Taming the Matthew Effect in Online Markets with Social Influence

Transcription

1 Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence AAAI-17 Taing the Matthew Effect in Online Markets with Social Influence Franco Berbeglia Carnegie Mellon University Pascal Van Hentenryck University of Michigan Abstract Social influence has been shown to create a Matthew effect in online arkets, increasing inequalities and leading to winner-take-all phenoena Matthew effects have been observed for nuerous arket policies, including when the products are presented to consuers by popularity or quality This paper studies how to reduce Matthew effects, while keeping arkets efficient and predictable when social influence is used It presents a arket strategy based on randoization and segentation, that ensures that the best products, if they are close in quality, will have reasonably close arket shares The benefits of this arket strategy is justified both theoretically and epirically and the loss in arket efficiency is shown to be acceptable Introduction Salganik, Dodds, and Watts 2006 wrote a seinal paper on the negative influence of social influence in cultural arkets They created an experiental virtual arket, called the MU- SICLAB, in which participants can listen to songs and then download the if they like the The songs are organized in a list or atrix for, giving different visibilities to the various songs, as is typically the case in online advertiseent, online stores, or physical retail stores eg, Craswell et al 2008; Li, Rodrigues, and Zhang 2004 Each song was also associated with a popularity signal eg, Engstro and Forsell 2014; Viglia, Furlan, and Ladrón-de Guevara 2014, ie, the nuber of downloads of the song by earlier arket participants In addition, all the songs were ranked by popularity, which eans that the song with the ost downloads received the ost visible position ie, the top of the list, the second ost popular song the second ost visible position, and so on Salganik, Dodds, and Watts 2006 showed that social influence creates unpredictable arkets, as well as significant Matthew effects Rigney 2010 Indeed, ultiple realizations of the MUSICLAB with various sets of participants show significantly different outcoes and strong inequalities in arket shares between the songs Those results have been reproduced by any authors in subsequent experients or siulations eg, Hu, Milner, and Wu 2016; Leran and Hogg 2014; Muchnik, Aral, and Taylor 2013; van de Rijt et al 2014 The MUSICLAB is an exaple of Copyright c 2017, Association for the Advanceent of Artificial Intelligence wwwaaaiorg All rights reserved trial and offer arkets which are particularly relevant in online settings The popularity ranking is also widely used by firs running such arkets in practice In recent years, soe authors have started revisiting soe of the assuptions in the MUSICLAB experient and, in particular, the ranking policy which is the ain controllable action of the fir running such a arket Van Hentenryck et al 2016 showed that, when the products are ranked by quality instead of popularity, the arket becoes asuyptotically predictable and optial: It leads a onopoly for the product of highest quality Abeliuk et al 2016 also proved that the dynaic ranking that optiizes the nuber of downloads at each step, which they called the perforance ranking Abeliuk et al 2015, leads to the sae outcoe These two ranking policies address the unpredictability and inefficiency of the arket under social influence identified by Salganik, Dodds, and Watts 2006 However, they fail to address the arket inequalities created by social influence: Two products of siilar appeals and qualities will have fundaentally different outcoes with one becoing a onopoly in the long run This winner-takes-all phenoena, although optial fro an efficiency standpoint, is typically considered undesirable This paper proposes a novel strategy that ais at addressing the three probles identified by Salganik, Dodds, and Watts 2006 siultaneously: unpredictability, inefficiencies, and inequalities The strategy is a randoized segentation protocol and is siple to deploy in online settings The paper also analyzes its properties both theoretically and experientally and shows that the protocol draatically reduces inequalities aong the best products in the arket, while preserving high predictability and efficiency The rest of the paper briefly reviews trial and offer arkets, presents the new randoized segentation protocol, and analyzes its properties both theoretically and experientally Trial and Offer Markets Following Krue et al 2012, this paper considers trial and offer arkets odeled with an extended ultinoial logit that includes product visibilities and social influence In these arkets, an incoing consuer observes n products in a list and a social influence vector d t, where d t i is the nuber of purchases of product i at tie t She can then saple one product eg, listening to a song and then de- 10

