Universality in movie rating distributions

EPJ man uscript No. (will be inserted b y the editor) Universalit y in movie rating distributions Jan Lorenz Chair of Systems Design, ETH Z ¨ urich, Kreuzplatz 5, 8032 Z ¨ urich, Switzerland, jalorenz@ethz.ch Received: date / Revised versio n: date Abstract. In t h is pap er histograms of user ratings f or mo vies (1 ⋆ , . . . , 10 ⋆ ) a re analysed. The ev olving stabilised shapes of histograms fol low the rule that all are either dou b le- or triple-p eaked. Moreov er, at most one p eak can b e on the central bins 2 ⋆ , . . . , 9 ⋆ and the distribution in these bins looks smooth ‘Gaussian-lik e’ while c hanges at the extremes (1 ⋆ and 10 ⋆ ) often look abrup t. I t is shown that this is w ell approximated under the assumption that histograms are confined and discretised probab ility density functions of L´ evy skew α -stable distributions. These distributions are the only stable distributions which could emerge due to a generalized central limit theorem from a vera ging of v a rious indep endent random v ariables as which one can see th e initial opinions of users. Averaging is als o an appropriate assumption abou t the social process whic h underlies the pro cess of contin uous opinion formation. Surprisingly , n ot the normal distribution achiev es the b est fit o ver histograms obsev ed on the web, but distribu t ions with fat tails whic h d eca y as p ow er-la ws with exp onent − (1 + α ) ( α = 4 3 ). The scale and skewness parameters of the L´ evy skew α -stable distributions seem to dep en d on the dev iation from an av erage movie (with mean ab out 7 . 6 ⋆ ). The histogram of such an av erage m ovie has n o sk ewness and is t h e most narrow one. If a mo vie d ev iates from a ve rage the distribution gets broader and skew. The skewness pronounces the deviation. This is used to construct a one parameter fit which giv es some evidence of univ ersalit y in processes of contin uous opinion dyn amics ab out taste. P ACS. 89.20.Hh W orld Wide W eb, Internet – 89.75.Da S ystems ob eying scaling laws 1 In tro duction Are there universal laws underlying the dynamics of opin- ion formation? Understanding opinion forma tion is tackled cla s sically by so cial psyc hologists and so ciolo gists with exper iment s (see e.g. (1; 2; 3; 4; 5; 6; 7)) , but also by the socia l simu- lation (see e.g . (8 ; 9; 10; 11; 1 2; 1 3; 14)) a nd so ciophysics (see sur veys (15 ; 16; 17)) communities. O ften s tudies are either empirical but on small exp erimental samples or con- trary they analyse models analytically or by simulation but without empirica l v alidatio n. Both r estricts the p os- sibility to draw conclusions on universality in real world opinion for mation. This is to a larg e extent due to the difficulties in getting large scale data on human o pinions. But this situatio n changes r apidly now ada ys thanks to the world wide web. The existence of rating mo dules is al- most ubiquitous. (In the meant ime the ubiquity of ra tings has raised the ques tion how to standardise rating mo dules (18).) This pa p e r is an attempt to explo it r ating data to ex- tract universal prop erties in opinion fo r mation pro ces ses. Spec ific a lly , the fo cus he r e is on opinions ab out the qua lity of movies, as expr essed by user s on movie ra ting sites. Rat- ings stand as a proxy fo r any opinio n re la ted to taste which is one-dimensiona l and of a contin uous nature (‘contin u- ous’ means expressible a s a real num ber and a lso g r adually adjustable (at least to some ex ten t)). Apparently , poss ible user r atings are discrete (1 ⋆ (awful), . . . , 10 ⋆ (excellent)), but the contin uous nature (in the sense of ordered num- ber s) is also o bvious. Thu s, this pap er is no t abo ut discrete opinion dynam- ics without a contin uous nature (lik e e.g. with respect to decision: ‘yes’ or ‘no’) as o ften studied in physics b e cause of the analogy to spin s ystems. This paper is also not on m ultidimensional man y-faceted opinio ns (as e.g. (19; 14)) but on issues which are broken down to one v ariable: the quality of a movie. It is also imp or tant to distinguish the t yp e of opinio n. Mo vie ratings are ab out taste. There is no true v alue a s for example in issues of fact-finding ab o ut an unknown quan tit y . F urther on, there is no rea l physical constraint for opinions. It is alwa ys p ossible to like a movie more than s omeone else. This is for example not the case in opinions ab out budgeting in the p olitical re a lm, wher e opinions ha v e to b e within certain bounds. Finally , taste differs from issues ab o ut negotiations wher e there is a clear incent ive of agreeing on a common v alue (as e.g. for pr ic e s in trade, or forming a po litcal party in p o litical issues). In issues of tas te there is nevertheless a weak er forc e to adjust tow ards the opinions of p eer s, e.g . for normative reasons (‘I’d lik e to lik e wha t m y peer s lik e.’). But there might also b e a force to adjust awa y from the o pinions of others to pronounce individuality . 