Post on 21-Dec-2015
Competitive Generalized Auctions
Paper by Amos Fiat, Andrew Goldberg, Jason Hartine, Anna Karlin
Presented byChad 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 = { 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
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
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 pM(b) be the profit for bid vector b using truthful mechanism M
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
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
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
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.
Competitive Ratios
• A single dominant bidder causes problem– Let’s give the bidder an opponent
– F(2)(b) = maxi≥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{pM(b)} ≥ (1/β) ×
maxr{0,1}m
(1≤j≤m rj × F(2)((b[sj]) – βc(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
CostShareC(b) : find largest k bidders such that they can share the price C/k
• CostShareC(b) is truthful
Sample Cost Sharing Auction(Algorithm)
• Given m=1 and c(r)=01. Partition bids b into s’ and s’’ via fair coin2. Compute each partitions optimal fixed
price F’ = F(b[s’]) and F’’=F(b[s’’])3. Compute auction results as
1. CostShareF’’(b[s’])
2. CostShareF’(b[s’’]) if CostShareF’’(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{pscs(b)} ≥ (1/4) × F(2)(b) or
– E{pscs(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’’• CostShareF’’(b[s’]) = 0 (rejects all bids in s’)• CostShareF’(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{pSCS} = 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)(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
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