2 cides whether she wants to purchase it Each position j in the list is associated with a visibility v j that represents the inherent probability of sapling a product in position j or, intuitively, how uch custoers are attracted by a given position in the list Each product i is characterized by two values: an appeal a i which represents the probability of sapling product i and a quality q i which captures the probability of purchasing product i after it has been sapled Note that the appeals can capture soe for of externalities such as arketing capaigns The fir running the arket controls how to present the products to consuers, ie, where the products are positioned in the list In particular, the fir ust choose a ranking θ that assigns a position θi to each product i Asaresult, the probability that a custoer saples product i given ranking θ and social signal d is given by p i θ, d v θi a i + d i n j1 v θja j + d j The probability that a custoer purchases product i is then given by p i θ, d q i The fir is interested in axiizing the total nuber of purchases over tie, ie, n p i θ t,d t q i t i1 where θ t and d t are the ranking and social signals at tie t Prior work by Van Hentenryck et al 2016 has shown that the quality ranking, a static ranking which ranks the products of highest quality to the highest visibilities ie, q i >q j v θi v θj, leads to arkets where the product of highest quality becoes a onopoly asyptotically 1 The sae outcoe holds for the so-called perforance ranking that deterines the optial ranking θ at each step t, ie, θ arg-ax θ n i1 p iθ, d t q i These two ranking policies are optial and predictable asyptotically but they obviously create ajor Matthew effects: A sall difference in quality leads to a product losing its entire arket share in the long run Randoized Segentation We now propose a Randoized Segentation Protocol RSP to address the Matthew effect, while preserving predictability and efficiency The RSP is presented in Figure 1: It uses two key ideas to tae the Matthew effect First, it segents the arket in subarkets that we call worlds These worlds evolve independently with their own social influence signals and ranking policies This segentation itself has no ipact on the Matthew effect if the quality or perforance ranking is used, since the best-quality product will still obtain a onopoly in each world asyptotically and a large arket share over a finite horizon To counteract the Matthew effect, the RSP perturbs the product qualities slightly with soe rando noise of zero eans and then applies the quality ranking on these perturbed qualities 1 If several products have the sae highest product v θi q i, then the arket becoes a beta distribution aong the Overview of The Randoized Segentation Protocol 1 The arket is organized in sub-arkets called worlds 2 In each world, the product qualities are perturbed with a rando noise of zero ean The products are displayed using the quality ranking over the perturbed qualities 3 Each world aintains social influence signals independently of the other world 4 When a custoer enters the arket, she is randoly assigned to a world and presented with the ranking and the social signals of that world 5 If the custoer buys the product she sapled, the social signal of that product in that world is increased Figure 1: The Randoized Segentation Protocol RSP The qualities obviously reain the sae for the purpose of downloading the products As a result, each world ay now use a different ranking and a different product ay now take a substantial arket share Indeed, if the product with the second highest quality is placed in first position, it ay becoe the ost attractive product through the position visibilities, ie, v 1 q 2 >v 2 q 1 As a consequence, the RSP will balance the arket shares across the different worlds, ensuring a fairer distribution of the arket shares Theoretical Analysis This section foralizes the RSP ore precisely and presents a theoretical analysis of its behavior both in ters of inequalities, efficiency, and predictability Given the coplexity of the forulas, the analysis is only carried out for the perturbations of the first three products, but it generalizes naturally