Fleder and He ar: The Impact of Recommender on Sales Diversity 12, @2009 INFORMS Figure 5 ing the p x r Space to Concentration Effects probability of arrivin This, in turn allow ers Refer to Cases in Proposition 2) us to calculate the expected effect on concentration. PROPOSITION 3. The distribution of lim ooX. P(lim oX,= P(lim a X, supp Case point 1 point 2 int 1) ≈0.5 2 h (1-m)·(1-l/(1-D) y m:(1-(1-h)/h)+(1-m):(1-1/(1-D∈(0,1 Case 2A occurs where the recommender's influence (r) is high relative to the initial probability (p). This his proposition will be applied subsequently two implications, one at the sample-path level 4. 2.2. Graph ical Example. A aphical example and one at the aggregate leveL. At the sample-path helps illustrate the results For the sake of illustration, level, either product can win the market, regardl take p=0.70 and r=0.50. Figure 6 plots 10 realiza of p. For example, p=0.55 and r=0.75 imply limit- tions of this process over time. The top part of the fig- In the first outcome w wins the market. In the sec- ond, b wins, even though p=0.55 initially favored w One sees that the limits are in accord with Propos (cf Corollary 1). This occurs because r is large rela- tion 1, which says the process converges to a random tive to p, and the recommender reinforces whichever Figure 6 The Two Limiting Outcomes for Our Example f(x) product does well early on without too much resis- tance from p. This leads to the finding that recom- menders can create hits. Some product will become a winner with a permanent, majority share, but we can- not say which beforehand. At the aggregate level,con centration always increases. We do not know which b will win but we know that one will, and whichever does will be an outcome with greater con centration. Although they start with different mode a similar phenomenon occurs in other contexts( e.g studies of firm location). Arthur(1994)provides an overview of applications, whereas earlier mathemati cal results are in Hill et al. (1980) Last, in Case 2B, neither the initial probability G Time nor the recommender's influence(r)is strong relative to one another. As a result, two outcomes are possible Frequency of outcome The tendency p can be reinforced by the recommender Increases concentration can give whichever product was not favored a small majority. This decreases concentration. For example, if P=0.60, which is mild, and r=0. 25, the limit points are 0.70 and 0.45. Often w has more early successes nd the recommender reinforces this, leading to less- diverse 0.70 outcome. In some cases, if b is chosen nough early on, the recommender reinforces b, lead- &ig to the 0.45 outcome, which entails less concentra- n than the initial share of 0.40 Although both outcomes are possible in Case 2B they are not equi likely. Next we determine the Limiting P(white)
Fleder and Hosanagar: The Impact of Recommender Systems on Sales Diversity 702 Management Science 55(5), pp. 697–712, © 2009 INFORMS Figure 5 Relating the p × r Space to Concentration Effects (Numbers Refer to Cases in Proposition 2) 0.0 0.5 1.0 0.0 0.5 1.0 r p 3 1 2B 2B 2A Case 2A occurs where the recommender’s influence (r) is high relative to the initial probability (p). This has two implications, one at the sample-path level and one at the aggregate level. At the sample-path level, either product can win the market, regardless of p. For example, p = 055 and r = 075 imply limiting market shares of w b ∈ 089 011 014 086 . In the first outcome, w wins the market. In the second, b wins, even though p = 055 initially favored w (cf. Corollary 1). This occurs because r is large relative to p, and the recommender reinforces whichever product does well early on without too much resistance from p. This leads to the finding that recommenders can create hits. Some product will become a winner with a permanent, majority share, but we cannot say which beforehand. At the aggregate level, concentration always increases. We do not know which of w or b will win, but we know that one will, and whichever does will be an outcome with greater concentration. Although they start with different models, a similar phenomenon occurs in other contexts (e.g., studies of firm location). Arthur (1994) provides an overview of applications, whereas earlier mathematical results are in Hill et al. (1980). Last, in Case 2B, neither the initial probability (p) nor the recommender’s influence (r) is strong relative to one another. As a result, two outcomes are possible. The tendency p can be reinforced by the recommender. This increases concentration. Or, the recommender can give whichever product was not favored a small majority. This decreases concentration. For example, if p = 060, which is mild, and r = 025, the limit points are 0.70 