social influence is ubiquitous in cultural markets . from book recommendations in amazon ,to song popularities in itunes , and article rankings in the online version of the new york times , social influence has become a critical aspect of the customer experience .yet there is considerable debate about its benefits and its effects on the market .  in their seminal paper , salganik , dodds , and watts argued that social influence makes markets more unpredictable , providing an explanation about why prediction in cultural markets is a wicked problem . to investigate this hypothesisexperimentally , they created an artificial music market called the musiclab .participants in the musiclab were presented a list of unknown songs from unknown bands , each song being described by its name and band .the participants were divided into two groups exposed to two different experimental conditions : the _ independent _ condition and the _ social influence _ condition . in the first group ( independent condition ) , participants were provided with no additional information about the songs . each participant would decide which song to listen to from a random list . after listening to a song , the participant had the opportunity to download it . in the second group ( social influence condition ) , each participant was provided with an additional information : the number of times the song was downloaded by earlier participants . moreover ,these participants were presented with a list ordered by the number of downloads . additionally , to investigate the impact of social influence , participants in the second group were distributed in eight `` worlds '' evolving completely independently .in particular , participants in one world had no visibility about the downloads and the rankings in the other worlds .the musiclab is a trial - offer market that provides an experimental testbed for measuring the unpredictability of cultural markets . by observing the evolution of different worldsgiven the same initial conditions , the musiclab provides unique insights on the impact of social influence and the resulting unpredictability .in particular , salganik et al suggested that social influence contributes to both unpredictability and inequality of success , with follow - up experiments confirming these initial findings  ( see also for a perspective on the musiclab paper ) .  in recent work , abeliuket al reconsidered the musiclab setting by exploiting the generative model that reproduces the experimental data .they showed that social influence always benefits market efficiency , if participants are presented with the optimal ranking , i.e. , the ranking that maximizes the probability of a download at each step .computational experiments also showed that the optimal ranking significantly outperforms the ranking in which products are displayed by decreasing popularity ( known as popularity ranking ) .  in this paper, we analyze the properties of a simple ranking policy for trial - offer cultural markets : the _ quality ranking _ that simply orders the products by quality .we show a number of positive results about the quality ranking , including the fact that social influence always benefits market efficiency when it is used and that the benefits of social influence and position bias are cumulative .most importantly , we show that , under the quality ranking , the market almost surely converges to a monopoly , assigning the entire market share to the product of highest quality . as a corollary , this implies that the market is optimal asymptotically .these results are robust and do not depend on specific values for the appeal and quality of the products . moreover ,computational experiments using the generative model of the musiclab indicate a fast convergence to the asymptotic predictions : the quality ranking quickly assigns the largest proportion of the market share to the product of highest quality and outperforms other ranking policies . also , there is only minor unpredictability in the market under the quality ranking policy .our results provide a sharp contrast with the conclusions in . under the quality ranking policy ,the market is entirely predictable : it converges to `` blockbusters '' and this convergence is due to social influence . in other words ,_ it is not social influence that makes markets unpredictable : unpredictability depends on how social influence is used . _it is also important to emphasize that the benefits of popularity , the social influence signal used in this paper , have been validated experimentally .indeed , two recent studies were conducted to understand the relative importance of the different social signals on consumer behaviour .the first study surveyed the downloads of more than 500,000 from the android marketplace google play , while the second study conducted an online experiment ( ) where participants were asked to rank hotels based on the number of reviews and the average rating .both studies arrived at the same conclusion , namely that the popularity signal ( i.e. , social preference ) has a much stronger impact than the rating signal ( i.e. , a state preference ) .these two studies provide evidence that quantitative models with popularity as the social signal are of significant importance .the relationships between quality , popularity , and appeal have also been studied experimentally .stoddard studied the relationship between the intrinsic article quality and its popularity in the social news sites reddit and hacker news .the author proposed a poisson regression model to estimate the demand for an article based upon its quality , past views and age among others .the results obtained after an estimation of each intrinsic article from these social news site showed that the most popular articles are typically the articles with the highest quality .another study of social influence was carried out by .the authors conducted a field experiment which showed that popularity information may benefit products with narrow appeal significantly more than those with a broad appeal .the rest of this paper is organized as follows .section [ section : market ] introduces trial - offer markets and section [ section : rankings ] surveys the ranking policies used in this paper .section [ section - properties ] presents the main properties of the quality ranking , including the benefits of position bias and social influence and the asymptotic convergence .section [ section : experiments ] describes the computational experiments and section [ section : conclusion ] presents the final discussion .all the proofs are in the appendix .this paper considers a trial - offer market in which participants can try a product before deciding to buy it .such models are now pervasive in online cultural markets ( e.g. , books and songs ) .the market is composed of products and each product is characterized by two values :  1 .its _ appeal _ which represents the inherent preference of trying product ; 2 .its _ quality _ which represents the conditional probability of purchasing product given that it was tried .each market participant is presented with a product list : she then tries a product in and decides whether to purchase with a certain probability .the product list is a permutation of and each position in the list is characterized by its _ visibility _ which is the inherent probability of trying a product in position .since the list is a bijection from positions to products , its inverse is well - defined and is called a ranking .rankings are denoted by the letter in the following , denotes the product in position of the list , and denotes the position of product in the ranking . hence denotes the visibility of the position of product .this paper examines a number of questions about these markets , including  * what is the best way to allocate the products to positions ? *is it beneficial to display a social signal , e.g. , the number of past purchases , to customers ?