Mechanism Design for Personalized Recommender Systems


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1 Mechanis Design for Personalized Recoender Systes Qingpeng Cai Tsinghua University, China edu.cn Aris FilosRatsikas University of Oxford, UK Aris.Filos Pingzhong Tang Tsinghua University, China Chang Liu Alibaba Group, China ABSTRACT Strategic behaviour fro sellers on ecoerce websites, such as faking transactions and anipulating the recoendation scores through artificial reviews, have been aong the ost notorious obstacles that prevent websites fro axiizing the efficiency of their recoendations. Previous approaches have focused alost exclusively on achine learningrelated techniques to detect and penalize such behaviour. In this paper, we tackle the proble fro a different perspective, using the approach of the field of echanis design. We put forward a gae odel tailored for the setting at hand and ai to construct truthful echaniss, i.e. echaniss that do not provide incentives for dishonest reputationaugenting actions, that guarantee good recoendations in the worstcase. For the setting with two agents, we propose a truthful echanis that is optial in ters of social efficiency. For the general case of agents, we prove both lower and upper bound results on the effciency of truthful echaniss and propose truthful echaniss that yield significantly better results, when copared to an existing echanis fro a leading ecoerce site on real data. CCS Concepts Inforation systes! Recoender systes; Theory of coputation! Algorithic echanis design; Applied coputing! Ecoerce infrastructure; Keywords Mechanis design; Reputation systes; Approxiation 1. INTRODUCTION When a buyer signs in an ecoerce website (e.g., Aazon or ebay or Taobao), the website returns a list of recoended productseller pairs that the buyer ight be interested in. This recoendation is usually personalized, i.e. it is based on several factors related to the buyer, such as the buyer s deographic and past Perission to ake digital or hard copies of all or part of this work for personal or classroo use is granted without fee provided that copies are not ade or distributed for profit or coercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for coponents of this work owned by others than ACM ust be honored. Abstracting with credit is peritted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific perission and/or a fee. Request perissions fro RecSys 16, Septeber 1519, 2016, Boston, MA, USA c 2016 ACM. ISBN /16/09... $15.00 DOI: browsing or purchase history. The appropriate choice of productseller pair to be suggested to a buyer of certain characteristics is selected by a ranking algorith, which can be thought of as a systeatic way to allocate the whole aount of buyer ipressions. It is in the platfor s best interest to allocate the buyer ipressions in a way that yields high clickthrough rates (CTRs) and high clickconversion rates (CVRs), typically by giving better display slots (i.e., higher rankings on the webpage) to sellers with higher reputation, ore historical transactions or those that best atch the buyer s characteristics. As a result, all these websites incorporate a reputation syste (e.g. see [6]) in their designs, that records the sellers reputation and historical transactions and rewards those with higher scores via their ranking algoriths. We will refer to such scores as recoendation scores. A welldesigned reputation syste encourages sellers to increase their quality of service, and in turn attracts ore businesses [18]. It takes tie and effort for sellers to build up their reputation; in Aazon for exaple, soe trusted, wellknown sellers have accuulated ore than one illion reviews with positive scores as high as 97%. As a result, as it is also observed often in the industry, dishonest sellers ay take a shortcut and hire buyers to conduct fake transactions with the as a fast way to accuulate positive feedback and increase their reputation scores and nuber of historical transactions. The severity of the proble is also highlighted by Aazon s recent lawsuit against sellers that were allegedly using fake reviews to boost their profits. In fact, there has even been an eerging underground industry that provides sophisticated solutions for the sellers who want to quickly boost their reputations. Xu et al. [20] refer to such enterprises as sellerreputationescalation (SRE) arkets. Their study shows that online sellers using SRE services can increase their stores reputation at least 10 ties faster than legitiate ones, while only 2.2% will be detected and penalized. Current approaches, which are reflected in ost of the existing literature [11, 4, 21] ai to tackle the proble by training achine learning predictive odels using features of the review texts, to detect and punish fake reviews. However, Ott et al. [15] show that such deceptive stateents are not easily identified either by learning algoriths or even by huan readers. For exaple, Aazon recently sued ore than one thousand sellers for conducting fake transactions, each of which was involved in several purchases; it is easily conceivable that this is only a sall fraction of the nuber of sellers that eploy such reputationaugenting strategies. Also, in the current design of Taobao, the world s largest ecoerce website in ters of gross volue, according to a third party estiation
2 (which is reinforced by inference fro data) even after applying such a anipulationdetection engine, there is still ore than 10% of the total Taobao orders that are fake. Finally, such detection ethods also suffer fro the possibility of penalizing honest sellers, decreasing their overall experience of using the website as a platfor for their transactions. A echanis design approach In this paper, we ai to tackle this proble fro a different perspective, using the tools fro the fields of gae theory and ore specifically, echanis design. Gae theory is predictive in the sense that it is concerned with what the selfish or rational actions of the people involved in a syste will lead to. For our proble, the participants or agents are the sellers who ai to boost their recoendation scores. The field of echanis design, which has its roots in the pioneering works of Maskin [9] and Myerson [13] is preventive, in the sense that the rules of the syste are designed appropriately, in such a way that selfish behaviour is either copletely discouraged or at the very least, it is handled carefully