and 0.45. Often w has more early successes and the recommender reinforces this, leading to lessdiverse 0.70 outcome. In some cases, if b is chosen enough early on, the recommender reinforces b, leading to the 0.45 outcome, which entails less concentration than the initial share of 0.40. Although both outcomes are possible in Case 2B, they are not equally likely. Next we determine the probability of arriving at each. This, in turn, allows us to calculate the expected effect on concentration. Proposition 3. The distribution of limt→ Xt is P limt→ Xt = P limt→ Xt = Support Support support support Case point 1 point 2 point 1) point 2) 1 l 1 0 2 lh 1 − 3 h 1 0 where = 1 − m·1 − l/1 − l m ·1 − 1 − h/h + 1 − m·1 − l/1 − l ∈ 0 1 This proposition will be applied subsequently. 4.2.2. Graphical Example. A graphical example helps illustrate the results. For the sake of illustration, take p = 070 and r = 050. Figure 6 plots 10 realizations of this process over time. The top part of the figure shows these paths converging to two outcomes. One sees that the limits are in accord with Proposition 1, which says the process converges to a random Figure 6 The Two Limiting Outcomes for Our Example f x 1,000 1,500 0 0.35 0.85 1.00 P(white) 0 0.35 0.85 1.00 0 1 0.27 0.73 Frequency of outcome 10 sample paths Time Limiting P(white)
r and Hosanagar: The Impact of Recommender Systems on Sales Diversity gement Science 55(5), pp 12,⑥2009 INFORMS variable whose support is (0.35, 0.85]. The bottom part artifact of the initial conditions assumed for urn 2, of the figure shows the frequencies of arriving at the which places one w and one b in a high r recom- lower versus upper outcome approach 0.27 and 0.73, mender even when p a0 or a1.5 in accord with Proposition ummarizing, under recommendations the shares 4.2.3. Net effect on sales Concentration With converge to either one or two limiting outcomes the limiting distribution of (x, known, we complet depending on(p, r). When there is one outcome, the connection to sales concentration. For two prod- it always reflects increased concentration: the rec- ucts with shares p and 1-P, the Gini coefficient is ommender reinforces the popularity of the initially proportional to(Sen 1976): preferred product. In the two outcome cases, either G()=|-l both outcomes have greater concentration or one has greater concentration and the other has less. For the With recommendations, we define latter, a net effect must be calculated. This typically has greater concentration, although for extreme(p, Gp. -EG(lim x)p, rl as discussed, increased diversity may occur. Thus, the recommender seems to increase concentration among G(P(limx,=)+G(h)P(lim X, =h).(4) a set of similar users. The net effect on concentration is given by Gp, -Gp, 0, 5. Simulations which is >0(<0) when concentration increases (decreases). Substituting into(3)and(4)terms from 5.1. Rationale for Simulation the previous propositions gives Simulation offers three benefits for this problem. First, although actual recommender algorithms are difficult Case G G to represent analytically, they can be implemented in simulation. Second, heterogeneity in user preferences Ip-2 is easily accommodated. Third, more complex choice y+ h-2 5.2. Choice Model and Simulation Design p-i We now turn to a simulation that combines a choice model with actual recommender systems. Repeat pur- The above gives a closed-form expression for the chases are permitted in the simulation. Examples of change in Gini coefficient. Figure 7 shows this graph- contexts with repeat purchases could include music ically. For most of the p x r square, concentration and video streaming from a subscription service(e.g increases. This is true, of course, for areas under Rhapsody) Cases 1, 2A, and 3, where the only possibility was An overview of the process is as follows. There are increased concentration. It is also true for most areas I consumers and products positioned in an attribute where both outcomes were possible( Case 2B). In space. Consumers are not aware of all products. Each extreme cases, it is possible for a net decrease to occur, consumer knows most of the center products and a as shown by the shading. These areas are largely an small number of products in his own neighborhood Figure 7 Concentration Increases in White Areas and Decreases in Shaded ones the products or makes no purchase at all. To model this choice, a multinomial logit is used for J products plus an outside good. Just before choosing a product, a recommendation is generated. The recommender has two effects. First, the consumer becomes aware of the recommended product if he was not already This increase in awareness is permanent. Second, the .0.5 An example illustrates how this is related to initial conditions. uppose p=0.99, and r=0.99, which is in the shaded region Because X,=0.50, P(b on first purchase)N0.50. If b is chosen, the recommender next suggests b; because r=0.99 the next consumer is almost certain to pick b too, and so on for the con- sumers, even though p=0.99 favors w. If the initial conditions are determined by k Bernoulli(p) trials, diversity decreases even more: the shaded areas of Figure 7 begin to turn white even for small k. These additional experiments are available from the authors on