to ore products For siplicity, assue that the product qualities sastify q 1 q n and that the qualities of the top three products are reasonably close, ie, q 2 q 1 ɛ 2 1 q 3 q 1 ɛ 3 2 where ɛ 3 ɛ 2 0 More precisely, we assue that ɛ 2 and ɛ 3 are sall enough such that v 1 q 3 >v 2 q 1 As a result, a ranking 3, 1, 2 ensures that product 3 has the best noralized quality ie, its inherent quality ties the visibility of its position and the arket will go to a onopoly for product 3 asyptically Our proposed randoized segentation protocol partitions the arket in worlds and defines pertubed qualities ˆq i,w for each product i and world w as ˆq i,w q i + η i,w where η i,w is a norally distributed noise with zero ean and standard deviation σ In other words, η i,w is a rando variable distributed under the noral distribution f σ x 1 σ 2π exp x2 2σ

3 whose CDF F σ x is given by F σ x 1 [ ] x 1+erf 2 σ 2 Cobining this with the fact that the highest qualities are close, the perturbed qualities can thus be rewritten as ˆq 1,w q 1 + η 1,w, ˆq 2,w q 1 ɛ 2 + η 2,w, ˆq 3,w q 1 ɛ 3 + η 3,w In each world w, the products are ranked in ters of their perturbed qualities ˆq i,w instead of their intrinsic quality q i Hence, depending on the outcoes of the rando variables η 1,w, η 2,w and η 3,w, the products ay be ranked in the sequences 1,2,3, 1,3,2, 2,1,3, 2,3,1, 3,1,2, or 3,2,1 As a result, asyptically, the different worlds will converge to different onopolies: In soe of the, the arket will becoe a onopoly for product 1 while, in other worlds, the arket will becoe a onopoly for product 2 or product 3 To derive the theoretical results, we will use the following notations: η i,w η + i,w x + i x i x + ɛ i, x ɛ i, η i,w + ɛ i and η i,w ɛ i for i {2, 3} Theore 1 When the perturbed quality ranking is perfored in different worlds, the nuber of worlds in which products 1, 2 and 3 are ranked first denoted by s 1,, s 2, and s 3, respectively are given by the following binoial distributions Ps 1, k P k 1 k P 2 + P 3 k Ps 2, k P k 2 k P 1 + P 3 k Ps 3, k P k 3 k P 1 + P 2 k where P 1 P 2 P 3 f σ xf σ x + 2 F σx + 3 dx 4 f σ x + 2 F σxf σ x + 3 dx 5 f σ x + 3 F σxf σ x + 2 dx 6 Proof Let P 1 be the probability that ˆq 1,w be greater than axˆq 2,w, ˆq 3,w, ie, P 1 Pˆq 1,w > axˆq 2,w, ˆq 3,w P η 1,w > axη 2,w,η 3,w f σ xpx >η 2,w Px >η 3,w dx f σ xf σ x + 2 F σx + 3 dx Now let P 2 be the probabilty that ˆq 2,w be greater than axˆq 1,w, ˆq 3,w, ie, P 2 Pˆq 2,w > axˆq 1,w, ˆq 3,w P η 2,w > axη 1,w,η 3,w f σ x + 2 Px >η 1,wPx >η 3,w dx f σ x + 2 F σxf σ x + 3 dx Finally, let P 3 be the probability that ˆq 3,w be greater than axˆq 1,w, ˆq 2,w P 3 Pˆq 3,w > axˆq 1,w, ˆq 2,w P η 2,w > axη 1,w,η 3,w f σ x + 3 Px >η 1,wPx >η 2,w dx f σ x + 3 F σxf σ x + 2 dx Now the nuber of ties s 1, that product i is ranked first aong worlds is binoially distributed since, in each world, there is a probability P i that product i is ranked first Ps 1, k P k 1 k P 2 + P 3 k Ps 1, k P k 2 k P 1 + P 3 k Ps 1, k P k 3 k P 1 + P 2 k Corollary 2 The expectation and variance of s i, for i {1, 2, 3}, is given by E[s i, ]P i Var[s i, ]P i 1 P i Furtherore, the probabilities that the s i, s fall between their expected value ±X% are given by: P s 1, E[s 1, ] X 100 E[s 1,] 1+ X 100 E[s1,] k P 1 X 100 E[s1,] k P k 1 P 2 + P 3 k, s 2, E[s 2, ] X 100 E[s 2,] 1+ X 100 E[s2,] k P 1 X 100 E[s2,] 7 k P k2 P 1 + P 3 k 8 s 3, E[s 3, ] X 100 E[s 3,] 1+ X 100 E[s2,] k 1 X 100 E[s3,] k P k 3 P 1 + P 2 k 9 12

4 Figure 2: Probabilities that the nuber of worlds in which products 1, 2 and 3 becoe a onopoly fall in the expected value ± X % of the total nuber of worlds, as a function of the nuber of worlds Figure 3: Probabilities that products 1, 2, and 3 becoe a onopoly as a function of ɛ 2 where ɛ 3 2ɛ 2 and σ 01 We now quantify the loss ratio in purchases induced by the RSP copared to the asyptotically optial quality ranking, ie, d t q d t rsp l t d t, 10 q where d t q is the total nuber of purchases under the quality ranking at tie t and d t rsp is the total nuber of purchases of the RSP at t Expression 10 can be approxiated asyptotically with d t q q 1 t 11 and d t rsp q 1 P 1 + q 2 P 2 + q 3 P 3 t, 12 Equations 11 and 12 together with definition 10 can be used to characterize the asyptotic loss ratio, ie, l t ɛ 2P 2 + ɛ 3 P 3 q 1 13 Observe that, when q 2 and q 3 are equal to q 1, the RSP is optial but has no Matthew effect contrary to the quality and perforance ranking We now investigate the consequences of these results Figure 2 plots the probability that the nuber of worlds in which product i becoes