and without severe consequences. We odel the proble described above as a variant of the resource allocation setting [Chapters 10,11 fro [17]] where the designer (i.e., the platfor) has to allocate one unit of a single divisible good. This unit can be interpreted as the nuber of total ipressions of buyers with certain characteristics that have to be allocated aong sellers, 1 or the probability that a seller is recoended to a single buyer when the buyer visits the website, or even the fraction of tie for which the seller will be suggested to the appropriate buyers over a specified period of tie. For exaple, given the first interpretation, if a seller receives an allocation of 1/3, it eans that he will receive 1/3 of the total recoendation slots for buyers of a certain kind. In traditional echanis design settings, each agent has an associated type, which conveys inforation about the preferences of the agent and is reported to the echanis designer, which then runs the allocation rule or the echanis with the types as inputs. The type does not necessarily contain the true preferences of the agent; if a rational agent can force a better outcoe by feigning a fake reported preference, he will do so. Central to the field of echanis design is the notion of truthfulness, i.e. a guarantee that under any circustances and regardless of the choices of the other participants, an agent will never have an incentive to report anything but his true type. The preferences of the agents are easured through utility functions [20] and a truthful echanis ensures that an agent receives the highest possible utility by telling the truth. In fact, ensured by the wellknown revelation principle [12], it is without loss of generality to consider only truthful echaniss. 2 This is also the reason why we can restrict attention to truthful echanis design throughout the paper. Our setting is slightly different fro the traditional odel, in the sense that the types of the agents are the recoendation scores, which are aintained by the syste and are established through the process of carrying out transactions and obtaining positive feedback. One key feature of our odel is the cost of anipulation for 1 We odel the total nuber of ipressions as a continuous unit rather than an integer. Considering that in ost ecoerce websites, this nuber is rather large, this is not an unrealistic assuption; in fact our guarantees will hold approxiately with very sall approxiation error for any large nuber of discrete ipressions. 2 The revelation principle states that any objective ipleentable in doinant strategies can be ipleented by a truthful echanis. Other coonly used naes for truthfulness are incentivecopatibility or staregyproofness. the sellers. Each seller can report any possible type, however, he suffers fro a cost by isreporting, which is the cost of hiring people to write fake reviews or using services like the SRE arket entioned earlier or even the probability of getting caught and being penalized. We odel the cost function to be explicitly correlated with the distance of anipulation (reported type inus the true type) as well as the current value of the true type. It is natural to assue that the higher one s current reputation, the harder the anipulation, especially since it involves the risk of detection; being severely penalized or being reoved fro the arket ight be catastrophic for a highlyrespected seller. A seller s utility is the difference between how uch he values the current allocation and his cost if he chooses to anipulate. We note here that while our utility functions are quasilinear in the cost, they are different fro standard quasilinear utilities in ost of the work in echanis design; the payent function in the standard quasilinear settings is iposed exogenously by the echanis in order to produce good incentives (like the wellknown VCG echanis [19, 3, 1]) whereas here, the cost function is associated with the anipulation only and the allocating echaniss do not use payents. In that sense, we can view our approach as following the agenta of approxiate echanis design without oney put forward by Procaccia and Tennenholtz [16]. Mechanis design approaches have been eployed in the past in recoendation systes, but ost of the [5, 7, 8, 2] are concerned with how to design reputation systes that incentivize buyers to report honest and constructive feedbacks rather than considering sellers as the selfish participants. As a notable exception, Zhang et al. [22] consider both strategic buyers and sellers, but their odel incorporates a social networktype graph and reputation systes for both buyers and sellers and is, in ost regards, quite different fro ours. Our results Our goal will be to design truthful echaniss, i.e. echaniss that do not encourage sellers to engage is reputationaltering anipulations and at the sae tie axiize the socially desired outcoe, i.e. ake sure that buyers receive recoendations for sellers with high recoendation scores. Following the usual echanis design terinology, we will refer to our objective as the social welfare. 3 We will easure the perforance of truthful echanis by its efficiency, i.e. worst case ratio between the social welfare achieved by the echanis over the optial social welfare, achieved by recoending the sellers with the highest recoendation scores, ignoring potential anipulations and strategic behaviour. Our results can be suarized as follows. For the case of two sellers, we design truthful echaniss that are optial aong all such echaniss in ters of efficiency for both regular cost functions and general cost functions. For the general case of any sellers, we design two truthful echaniss. We provide a worstcase guarantee for the efficiency of the first one together with a general upper bound on the efficiency of any truthful echanis, establishing that under soe assuptions on the valuation and cost functions, the efficiency of the echanis is quite close to the efficiency of the best truthful echanis. We evaluate both echaniss on reallife data fro Taobao and show that our echaniss significantly outperfor the echanis that Taobao currently uses. We 3 We note here that usually the social welfare refers to the aggregate happiness of the participanting agents. In our case however, although the strategic entities are the sellers, the real welfare objective is the aggregate satisfaction of the buyers which is also aligned with the interests of the ecoerce platfor.