2 Jan Lorenz: Universalit y in movie rating distributions User ra tings on the world wide web hav e alrea dy been sub ject of resea r ch. Dellaro cas (2 0) sketc hes their role fo r digitising W ord-of-Mo uth (with the main fo cus o n reputa- tion mechanisms). Ratings play a key role in so me recom- mendation algorithms, s ee Goldb erg et al (21), C he ung et al (22), and Umay arov et al (23) which work by compa r- ing the ra ting profiles o f different us ers. They als o play a r ole in a recent method of pricing an option on movie reven ues, see Chance et al (24). Salg anik et al (6 ) study the emerging po pula rity of song s measur ed by downloads under the impact of the visibility of the n umber o f down- loads. They used ratings to chec k if liking corresp onds to downloads, which is the case. But which movie g ets p op- ular is to some extend arbitrar y . Jiang and Chen (25) arg ue eco nomically that the im- plement ation of online r ating s ystems ca n enhance co n- sumer surplus, vendor profitabilit y and so cial welfare. But they also arg ue, that this could work better in a monop o- listic market than a duop olistic market. Cosley et al (author?) (26) chec k ed how users r e-rate movies espe cially if they a r e confro nted with a prediction of the quality (lik e the mean of other ratings). They found a tendency to adjust tow ards the presented prediction. They also show that users rate quite consistently when they re-ra te on other scales (like 5 ⋆ compar ed to 10 ⋆ ). Li a nd Hitt (2 7) ana ly sed the time evolution of the user r e views ar riving. (A review is a text but it is accom- panied b y a rating is assig ned b y the writer.) They pres e nt an economic mo de l where the utility of a pro duct for a user is deter mined by individual sea r ch attr ibutes whic h are known before purchase and individual qualit y which can only b e chec k ed after purchase. Both attributes are heterogeneo us acr oss the popula tio n and purchasing deci- sions are made with resp ect to ex pe c ted quality . E xp ecta- tions can be influenced b y user reviews. P ositive r eviews of early adopter s pr o duce high av erage ratings and thus to o high exp ected quality . This trig g ers purc hases of other consumers which then ge t disapp ointed and write bad re- views. If individual sear ch attributes tow ards a pro duct are p os itively corre la ted with individual q ua lity then this may imply a declining trend of rev ie ws. This is called p os- itive self selection bias. Neg ative correlations imply neg- ative self selection bias and thus an incr e asing trend of reviews. These trends are confirmed empirically b y bo ok review da ta on amazon. com with the ma jo rity of pro ducts (70%) showing p ositive self selection. The phenomenon of declining av erage v otes has been explained in a different way in (2 8). They a r gue fro m the po int of v ie w of the writer in fr ont of a computer. W riting a review is costly (in terms of time) and writers wan t to impact the av erage v ote. While the av erage v ote ov er all b o oks is more p o sitive one can only ma ke a differenc e with a neg ative review, so writers with a p ositive attitude hesitate to write a review. (If ther e are alrea dy a lot, so why write a nother?) They also emphasize that internet reviews do not show a group po larizatio n effect whic h is known to app ear in small g roups discussing in the same ph ysical ro o m(5). There are few studies on characterising the e mpir ical distributions of r a tings. In (29) histogr a ms of user-ra tings (on 5 ⋆ -scale in movies. yahoo. com ) are c haracterised a s U-shap ed, while professional critics have a single-p eaked usage o f the v otes (peak is at 4). Other studies co ncentrate either on user profile co mparison or only on the av erage vote and how it could impact further votes and sales. In mo dels idiosyncra tic opinions ar e very often thought to be normal distributed (30; 31; 23). In the mo del of (27) the b eta distr ibutio n is us e d which lives on a b ounded int erv al. Normal distribution, Beta distribution, and U-shap e all do not coincide with the obse r v a tion of ra ting his- tograms studied which are very often triple p ea ked. In the following, the idea is intro duced that a