a onopoly falls in between its expected value ± X % as a function of the nuber of worlds The plots are obtained fro Equations 7 9 for various values of paraeters ɛ 2 and ɛ 3 given σ 01 The figure also reports the forula for the expected arket shares that show a nice distribution of the purchases aong the products The plots also show a rapid convergence around the ean as the nuber of worlds increases Figure 3 depicts the probabilities that a given product becoes a onopoly as a function of ɛ 2 assuing that ɛ 3 2ɛ 2 and σ 01 The results show that, as the distance in quality increases, the arket shares also grow apart Figure 4: Probability that product 1,2 and 3 becoe a onopoly as a function of σ where ɛ 3 2ɛ but they reain nicely distributed aong the products as desired Figure 4 depicts the probabilities that a given product becoes a onopoly as a function of the standard deviation σ 01 given ɛ and ɛ 3 2ɛ 2 The figure shows how to use the standard deviation to balance the arket shares between the products It shows that sall standard deviations already balance the arket shares well, opening significant opportunities for controling the arket towards a desired outcoe Figures 5 plots the loss ratio as a function of ɛ 2 The ratio initially increases as ɛ 2 increases, since a lower quality product ay becoe a onopoly in soe worlds Once ɛ 2 approaches the standard deviation, the loss ratio starts to decrease since this sub-optial behavior is increasingly less likely to occur Experiental Results We now describe experiental results on the MusicLab settings Salganik, Dodds, and Watts 2006 Like previous stud- 13

5 Figure 7: Nuber of Purchases of the First 3 Products Figure 5: The Loss Ratio 13 as a Function of ɛ 2 When ɛ 3 2ɛ 2 and σ 01 Figure 8: Nuber of Purchases For the First 3 Products Figure 6: The Inequality Measure 14 over 50 Experients The largest possible Matthew effect is 2 ies, the experients use the setting in Krue et al 2012 which gives specific values for appeals, qualities, and visibilities The qualities of the second and third best products are odified to obtain a setting in which ɛ and ɛ the quality of the best product is q 1 08 This akes the Matthew effect particularly draatic, since a sall difference in quality induces large arket inequalities The experients ai at confiring the theoretical results within a finite horizon and at balancing the arket shares between the top three products The experients also use a standard deviation of σ 01 The presentation discusses the effectiveness of the RSP to tae the Matthew effect and the cost involved in doing so We also present soe results obtained on the original MUSICLAB data for copleteness Taing the Matthew Effect Figure 6 depicts the effectiveness of the RSP to tae the Matthew effect It reports the inequality easure d t 1 d t 2 2 +d t 1 dt 3 2 +d t 3 dt 2 2 d t 1 + dt dt 3 for the quality ranking and the RSP as the nuber of custoers increases The plot reports the ean solid and the ean ± the variance dashed lines for 50 siulations The largest possible Matthew effect with this easure is 2 The figure shows that the quality ranking converges towards 1 in average, while the RSP converges towards a value around 045 This represents a draatic reduction in inequalities aong the best products Figure 7 presents box plots for the total purchases of the top three products and clearly explains why there is such a reduction in the Matthew effect The figures suarize 50 experients, each with 2 illion custoers They show that the quality ranking ay produce significant Matthew effects since the third quartile exhibits purchases up to the order of and for the first two products This severe Matthew effect arises despite the fact that the products have alost the sae quality, showing the strong need of taing this effect in practice to obtain reasonably fair arkets In contrast, the RSP has a third quartile in the order of and for the sae products Observe also the higher edians and the saller variances in the RSP Figure 8 depicts the total nuber of purchases for the first three products in the quality ranking and in the RSP The plots show that the quality ranking has ore aggregate purchases We will coe back to this when we discuss the cost of fairness, ie, the cost of taing the Matthew effect Note also that the variance also increases in the RSP when aggregating the purchases of the three products, which is due to the added Gaussian noise and the weaker social influence since the arket is split in several words Of course, the variance for each product independently decreases in a substantial way, as discussed earlier 14

