A Truthful Mechanism for Offline Ad Slot Scheduling

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A Truthful Mechanism for Offline Ad Slot
Scheduling
Jon Feldman                       S.
MuthukrishnanEddie NikolovaMartin Pál
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ads
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Slot Auctions

• "Generalized Second Price (GSP)" auction
• score bid Prclick. (ctr)  
• Rank by score, charge scorei1 /ctri
• Analysis V, AGM, EOS
• Position j receives DJ clicks
• Each bidder has private valuation vi per click,
  tries to maximize profit.
• Equilibrium  same allocation/prices as VCG
• VCG asks for vi directly
• "Truthful"  (bidder i incented to report vi)

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GSP is useful, but incomplete

• Bidding strategy at campaign-level (days) not
  query level (microseconds).
• Campaign goals are complex, cross-media.
• Metrics impressions, clicks, conversions, reach,
  frequency, view-through conversions, assists,
• Bidders declare budgets
• Makes auctions non-independent.
• Even in the cleanest model, reporting vi may not
  be optimal.

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(theory day campaign slide)
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Our work

• Suppose we could auction off all the impressions
  at the beginning of the day. 
• What should the schedule (ad, slot, time) be?
• How much should we charge?
• How would bidders behave in such a system?
• Efficiency, revenue of equilibria?

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Our results

• Natural descending-price auction mechanism
• Truthful under a certain utility model
• Equivalence with greedy mechanism equilibrium
• Technical preview
• Proportional sharing (bandwidth) auction
• Machine scheduling ( Q pmtn Cmax )

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• Problem Setup
• Bidders declare budget Bi , bid bi.
• Page has m ad slots, with known click supply
• D1 gt D2 gt ... gt Dm
• Need a mechanism
• Gives feasible schedule allotting clicks ci
• Sets prices pi such that
• pi bi and pi ci
  Bi.
• Could refer to arbitrary exposure weights
• Clicks can scaled by bidder-specific ctris
•  

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Single Slot
D clicks
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Single slot, budgets only

• Input is B1, B2, ..., Bn.
• Mechanism Kelley 97, Johari/Tsitsiklis 04
  Split D clicks according to budget.
• Single price
• Allotment  

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Single slot, bids and budgets

• Input bi and Bi for all i. 
• Descending-price mechanism
• Maintain set A i bi p
• Reduce p until bidders in A can buy all clicks
• (If adding to A causes gt, reduce Bk until .)
• Allocate Bi / p clicks to all bidders in A.

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Example
D100
0
clicks
20
8
64
16
20
80
120
100
100
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Analysis

• If bidders act in their own self-interest, will
  they tell the truth?
• No budgets are problematic in general
• Borgs et al., EC 05 Reasonable, truthful
  efficient mechanisms with budgets are impossible.
• Dobzinski, Lavi, Nisan, FOCS 08 Truthful
  mechanisms cannot even produce Pareto-optimal
  (maximal) allocations.

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Utility Model with Budgets

• Profit maximization 
•      
• ...but if the bidder knows her value vi,    
  why would she give a budget?
• Click maximization
•  

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Our mechanism under click maximization -
truthful (? monotone in declared bid, budget)
- efficient (? allocates all clicks)

• Budgets only
• If all other bids fixed, ci is monotone in Bi
• Bids and budgets
• Want to participate exactly when pi vi. So
  will set bi vi... reduces to budgets-only
  case.

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Greedy Schedule

• In decreasing order of bi, until no more clicks
• Allocate Bi/bi clicks to  bidder i at price pi
  bi.
• Not truthful                                
      value   budget              Bidder A  
       2          100                D120      
         Bidder B        1          
  80Greedy   A 50 clicks _at_2, B 70 clicks
  _at_1If A bids 1.01 A 100 clicks _at_1, B 20
  clicks _at_1

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Greedy Schedule

• In decreasing order of bi, until no more clicks
• Allocate Bi/bi clicks to  bidder i at price pi
  bi.
• But...Theorem   There is a Nash Equilibrium
  of the greedy mechanism whose outcome is
  identical to our mechanism. Equilibrium All
  bidders bid min(pmech, vi)

                                                 
  I've left out some epsilons
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Multiple Slots
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Multiple Slots

• How do we characterize feasible schedules?
• Feasible schedule Allocation of ads to (time,
  slot) pairs such that no ad is assigned to the
  same slot at the same time.