3 also observe that the perforence of our two echaniss scales differently with the nuber of sellers and the choices for the cost and valuation functions, showing that both are useful, for different input paraeters and sizes. 2. THE MODEL In our odel, there are sellers and one divisible unit of ipression or ite to allocate between the. Each seller is associated with a nonnegative recoendation score v i which is a function of a sellers reputation and propriety or fitness with respect to a buyer of certain characteristics. Let r i denote the recorded recoendation score of seller i, i.e. the recoendation score that is stored in the platfor s database for this seller. 4 Note that the recorded recoendation score ight be different fro the real, inherent recoendation score of a seller, since the forer ight have been acquired through fake transactions. Let r denote the vector of recorded recoendation scores of all sellers and let r i denote the vector of recorded scores of all sellers besides seller i. We will call r a recoendation score profile. The definitions for the vectors of real recoendation scores are siilar. A echanis f is a function that inputs a recorded score profile r, and outputs an allocation q, i.e. a apping fro r to q = (q 1,...,q ), where q i(r) denotes the fraction of the ite seller i gets, which as entioned in the introduction, can have different interpretations. Clearly, an allocation f is feasible if and only if 8r8i, q i(r) 0, and 8r, P i=1 qi(r) apple 1. Note that any feasible allocation of sellers to ad slots can be realized by an appropriate convex cobination of perutations of sellers to those slots. Each seller i has an intrinsic positive valuation g(v i) towards receiving the ite that denotes how happy the seller would be if he were allocated the whole unit of ipression according to his real recoendation score. In general, we odel the valuation g to be a positive function that aps recoendation scores to valuations; this allows us to consider cases where the value is positively or negatively correlated with the recoendation score. A natural choice would be to set g(v i) = for soe constant, which iplies that all sellers would be equally happy if they received the whole unit of buyer ipression. 5 In order to feign a fake recoendation score, the seller has to incur a cost for anipulating. DEFINITION 1. The cost for a seller with (real) recoendation score v to obtain a recorded score r is c(v, r) = r v h(v), where h(v) is a positive continuous increasing function. Intuitively, the higher a seller s recoendation score, the ore costly the anipulation. The for of the cost function also assues that it linearly depends on the extent to which a seller can increase his recorded score. We will soeties say that a seller reports a recoendation score of r, but it should be understood that he obtains that score through costly anipulations, according to the cost function defined above. Our odel assues that the shape of the cost functions is public inforation; this is not an unrealistic assuption since the cost of hiring fake reviewers or using SRE 4 Or ore precisely, the vector of scores, since each score depends on a group of buyers of certain characteristics. 5 In our odel, it is iplicitly assued that sellers are indifferent between different advertiseent slots on a website. We discuss the added difficulties introduced when considering different values g(v i) j for different slots j when it coes to achieving truthfulness in conjuction with efficiency in Section 6; besides, there are several websites (e.g. soe ajor flightcoparing websites) that only have one single advertiseent slot or several slots that are not favourable over others. services can be calculated or estiated to a high degree. Furtherore it is not hard to see that without any knowledge of the cost function, we can not hope to do uch in ters of truthfulness. We now definte the utility 6 of a seller. DEFINITION 2. The utility of seller i with (real) recoendation score v i when the profile of recorded recoendation scores is r is defined as u i(v i,r)=q i(r)g(v i) c(v i,r i). As we entioned earlier, it is without loss of generality to restrict attention to truthful echaniss. DEFINITION 3. A echanis f is truthful if for each seller i and for all recorded scores of all other sellers r i and for each report r i of seller i it holds that u i(v i, (v i,r i)) u i(v i, (r i,r i)), i.e. the seller does not have any incentives to try to fake his real recoendation score. By the definition above, when analyzing a truthful echanis, we will use v to denote the input to the echanis, since the recorded scores and the real scores are the sae. Ideally, one would be interested in finding a truthful allocation echanis which axiizes the social welfare (aong all such echaniss) for every instance of the proble. There are several obstacles to doing this. First, the space of available scores (the type space) is continuous and hence there are infinitely any input instances that one would have to consider. Secondly, if we assue that the recoendation scores coe fro soe discrete set, then one idea would be to adopt a linear prograingbased approach where the truthfulness constraint of Definition 3 would be a constraint of the linear progra. However, such an approach would require us to write one constraint for each possible pair of scores, resulting in a nuber of constraints not anageable even for a relatively sall nuber of recoendation scores. Given that in platfors with any sellers, we need to aintain any different possible scores in order to distinguish between the, it does not see that such an approach would work. Instead, we will ai to design echaniss that perfor well with respect to all possible inputs. As we will see, the perforance of our echaniss will be liited by the worstcase instances but the experiental evaluation suggests that they perfor uch better on typical inputs. We define the efficiency of a echanis f as the ratio between its social welfare and the algorithic optiu (i.e, the best welfare one can achieve without iposing truthfulness constraints) in the worst case, i.e., E(f) =in v>0 P i=1 qi(v)vi v (1), We reark here that our efficiency notion is the sae one as the approxiation ratio for truthful echaniss used in the literature of algorithic echanis design [14, 16]. We will be interested in designing echaniss that have the axiu efficiency aong all truthful echaniss. 3. MECHANISMS FOR 2 SELLERS In order to explain our approach better, we will start fro the design of truthful echaniss for two sellers; this will allow us to deonstrate soe of the concepts of echanis design in a sipler environent plus, the echaniss that we will present in the 6 We reark here that while the cost function is singlepeaked [10] in the recoendation score, the dependence of the utility on the allocation and the cost ight give rise to ore coplicated structures.
