Does Efficiency Pay

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Does Efficiency Pay?

• Mukund Sundararajan
• Joint work with
• Shaddin Dughmi, Tim Roughgarden

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Auction Setting

• Focus on multi-item auctions
• n players, k items, unit demand
• Buyers values vis
• Allocation rule bis!xis (xi is 0/1)
• Payment rule bis ! pis
• Focus on truthful auctions
• bis vis

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Auction Objectives

• Revenue ?i pi
• (Social) Efficiency ?i xivi

seller
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Why Efficiency?

• Social Welfare Efficiency (?i xivi)
• Simple, prior-free auction
• Eff Allocate to top k bidders, charge k1th
  highest bid
• Cannot squeeze revenue in settings with competing
  sellersMcAfee 93

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What Fraction of Optimal Revenue Does
Efficient Auction Make?
What is the Optimal Revenue?
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Optimal Auction with Prior

• Assume values drawn i.i.d from dist. F
• Expected revenue
• Opt
• lbidders with bid r
• Allocate to top min(k,l) bidders charge max(
  k1th highest bid, r )
• Prior used to calculate reserve
• r 0.5 if F Unif 0,1

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Warm Up

• 1 bidder, 1 item, FUnif0,1, r1/2
• Eff 0, Opt r(1-F(r)) ¼
• (No approximation at all)
• What is the convergence by trivial argument?
• 2 bidders, 1 item, FUnif0,1, r1/2
• Eff 1/3, Opt 13/24
• (1/2 approximation!)

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Talk Outline

• Main Result Efficient auction yields good
  approximation of optimal revenue under modest
  competition
• (technical point 1 virtual values)
• Applications to sponsored search
• (technical point 2 multi-unit auctions)
• Discuss novel market expansion problem

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Matroid Auctions

• Matroid specifies sets of buyers that can be
  simultaneously serviced
• Matroid M(universe U, collection of sets I)
• Closure S2 I implies S2 I for all Sµ S
• Exchange S2 I, S2 I, S lt S implies
  exists i 2 S/S, Si 2 I
• Multi-item auctions I is all subsets of size at
  most k (uniform matroid)

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Main Result

• For any matroid auction, buyer values drawn
  i.i.d from dist. F, efficient auction extracts
  at least a 1-1/P fraction of the optimal revenue
• -P disjoint bases of matroid
• -For multi-item auctions P n/k and we have
  (1-k/n) approximation

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Why is result interesting?

• Result Eff is (1-k/n) approximate
• Non-asymptotic ½ approximation if n 2k
• Not the folk result
• Not distribution specific
• For fixed dist. Easy to see when r does not
  apply
• Unit demand matching, multi-item auctions,
  segmented markets search auctions

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Why is Proof challenging?

• Recall auctions allocate to top min(k,l) bidders
  charge max(k1 th highest bid,r)
• For Unif0,1, Eff r 0, Opt r 1/2
• n10, k1, for all i, bi 1/2-
• Eff 1/2, Opt 0
• Cannot compare auctions point-wise

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Proof Structure

• Step 1 Revenue of any truthful, single parameter
  auction is expected virtual value served (Myerson
  81)
• Step 2 Bi-criteria Result Eff auction with k
  extra bidders outdoes Opt
• Step 3 Revenue of Opt is sub-modular in bidder
  set

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Step 1 Myerson's Lemma

• Expected revenue is expected virtual value
  served Myerson 81
• Fix i, v-i,
• Truthfulness i sees take-it-or-leave it price t
• Virtual value ?(v) v (1-F(v))/f(v)
• Revenue t(1-F(t)) ?t?v (1-F(v))/f(v) f(v)
  dv
• Evi?(vi) xi

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Virtual Valuations can be -ve

• e.g. Uniform 0,1 ?(v) 2v 1, vlt 1/2
• Fact Expected virtual value is 0

E?(v) Ev E(1-F(v))/f(v) Ev
Ev 0
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Virtual Value View

• Optimal auction picks top min(k,l) virtual values
• l non-negative vvs bids at least r
• Assume v.v. is monotone in value
• Satisfied by most distributions you can name not
  satisfied by Zipfians
• Efficient auction picks top k virtual
  values/values

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Step 2 Bi-criteria result

• Revenue of VCG with nk bidders
• ³
• Revenue of Optimal auction with n bidders

Single item result by Bulow-Klemperer96
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Hybrid Auction with nk bids

• Partition bidders in 2 groups of sizes n, k.
• For fixed bids of first group
• 1.Allocate top min(k,l) bidders of group 1
• 2.Allocate to arbitrary k-l bidders of group 2
  (Expected vv 0)

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Step 3 Diminishing returns

• Expected revenue of optimal auction is concave
• In contrast, this is not true for efficient
  auctions
• Proof idea point-wise using virtual values

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Completing the proof

• must show Eff(n) ³ (1-k/n) Opt(n)
• Bi-criteria bound Eff(n) ³ Opt(n-k)
• w.l.og. n ³ k
• Diminishing returns
• Opt(n-k) ³ (1- k/n) Opt(n)

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Pay-Per-Click Auctions

• n bidders, k slots
• Bidders have value/click.
• Bids represent maximum willingness to pay per
  click
• Advertisement of bidder i in slot j has
  click-through-rate is ?j
• Naturally, ?j ³ ?j1

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Key Lemma

• Revenue of Eff is weighted sum of revenues of k
  efficient auctions
• Revenue of Opt is weighted sum of revenues of k
  optimal auctions
• ith multi-unit auction sells i items, weights
  are the same

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One Auction Many Multi-unit
ctr
?1
?2
?3
?4
slots
4 3 2 1
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GSP not truthful

• Canonical nash equilibrium has revenue equal to
  that of efficient auction
• Varian, Edelman-Ostrovsky-Schwarz, Agarwal Goel
  Motwani
• Similar result for optimal auctions

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Geometrically falling CTRs

• Typically, ?j1/?j c 0.7
• Clarify
• (1 ck ) (1-k/n) approximation for any k
• Roughly, (0.9)(1-6/n) approximation for c0.7

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Market Expansion

• So far choice of auction (Eff/Opt) not critical
  under modest competition
• Suppose we could add m buyers to the current
  market. Which m buyers would increase revenue the
  most?
• Bidder values are i.i.d

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Greedy Market Expansion

• For matroid auctions, greedy market expansion is
  1-1/e approximate if initial market is minimally
  competitive
• Greedy Repeatedly add buyer that maximizes
  increase
• Also have results for an additive cost model

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Example Segmented Market

• Advertising Segments Automobile, Health, Sports
  with disjoint bidder sets
• Minimally competitive each market initially has
  at least slot bidders
• Implementation Think duplication

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Conclusions

• Efficient auction is simple Optimal auction uses
  prior and assumes monopoly
• Efficient auctions have good revenue under modest
  competition
• Market expansion can be studied formally

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Thanks!