An Expressive Auction Design for Online Display Advertising. AUTHORS: Sébastien Lahaie, David C. Parkes, David M. Pennock

Transcription

1 An Expressive Auction Design for Online Display Advertising AUTHORS: Sébastien Lahaie, David C. Parkes, David M. Pennock Li PU & Tong ZHANG

2 Motivation Online advertisement allow advertisers to specify what kinds of users they are targeting: geographical, user profiles, behavioral Targeting slots targeting users

3 Bidding language: Motivation Give me a number Please tell me your values of all outcomes Expressive about the preferences of advertisers, and succinct as well 3

4 Motivation Allocation and Pricing algorithms: Computing efficient allocation in combinatorial auctions can be NP hard Good design of bidding language leads to tractable allocation and pricing algorithms Incentive compatibility: Minimize the incentives of bidders to manipulate 4

5 Goals Efficient and scalable allocation algorithm Unique market clearing prices that minimize the bidders incentives to game Efficient, combinatorial algorithm for computing market clearing prices Market clearing: be cleared of all surplus and shortage (supply = demand) 5

6 Overview Advertisers are charged by impressions At the beginning of a period, all advertisers report their bids for different types of impressions Supply estimation Advertisers Allocation Algorithm Supply Estimator Scheduler prices Pricing Algorithm

7 Definition of Impression ² attributes ² values ² impressions A = fa 1 ;:::;A s g A 1 = fca;ma;ny;:::;undefinedg A s = f :» 1:;:::;undefinedg ha 1 ;:::;a s im hca;:::;:» 8:i ² number of all impressions m = jmj = sy ja t j t=1

8 Impressions and Prices ² bundle of impressions x i =[x i (1) x i () ::: x i (m)] Z M + x i (j) f; 1; ;:::g number of times an impression was displayed ² price for each impression p =[p(1) p() ::: p(m)] ² total price of bidder i : p x i = X jm p(j)x i (j) ² utility of bidder i : u i (x i )=v i (x i ) p x i 8

9 Allocation of Impressions ² supply of impressions z =[z(1) z() ::: z(m)] ² feasible: allocated supply nx x i (j) z(j) ; 8j M i=1 ² feasible allocation x =[x 1 x ::: x n ] ² e±cient: x arg max y nx v i (y i ) i=1 ² why e±cient? long term goals 9

10 Market clearing Prices ² market-clearing prices are also called competitive equilibrium (CE) prices ² demand set: D i (p) =argmax x i v i (x i ) p x i CE = hx; pi a) x i D i (p) ; 8i N b) p(j) =if X in x i (j) <z(j) ; 8j M ² if p are CE prices x is e±cient, hx; pi is a CE 1

11 Bid Tree CA $. FL CA $. FL $. $. {auto, sports} $.5 news blog {auto, sports} news $.5 $. $. $. blog fashion 5 capacity capacity $.1 $.1 $.1 $.1 value = $.+$.+$.5+$.1 = $. each node of the tree is a subset T k μ M, T i = ft k g 11

12 Properties of Bid Tree ² Each node of the tree is a subset T k μ M, T i = ft k g ²T i is laminar : 8 T;T T i,eithert μ T, T μ T, or T \ T = ; CA $. FL $. $. {auto, sports} fashion $.5 $. news blog $.1 $.1 5 c it b it v it (r) = ( bit r;if r c it 1 ; otherwise! v i (x i )= X Ã X v it x i (j) T T i jt hca;autoi hca;sports;:» 8:i hca;sports;blog;:» 8:i 1

13 Example of Bidder Value CA $. FL $. $. {auto, sports} fashion $.5 $. news blog $.1 $.1 5 Bidding tree T i of bidder i hca;auto;nighti :5 5 $:5 =5 hca;sports;news;nighti :1 1 $:5+1 $ :1 =4 hfl;auto;newsi :4 4 $: =8 Total value of bidder i : v i (x i ) = = 58 13

14 Queries: Value and Demand Value: input allocation and bidding tree output value of the bidder Demand: input prices of each type of impression and bidding tree v i (x i ; T i ) p, T i x i, T i output allocation to maximize utility x i Demand Query Algorithm: 1. Compute marginal surplus ¼ i (j) = P ft T i jjt g b it p(j). Pick out the impressions that have positive surplus (profit) and sort them in descending order 3. Assign a number as large as possible to the current most profitable impression, update the bidding tree and repeat 14