* is the market predictable ?the primary objective is to maximize the market efficiency , i.e. , the expected number of purchases .note also that the higher this objective is , the lower the probability that consumers try a product but then decide not to purchase it .thus , if we interpret this last action as an inefficiency , maximising the expected efficiency of the market also minimizes unproductive trials .[ [ static - market ] ] static market + + + + + + + + + + + + +  in a static market , the probability of trying product given a list is and the static market optimization problem consists of finding a ranking maximizing the expected number of purchases , i.e. ,    where represents the symmetry group over .observe that consumer choice preferences for trying the products are essentially modelled as a discrete choice model based on a multinomial logit in which product utilities are affected by their position .abeliuk et al . studied a generalization of the maximization problem where it is not required to show all products to consumers .the authors interpreted the problem as a optimal assortment problem with position - bias and they proved that it can be solved in polynomial time .[ [ dynamic - market - with - social - influence ] ] dynamic market with social influence + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the paper also considers a dynamic market where the appeal evolves over time according to a social influence signal .given a social signal , the appeal of product becomes and the probability of trying product given a list becomes the dynamic market uses the number of purchases of product at a time as the social signal .hence , in a dynamic market with steps , the expected number of purchases is defined by the following recurrence :  & u_t(d ) = _ s_n _ i=1^n p_i(,d ) ( q_i ( 1 + u_t+1(d[d_i d_i + 1 ] ) ) + ( 1 - q_i ) u_t+1(d ) ) & ( t 1 .. t ) + & u_t+1(d ) = 0 &  where  ] , the expected number of purchases at time 1 given ranking is the probability that product i is purchased in time 1 is given by which yields at time 2 , there are four possible states , depending upon whether some product was purchased at time 1 .let be the state where no product was purchased and let be the state where product was purchased at time 0 .the performance rankings for each state are :  ] .the expected numbers of purchases for each state are thus given by therefore , the expected number of purchases of the performance ranking for the first two steps is where the first term is the expected purchases at time 1 and the second and third terms are the expected number of purchases at time 2 given state times the probability of state .consider now the quality ranking which corresponds to the permutation $ ] which is fixed for all possible states .the expected number of purchases for ranking in each of the four states are the probabilities that product is purchased at time 1 are given by as a result , the expected performance of the quality ranking in two periods is , outperforming the performance ranking by about 3% in this case .this section describes the main properties of the quality ranking .  [ [ the - benefits - of - position - bias ] ] the benefits of position bias + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  we first show that position bias always increases the expected number of purchases when quality ranking is used .[ thm : q - position - bias ] position bias increases the expected number of purchases under the quality - ranking policy , i.e. , for all visibilities , appeals , and qualities  , we have   let be the expected number of purchases for the quality ranking .we have consider the index such that and .since , we have and , since , it follows that   [ [ the - benefits - of - social - influence ] ] the benefits of social influence + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  an important question in cultural markets is whether the revelation of past purchases to consumers improves market efficiency .this section shows that , under the quality ranking policy , the expected number of purchases in the studied trial - offer model increases when past purchases are revealed .this indicates that both social influence and position bias improve the market efficiency and their benefits are cumulative .the proof uses a lemma from specifying a sufficient condition for a ranking to benefit from social influence .the expected number of purchases is non - decreasing over time for the quality ranking under social influence .let =\frac{\sum_{i}v_{i}a_{i}q_{i}}{\sum_{i}v_{i}a_{i}}=\lambda.\ ] ] denote the expected number of purchases at time .the expected number of purchases in time conditional to time is =\sum_{j}\left(\frac{v_{j}a_{j}q_{j}}{\sum v_{i}a_{i}}\cdot\frac{\sum_{i\not = j}v_{i}a_{i}q_{i}+v_{j}(a_{j}+1)q_{j}}{\sum_{i\not = j}v_{i}a_{i}+v_{j}(a_{j}+1)}\right)+\left(1-\frac{\sum_{i}v_{i}a_{i}q_{i}}{\sum_{i}v_{i}a_{i}}\right)\cdot\frac{\sum_{i}v_{i}a_{i}q_{i}}{\sum_{i}v_{i}a_{i}}\ ] ] we need to prove that \geq \mathbb{e}[d_{t}],\label{eq:1}\ ] ] which amounts to showing that which reduces to proving \geq 0\ ] ] or , equivalently , \geq 0.\ ] ]  let , i.e. , the smallest index , such that .we have that = \sum_{j=1}^k \left[\frac{v_{j}q_{j}}{\sum_{i}v_{i}a_{i}+v_{j } } a_{j}v_{j}\left(q_{j}-\lambda\right)\right]+ \sum_{j = k+1}^n\left[\frac{v_{j}q_{j}\left(q_{j}-\lambda\right)}{\sum_{i}v_{i}a_{i}+v_{j}}a_{j}v_{j}\left(q_{j}-\lambda\right)\right].\ ] ] by definition of , the terms in the summation on the left are positive and the terms in the summation on the right are negative .moreover , for any and , we have that since , moreover , since the quality ranking rankes the products by quality and , equation ( [ eq : property1 ] ) and the definition of implies that  this observation , together with the fact that the left - hand ( resp .right - hand ) terms are positive ( resp .negative ) , produces a lower bound to the right - hand side of inequality ( [ eq : condition1 ] ) : + \sum_{j = k+1}^n\left[\frac{v_{j}q_{j}}{\sum_{i}v_{i}a_{i}+v_{j}}a_{j}v_{j}\left(q_{j}-\lambda\right)\right]\ ] ] + \frac{v_k q_k}{\sum_{i}v_{i}a_{i}+v_k}\sum_{j = k+1}^n\left[a_{j}v_{j}\left(q_{j}-\lambda\right)\right].\ ] ] now , by definition of , which implies that =0\ ] ] concluding the proof .[ [ asymptotic - behavior - of - the - quality - ranking ] ] asymptotic behavior of the quality ranking + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  we now show the key result of the paper : the trial - offer market becomes a monopoly for the best product when the quality ranking is used at each step . as a consequence, the quality ranking is optimal asymptotically since the best product has the highest probability to be purchased .the result also indicates that trial - offer markets are predictable asymptotically . to prove the result, we first characterize the probability that the next purchase is product . then we reduce a trial - offer market with two products to a generalized plya scheme , a urn and ball model with replacement .the result is then generalized to many products .the first result characterizes the probability of the next purchase .[ p_i ] the probability that the next purchase ( after any number of steps ) is product is   the probability that product is purchased in the first step is given by the probability that product is purchased in the second step and no product was purchased in the first step is given by more generally , the probability that product is purchased in step while no product was purchased in earlier steps is given by defining , equation [ noarreg ] becomes hence the probability that the next purchased product is product is given by since the probability that the next purchase is product is given by   consider first a trial - offer market with two products .since the steps in which no product is purchased can be ignored , by lemma [ p_i ] , we can use the following variables to specify the market :  where .the trial - offer market can thus be modeled as generalized plya scheme , where represents the number of balls of type at step and is the probability that a ball of type is drawn at step . since product is purchased , the generalized plya scheme add balls , each time product is purchased . as a result , the plya scheme uses the replacement matrix   this plya scheme is known to converge almost surely to a monopoly whenever ( see theorem 6 in , whose proof uses stochastic approximation techniques ) .