r a ting of a user is derived from an originally contin uous o pinion from the whole real axis . The opinio b eco mes a rating by discretis- ing and confinig it to the ratings sca le. F urther on, w e as - sume that user’s orig inal opinions when it comes to rating are already arithmetic a v erag es of the express ed o pinions of peer s , opinions of professional critics and poss ibly the existing average (similar to the appr oach in (31)). This implies that limit theorems for sums of ra ndom v ariables play a role. 2 Empirical rating distributions and a s imple mo del The aim of this pap er is to characterise the distribution of ratings tow ards a certain movie when the ra ting histogr am contains a lot of rating s . F or a first analysis of the question some histog rams of movie rating have b een co lle c ted (32) A brief insp ection of a co uple of histogra ms reveals the following pictur e : Alm ost ev ery his to gram has either t w o or three pe aks. (A ‘pe ak’ is a bin where all neighbo ur bins are less in size. It is a loca l maxim um (or mode) o f the probability mass function of the distribution.) In the case of tw o p eaks at lea st o ne is at 1 ⋆ or 10 ⋆ . . In the case o f three pe a ks one is at 1 ⋆ and one at 10 ⋆ . The histog r am at the cent ral bins 2 ⋆ , . . . , 9 ⋆ has a ‘Gaussian- function like’ shap e with a p eak and exp o nent ially looking deca y . This gives rise to the idea that the histogram is a discr etised version of a proba bilit y densit y function on the real axis which is confined to the interv al of p oss ible r atings. Sp ecif- ically , we co nsider the opinion ab out a mo vie from cinema- go ers to be a real-v alued ra ndom v a riable which is some- how distributed. When it comes to ass ig n stars the v oter has to dis cretise her opinion to the bins 1 ⋆ , . . . , 1 0 ⋆ . Nat- urally , the voter would discr etise according to the interv a ls ] − ∞ , 1 . 5 ] , ]1 . 5 , 2 . 5] , . . . , ]8 . 5 , 9 . 5 ] , ]9 . 5 , + ∞ [. If a ll voters draw their vote from the same distributio n the histo gram will have bins with masses propo rtional to the in tegrals of the probability density function (p df ) of that distribution ov er the a b ove in terv a ls. Figure 1 sho ws how a contin uous distribution is confined a nd discretised to a proba bility mass function on 1 ⋆ , . . . , 1 0 ⋆ . The q ue s tion is now: What is this dis tr ibution a nd how univ ersa l can it b e para meterised? Before trying to Jan Lorenz: Universalit y in movie rating distributions 3 pdf for Levy skew α −stable S( α , β , γ , µ ;1) α =1.3202 β =0.045661 γ =1.2454 µ =7.7312 −5 0 1 2 3 4 5 6 7 8 9 10 11 15 0.1 0.2 pdf for Levy skew α −stable S( α , β , γ , µ ;1) α =1.2669 β =−0.00073142 γ =1.1933 µ =9.3817 −5 0 1 2 3 4 5 6 7 8 9 10 11 15 0.1 0.2 S( α , β , γ , µ ;1) pdf in bins 1,...,10; mass in tails on top of 1 and 10 −5 0 1 2 3 4 5 6 7 8 9 10 11 15 0.1 0.2 S( α , β , γ , µ ;1) pdf in bins 1,...,10; mass in tails on top of 1 and 10 −5 0 1 2 3 4 5 6 7 8 9 10 11 15 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10 5K 10K 15K 20K rating histogram "I Am Legend" (2007) ∗ is S( α , β , γ , µ ;1)−pdf scaled with #votes 1 2 3 4 5 6 7 8 9 10 20K 40K 60K 80K 100K 120K 140K rating histogram "Pulp Fiction" (1994) ∗ is S( α , β , γ , µ ;1)−pdf scaled with #votes Fig. 1. Exp lanation of confined L ´ evy sk ew α -stable distributions transformed to rating histograms. S how n are b est fits for tw o examples movies. answer this q uestion by loo k ing a t the data we formulate a simple so cial theory which limits the possible distributions to ’Ga ussian-like’ shap es. It is natural to assume that p eople make their mind ab out a mo vie not independent of the opinions of others. Each cinemago er might a djust her initial impressio n to- wards the o pinio ns o f others, to wards the existing mean rating or tow ards ratings of pro fessional critics. This is mo delled b y taking an average of several opinions as the final opinion of a cinemago er. Here, several asp e cts might be impor tant lik e so c ia l netw orks including corr elations of links and initial impr e ssions, opinion le a ders, timing ef- fects and so on. But if we assume that initial impressions are drawn from a random v ariable with finite v ariance, av eraging of a large eno ugh n umber of opinions implies a distribution of av eraged opinions clos e to a normal distri- bution due to the central limit theore m. This holds also when individual ra ndom v ariables a r e different under some additional mild assumptions. Also for contrasting forces like ’if I o bserve the av erage to b e 1 ⋆ higher then my opinion, I low er my opinion 1 ⋆ ’ the limit