15 Example of Demand Query CA $. $.1 $. sports fashion 1 $.4 $. FL 5 M = 4 3 hca;sportsi hca; f ashioni hca;autoi hfl;sportsi 5 hfl;fashioni p = 3 $:1 $:5 $:5 4 $:1 5 $:5 ¼ i (j) = P ft T i jjt g b it p(j) 3 3 $:1+$:4 $:1 $:4 $:1 $:5 $:5 ¼ i = $:1 $:5 = $:5! (sort)! M = 4 $: $:1 5 4 $:1 5 $: $:5 $:15 3 hca;sportsi hfl;fashioni hf L; sportsi 4 hca;fashioni 5 hca;autoi 15

16 Example of Demand Query 3 $. hca;sportsi CA FL hfl;fashioni $.1 $. 5 M = hfl;sportsi x i = sports fashion 4 hca;fashioni5 1 $.4 $. hca;autoi $. hca;sportsi CA FL hfl;fashioni 1 $.1 $. 5 M = hfl;sportsi x i = fashion 4 hca;fashioni5 $.4 $. hca;autoi sports

17 Example of Demand Query 3 $. hca;sportsi CA FL hfl;fashioni 1 $.1 $. 5 M = hfl;sportsi x i = fashion 4 hca;fashioni5 $.4 $. hca;autoi sports 3 $. hca;sportsi CA FL hfl;fashioni 1 $.1 $. M = hfl;sportsi x i = fashion 4 hca;fashioni5 $.4 $. hca;autoi sports

18 Example of Demand Query 3 $. hca;sportsi CA FL hfl;fashioni 1 $.1 $. M = hfl;sportsi x i = fashion 4 hca;fashioni5 $.4 $. hca;autoi sports 3 $. hca;sportsi CA FL hfl;fashioni 1 $.1 $. M = hfl;sportsi x i = fashion 4 hca;fashioni5 $.4 $. hca;autoi sports

19 Example of Demand Query 3 $. hca;sportsi CA FL hfl;fashioni 1 $.1 $. M = hfl;sportsi x i = fashion 4 hca;fashioni5 $.4 $. hca;autoi sports sports 3 $. hca;sportsi CA FL hfl;fashioni $.1 $. M = hfl;sportsi x i = fashion 4 hca;fashioni5 $. hca;autoi $

20 Allocation ² Take into account forecasted supply z(j). The allocation problem is a linear program: X X X max b it x i (j) x in T T i jt 8 X x i (j) c it (T T i ;i N) >< jt X s.t. x i (j) z(j) (j M) in >: x i (j) (i N; j M) ² Proposition 1 : When agents submit their valuations as bid trees, the corresponding allocation LP has an integer optimal solution.

21 Allocation ² Dual of the allocation LP: X X X min ¼ it c it + p(j)z(j) ¼;p in T T i jm 8 X ¼ it X b it p(j) (i N;j M) >< ft T i jjt g ft T i jjt g s.t. ¼ it (i N;T T i ) >: p(j) (j M) ² Use subgradient method on the dual of the allocation LP: p k+1 = p k + kg k g k = y k P i xk i,wherey k is a revenue-maximizing allocation at price p k. 1

22 Pricing and Incentives ² Proposition : If x is an optimal primal solution to the allocation LP and p is an optimal dual solution, then hx; pi is a competitive equilibrium. ² Proposition 3 : The set of competitive equilibrium prices is a lattice when valuations are described by bid trees. ² unique minimal element p! most possible surplus ² unique maximal element ¹p! most possible revenue ² VCG payment is not feasible because of linear pricing ² Choose p to minimize the incentives of misreporting (minimize the improvement of payo when misreporting)

23 Proposition 1 & LP solution <x,p> is a CE Write down the dual LP A fractional CE exists M concave valuations: impressions are substitutes By complementary slackness conditions, p(j)> implies all supply are allocated revenue maximized A fractional CE an integer CE xi(j)> and πit(j)> implies x maximize bidder i s utility at price p Proposition 1 Proposition 3

24 Bidder Feedback ² If advertiser i wants to get at least d it impressions from sites in T μ M, he has to ensure that c it d it and increase the bid b it to get the desired volume. ² Add a new constraint : P jt x i(j) d it Let be the optimal value of the dual variable corresponding to this constraint. ² Proposition 4 : Suppose bidder i increases b it to b it + in its bid tree, and leaves the bids in the other nodes unchanged. Then there exists an e±cient allocation, with respect to the new pro le of bid trees, in which i receives at least d it units of impressions from T μ M. 4

25 Conclusion A bidding language (bid tree) that allows advertisers to specify values for different types of impressions, and ignore the irrelevant attributes Scalable allocation and pricing algorithms Bidder feedback to help advertisers assess the value of some nodes to get desired number of impressions Since the number of impressions may be very large, what attributes the seller should choose? Since the mechanism is not incentive compatible, what is the dynamics of a series of bidding? 5