[ monopoly-2 ] consider a trial - offer model with two products of quality and ( ) .the market converges almost surely to a monopoly for product 1 .this lemma can be generalized when there are many products but only two values of .[ monopoly - j ] consider a market with products such that and ( ) with .then , the market share of products converges almost surely to zero .the probability of purchasing one of the first ( resp .last ) product next is hence , we can define a 2-product market in which product 1 has an appeal and a quality and product 2 has an appeal and a quality , where by lemma [ monopoly-2 ] , this market converges almost surely to a monopoly for product 1 .hence the market share for products converges almost surely to zero .we are now in position to prove the main result of this section .consider a trial - offer markets where .then the market converges almost surely to a monopoly for product 1 .consider the generalized plya scheme associated with the market with types of balls .define a new generalized plya scheme which is similar to except that are replaced by . at the limit ,the fraction of balls of type in scheme can not be smaller than in scheme .consider now the market that corresponds to . by lemma [ monopoly - j ] , the market share of product converges almost surely to zero .the result follows from lemma [ monopoly-2 ] after repeating the process for products .this result has some key corollaries . in particular , the quality - ranking is optimal asymptotically , since only the best - quality song is left .the quality - ranking is asymptotically optimal in trial - offer markets .it is also important to point out that the market is completely predictable at the limit .  for completeness, it is useful to consider the case where the products have the same quality and there are no position bias . in this case, the market share of each product converges almost surely to a random variable following a beta - distribution .we prove this result for the case of two products . by lemma [ p_i ] , the probability that the next purchase is product becomes andthe replacement matrix becomes the diagonal matrix .now is a martingale sequence .indeed , =\frac{a_{i , k}+q}{a_{1,k}+a_{2,k}+q}z_{i , k}+\frac{a_{i , k}}{a_{1,k}+a_{2,k}+q}(1- z_{i , k } ) \\= \left(\frac{a_{i , k}+q}{a_{1,k}+a_{2,k}+q}\right)\frac{a_{1,k}}{a_{1,k}+ a_{2,k}}+\left(\frac{a_{i , k}}{a_{1,k}+a_{2,k}+q}\right)\frac{a_{2,k}}{a_{1,k}+ a_{2,k } } \\= \frac{a_{i , k}}{a_{1,k}+ a_{2,k}}\left(\frac{a_{1,k}+q}{a_{1,k}+a_{2,k}+q}+\frac{a_{2,k}}{a_{1,k}+a_{2,k}+q } \right )         \hspace{10 mm } \\ = \frac{a_{i , k}}{a_{1,k}+ a_{2,k}}=z_{i , k } \hspace{40 mm } \end{split}\ ] ]  as a result , the market is equivalent to a classical plya - eggenberger urn model where the variable converges almost surely to a random variable obeying a beta distribution ( theorem 3.2 in ) .  consider a market with two products of the same quality and equal visibilities .the market share of product converges almost surely to a random variable obeying a beta distribution with parameters and .[ [ static - performance - of - quality - ranking ] ] static performance of quality ranking + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the previous section has shown that the quality - ranking is optimal asymptotically in the dynamic market .we also know that the performance ranking is optimal for the static market .this section shows that the quality ranking has some performance guarantees even for the static market .in particular , it presents a bound on the performance of the quality ranking which depends only the visibility coefficients .the bound is also tight .[ thm : approx ] the quality ranking is an -approximation of the static market optimization problem , where .moreover , the -approximation of the quality ranking is tight .let be the optimal sorting and its expected number of purchases .we have let be the expected number of purchases for the quality ranking , i.e. , by theorem [ thm : q - position - bias ] , combining both bounds yields   we now show that the approximation is tight .consider 3 products with qualities and appeals and let the visibilities be .the quality ranking is and the optimal performance ranking is .the expected number of purchases for the quality ranking is while it is for the performance ranking .when tends to zero , the ratio between the performance and quality ranking becomes hence , when is large enough , the ratio is approximately , i.e. , previous sections have derived theoretical results for the quality ranking . in particular , they have shown that social influence and position bias always increase the expected performance of the quality ranking .these results hold for all possible values of the product quality , appeal , and visibility inputs .in addition , the results have provided strong static and asymptotic guarantees for the quality ranking .this section aims at illustrating these results on settings that model the musiclab experiments discussed in . as mentioned in the introduction ,the musiclab is a trial - offer market where participants can try a song and then decide to download it .it is important to emphasize that the generative model of the musiclab uses the same model for consumer choice preferences .[ [ the - experimental - setting ] ] the experimental setting + + + + + + + + + + + + + + + + + + + + + + + +   ( y - axis ) of position in the song list ( x - axis ) where is the top position and is the bottom position of the list which is displayed in a single column . ]   ( blue ) and appeal ( red ) of song in the four settings . in the first setting ( top left ) , the quality and the appeal were chosen independently according to a gaussian distribution .the second setting ( top right ) explores an extreme case where the appeal is negatively correlated with the quality used in setting 1 . in the third setting ( bottom left ) , the quality and the appeal were chosen independently according to a uniform distribution.the fourth setting ( bottom right ) explores an extreme case where the appeal is negatively correlated with the quality used in setting 3 . ]  the experimental setting uses an agent - based simulation to emulate the musiclab .each simulation consists of iterations and , at each iteration ,  1 .the simulator randomly selects a song according to the probabilities , where is the ranking proposed by the policy under evaluation and is the social influence signal ; 2 .the simulator randomly determines , with probability , whether selected song is downloaded ; in the case of a download , the simulator increases the social influence signal for song , i.e. , . otherwise , .every iterations , a new list is computed using one of the ranking policies described above . for instance , in the social influence condition of the original musiclab experiments , the policy ranks the songs in decreasing order of download counts , i.e. , the d - rank policy .the parameter is called the refresh rate .  the experimental setting , which aims at being close to the musiclab experiments , considers 50 songs and simulations with 20,000 steps .the songs are displayed in a single column .the analysis in indicated that participants are more likely to try songs higher in the list .more precisely , the visibility decreases with the list position , except for a slight increase at the bottom positions .figure [ fig : visibility ] depicts the visibility profile based on these guidelines , which is used in all computational experiments . under this setting , the ( worst - case ) approximation factor of the quality ranking ( see theorem [ thm : approx ] ) is .the paper also uses four settings for the quality and appeal of each product , which are depicted in figure [ fig : qualityappeal ] :  1 .  