4 following section for the general case of sellers will be very siilar in spirit. We will first consider the case of regular cost functions and then extend our analysis to the case of general cost functions. DEFINITION 4. A cost function c is regular if h(v)/g(v) is nonincreasing and integrable, i.e. if we let H(v) = R h(v)/g(v)dv, then c is regular if H(v) is concave. By the definition above, since H 0 (v) =h(v)/g(v) > 0, it holds that H(v) is an increasing function. Optial echanis for regular cost functions In this section we present a truthful echanis for two sellers, prove that it is optial aong all truthful echaniss for the case of regular cost functions and actually the echanis is derived fro the deduction in the proof. The echanis is the following one. MECHANISM 1. Consider the recoendation score profile v and let v l denote the larger value, and v s denote the saller value. Let q j(v) be the allocation of the seller with value v j for j 2{l, s}. Then allocate the ite as follows: q l (v) =in H(v l ) H(v 1 s)+ 2, 1, qs(v) =1 q l(v) As an exaple, when v l = v s, q l (v) =q s(v) =1/2 and each seller receives half of the total ipression. It is easy to see that the echanis is feasible, and the intuition why this echanis is truthful is that anipulations are not desirable because H is a concave function. We foralize this intuition in the following theore. THEOREM 1. Mechanis is truthful. PROOF. Without loss of generality, (by syetry), we prove that seller 1 does not have an incentive to report a fake score, given an arbitrary recorded score of seller 2. Clearly, seller 1 has no incentive to report a score saller than his real score because he will receive a saller fraction of the ite in that case and therefore we consider two cases. Case 1. v 1 apple r 2: The utility of seller 1 by reporting truthfully is u 1(v 1, (v 1,r 2)) = ax H(r 1 2)+H(v 1)+ 2, 0 g(v 1). If seller 1 reports r 1 such that v 1 apple r 1 apple r 2, then his utility u 1(v 1, (r 1,r 2)) is ax H(r 1 2)+H(r 1)+ 2, 0 g(v 1) (r 1 v 1)h(v 1). We have that the difference in utility u = u 1(v 1, (r 1,r 2)) u 1(v 1, (v 1,r 2)) is at ost u apple (H(r 1) H(v 1))g(v 1) (r 1 v 1)h(v 1), H(v) is a concave function, and the derivative of it is h(v)/g(v). By concavity and by the inequality above, we get that u apple 0. If seller 1 chooses to report r 1 such that r 1 > r 2, then his utility u 1(v 1, (r 1,r 2)) is in H(r 1) H(r 12 2)+, 1 g(v 1) (r 1 v 1)h(v 1). Then, cobining the forulas for the utility of truthful reporting and of isreport r 1, we obtain again that the difference in utility 0 u = u 1(v 1, (r 1,v 2)) u 1(v 1, (v 1,r 2)) is at ost 0 u apple (H(r 1) H(v 1)g(v 1) (r 1 v 1)h(v 1) apple 0. Case 2. v 1 >r 2. The utility of seller 1 by reporting truthfully is u 1(v 1, (v 1,r 2)) = in H(r 1 2)+H(v 1)+ 2, 1 g(v 1). If seller 1 reports r 1 such that v 1 <r 1, his utility u 1(v 1, (r 1,r 2)) is in H(r 1 2)+H(r 1)+ 2, 1 g(v 1) (r 1 v 1)h(v 1). Siilarly to before, we get that the difference in utility u = u 1(v 1, (r 1,r 2)) u 1(v 1, (v 1,r 2)) is at ost u apple (H(r 1) H(v 1))g(v 1) (r 1 v 1)h(v 1) apple 0. This copletes the proof. In the following, we will prove the worstcase efficiency guarantee of Mechanis 3 and the fact that it is optial aong all truthful echaniss. For the latter part, we will need the next lea, that provides a necessary condition for a echanis to be truthful. LEMMA 1. Let f be a truthful echanis. It holds that  for all v 1 v 2,q 1(v 1,v 2) apple q 1(v 2,v 2)+H(v 1) H(v 2).  