6 Figure 10: Purchase Loss Ratio 10 Worlds Figure 9: Purchase distribution for 500 experients with 2 illion custoers for the quality ranking and the RSP with 10 worlds right Figure 11: Purchase Loss Ratio 10 Worlds on the Actual MusicLab Data Figure 9 describes an interesting side-effect of the RSP: a significant decrease in outliers and variances for the total purchases of all products Observe how the top three products left side of the botto plot now have very siilar distributions and how the products in the range 4 15 have uch saller variances The segentation of the arket has strong benefits on the predictability of the arket as well that the loss slowly increases with the nuber of worlds Conclusion This paper presented a randoized segentation protocol RMS to jointly address the unpredictability, inefficiency, and inequalities created by social influence in trial and offer arkets The RMS was shown, both through theoretical analysis and siulations, to reove the Matthew effect aong the best products, balancing the arket shares as a function of product qualities The RMS also leads to predictable and efficient arkets as the loss in efficiency is shown to be very sall and tends to zero as the product qualities tend to the sae values As a result, the RMS sees to be a appealing, easy to deploy arketing strategy to leverage social influence in trial and offer arkets The Cost of Fairness Intuitively, the RSP has two net effects that ake the arket less efficient First, it reduces the ipact of social influence which is spread over any worlds, increasing the randoness in the early stage of the arket Second, it allows the arket to achieve onopolies that are not optial ie, onopolies of products whose quality is not the best aong all products Figure 10 quantifies this loss of efficiency: It depicts the average loss in purchases when using the RSP with 10 worlds As can be seen, the loss is sall and decreases significantly as the nuber of custoers increases, atching the theoretical predictions Figure 12 depicts the loss for the actual M USIC L AB data for various values of σ and the noise added to all songs Figure 12 shows how the loss varies at the nuber of worlds increases for different nuber of custoers The figure shows References Abeliuk, A; Berbeglia, G; Cebrian, M; and Van Hentenryck, P 2015 The benefits of social influence in optiized cultural arkets PLoS ONE

7 Abeliuk, A; Berbeglia, G; Maldonado, F; and Hentenryck, P V 2016 Asyptotic optiality of yopic optiization in trial-offer arkets with social influence In Kabhapati, S, ed, Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI 2016, New York, NY, USA, 9-15 July 2016, IJCAI/AAAI Press Craswell, N; Zoeter, O; Taylor, M; and Rasey, B 2008 An experiental coparison of click position-bias odels In Proceedings of the 2008 International Conference on Web Search and Data Mining, ACM Engstro, P, and Forsell, E 2014 Deand effects of consuers stated and revealed preferences Available at SSRN Hu, M; Milner, J; and Wu, J 2016 Liking and following and the newsvendor: Operations and arketing policies under social influence Manageent Science 623: Krue, C; Cebrian, M; Pickard, G; and Pentland, S 2012 Quantifying social influence in an online cultural arket PloS one 75:e33785 Leran, K, and Hogg, T 2014 Leveraging position bias to iprove peer recoendation PloS one 96:e98914 Li, A; Rodrigues, B; and Zhang, X 2004 Metaheuristics with local search techniques for retail shelf-space optiization Manageent Science 501: Muchnik, L; Aral, S; and Taylor, S J 2013 Social influence bias: A randoized experient Science : Rigney, D 2010 The Matthew Effect Colubia University Press Salganik, M J; Dodds, P S; and Watts, D J 2006 Experiental study of inequality and unpredictability in an artificial cultural arket Science : van de Rijt, A; Kang, S M; Restivo, M; and Patil, A 2014 Field experients of success-breeds-success dynaics Proceedings of the National Acadey of Sciences 11119: Van Hentenryck, P; Abeliuk, A; Berbeglia, F; Maldonado, F; and Berbeglia, G 2016 Aligning popularity and quality in online cultural arkets Tenth International AAAI Conference on Web and Social Media ICWSM-16 Viglia, G; Furlan, R; and Ladrón-de Guevara, A 2014 Please, talk about it! when hotel popularity boosts preferences International Journal of Hospitality Manageent 42: Figure 12: Purchase loss ratio as a function of the nuber of worlds for several nuber of trials The plot shows the average of 100 experients and the variance is shown with the error bars The paraeters chosen where σ 01, ɛ 3 2ɛ