A Truthful Mechanism for Offline Ad Slot Scheduling

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

4. ads

5. Slot Auctions • "Generalized Second Price (GSP)" auction: • score = bid * Pr[click]. (“ctr”)   • Rank by score, charge scorei+1 /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)

6. 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.

7. (theory day campaign slide)

8. 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?

9. 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 )

10. Problem Setup: • Bidders declare budget Bi[, bid bi]. • Page has mad slots, with known click* supply D1 > D2 > ... > 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

11. Single Slot D clicks

12. Single slot, budgets only • Input is B1, B2, ..., Bn. • Mechanism [Kelley ’97, Johari/Tsitsiklis ‘04]: Split D clicks according to budget. • Single price: • Allotment:  

13. 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 >, reduce Bk until =.) • Allocate Bi / p clicks to all bidders in A.

14. Example D=100 0 clicks $20 8 16 20 64 80 100 120 $100

15. 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.

16. Utility Model with Budgets Profit maximization:  ...but if the bidder knows her value vi,     why would she give a budget? Click maximization:

17. 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.

18. 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   budgetBidder A:        $2          $100                D=120Bidder B:        $1           $80Greedy:   A: 50 clicks @$2,B: 70 clicks @$1If A bids $1.01: A: 100 clicks @$1, B: 20 clicks @$1

19. 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

20. Multiple Slots

21. 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

22. 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

23. Multiple Slots -  budgets only • Suppose we want to allocate c1 > c2 > ... > 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.

24. 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* @ price p. • Perfect packing at current price. • Continue on remaining bidders and slots.

25. Our Mechanism (budgets) Fact: prices for each "block" decrease:Proof: When first block allocated,    c1 + ... + ck*    =  D1 + ... + Dk*.Since    c1 + ... + ck     ≤  D1 + ... + Dk      for all k,we get   ck*+1 +...+ ck  ≤  Dk*+1 +...+ Dk     for all k > k*.Thus, if we keep the same price p, all constraints still satisfied, and we can continue to descend.

26. Example Price hits $1:      80                           < p 100  80 + 70                   = p (100 + 50)    80 + 70 + 20            < p (100 + 50 + 25)    80 + 70 + 20 + 1       < p (100 + 50 + 25 + 0)

27. Example Price hits $0.84:      20                       < p 2520 + 1                  = p (25 + 0)

28. Our Mechanism (general) • Input: bi and Bi.  Assume b1 > ... > 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 > : 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.

29. Example p = $1.20:   Active = { A }B(A) = 80 < p D(1)

30. Example p = $1.00:   Active = { A, C }B(A)                      = 80  < p D(1)B(A) + B(C)            = 100 < p(D(1) + D(2))

31. Example p = $0.80:   Active = { A, C }B(A)                      = 80  = p D(1)B(A) + B(C)            = 100 < p(D(1) + D(2)) ...allocate all clicks from slot 1 to bidder A

32. Example p = $0.75:   Active = { B, C }B(B)                      = 70 > p D(2)B(B) + B(C)            = 90 > p (D(2) + D(3)) ....reduce B(B) to $36.25 so B(B) + B(C) = p (D(2) + D(3))

33. 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.

34. 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.

35. 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 (Σvici). • Do we match ¾-opt for multiple slots?

36. 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.

37. Using Σvicifor efficiency 1 << x << D OPT efficiency = D x (give all clicks to Y) ... but revenue ≤ x Our mechanism: efficency = D + x2 - x revenue = D