theorem holds, as long as the force s are linea r. According to this theory of opinio n making the histogr am of ra ting s should b e a discretised and confined probability density function of a normal distribution. The nor mal distribution do es not fit well, as it will turn out. Either the highest p eak is not achiev ed or the decay of bin size with distance from the highest pea k is to o fas t. Alternatively , we might as s ume, that initial impres- sions a re drawn from fat- ta iled distributions. This implies that distributions do no t hav e a finite v ariance. The prob- ability of extreme initia l impre s sions mig ht not v anish ex- po nentially but a s a pow er law with exp onent − (1 + α ). If this is the case a g eneralisa tion of the central limit the- orem sa ys that an av erage of these r andom v ariables has a distribution clo se to a L´ ev y skew α -stable distribution (the pa rameter α m ust indeed b e univ ersal for this theo - rem). So, we can keep the theory of averaging, but extend from the normal distribution to the wider c la ss of L´ evy skew α -stable distributions. The L´ evy sk ew alpha-stable distr ibutions are the o nly stable distributions (see (33)). It has four para meters α, β , γ , µ and is a bbreviated S ( α, β , γ , µ ). (There are several parametr i- sations of the L´ evy sk ew α -stable distribution. The one used here is S ( α, β , γ , µ ; 1) a s explained in (33).) Its pro b- ability density function is f S ( α,β ,γ ,µ ) ( x ) = 1 2 π Z + ∞ −∞ ϕ ( t ; α, β , γ , µ ) e − itx dt (1) with ϕ ( t ; α, β , γ , µ ) b eing its characteristic function given by ϕ ( t ; α, β , γ , µ ) = exp [ µ − | γ t | α (1 − i β sign( t ) Φ ) ] . (2) and Φ = t an( π α 2 ) if α 6 = 1 and Φ = − 2 π log | t | . The four para meters are α ∈ ]0 , 2 ], , β [ − 1 , 1], γ ∈ [0 , ∞ [, and 4 Jan Lorenz: Universalit y in movie rating distributions α =4/3 α =2 (Gaussian) α =2/3 β =0 β =1 β =−1 γ =1 γ =2 γ =0.7 Fig. 2. Role of parameters α, β , γ for the shap e of the p robabilit y densit y f unction of L´ evy skew α -stable distributions. The base line case in black is the probability density function of S ( 4 3 , 0 , 1 , µ ) with m u marked on the x-axis. µ ∈ ] − ∞ , ∞ [. The first tw o par ameters ar e shap e pa- rameters, where α represents the p eakedness and β the skewness; µ and γ are lo catio n and scale para meters. (But notice that β is not the s kewness in terms of the third moment, and α is not the pe akedness in terms of kurto- sis.) F or α > 1 , µ also represen ts the mean o f the distri- bution (otherwis e not defined). Figur e 2 shows how the parameters mo dify the sha p e o f the pro ba bility density function. Small α represents a sharp pea k but heavy ta ils which asymptotica lly decay as p ower laws with e xp o nent − ( α + 1). Maximal α = 2 is the no rmal distribution with exp onential deca y at the tails . Scale para meter γ corr e - sp onds to the v ariance σ 2 by the relation σ 2 = 2 γ 2 only for α = 2. F or lower α the v a riance is infinite. Skewness β = 0 gives a distribution s ymmetric aro und the mean, a positive β implies a heavier left ta il, a negative β a heavier r ight tail, but with the same decay on both sides. Only in the case β = ± 1 one tail v anishes completely . If α = 2 then β has no effect. Only the s pe c ia l cases o f the nor mal distribu- tion ( S (2 , 0 , σ, µ )), the Cauch y distribution ( S (1 , 0 , γ , δ )) and the L´ evy distribution ( S ( 1 2 , 1 , γ , δ )) hav e close d form expressions . In the following the L´ ev y skew α -stable distributions (discretised and confined) will b e fitted for each empirical rating distributions in the data set. 3 Fitted L´ evy skew α -stable distributions Fitting is done by minimising least squares of the differ- ence of the no rmalised empirical rating histogram to the confined and discretised probability density function of L´ evy skew α -s table distributions with par ameters ( α, β , γ , µ ). (F or numerical r easons, fitting has been done with a dif- ferent parameterisation S ( α, β , γ , δ ; 0) (see (3 3)). The pa- rameters α, β , γ ar e equal to the fo r mer parameteris ation and µ = δ − β γ tan( π α/ 2 ).) Computation w as p erformed as fo llows: The v alues of the probability density function f S ( α,β ,γ ,µ ) ( x ) are computed for x = − 20 , − 19 , . . . , 29 , 30 by computing and in tegrating the characteristic function (Eq. 2) on t = − 20 . 00 5 , +0 . 01 . . . : 20 . 0 05. Then v alues for x = − 20 , . . . , 1 are summed up and se t on bin 1 and v al- ues for x = 10 , . . . , 30 ar e summed up a nd set on bin 10. This pr o duces a probability mass function on 1 , . . . , 10 for ( α, β , γ , µ ). Results w ere rea sonably go o d, the missing mass of the tails (b elow -20 and ab ove +30) was mostly below 0 . 