Fleder and Hosanagar: The Impact of Recommender Systems on Sales Diversity Management Science 55(5), pp. 697–712, © 2009 INFORMS 703 variable whose support is 035 085 . The bottom part of the figure shows the frequencies of arriving at the lower versus upper outcome approach 0.27 and 0.73, in accord with Proposition 3. 4.2.3. Net Effect on Sales Concentration. With the limiting distribution of Xt known, we complete the connection to sales concentration. For two products with shares p and 1 − p, the Gini coefficient is proportional to (Sen 1976): Gp = p − 1 2 (3) With recommendations, we define Gp r = E G limt→ Xt p r = GlP limt→ Xt = l + GhP limt→ Xt = h (4) The net effect on concentration is given by Gp r − Gp 0, which is >0 (<0) when concentration increases (decreases). Substituting into (3) and (4) terms from the previous propositions gives Case Gp r Gp 0 1 l − 1 2 p − 1 2 2 l − 1 2 + h − 1 2 1 − p − 1 2 3 h − 1 2 p − 1 2 The above gives a closed-form expression for the change in Gini coefficient. Figure 7 shows this graphically. For most of the p × r square, concentration increases. This is true, of course, for areas under Cases 1, 2A, and 3, where the only possibility was increased concentration. It is also true for most areas where both outcomes were possible (Case 2B). In extreme cases, it is possible for a net decrease to occur, as shown by the shading. These areas are largely an Figure 7 Concentration Increases in White Areas and Decreases in Shaded Ones 0.0 0.5 1.0 0.0 0.5 1.0 r p artifact of the initial conditions assumed for urn 2, which places one w and one b in a high r recommender even when p ≈ 0 or ≈1.5 Summarizing, under recommendations the shares converge to either one or two limiting outcomes depending on (p r). When there is one outcome, it always reflects increased concentration: the recommender reinforces the popularity of the initially preferred product. In the two outcome cases, either both outcomes have greater concentration or one has greater concentration and the other has less. For the latter, a net effect must be calculated. This typically has greater concentration, although for extreme (p r), as discussed, increased diversity may occur. Thus, the recommender seems to increase concentration among a set of similar users. 5. Simulations 5.1. Rationale for Simulation Simulation offers three benefits for this problem. First, although actual recommender algorithms are difficult to represent analytically, they can be implemented in simulation. Second, heterogeneity in user preferences is easily accommodated. Third, more complex choice processes can be represented. 5.2. Choice Model and Simulation Design We now turn to a simulation that combines a choice model with actual recommender systems. Repeat purchases are permitted in the simulation. Examples of contexts with repeat purchases could include music and video streaming from a subscription service (e.g., Rhapsody). An overview of the process is as follows. There are I consumers and J products positioned in an attribute space. Consumers are not aware of all products. Each consumer knows most of the center products and a small number of products in his own neighborhood. Every period, a consumer either purchases one of the products or makes no purchase at all. To model this choice, a multinomial logit is used for J products plus an outside good. Just before choosing a product, a recommendation is generated. The recommender has two effects. First, the consumer becomes aware of the recommended product if he was not already. This increase in awareness is permanent. Second, the salience of the recommended product is increased 5 An example illustrates how this is related to initial conditions. Suppose p = 099, and r = 099, which is in the shaded region. Because X1 = 050, P(b on first purchase) ≈ 050. If b is chosen, the recommender next suggests b; because r = 099 the next consumer is almost certain to pick b too, and so on for the remaining consumers, even though p = 099 favors w. If the initial conditions are determined by k Bernoulli(p) trials, diversity decreases even more: the shaded areas of Figure 7 begin to turn white even for small k. (These additional experiments are available from the authors on request.)