for all v 2 v 1,q 2(v 1,v 2) apple q 2(v 1,v 1)+H(v 2) H(v 1). PROOF. By syetry, we only prove the first stateent of the lea. If seller 1 has score v 2 and seller 2 reports score r 2, we have that for any >0 it ust hold that (q 1(v 2 +, r 2) q 1(v 2,r 2)g(v 2) apple h(v 2), otherwise seller 1 will have an incentive to isreport v 2 +, i.e., 8 >0,v 2 > 0: q1(v2 +, r2) q1(v2,r2) apple h(v2) g(v 2). Because the cost function c is regular, the function h(v)/g(v) is integrable, and hence q 1(v 1,v 0 2) q 1(v 2,v 0 2) apple R v 1 v 2 h(v)/g(v)dv, i.e.,8v 1 v 2,q 1(v 1,v 2) apple q 1(v 2,v 2)+H(v 1) H(v 2). We are now state the following theore. The proof is oitted due to lack of space; we refer the reader to the full version. THEOREM 2. Let E(H) = in v 1 v 2 v2 v 1 + v1 v 1 v2 1 + H(v1) H(v2) 2 The efficiency of Mechanis is E(M 1)=in{E(H), 1}, which is optial aong all truthful echaniss. Optial echanis for general cost functions In this section we present a truthful echanis with two sellers and general cost functions, and prove it is optial aong all truthful echaniss. The idea is to extend the idea we used in the previous section and find a decreasing function that is below h(v)/v that is as large as possible. For ease of reference, we will use say that a function g 1 is not larger than function g 2 if for all v 2 (0, +1) it holds that g 1(v) apple g 2(v). Given the cost function c(v, r), for all v>0, we define the function h(t) h 1(v) = in 0<tapplev g(t). Let H 1(v) = R h 1(v)dv and hence H 1(v) is a concave function. Note that h 1(v) is a decreasing function not larger than h(v)/g(v), and oreover, it holds that h 1(v) =h(v)/g(v) when h(v)/g(v) is decreasing. The following lea states that function h 1(v) is the largest decreasing function which is not larger than h(v)/g(v). (1)
5 LEMMA 2. For any decreasing function h 2(v) not larger than h(v)/g(v), for all v>0, it holds that h 2(v) apple h 1(v). PROOF. For any score v 1 such that 0 <v 1 apple v, we have that h 2(v) apple h 2(v 1) apple h(v 1)/g(v 1), i.e. h(t) h 2(v) apple in 0<tapplev g(t) = h1(v). Using the concave function H 1(v) defined above and the sae intuition of the design of Mechanis, we obtain the following optial truthful echanis for general cost functions. MECHANISM 2. Consider the recoendation score profile v and let v l denote the larger value, and v s denote the saller value. Let q j(v) be the allocation of the seller with value v j for j 2{l, s}. Then allocate the ite as follows: q l (v) =in H 1(v l ) H 1 1(v s)+ 2, 1, qs(v) =1 q l(v). The following theore establishes the truthfulness of the echanis. THEOREM 3. Mechanis 2 is truthful. The proof of Theore 3 follows fro very siilar arguents as the ones used in the proof of Theore (see full version) and the following lea. LEMMA 3. Recall that H 1(v) = R h 1(v)dv. For any 0 < v 1 apple r 1, we have that H 1(r 1) H 1(v 1) apple h(v1) g(v 1) (r1 v1). PROOF. Because H 1 is concave, for any 0 < v 1 apple v 0 1, it holds that H 1(r 1) H 1(v 1) apple h 1(v 1)(r 1 v 1), and furtherore h 1(v 1) apple h(v 1)/g(v 1), which proves the lea. Finally, the following theore establishes that Mechanis 2 is optial aong all truthful echaniss for two agents, for general cost functions. THEOREM 4. The efficiency of Mechanis 2 is equal to in{e(h 1), 1} which is optial aong all truthful echaniss. Again, Theore 4 can be proved using arguents very siilar to those used in the proof of Theore 2, together with the following lea. The proof of the lea is siilar to the proof of Lea 1 and we oit it due to lack of space. LEMMA 4. Let f be a truthful echanis. It holds that  for all v 1 v 2,q 1(v 1,v 2) apple q 1(v 2,v 2)+H 1(v 1) H 1(v 2).  for all v 2 v 1,q 2(v 1,v 2) apple q 2(v 1,v 1)+H 1(v 2) H 1(v 1). 4. MECHANISMS FOR MANY SELLERS In this section we consider the ore general setting where we have sellers (with > 2) that we are allocating the unit of ipression to. Our ain contribution of this section is the design of a truthful echanis whose efficiency (a) approaches optiality aong truthful echaniss when the cost of anipulation approaches 0 and (b) is strictly better than the obvious truthful echanis, that allocates the unit uniforly to the sellers. As we will see in Section 5, in typical instances of the proble, our echanis will significantly outperfor the unifor allocation, which akes the extra effort of analyzing