in the first setting , the quality and the appeal were chosen independently according to a gaussian distribution normalized to fit between and .2 .  the second setting explores an extreme case where the appeal is negatively correlated with quality .the quality of each product is the same as in the first setting but the appeal is chosen such that the sum of appeal and quality is exactly 1. 3 .  in the third setting ,the quality and the appeal were chosen independently according to a uniform distribution .the fourth setting also explores an extreme case where the appeal is negatively correlated with quality .the quality of each product is the same as in the third setting but the appeal is chosen such that the sum of appeal and quality is exactly 1 .the experimental results were obtained by averaging the results of simulations .[ [ performance - of - the - market ] ] performance of the market + + + + + + + + + + + + + + + + + + + + + + + + +     figure [ fig : down ] depicts computational results on the expected number of purchases for the various rankings and settings .they highlight two significant findings :  1 .the quality ranking exhibits a similar performance to the performance ranking and provides substantial gains in expected downloads compared to the download ( purchase ) and random rankings . on settings with a negative correlations between appeal and quality , the quality ranking performs better than the performance ranking .the benefits of social influence and position bias are complementary and cumulative as predicted by the theoretical results .both are significant in terms of the expected performance of the market .[ [ predictability - of - the - market ] ] predictability of the market + + + + + + + + + + + + + + + + + + + + + + + + + + + +      figures [ fig : q1 ] and [ fig : q2 ] depict computational results on the predictability of the market under various ranking policies .the figures plot the number of downloads of each song for the 400 experiments . in the plots ,the songs are ranked by increasing quality from left to right on the x - axis .each dot in the plot shows the number of downloads of a song in one of the 400 experiments .figure [ fig : q1 ] presents the results for the first setting where the quality and the appeal were chosen independently according to a gaussian distribution .figure [ fig : q2 ] presents the result for the second setting where the appeal is negatively correlated with quality and the quality of each song is chosen according to a gaussian distribution .the computational results are compelling and confirms the theory .figure [ fig : q1 ] shows that the best song always receives the most downloads in the quality ranking ( with social influence ) and that the variance in its number of downloads across the experiments is very small .the performance ranking ( with social influence ) also performs well but the best song does not necessarily receive the most downloads and the variance in its download is significantly larger .the download ranking is highly unpredictable , while the random ranking is highly predictable as one would expect .it is also interesting to note that these observations continue to hold even when the appeal is negatively correlated with quality , as figure [ fig : q2 ] indicates .the contrast between the popularity ranking used in and the quality ranking in this setting is particularly striking .  in summary ,the computational results confirm that , under the quality ranking policy , the market is highly predictable .the song with the highest quality emerges and the variance in its number of downloads is very small .this paper studied trial - offer cultural markets , which are ubiquitous in our societies and involve products such as books , songs , videos , clothes , and even newspaper articles . in these markets ,participants are presented with products in a certain ranking .they can then try the products before deciding whether to purchase them .social influence signals are widely used in such settings and help promote popular products to maximize market efficiency .however , it has been argued that social influence makes these markets unpredictable . as a result, social influence is often presented in a negative light .  in this paper, we have reconsidered this conventional wisdom .we have shown that , when products are presented to participants in a way that reflects quality , the market is both efficient and predictable .in particular , a quality ranking tends to a monopoly for the product of highest quality , making the market both optimal and predictable asymptotically .moreover , we have also shown that both social influence and position bias improve market efficiency .these results are robust and do not depend on the particular values for the appeal and quality of the products .in addition , computational experiments using the generative model of the musiclab show that there is a fast convergence to our asymptotic theoretical results about the performance and the predictability of the quality ranking .these results are in sharp contrast with the popularity ranking studied in , where the products are ranked by the strength of the social influence signals . in these conditions ,the market is indeed unpredictable and less efficient .there are some important lessons to draw from these results .* it appears that _ unpredictability is not an inherent property of social influence _ : whether a market is predictable or not really depends on how social influence is used .* with the quality ranking , it is not hard to predict popularity as the market will converge to a monopoly .the computational experiments also show that `` blockbusters '' are quickly identified , even when the appeal is negatively correlated with quality .* whenever the quality of the products can be approximated easily ( e.g. , by running some experiments with a small number of participants or learning it on the fly as suggested in ) , the quality ranking is the policy of choice : it is predictable and optimal asymptotically and is comparable , or outperforms , the performance ranking , i.e. , greedy policy that continuously optimizes the ranking based on the appeal , the quality , and the social influence signal of each product . * with the quality ranking , a high - quality product overcomes a poor appeal but the opposite does not hold .a product of poor or average quality never become popular even with a great appeal . obviously , aligning appeal and quality makes the market more efficient but it does not change the asymptotic outcome .it is also interesting to contrast our results with the study in which uses a different model for consumer choice preferences . in their model ,the probability to purchase a product is given by where is a constant and denotes the market share of product . with this choice model , with is large , monopoly occurs and _ eventually any of the products can get the largest market share , so the number of equilibria is _this comes from the fact that the social influence signal is much stronger in this model , illustrating again that it is how social influence is used , not social influence per se , which makes markets unpredictable and less efficient . in other words ,the popularity ranking reinforces the social signal too much while , in the model from , the social signal may be too strong .recall also that the choice model used in this paper was shown to reproduce the original experiments of the musiclab .overall , these results paint a very different picture of the benefits and weaknesses of social influence . social influence does not make the market unpredictable : when used with the quality ranking, it creates a predictable market for our trial - offer model .our current goal is to obtain a broader understanding of social influence by generalizing these results to different types of markets ( e.g. , markets with heterogeneous customers ) and different types of social signals .10  a. abeliuk , g. berbeglia , m. cebrian , and p. van hentenryck . . , 2014a. abeliuk , g. berbeglia , m. cebrian , and p. van hentenryck .the benefits of social influence in optimized cultural markets . , 10(4 ) , 2015 .s. ceyhan , m. mousavi , and a. saberi . ., 7(2):107134 , 2011 .  