D1 200 D2 150 D3 20 D4 5
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Related Scheduling Theory Problem

• Suppose we want to allocate c1, c2,  ..., cn
• Equivalent to "Related Machine Scheduling with
  Preemption"( Q pmtn Cmax )
• "Speed" of machine j DJ
• "Size" of job i ci
• "Makespan" 1

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Multiple Slots -  budgets only

• Suppose we want to allocate c1 gt c2 gt ... gt cn
• Theorem Horvath, Lam, Sethi, 1977 The optimal
  schedule for Q pmtn Cmax completes in time
  Corollary There is a feasible schedule
  iff  
  for all k 1, ..., n.

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Our Mechanism (budgets)

• Descending price p.   Maintain ci Bi / p.
• Job lengths increase... Maintain feasible
  schedule 
•          
  for all k 1,...,n
• Reduce p until some constraint k is tight
•          
• Schedule bidders 1...k to slots 1...k _at_ price
  p.
• Perfect packing at current price.
• Continue on remaining bidders and slots.

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Our Mechanism (budgets)
Fact prices for each "block" decreaseProof
When first block allocated,    c1 ... ck   
  D1 ... Dk.Since    c1 ... ck      
D1 ... Dk      for all k,we get   ck1
... ck    Dk1 ... Dk     for all k gt
k.Thus, if we keep the same price p, all
constraints still satisfied, and we can continue
to descend.
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Example

• Price hits 1      80                          
  lt p 100    80 70                   p (100
  50)    80 70 20            lt p (100 50
  25)    80 70 20 1       lt p (100 50 25
  0)

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Example

• Price hits 0.84      20                      
  lt p 25    20 1                  p (25
  0)  

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Our Mechanism (general)

• Input bi and Bi.  Assume b1 gt ... gt bn.
• Descending-price-like mechanism
• Maintain set A i bi p .            
• Bi i-th largest budget in A
• Let ci Bi/p.
• Reduce p until, for some k
•              ci ... ck D1 ... Dk
• (If gt reduce Bi for added bidder i until .)
• Allocate Bi/p clicks to all bidders in 1...k
  using slots 1...k.
• Continue on remaining bidders/slots.

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Example

• p 1.20   Active A B(A) 80 lt p D(1)

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Example

• p 1.00   Active A, C B(A)              
          80  lt p D(1)B(A) B(C)           
  100 lt p(D(1) D(2))

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Example

• p 0.80   Active A, C B(A)              
          80  p D(1)B(A) B(C)           
  100 lt p(D(1) D(2)) ...allocate all clicks
  from slot 1 to bidder A

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Example

• p 0.75   Active B, C B(B)              
          70 gt p D(2)B(B) B(C)            90
  gt p (D(2) D(3)) ....reduce B(B) to 36.25 so
  B(B) B(C) p (D(2) D(3))

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• Theorem Our mechanism is truthful under click
  maximization.Intuition
• Each "price block" divides up its clicks in
  proportion to budget  thus local changes do  not
  bring more clicks (like single-slot case).
• Dropping to a lower group gives lower price, but
  also fewer clicks in the block...  turns out to
  be not worth doing.
•      

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Revenue-maximizing Mechanism

• Revenue-maximizing compute feasible schedule to
  maximize revenue, charging each bidder bi. 
• (Simple LP does it, we also give greedy
  algorithm.)

• Theorem  If the revenue-maximizing mechanism is
  used, there is a Nash equilibrium whose outcome
  is equivalent to our mechanism.Proof Outline
• Equilibrium Bidders in price block i bid pi.
• Allocation ends up the same
• Allocation is tight, so lowering bid cannot give
  you more clicks.

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Extensions

• Budgets across keywords
• Mechanism generalizes... equilibrium properties?
• More general mechanism on set of linear
  constraints?
• Extend Johari, Tsitsiklis to this setting
• JT, 04 (Single slot, profit maximization)
  Unique Nash equilibrium attains 3/4 of the
  optimal efficiency (Svici).
• Do we match ¾-opt for multiple slots?

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Future study in ad slot assignment

• (Quality, Efficiency, Revenue) tradeoffs
• Number/size of slots, advertiser weights, user
  click behavior.
• Offline scheduling/planning
• Supply estimates informing online assignments.
• Online algorithm analysis
• Secretary models, Comp. ratio, Stochastic.
• Incentives
• Payment schemes, refunds, cancellations.

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Using Svici for efficiency
1 ltlt x ltlt D

• OPT efficiency D x (give all clicks to Y)
• ... but revenue x
• Our mechanism efficency D x2 - x
• revenue D