its properties clearly justified. MECHANISM 3. Let i be the seller with the highest recoendation score, i.e. i = arg ax j v j and j be the seller with the second highest recoendation score. The allocation of is: q i(v) =in +(H(vi) H(vj)), 1 and q k (v) = 1 qi(v) for all k 6= i. It is easy to check that the echanis is feasible. Note that Mechanis 3 is in fact a generalization of Mechanis; the fraction of the ite that the seller with the highest score receives is deterined by the difference between the highest score and the second highest score, and other sellers split the reainder of the ite evenly. For ease of exposition, we will analyze the echanis in the setting of regular cost functions. We can obtain siilar echaniss with analogous efficiency guarantees for general cost functions by using siilar leas as the ones that we eployed in Section 3. THEOREM 5. Mechanis 3 is truthful. PROOF. Again, without loss of generality, it suffices to prove that seller 1 does not have an incentive to isreport his recoendation score. Obviously, seller 1 has no incentive to report a score saller than v 1 because then, he will get a saller fraction of the ite. Let r =(v 1,r 2,...,r ), and r 0 =(r 1,r 2,...,r ). We consider three cases. Case 1 v 1 truth is r i for all i 6= 1: The utility of seller 1 by telling the u 1(v 1,r)=in + H(v1) H(r2), 1 g(v1). If seller 1 reports a score r 1 such that (r 1 >v 1) to report, his utility u 1(v 1,r 0 ) becoes in + H(r1) H(r2), 1 g(v1) (r1 v1)h(v1). and for the difference in utility u = u 1(v 1,r 0 ) u 1(v 1,r) it holds that u apple (H(r 1) H(v 1))g(v 1) (r 1 v 1)h(v 1) apple 0. Case 2. r 2 >v 1 r 3... r Seller 1 s utility fro telling the truth u 1(v 1,r)= ax {0, ()/ + H(v1) H(r2)} g(v 1). If seller 1 reports r 1 such that r 2 >r 1 >v 1), his utility u 1(v 1,r 0 ) becoes ax {0, ()/ + H(r 1) H(r 2)} g(v 1) (r 1 v 1)h(v 1). and for the difference in utility u = u 1(v 1,r 0 ) u 1(v 1,r) it holds that u apple (H(r 1) H(v 1))g(v 1)/() (r 1 v 1)h(v 1) apple 0. If seller 1 reports r 1 such that r 1 r 2, his utility u 1(v 1,r 0 ) is in + H(r1) H(r2), 1 g(v1) (r1 v1)h(v1). and the difference u = u 1(v 1,r 0 ) u 1(v 1,r) is at ost H(r1 ) H(v 1 ) ( 2)H(r 2 ) g(v 1 ) (r 1 v 1 )h(v 1 ) () () Case 3. r 2 u 1(v 1,r)= apple (H(r 1 ) H(v 1 ))g(v 1 ) (r 1 v 1 )h(v 1 ) apple 0. r 3 >v 1. Seller 1 s utility fro telling the truth is ax {0, ()/ + H(r3) H(r2)} g(v 1).
6 By the construction of the Mechanis, seller 1 can only affect the allocation outcoe only when his reported score is the highest score or the second highest score. If he reports r 1 such that r 2 >r 1 >r 3, his utility u 1(v 1,r 0 ) becoes ax {0, ()/ + H(r 1) H(r 2)} g(v 1) (r 1 v 1)h(v 1). and the difference in utility u = u 1(v 1,r 0 ) (H(r 1) H(r 3))g(v 1) () apple (H(r1) H(v1))g(v1) () (r 1 v 1)h(v 1) u 1(v 1,r) is at ost (r 1 v 1)h(v 1) apple 0. If he reports r 1 such that r 1 >r 2 r 3, his utility becoes in + H(r1) H(r2), 1 g(v1) (r1 v1)h(v1). and the difference u = u 1(v 1,r 0 ) u 1(v 1,r) is at ost H(r1 ) H(r 3 ) ( 2)H(r 2 ) g(v 1 ) (r 1 v 1 )h(v 1 ) () () apple (H(r 1 ) H(v 1 ))g(v 1 ) (r 1 v 1 )h(v 1 ) apple 0. This concludes the proof of the theore. The following theore establishes the efficiency guarantee of Mechanis 3. Note that this a theoretical lower bound, over all possible instances of the proble. We will see in Section 5 that on reallife instances, Mechanis 3 outperfors the worstcase bound considerably. We oit the proof of the lea due to lack of space. THEOREM 6. Let a = h(0)/g(0). The efficiency of Mechanis 3 is at least 2 in, a. An upper bound for all truthful echaniss We conclude the section with an upper bound on the efficiency of any truthful echanis. To prove the bound, siilarly to the achinery needed for the proof of Lea 1, we firstly obtain a necessary condition for truthfulness for the case of sellers. Again, we oit the proof due to lack of space. LEMMA 5. Let f be a truthful echanis. For