per engstrom and eskil forsell .demand effects of consumers stated and revealed preferences . , 2014 .peter hedstrm .experimental macro sociology : predicting the next best seller ., 311(5762):786787 , 2006 .coco krumme , manuel cebrian , galen pickard , and sandy pentland .quantifying social influence in an online cultural market ., 7(5):e33785 , 2012 .h.j . kushner and g.g .springer verlag , 1997 .kristina lerman and tad hogg .leveraging position bias to improve peer recommendation ., 9(6):18 , 2014 .d. luce . .john wiley and sons , 1965 .hosam mahmound . .chapman & hall / crc texts in statistical science , 2008 .lev muchnik , sinan aral , and sean j. taylor . social influence bias : a randomized experiment ., 341(6146):647651 , 2013 .henrik renlund . . ,february 2010 .matthew j salganik , peter sheridan dodds , and duncan j watts .experimental study of inequality and unpredictability in an artificial cultural market ., 311(5762):854856 , 2006 .matthew j salganik and duncan j watts .leading the herd astray : an experimental study of self - fulfilling prophecies in an artificial cultural market ., 71(4):338355 , 2008 .matthew j salganik and duncan j watts .web - based experiments for the study of collective social dynamics in cultural markets . , 1(3):439468 , 2009 .greg stoddard .popularity and quality in social news aggregators : a study of reddit and hacker news ., 2015 .catherine tucker and juanjuan zhang .how does popularity information affect choices ? a field experiment ., 57(5):828842 , 2011 .arnout van de rijt , soong moon kang , michael restivo , and akshay patil . field experiments of success - breeds - success dynamics ., 111(19):69346939 , 2014 .giampaolo viglia , roberto furlan , and antonio ladrn - de guevara .please , talk about it ! when hotel popularity boosts preferences ., 42:155164 , 2014 .

Abstract: social influence is ubiquitous in cultural markets , from book recommendations in amazon , to song popularities in itunes and the ranking of newspaper articles in the online edition of the new york times to mention only a few . yet social influence is often presented in a bad light , often because it supposedly increases market unpredictability .  here we study a model of trial - offer markets , in which participants try products and later decide whether to purchase . we consider a simple policy which ranks the products by quality when presenting them to market participants . we show that , in this setting , market efficiency always benefits from social influence . moreover , we prove that the market converges almost surely to a monopoly for the product of highest quality , making the market both predictable and asymptotically optimal . computational experiments confirm that the quality ranking policy identifies `` blockbusters '' in reasonable time , outperforms other policies , and is highly predictable .   these results indicate that social influence does not necessarily increase market unpredictability . the outcome really depends on how social influence is used .


loop quantum cosmology is the cosmological sector of loop quantum gravity, a canonical quantization of general relativity using ashtekar s variables . in this formulation , gravity becomes a constrained gauge theory with the basic classical fields being an su(2)-connection , a sum of the spin connection and the extrinsic curvature , and its conjugate , the densitized triad .these fields are represented in the quantum theory by holonomies of the connection and fluxes of the triad , which allows a mathematically well - defines formulation of the full theory. nevertheless , the complexity of the full formalism makes it advantageous to consider simplified models in a minisuperspace quantization .since there is a well - developed mathematical structure behind the full theory , the reduction to symmetric models can be done in such a way that essential features of the full theory , e.g. , the discreteness of geometric spectra , are preserved .this is in contrast to the older wheeler dewitt quantization where the full theory would not even be known beyond a purely formal level .  technically , states of the full theory are functions on the space of connections which depend on only via holonomies ( cylindrical functions ) .symmetric states are by definition distributions on the space of connections which are supported only on the sub - manifold of invariant connections. it turns out that such a reduction even to the simplest cosmological models yields a quantum representation which is inequivalent to the one chosen in the wheeler dewitt quantization ( which is possible also in systems with a finite number of degrees of freedom since the weyl algebra is not represented continuously). this translates into a different point of view as to the representation of basic variables . while the wheeler dewitt quantization assumes that the extrinsic curvature can be promoted to a well - defined operator ,the background independent techniques employed in the loop quantization allow only a representation of holonomies as operators , not the connection itself which in the isotropic setting corresponds to the extrinsic curvature .also triad operators behave differently : they would have continuous spectra in a wheeler dewitt quantization while they have discrete spectra in a loop quantization . on the other hand ,these basic properties are the same in full loop quantum gravity and in loop quantum cosmology such that one can regard those cosmological models as reliable tests .these basic technical details have far - reaching physical consequences . for instance , the wheeler dewitt quantization can not eliminate the cosmological singularities that plague general relativity , while they do not occur in loop quantum cosmology. also the issue of boundary conditions , which has been the main issue in the wheeler dewitt quantization, appears in a different light. all these effects are intimately linked to the underlying discreteness of geometry .  in this contributionwe describe both the basic formulation of the theory , emphasizing differences to the wheeler dewitt quantization , and physical applications .both parts can be read independently of each other .loop quantum gravity is based on general relativity in ashtekar s canonical variables and on a spacelike slice , where is the spin connection associated with the densitized triad , the extrinsic curvature , and the barbero immirzi parameter which is a positive real number . for a background independent quantization it is important to represent these basic fields via holonomies where is a curve in with tangent vector field and are pauli matrices , and fluxes where is a surface in with co - normal and a test function . if all curves , surfaces and test functions are allowed , these quantities have the full information contained in the fields and .the variables are constrained by the usual su(2)-gauss constraint as well as the diffeomorphism constraint , which requires invariance under deformations of space , and the hamiltonian constraint which yields the dynamics of the theory . since the diffeomorphisms act directly on the basic fields , a representation of their classical poisson-*-algebra has to be diffeomorphism invariant if this classical symmetry is to be preserved in the quantization .there is in fact a natural diffeomorphism invariant representation of the holonomy / flux algebra which has recently been shown to be unique. a heuristic construction of this representation can be done as follows .since there is a rich mathematical