3%. The fitting was computed by minimising the squares of distances of the pro ba bility mass function for S ( α, β , γ , µ ) to the norma lis ed empirica l ra ting distr ibu- tion. The minima were found with the matl ab -function fminse arch . The sear ch c onv erged in 10 81 c a ses (99 . 5%) the remaining ca ses it terminated by maximum num ber of iteratio ns. Finding a glo bal minimum is no t guar anteed by this metho d, but r esults lo o ked convincing. (Exper i- men tally , some fits have bee n co mputed v ia minimising by gradient descent. It lead to very similar fits.) W e refer to this fit as fit( α, β , γ , µ ) Examples o f fits ar e shown in Figure 1. T a ble 1 shows the mea n v alues o f fitted pa rameters ov er all movies as well as go o dness -of-fit measures. The sum of squared error (SSE = P 10 i =1 ( r i − f S ( α,β ,γ ,µ ) ( i )) 2 with r i being the fraction of ratings for i ⋆ ) is on av erage very s mall, the co efficient of determinatio n R 2 is o n av er- age almost one. ( R 2 = 1 − SSE P 10 i =1 ( r i −h r i ) with r i the frac- tion o f r a tings for i ⋆ (therefore h r i = 0 . 1).) Both r eflects that indeed most fits also loo k impressively close to the empirical histogram. F urther on, a Kolmogo rov-Smirnov test has been perfo rmed for ea ch mo vie. (Done with the matlab -function kstes t2 on the vector of all ratings and a vector with the sa me num ber of ratings as exp ected ac- cording to the fit.) With level of significance 0 . 0 5 the null hypothesis that the exp ected fitted distribution and the empirical his to gram are drawn from the sa me distribu- tion could not b e re jected fo r 68 . 7% of the movies. The Kolmogo rov-Smirnov test is very hard, it rejects the null hypothesis very likely for la rge samplesizes . Given the high nu mber of ratings ( > 20 , 000 ) for each movie this r ate is still impressive. But it is als o clear tha t L ´ evy skew α -stable cannot fully explain all p o s sible rating histograms . F o r comparison T a ble 1 also con tains mean v alues for a fit with normal distributions S (2 , 0 , γ , µ ). The go o dness- of-fit pa rameters are worse. This is natur a l b eca use there are less free parameters , but clearly the no rmal dis tribu- tion is ruled out as an appropr iate candida te. Figure 3 shows the parameters o f b e s t fits for all mo vies as scatter plo ts. All four subplots show µ at the absciss a. Dark points indicate mo vies whic h fits have a small sum of squared errors (SSE), red s tars indicate medium SSE, and yellow stars indicate bad fits with high SSE. The fir st plo t shows µ with res pe ct to the original av- erage of r atings. It shows that µ is spread wider than the Jan Lorenz: Universalit y in movie rating distributions 5 h ratings i co rr-co ef h fit( α, β , γ , µ ) i h fit(2 , 0 , γ , µ ) i h fit( µ ) i mean 7.3464 0.5772 7.6326 7.6590 7.5862 µ std 1.9669 0.8883 1.2021 1.2456 1.1993 γ skew ness -1.0610 -0.2829 0.015 9 0 -0.0114 β kurtosis 1.8581 -0.0138 1.32 61 2 4 3 α 0.0002 0.0035 0.0035 SSE 0.9965 0.9404 0.9434 R 2 68.7% 0% 4.5% K-S X T able 1. Aggregated measures on the d ata set and on three confined L ´ evy skew α -stable fits (fit( α, β , γ , µ ), fit(2 , 0 , γ , µ ), fit( µ )). Mean, standard deviatio n, sk ewness, and k urtosis are computed for the histogra ms of eac h movie. Par ameters of fits are also computed for eac h mo vie. The table sho ws the mean v alues for all 1,08 6 movies. The correla tion co efficient is co mputed for the ‘analog’ measures for the ratings and fit( α, β , γ , µ ). T he lo w (and nega tive) correlation sk ewness v s. β and kurtosis vs. α sho w that the parameters of the fi t d eliv er information on t he distribution which is not extracted by th e ‘standard measures’ on the ra w data. Goo dness-of-fit parameter are computed for each fit for eac h m ovie. The mean ov er all mo vies is sho wn for sum of squared error (SSE) and co efficient of determination R-square. F or the Kolomogoro v-Smirnov t est (K-S) the rate of not rejecting of the hyp othesis of a common distribu tion of ratings and fi tted distribution is given. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 average vote µ 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 α µ 1 2 3 4 5 6 7 8 9 10 −1 −0.5 0 0.5 1 β µ 1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 γ µ Fig. 3. Parameters of best fit for confi ned Levy skew α -stable distributions for all movies. µ is the mean of the distribution, α mod ifi es p eakedness and tail exp onent, β sk ewness, and γ scales ho w broad th e distribution is. 6 Jan Lorenz: Universalit y in movie rating distributions original av erag e. So, µ can serve as a measure for mo vie quality which differen tiates b etter than the original av er- age. The remaining three subplo ts sho w the rela tions of µ to the other three parameters α, β , γ of the b est L´ evy sk ew α -stable fits. The blue dots in the tw o b ottom plots s how the averages o f the o rdinate v alues within the µ -reg ion marked by the g rid lines. The green lines represents α = 4 3 , for β the b est linea r fit for the blue dots, a nd the b est quadratic fit for γ . T he plots show that the p eakedness α concentrates to v a lues b etw een 1 . 2 a nd 1 . 5, which is clearly not normal distributed. The a verage v alue is h α i = 1 . 3261 ≈ 4 3 . F or the skewness β there is a clear tre nd with resp ect to µ . Interestingly , β = 0 is most likely almos t exactly at µ = 7 . 6326 which is equal to h µ i . F or b etter movies ther e is an additional p ositive s kewness (mea ning that the righ t tail is fatter). Respectively , for movies worse than h µ i there is additional neg a tive skewness (meaning that the left tail is fatter). F or the sca le parameter there is also a clear trend visible. The most narrow distr ibution is achiev ed also almost exac tly for movies with µ = h µ i . F or better and worse movies the distributio ns g et broader . It is no t apr iori clear and thu s remar k a ble that h µ i plays a ce ntral ro le for the deviations in β and γ with re- sp ect to µ . This gives rise to the sp eculation that h µ i is kind of universal mo dulo the scale of ra ting s (here 1 , . . . , 10). This is underpinned b y the finding o f (author?) (26) tha t users rate consistently in different ra ting schemes. If we rescale h µ i = 7 . 6326 to the scale 1 , . . . , 5 we get 4 . 0 633 which coincides almost exa ctly w ith 4 . 07 which is the av- erage mean rating of b o oks av eraged o ver all b o oks in the amazon .com - sample of (author?) (27). Rescale is done under the ass umption that each rating stands for a bin centred on the r ating with width equal to the distance o f successive ratings (here 1). Thus a 10 ⋆ -ra ting r 10 is c o n- verted to the 5 ⋆ -r ating b y r 5 = 5( r 10 − 0 . 5 10 ) + 0 . 5. This en- sures fo r examples that 1 ⋆ in a 5 ⋆ -ra ting cor resp onds to 1 . 5 ⋆ sta rs in a 1 0 ⋆ -rating, r esp ectively 5 ⋆ corresp onds to 9 . 5 ⋆ . It do es not coincide as go o d with 3 . 44 which was found b y (29) for mov ies.y ahoo.c om -data. The devia tion may come from tw o differences: First, in (2 7) and in this study the average rep o r ted is the average o f the av erag e ratings of movies, while (29) rep o rts the the pure average rating ov er all ratings in the da tabase. Second, (27) and this study select b o oks r esp ectively movies similar: this study by all having mor e than 20,000 r atings, and (27) by being on a b estse ller list and having a sufficien t num ber of reviews. Both sampling method imply a simila r selection bias which is different from (2 0) w hich co llects all movies released in 2002. T a king this speculatio n as true it means that an av- erage movie r eceives an average vote o f about 0 . 71 o n a generalise d sca le [0 , 1 ]. This indicates a universal strong po sitive bia s for the av erag e movie. The strong p o sitive bias may b e implied by an overall selection-bias, that user select movies or pro ducts they are lik ely to like or ev en they lik e movies and products just b eca us e they paid for them. Co nt ras ting , a neg ative bias is rep or ted on r atings for jokes in (2 1). F ollowing the re sults o f fit( α, β , γ , µ ) we further conclude that the dis tribution of rating s for a n av erage movie has no skewness ( β = 0) and the smallest scale parameter (here γ = 1 . 1). If a movie dev ia tes from av erage this implies higher deviations in the distribution ( γ > 1 . 1) and a s kewness whic h pronounces the deviation from the av erage movie. The latter observ ation can b e r e- garded a s a hint for a so cially implied p ositive feedback in deter mining opinio ns on movies which quality is ab ove (or below) a n average movie. T a king the trends displayed by the gr een lines in Fig- ure 3 o ne can co ns truct a one- parameter fit on µ with α = 4 3 , β determined by the linear fit and γ by the quadratic fit. The equations to co mpute β , γ from µ are β = b 1 µ + b 2 and γ = c 1 µ 2 + c 2 µ + c 3 with para meter s b 1 = 0 . 1 178 , b 2 = − 0 . 904 9 , c 1 = 0 . 0 5342 , c 2 = − 0 . 8388 , c 3 = 4 . 