every v 1 v 2, q 1(v 1,v 2,...,v 2) apple q 1(v 2,v 2,...,v 2)+H(v 1) H(v 2). Using Lea 5 we prove the following theore. Notice the iplications of the theore; when c =0, in which case there is no cost for anipulating, then the best thing that we can hope for with truthful echaniss is a unifor allocation between sellers. As c goes to infinity the incentive for anipulation is too sall and truthful echaniss that approach algorithic optiality are possible. Furtherore, notice that if the functions h( ) and g( ) are sooth enough (for exaple when h and g are constant functions) and their values at 0 are sall enough the efficiency guarantee of Mechanis 3 is very close to that of the best possible truthful echanis. THEOREM 7. Let c = h(1)/g(1). q The efficiency of any truthful echanis f is at ost 1 +2 c c + 2c. PROOF. Assue without loss of generality that v 1 =ax i v i. By definition, E(f) =in P v>0( i=1 qi(v)vi)/v1. Consider the profiles v =(v 1,v 2,...,v 2) and v 0 =(v 2,v 2,...,v 2), It holds that v2 (v1 v2) E(f) apple in + q 1(v). v 1 v 2 >0 v 1 v 1 ByLea 5, we have that q 1(v) apple q 1(v 0 )+H(v 1) H(v 2). Then v2 E(f) apple in + (v 1 v 2 ) (q 1 (v 0 )+H(v 1 ) H(v 2 )). v 1 v 2 >0 v 1 v 1 Siilarly, for all sellers i it holds that v2 E(f) apple in + (v 1 v 2 ) (q i (v 0 )+H(v 1 ) H(v 2 )). v 1 v 2 >0 v 1 v 1 By feasibility, we have that P i=1 qi(v0 ) apple 1 and therefore v2 (v1 v2) 1 E(f) apple in + v 1 v 2 >0 v 1 v 1 + H(v1) H(v2). Letting a = v 1/v 2, the righthand side of the inequality above can be written as 1 in v 1 >0,a 1 a + a 1 1 a + H(v1) H(v1/a). Now observe that H(v 1) H v1 apple a v 1 v 1 a h(v1/a) g(v 1/a), and hence the efficiency can be upper bounded as 1 E(f) apple in v 1 =a,a 1 a + a 1 1 +(a 1)c, a which iplies the bound of the theore. 5. EXPERIMENTS Up until now, we have been discussing the worstcase theoretical guarantees of the echaniss that we designed. In this section, we will evaluate the perforance of our echaniss epirically, using real data fro Taobao, the priary online arketplace in China and one of the biggest ecoerce websites in the world. In particular, because the nuber of sellers and buyers in Taobao is very large, we gather inforation about the transactions and buyers data fro 2047 randoly sapled sellers with respect to buyers of a certain deographic (feale buyers, of ages between 20 and 30) that occurred within the past year. The nuber of transaction orders after deleting buyers that have been detected to fake transactions in this dataset is , thus we think doing experients on this dataset is without loss of generality. As we explained in the odel section, we will interpret the ite as the total nuber of buyer ipressions for this buyer category. The recoendation scores of the sellers are calculated as follows. First, for each seller, we calculate the nuber of transaction orders he could have ade if he were allocated all buyer ipressions for this buyer category by achine learning ethods and then, we scale these nubers appropriately to ake sure they lie in the range [0, 1]; the latter is just a convention but it is in accordance with usual conventions in reality, where the scores are usually % percentages. The social welfare achieved by a echanis is the total (noralized) nuber of transactions resulted fro the ipressions being allocated by to the echanis. As we entioned earlier, there exist fake transactions in the input data that need to be taken into consideration. For this reason, first we copose a blacklist of buyers that have been detected to fake transactions in the past by the Alibaba group, the copany that owns Taobao. Then, we reove these fake orders fro the input data and estiate the real recoendation score for each seller. As a result, we can construct a data generator D for any seller in Taobao with associated recoendation scores, i.e, the distribution of real recoendation score is uniforly drawn fro the real scores of 2047 sellers.