structure on the space of connections , it is convenient to look for the state space in the connection representation .the simplest such state is the `` ground state '' * 1* which does not depend on the connection at all .all higher states can then be `` created '' by acting with holonomies as multiplication operators . since one can act with several holonomies , one at a time, the final states will depend on the connection non - trivially , but only along the edges in the holonomies chosen , and only via the holonomies ( cylindrical functions ) .labels of a given state , the edges of the holonomies used to construct it , can be arranged to a graph in which may have self - intersections and also knotting and linking information . furthermore , since one can act with the same holonomy several times , one also needs a label representing the number of times a given edge occurs in the graph . another way to includethis information is by using irreducible su(2)-representations since acting once with a holonomy creates dependence of the state via the fundamental representation while acting several times then creates dependence according to the su(2)-coupling rules .an independent set of states can be seen to be given by the spin network basis given by graphs whose edges are labeled by irreducible su(2)-representations , or half - integer spin labels .  while any such state depends on the connection only via finitely many holonomies , the full state space is obtained as a projective limit over the set of graphs. thus , there is no restriction on the dependence of states on connections and one obtains a faithful representation of the infinite dimensional space of connections .this state space carries a natural diffeomorphism invariant inner product , the ashtekar lewandowski inner product , which defines the hilbert space of loop quantum gravity .fluxes represent the conjugate of the connection and thus act as ( functional ) derivative operators in the connection representation .since the states depend on the connection via holonomies , the action of flux operators can be obtained from the chain rule . a flux operator associated with a surface has a non - trivial action on a state labeled by a graph only if and intersect each other transversally .if they do , the action is given by a sum over transversal intersection points , each contribution counting the multiplicity of the edge encoded in its spin label .thus , flux operators count the intersection number of the surface with the graph of a state , counting also multiplicities , which is an integer . in this way , flux operators obtain a discrete spectrum which translates to discrete spectra of other geometrical operators constructed from the densitized triad , such as area or volume.   in order to follow the reduction procedure , we need to know the sub - manifold of invariant connections for an isotropic symmetry group .any isotropic connection can be written in the form where are left - invariant 1-forms on for the action of the symmetry group , is an internal triad ( an -matrix ) which is purely gauge , and is the only gauge - invariant component of the connection .similarly , an isotropic triad has the dual form with left - invariant vector fields dual to .isotropic states are then distributions in the full state space which are supported only on isotropic connections .thus , gauge invariant isotropic states can depend only on the component , and an explicit reduction , e.g. by restricting spin network states to isotropic connections , shows that any isotropic state can be represented as where the sum is over a countable subset of .the isotropic states form an orthonormal basis since their inner product is given by which implies that the hilbert space is non - separable ( all are allowed ) .in this connection representation , holonomies act as multiplication operators and the flux as a derivative operator while the hilbert space and the action of basic operators are much simpler than in the full theory , their technical features are the same : as already noted , the hilbert space is non - separable .furthermore , only holonomies , not itself are promoted to well - defined operators ; it is impossible to derive an operator for from the holonomy operators by differentiation since the operator family is not continuous in at : also as in the full theory , the flux operator has a discrete spectrum : its eigenstates are normalizable ( even though a continuous range of eigenvalues is allowed ) .  the results ( [ states ] ) , ( [ inner ] ) , ( [ c ] ) and ( [ p ] ) can also be used as axioms to define isotropic loop quantum cosmology , without following the reduction from the full theory .this may then be seen as a quantization alternative to and , as we will see , different from the wheeler dewitt quantization ( see also [ ] for a similar discussion in terms of adm variables ) .in order to compare this basic framework with that of the wheeler dewitt quantization , we start with the formal hilbert spaces . in a wheeler dewitt quantization one would simply employ a schrdinger quantization of the single degree of freedom such that the hilbert space is and the conjugate momentum acts as a derivative operator . the loop quantization is based on a different , inequivalent hilbert space since it is non - separable .it can be written as where is the bohr compactification of the real line .it is a _ compact _ abelian group which contains as a dense subset .the measure of the isotropic loop hilbert space is the haar measure on this compact group .thus , the wheeler dewitt quantization and the loop quantization are based on inequivalent hilbert spaces such that they can not be isomorphic to each other .this is possible even with finitely many degrees of freedom since the weyl algebra is not represented continuously in the loop representation ; see ( [ noncont ] ) .this assumption of the stone - von neumann theorem , which is usually used to prove the uniqueness of the quantum mechanical representation , is thus violated . as a consequence we have the basic differences for the operators : in the wheeler dewitt quantization there is an operator , which does not exist in the loop case , and the operator of the wheeler dewitt quantization has a continuous spectrum compared to the discrete one in the loop quantization .  on the other hand , the loop quantization of the isotropic model shares basic features with full loop quantum gravity . in the full theory ,the quantum configuration space is a compactification of the classical space of connections which is contained in as a dense subset .this can be seen by analyzing the projective limit construction sketched above. the situation in the isotropic case is completely analogous : the isotropic space of connections is extended and compactified to the quantum configuration space which contains as a dense subset . here , the same mechanism occurs just in a simpler context . as consequences we have both in the full theory and in the isotropic case a non - separable hilbert space , no well - defined operators for connections , and flux operators with discrete spectra .there is a difference , though , namely that the isotropic flux operator has a continuous set of eigenvalues even though its spectrum is discrete .this property comes from a degeneracy manifest in the reduction procedure : the label of an isotropic state does not distinguish between the parameter length of an edge and the spin when we compare it to a holonomy .there are technical subtleties resulting from this issue that can be dealt with appropriately ( see [ ] for a detailed discussion of all these concepts ) .isotropic models , then , provide reliable tests of basic and important features of the full theory . calculations in these models are certainly much simpler such that explicit results can be obtained .this is already shown by the volume operator which is rather complicated in the full theory while its spectrum can be obtained explicitly in the