401. W e refer to this fit as fit( µ ). Mea n v alues and mean go o dness - of-fit measures are also shown in T able 1. The one-pa rameter fit gets better g o o dness than the tw o parameter fit(2 , 0 , γ , µ ). Figure 4 shows how fit( µ ) is able to approximate em- pirical histograms. The shape of empirical distributions is well captured but v aria tions fo r different movies ar e big enough to co nclude that fit( µ ) can only be seen as a base- line case. Movies can hav e some individual characteristics of their rating distribution which go b eyond the qualit y (captured in µ ). Devia tions fro m the bas eline case can b e used to classify mo vies in a new wa y to under stand what the cause o f deviations might b e. This is a task for further resear ch. Finally , Figure 5 sho ws a comparis on of theoretical his- tograms of fit( µ ) and the average empirical his tograms. The theoretical his tograms ar e for µ = 5 , +0 . 5 . . . , 9 a nd the av erage empirical histogra ms are over all movies with fit- ted v alue of µ w ithin the interv als µ ∈ [4 . 75 , 5 . 25 ], [5 . 25 , 5 . 75 ], . . . , [8 . 2 5 , 8 . 75], [8 . 75 , 9 . 25]. The simila rity underpins that fit( µ ) can really serve as a go o d baseline case. But s ome deviations fro m the baseline cas e seem to b e not to tally random. E.g. the residuals show that the s ize of the 1 ⋆ bin for low quality movies ( µ < 7 ) is on av erage pr edicted to o high, while the 2 ⋆ , 3 ⋆ bins a re o n average predicted to o low. 4 Con c lusion With some succ ess rating histograms were fitted to con- fined L´ evy s kew α - stable distributions. This clearly demon- strates that the a ssumptions that opinions are normally distributed, b eta distributed or U-shap ed ar ound the qual- it y of the mo vie is not v alid. Some histogr ams have of course a U-shap ed (or better J - shap ed) for m, e.g. rig ht- hand side in Figure 1. But a U-sha p e c an not approxiamte all histograms, e.g. left-hand side of Figure 1. If the as sumption that expressed opinions of users are weigh ted av erag es of formerly expressed opinions of others this implies that these opinions must come fro m distribu- tions with fat tails with a p ow er-law exp onent o f about 1 . 2 to 1 . 5 to get go o d fits. F urther on, the sca le and skewness parameter of the best fits c hange systematically with the deviation of its mean from the mea n of an av erage movie (with µ ≈ 7 . 6). A movie b etter than a verage shows rig ht Jan Lorenz: Universalit y in movie rating distributions 7 1 2 3 4 5 6 7 8 9 10 10% 20% 30% ratings µ −fit and histograms movies around µ =6 1 2 3 4 5 6 7 8 9 10 10% 20% 30% ratings µ −fit and histograms movies around µ =7.5 1 2 3 4 5 6 7 8 9 10 10% 20% 30% ratings µ −fit and histograms movies around µ =9 Fig. 4. The theoretical probabilit y mass functions according to the parameters of fit( µ ) for µ = 6 , 7 . 5 , 9 and all empirical histograms which receiv ed b est fitted v alues for µ ∈ [5 . 9 8 , 6 . 02] , [7 . 48 , 7 . 52] , [8 . 98 , 9 . 02] . 1 2 3 4 5 6 7 8 9 10 10% 20% 30% votes Levy skew α −stable pdf of µ −fit 1 2 3 4 5 6 7 8 9 10 10% 20% 30% votes average rating histograms 1 2 3 4 5 6 7 8 9 10 −2% −1% 0% 1% 2% votes residuals: average ratings minus fitted pdf Fig. 5. The theoretical p robabilit y mass functions according to th e parameters of fit( µ ) for µ = 5 , +0 . 5 . . . , 9 (left), empirica l a vera ge histograms for all movies with b est fitt ed v alues for µ ∈ [4 . 75 , 5 . 25], [5 . 25 , 5 . 75], . . . , [8 . 2 5 , 8 . 75], [8 . 75 , 9 . 25] (righ t), and residuals of b oth. The stars mark the underlying µ -v alues of curves. Colors are the same in all plots. skewness and a larg er sca le pa rameter. A movie worse than av erage shows left skewness and a als o a larger s cale pa- rameter. Thus, b e tter movies ha ve a ls o a heavier tail on the b etter side a nd worse movies hav e a heavier tail on the worse s ide. In gene r al, distributions get broader when deviating from the mean. Both o bserv ations s eem plausi- ble from a so ciolo gical point of view. The new measures of skewness ( β ) a nd pe akedness ( α ) are not the same as the cla ssical s kewness a nd exc ess kur to sis which ar e com- pute directly from the sample data (see T able 1 ). There is no corr elation of both measures, or ev en a negative one. This underpins, that fitting rating histogr ams a s confined distributions really delivers a new characterisation. A fur- ther adv ant age of this approach is, that the L´ e v y skew α - stable distribution defines a distr ibutio n co mpletely , whic h mean, standar d deviation, sk ewness and excess kurtosis do not. A o ne-parameter fit based on this observ ations shows to a pproximate the data well, but is no t able to esta blish a strict characterisation of mo vie histogra ms. Deviations from the constructed ba s eline case a re not neglectable. Nevertheless, it could b e useful to characteris e movies by their deviation from their baseline case . F urther o n, there might is a selection bias in the data, b ecause o nly movies with a larg e num ber of ratings were selected. The fit migh t not work fo r less rated movies. The method migh t b e us e d to detect attacks of en thusiastic fans (or movie co mpanies) which try to rate movies up. There seems to be some universalit y in movie rating distributions, which ma y b e implied b y p eople adjusting their opinions with p eers and other so urces of opinions. 