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

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?