isotropic context. since it plays an important role in constructing other operators such as the hamiltonian constraint, also the dynamical equations become more treatable .we now turn to the physical applications which can be derived in this way .the dynamical equation of isotropic cosmology is the friedmann equation which is the same as the hamiltonian constraint equation . in connection variables ,the isotropic gravitational part of the constraint reads with the gravitational constant and the curvature parameter which is zero for the spatially flat model and one for the closed model . in order to relate this to the usual form of the friedmann equation, we need to know the relation between the basic variables and the scale factor . for the triad component, one has ( can be negative , indicating the orientation of the triad ) , while the connection component is .if we insert this into the constraint equation with some matter hamiltonian , we obtain in fact the friedmann equation    the quantization of the hamiltonian constraint is rather complicated ( see [ ] for the expression ) since it is a composite operator in terms of the basic ones , holonomies and fluxes .to solve the constraint , we need to act on a general state where the coefficients depend on matter fields which we collectively denote as .it is more convenient to write down the constraint equation for the coefficients directly , i.e.to work in a triad rather than connection representation , which we can later compare to the wheeler dewitt constraint . in this way ,our dynamical law becomes the difference equation using the volume eigenvalues   a wheeler dewitt quantization , in contrast , would result in a differential equation which turns out to be a good approximation to ( [ disc ] ) at large volume , , where the differences can be taylor expanded. at small , however , the discrete formulation is very different from the continuous one .this is crucial when we consider the fate of the classical singularity which is not removed in the wheeler dewitt quantization : generically , curvatures and energy densities still diverge and the classical singularity presents a boundary to the evolution ( in some special models one can argue for a more regular behavior in a different sense , see e.g. [ ] ) . in the loop quantization , on the other hand , it turns out that there is no singularity; instead , the dynamical law provides a unique evolution from the wave function at positive to its values at negative .this can be seen directly when one starts with initial data at large positive and uses ( [ disc ] ) as a recurrence relation to evolve backward toward the classical singularity at .there are no problems at first as long as the lowest order coefficient , , in the difference equation is non - zero . for , however , this coefficient is zero which means that remains undetermined in this way .if this value would then appear in the next stages of the recurrence , the evolution would break down and the classical singularity would present a boundary also for the quantum evolution . as one can see , this does not occur : also for and , where would appear in ( [ disc ] ) at first sight , it drops out thanks to vanishing coefficients . herewe use the fact that quantum geometry hamiltonians imply the special feature which is independent of quantization ambiguities and comes from a general procedure in the full theory. thus , also drops out in the matter part of the constraint equation and we can follow the evolution completely to negative .( for a classical picture of the corresponding bounce in a closed model , see [ ] . )  the classical singularity , therefore , does not appear as a natural starting point where one could pose initial conditions .nevertheless , the issue of boundary conditions appears in a new light now. when following the evolution we noticed that remains undetermined and we ignored the part of the constraint where it dropped out the first time ( ) .however , this part of the constraint still has to be satisfied providing us with a linear relation between and .this , in turn , translates to a linear relation between the initial data we chose . in the isotropic casethis is just the right amount to specify the gravitational wave function completely up to its norm ( ignoring the matter field dependence ) as usually done in the wheeler - dewitt quantization . in order to compare with wheeler dewitt wave functions we have to assume that the solution we obtain is almost constant at large volume at least for positive , i.e. that it does not have wild oscillations at the planck scale .this condition ( pre - classicality ) ensures that the solution has a continuum limit to be compared with a wheeler dewitt wave function .there are two independent such solutions which , when using the linear relation , are reduced to a unique ( up to norm ) linear combination . in this way , the dynamical law implies the boundary condition ( _ dynamical initial condition _ ) which is not chosen separately .boundary proposals for the wheeler dewitt quantization have been studied for some time , mainly focusing on the tunneling and no - boundary proposals. we now have a very different formulation in loop quantum cosmology with a discrete evolution equation and derived boundary conditions . nevertheless , after evolving the wave function to large volume , where the wheeler dewitt quantization is a very good approximation , one can compare different boundary proposals and possible physical implications .for this purpose , we will only consider positive and for the rest of this section .the first proposal is due to dewitt who suggested that the wave function should vanish at the classical singularity, .the dynamical initial conditions in fact coincide with this one for simple systems such as de sitter space with flat slices ( see fig .[ desitter]). in more complicated cases , however , the discreteness at small volume implies differences in the behavior of the wave functions . in this way, the dynamical initial conditions even work for models where dewitt s condition would not be well - posed.   and thus deviate strongly from the discrete one.,width=453,height=377 ]  other boundary proposals have almost exclusively been studied for the closed model , which is in particular true for the tunneling and no - boundary proposals . in the formulation of the proposals ,one simply considers the closed model with a cosmological constant , where there is a classically forbidden region between values of the scale factor between zero and some minimum classically allowed value .there are detailed evaluations of their different implications , but the central difference we are concerned with here is as follows .the tunneling wave function starts with a large value at and then decays exponentially in the classically forbidden region , which is motivated as a tunneling process of the universe .after that , the wave function becomes oscillating in the classically allowed region .the no - boundary proposal , on the other hand , does have a non - vanishing exponentially increasing contribution to the wave function which will soon dominate .thus , the wave function starts with a small value at and grows much higher before reaching the oscillatory phase .this different behavior is claimed to have consequences for chaotic inflation when one computes the probability distribution for the initial inflaton .this issue is still disputed , but it is often said that the no - boundary proposal prefers small initial values for the inflaton which would result in an insufficient amount of inflation .  in order to see how the loop wave function fits into this situation it is sufficient to note that the tunneling wave function is very special .it requires that the exponentially growing part does not contribute to the wave function which can be achieved only in this one very special case .any other proposal would have a non - vanishing contribution of the growing mode and would soon differ from the tunneling wave function .thus , any proposal which is not tailored to be exactly the tunneling one , would be much closer to the no - boundary condition at large volume .this is in particular true for the loop wave function with its dynamical initial condition .a numerical analysis ( figs .