8 Jan Lorenz: Universalit y in movie rating distributions Clearly , o ther theories which may imply other underlying distributions need to b e develop e d and check ed a g ainst data and also this theory needs to b e check ed against data from other sources to clarify universality in co ntin uous opinion dynamics ab out taste. Acknowledgement The r esearch leading to these r esults has received funding from the Euro p e an Communit y’s Sev - ent h F ramework Prog ramme (FP7/2 007- 2 013) under grant agreement no. 231 323 (Cyb erEmotions pro ject). References 1. S. Asch, Scientific American 193 (5), 31 (195 5) 2. I. Lor ge, D. F ox, J. Davitz, M. Brenner, P sychological Bulletin 55 (6), 337 (1 958) 3. M. Sherif, C. Hovland, So cial Judgment: Assimilation and Contr ast Effe cts in Communic ation and A ttitude Change (Y ale University Press , New Hav en, CT, 196 1) 4. J. F org a s, Eur op ean Jo urnal of Social Psychology 7 (2), 1 75 (1 977) 5. N. F riedkin, American So cio logical Review 64 (6), 856 (1999) 6. M. Salga nik, P . Do dds, D. W atts, Scie nc e 311 (576 2), 854 (2006) 7. W. Maso n, F. Conrey , E . Smith, Persona lit y and So- cial Psychology Review 11 (3), 279 (20 0 7) 8. G. Deffuan t, D. Neau, F. Am blard, G. W eisbuch, Ad- v a nces in Complex Systems 3 , 87 (2000) 9. R. Hegselmann, U. Kr ause, Journal of Artificial So cieties and So cial Sim ulation 5 (3), 2 (2 002), http:/ /jass s.soc.surrey.ac.uk/5/3/2.html 10. D. Urbig, Journal of Artificial So cieties and So cial Simulation 6 (1), 2 (20 03), http:/ /www. jasss.surrey.ac.uk/6/1/2.html 11. G. Deffuant , D. Neau, F. Am blard, G. W eisbuch, Journal o f Artificial So cieties and So cial Sim ulation 5 (4) (2 002) 12. W. Jage r, F. Amblard, Co mputational & Mathema t- ical Organiza tion Theory 10 , 29 5 (20 04) 13. L. Salzarulo, Journal of Artificial So ci- eties and So cial Sim ulation 9 (1), 13 (2006), http:/ /jass s.soc.surrey.ac.uk/9/1/13.html 14. D. Balda ssarr i, P . Bea rman, Amer ic an So c iological Review 72 (5), 784 (200 7) 15. D. Stauffer, AIP Conference Pro ceedings 779 , 56 (2005) 16. J. Lorenz, Int. Journal of Modern Physics C 18 (1 2), 1819 (20 07) 17. C. Castellano, S. F or tunato, V. Lo reto, Review of Mo dern Physics 81 , 591 (2 0 09) 18. D. T urnbull, in CHI ’07: CHI ’07 exten de d abstr acts on Human fa ctors in c omputing systems (A CM, 2007), pp. 2705– 2710 19. J. Lorenz, in Managing Complexity: Insights, Con- c epts, Applic ations , edited by D. Helbing (Springer, 2008) 20. C. Dellaro cas, Management Science 49 (10 ), 1407 (2003) 21. K . Goldber g, T. Ro eder, D. Gupta, C. Perkins, Infor- mation Retriev a l 4 (2), 133 (2 001) 22. K . Cheung, J. Kwok, M. Law, K. Tsui, Decision Sup- po rt Systems 35 (2), 231 (2003 ) 23. A. Um yarov, A. T uzhilin, L ever aging aggr e gate r atings for b etter r e c ommendations , in R e cSys ’07: Pr o c e e d- ings of the 2007 ACM c onfer enc e on R e c ommender systems (A CM, 2007 ), pp. 16 1–16 4 24. D.M. Chance, E. Hillebra nd, J.E. Hilliard, Ma na ge- men t Science 54 (5 ), 101 5 (20 08) 25. B.J . J iang, P .Y. Chen, SSRN eLibrary (2007) 26. D. Cosley , S. Lam, I. Alb ert, J. K onstan, J. Riedl, P ro- ceedings o f the SIGCHI confer ence on Human factors in computing systems pp. 585 –592 (2003 ) 27. X. L i, L.M. Hitt, Informa tion Sy stems Res earch 19 (4), 456 (2008) 28. F. W u, B.A. Huberman, Arxiv prepr int arXiv: 0805.3 537 (2008 ) 29. C. Dellaro cas, R. Narayan, Statistical Science 21 (2), 277 (2006) 30. C. Borghes i, J .P . Bouchaud, Qualit y and Quantit y 41 (4), 557 (2007 ) 31. B. Gu, M. Lin, The Dynamics of Online Consum er R eviews , in Workshop on Information Systems and Ec onomics (WISE) (2006) 32. I looked at all mo vies with mor e than 2 0 , 000 ratings on IMD b.com in March 20 08. These were identified by IMDb “Pow er Se a rch”-function. These w ere 1 , 08 6 his- tograms. 33. J .P . Nolan, Stable D ist ributions - Mo dels for He avy T aile d Data (Birkh¨ auser, 2010 ), in prog ress, Chapter 1 online at academic2.amer ican.edu/ ∼ jpnola n