.Compute auction results as 1.CostShare F’’ (b[s’]) 2.CostShare F’ (b[s’’]) if CostShare F’’ (b[s’]) does have any winners 4.Profit !??!

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{p scs (b)} ≥ (1/4) × F (2) (b) or –E{p scs (b)} / F (2) (b) ≥ (1/4)

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’’ CostShare F’’ (b[s’]) = 0 (rejects all bids in s’) CostShare F’ (b[s’’]) = F’

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’’

Sample Cost Sharing Auction (Proof) Now let us look at the expected value of SCS –E{p SCS } = E{R} =  1≤i≤k-1 (min(i,k-i)( k i )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)( k i )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

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

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

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?