[ wavefunc ] and [ wavegamlog ] ) shows that this wave function also increases exponentially in the classically forbidden region and thus is closer to the no - boundary proposal .whether or not there are physically significant differences to the no - boundary wave function also at large volume remains to be studied .  ) for de sitter space with positive curvature spatial slices ( dots ) compared to a solution of the continuous wheeler dewitt equation which is regular at ( solid line).,width=453,height=377 ]    this conclusion may seem bad if the no - boundary proposal does in fact not provide enough inflation .however , when we consider the evolution of a scalar or other matter field coupled to the discrete quantum geometry , there exist small - volume quantum modifications to the kinetic term of the matter hamiltonian which automatically lead to inflation. the modifications arise from the quantization of the inverse volume which removes the classical divergence in the situation of chaotic inflation where one couples an inflaton , the modified inflaton dynamics implies that is driven up its potential at early stages ( see fig .[ scalarfield ] ) . before scales much larger than the planck lengthare reached , the inflaton turns around and enters a slow - roll phase of chaotic inflation .thus , the traditional considerations of probability distributions for the inflaton are not even needed here since there is a dynamical mechanism which results in large inflaton values .this occurs for a wide range of initial conditions with the initial value of the scalar field and its conjugate momentum being constrained by the uncertainty principle. the particular behavior may even have observable consequences since the slow - roll condition is violated around the turning point , i.e. at early stages of the slow - roll phase translating into large current scales. overall picture we have seen from loop quantum cosmology is very different from the traditional wheeler dewitt quantization , and its structure is much richer .there are many phenomenological effects : a cut - off of classically diverging densities, which is well - studied in flat models , implies inflation with possibly observable effects when coupled to chaotic inflation . also without coupling an inflaton ,the universe would still expand with positive acceleration .whether or not this is physically viable is undecided at this time since the usual perturbation theory becomes inapplicable.  in the closed model with its classically forbidden region there are even more diverse phenomenological possibilities which have not yet been studied in detail . in this case ,not just densities , or extrinsic curvature , experience a cut - off by quantum geometry effects , but also the intrinsic curvature can become suppressed at small volumes. there the closed model behaves similarly to the flat model with intrinsic curvature close to zero and a classically allowed region can be created near the singularity .it remains to be seen what physical consequences these effects have as the classically allowed regions lies deep within the quantum geometry regime at small volume .other open issues are concerned with details of the evolution through the classical singularity . since this involves the dynamics deep inside the planck regime , results are very sensitive to quantization choices .this may be used to find restrictions among possible choices , but there are also many different cases which have to be studied .the equation ( [ disc ] ) used here implies a symmetric solution which leads to equal behavior at positive or negative .but there is a more complicated version which is closer to the constraint in the full theory. the properties discussed here are insensitive to which one of those two forms for the constraint is used .but if we compare the wave function at positive and negative obtained from the more complicated constraint , then there will usually be significant small - scale oscillations at one side if they are constrained to be small at the other ( see fig . [ flatlorentz]). these oscillations arise because the more complicated constraint is of higher order in internal time than the simpler one ( also are involved ) .being discrete , the evolution equation may contain coefficients which do not behave smoothly especially near the classical singularity . thus passing through the singularitya smooth wave function can have the planck scale frequency modes excited whose amplitudes can be very large in some models. particular quantizations choices can be favored if they provide an evolution equation with smooth coefficients , which may be exploited to reduce the freedom in the full quantization .   from the continuum picture point of view , the freedom for solutions to the discrete equation is reduced compared to the wheeler dewitt equation .since one has to require the existence of a continuum limit ( or at least a good continuum approximation ) for the comparison , equation ( [ disc ] ) is reduced to second order ( even for the more complicated version mentioned in the preceding paragraph ) .the dynamical initial condition then leads to a reduction in the number of free parameters as seen before . from the discrete point of view , however , there are many more solutions ( infinitely many since is continuous ) with planck scale oscillations .their role and physical interpretation are open issues , and it is expected that an understanding requires the so far unknown physical inner product and the consideration of quantum observables .tentative ideas in this direction can be found in [ ] , and further investigations are in progress .there are promising possibilities for future developments .it is possible to extend the results to less symmetric models , which has already been completed for anisotropic but still homogeneous ones resulting in new physical effects. for cosmological considerations it is also important to add inhomogeneities at least as perturbations or in midisuperspace models .this will be done in the future in order to complete the picture provided by loop quantum cosmology , but even before these technical developments much remains to be studied , as indicated above , even in isotropic models .m. b. is grateful to c. kiefer for an invitation to a talk at the xth marcel grossmann meeting , july 2026 , 2003 , rio de janeiro , on which this paper is based .this work was supported in part by nsf grant phy00 - 90091 and the eberly research funds of penn state .d. l. wiltshire , an introduction to quantum cosmology , in b. robson , n. visvanathan , and w. s. woolcock , editors , _ cosmology : the physics of the universe _ , pages 473531 , world scientific , singapore , 1996 , [ gr - qc/0101003 ]

Abstract: for many years , the most active area of quantum cosmology has been the issue of choosing boundary conditions for the wave function of a universe . recently , loop quantum cosmology , which is obtained from loop quantum gravity , has shed new light on this question . in this case , boundary conditions are not chosen by hand with some particular physical intuition in mind , but they are part of the dynamical law . it is then natural to ask if there are any relations between these boundary conditions and the ones provided before . after discussing the technical foundation of loop quantum cosmology which leads to crucial differences to the wheeler  dewitt quantization , we compare the dynamical initial conditions of loop quantum cosmology with the tunneling and the no - boundary proposal and explain why they are closer to the no - boundary condition . we end with a discussion of recent developments and several open problems of loop quantum cosmology .