7 For our experients, we will evaluate the efficiency of Mechanis 3, the echanis used by Taobao and the following echanis which we can show to be truthful (proof oitted due to lack of space). MECHANISM 4. Recall the definition of H. Let H(0) = 1/. P Seller i is allocated the fraction q i(v) =H(v i)/ j=1. H(vj) Note that while we do not provide a worstcase lower bound for the efficiency of Mechanis 4, as we will see shortly, the echanis actually perfors very well on the reallife inputs that we generate. The recoendation algorith that Taobao uses works as follows: when a single buyer visits the syste, the algorith ranks sellers according to their recoendation scores associated to buyers of the visitor s characteristics. Then, it picks a certain nuber of sellers fro the top of the ranking and suggests these sellers to the buyer. Unfortunately, the exact allocation rule to the selected sellers based on their scores is not public inforation. However, we can infer the allocation rule using achine learning ethods fro the data, and we can siulate the Taobao echanis with input scores of any sellers by this rule; since we are interested in this particular data set, our ipleentation will be an accurate approxiation. We copare the echaniss for different saple sizes. For each saple size, we first use our data generator to generate recoendation scores of artificial sellers, i.e, the score of each seller is i.i.d drawn fro D. We then copare our two echaniss against the Taobao echanis, as well as the unifor echanis that gives each seller a1/ fraction of the ite; the later coparison is useful to deonstrate that although the worstcase bounds of Mechanis 3 are coparable to those of the unifor echanis, in reality, Mechanis 3 significantly outperfors the unifor allocations. We repeat those experients 3000 ties and we calculate the average efficiency (i.e. the average ratio between the social welfare of the echaniss and the welfare of the algorithic optial). We consider different choices for the valuation functions g(v) and the function h(v) associated with the cost function. Figure 1 shows the coparisons of the average efficiency as a function of the saple size, for the case when g(v) =1and h(v) =1. This siple case corresponds to the assuption that all sellers would be equally satisfied by receiving the whole aount of ipressions (g(v) =1) and that a sellers cost is siply the distance between his recorded score and his real score. As we can see, both of our echaniss outperfor Taobao s algorith for up to 4000 sellers and Mechanis 4 is actually uch better for all input sizes we consider. Both Mechanis 3 and Mechanis 4 outperfor the unifor allocation by a lot although, as the nuber of sellers grows large, Mechanis 3 sees to converge to its theoretical guarantee. Figure 2 akes the sae assuption on the valuation function g(v) but assues a steep scaling of the cost function, to odel instances where the cost of anipulation is uch lower than the valuation of a seller. In this case, Mechanis 3 exhibits a poorer perforance; on the other hand, Mechanis 4 perfors exceptionally well, relatively to the other echaniss. The high perforance of Mechanis 4 can be explained by the fact that the functions g(v) and h(v) are constant functions, and the echanis perfors better when sooth enough functions are considered. To deonstrate this even ore clearly, we consider the case where g(v) =v and h(v) = v + where is either 1 or 10; the results of the experients are suarized in Figure 3. As we can see, in the first case Mechanis 3 outperfors Mechanis 4 for input sizes up to roughly 3000 sellers and the Taobao algorith for sizes up to slightly less than 6000 sellers. Note that in the second case, when the function h(v) is ore steep, Mechanis Figure 1: The average efficiency of the four echaniss with h(v) =1and g(v) =1. Figure 2: The average efficiency of the four echaniss with h(v) =10 4 and g(v) =1). 4 perfors worse and is outperfored by Mechanis 3 for input sizes up to roughly 6000 sellers. It is also worth pointing out that both echaniss outperfor Taobao s algorith, which sees to not scale well with quickly increasing functions h(v) either. Based on our observation, we can draw the following conclusions for reallife instances: (a) Both of our echaniss outperfor Taobao s algorith for the ost part and the (truthful) unifor echanis on all occurrences and (b) Mechanis 4 sees to be uch better in ters of scaling with the nuber of sellers when the odel functions are relatively sooth and Mechanis 3 is preferable when these functions are quickly increasing. Overall, the experients see to be indicative that the truthful echaniss we present can yield better results than existing approaches. 6. CONCLUSION AND FUTURE WORK There are any interesting directions for future work. As we entioned earlier, we assue that the value g(v) denotes the satisfaction that a seller would experience for receiving any slot on a webpage. This is a fair assuption in any personalized recoender systes with liited adslots for instance, but ideally, we would like to assue that sellers also have preferences over the different slots. Then, we would have to odel each slot as a different ite j and each seller would have a different value v ij for each one of the. However, this ultidiensionality introduces added dif
8 Figure 3: The average efficiency of the four echaniss with h(v) =v+1 and,g(v) =v and h(v) =10v+10 and,g(v) =v. ficulties in the design of truthful echaniss with good efficiency guarantees. The task is challenging but certainly worth exploring. Another interesting furure direction is to ipose certain allocative constraints on the aount of ipressions. Currently, we assue that the social optiu would be to give the unit to the seller with the highest score; it sees natural to assue that the platfor iposes soe fairness constraints as well, aking sure that at least respected sellers receive a certain fraction of an ipression. The challenge then would be to design truthful echaniss that obey the allocative constraints (which can render the quest for truthfulness quite harder) and those echaniss would be copared to the best possible allocation aong those that respect the constraints. Finally, since Mechanis 4 sees to outperfor the other echaniss significantly, at least for the case of sooth functions, a future task would be to provide a worstcase theoretical guarantee on the efficiency of the echanis. Acknowledgents We thank Yao Lu fro Alibaba Group for the advice on revising the paper and her contribution to the experiental section. Qingpeng Cai and Pingzhong Tang were supported by the National Basic Research Progra of China Grant 2011CBA00300, 2011CBA00301, the NSFC Grant , , , a Tsinghua Initiative Scientific Research Grant and a National Youth talent progra. 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