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 limit￾ing market shares of w b ∈ 089 011 014 086 . In the first outcome, w wins the market. In the sec￾ond, b wins, even though p = 055 initially favored w (cf. Corollary 1). This occurs because r is large rela￾tive to p, and the recommender reinforces whichever 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 of w or 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 models, 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 (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 less￾diverse 0.70 outcome. In some cases, if b is chosen enough early on, the recommender reinforces b, lead￾ing to the 0.45 outcome, which entails less concentra￾tion 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 realiza￾tions of this process over time. The top part of the fig￾ure shows these paths converging to two outcomes. One sees that the limits are in accord with Proposi￾tion 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)