Fleder and He he Impact of Recommender Systems on Sales ORMS ASSUMPTION 2. We assume there are only two products, Figure 3 A Two-Urn Model for Recommender Systems w and b(white and black This assumption is for tractability, but it still allows us to illustrate how the use of sales information affects AsSUMPTION 3. Consumers have purchase probabilities (P, 1-p) for(w, b) in the absence of recommendations We do not model the decision process that gener- ates these purchase probabilities ASSUMPTION 4. At each occasion, the firm recommends a product, which is accepted with probability r AssUMPTION 5. The firm's recommendation is gener ated using a function 8(X,Elw, b, where X, is the seg ment share of w just before purchase t This segment of similar consumers is identified based on past behavior, possibly from purchases of The recommender's inputs are segment shares products in other categories. The assumption that (market shares within a segment of similar users), the group does not evolve is for tractability, but it nd its output is a product. The system modeled rec- does have a parallel with business practice. In indus- ommends the product with higher segment share. try, real-time updating of segments is often computa This choice of g has a parallel with collaborative fil- tionally prohibitive, so many firms update segments ers, which identify similar customer segments and periodically recommend the most popular item within them The process defined by these assumptions can be (e. g , "people who bought X also bought Y"). This illustrated by an urn model. Consider the two-urn recommender can be represented by the step function system of Figure 3. Urn 1 contains balls represent- ing products w and b. A fraction p of the balls in urn 1 are white; this fraction is the consumer's pur- chase probability for w in the absence of recommen 8( ) =P(w recommended (x)=1 X,=2 dations. Urn 2 is the recommender: its contents reflect the sales history within the segment, and it produces 1X2>2 recommendations according to 8(X,), where X, is the fraction of w in urn 2 just before t. To start,urn where X, E [0, 1]. Figure 2 plots this. When X,=5 2 contains one wo and one b. At time t=l, a ball and the products have equal shares, the recommen- is drawn with replacement from urn 1 representing dation is determined by a Bernoulli()trial. To start, the consumer's choice before seeing the recommenda the recommender does not favor either product, and tion. Next, a ball is drawn with replacement from urn we assume X1=2- 2 according to 8(X,), representing the recommended product. With probability r, the consumer accepts the ASsUMPTION 6. The segment of consumers constitut- recommendation, and with probability(1-r)the con- ing X is preselected and does not change over time sumer retains the original choice. The ball chosen represents the actual product purchased; a copy of Figure 2 Recommender g(X,) it is added to urn 2, which is equivalent to updat ing the recommender's sales history(e.g, the firms database). Consumer 2 then arrives, and the process repeats(p and r are the same, but X2 is used instead of X,, and so on for other customers From these assumptions, the probability that w is purchased at time t is f(X,): P(w chosen on occasion tIX, =p(1-r)+g(X)r Section 5 presents an alternate approach where we relax assumptions. Specifically, we model the consumer's decision cess, consider multiple products with a no-purchase option, allow segments to evolve over time.Fleder and Hosanagar: The Impact of Recommender Systems on Sales Diversity 700 Management Science 55(5), pp. 697–712, © 2009 INFORMS Assumption 2. We assume there are only two products, w and b (white and black). This assumption is for tractability, but it still allows us to illustrate how the use of sales information affects diversity. Assumption 3. Consumers have purchase probabilities (p 1 − p) for (wb) in the absence of recommendations. We do not model the decision process that gener￾ates these purchase probabilities. Assumption 4. At each occasion, the firm recommends a product, which is accepted with probability r. Assumption 5. The firm’s recommendation is gener￾ated using a function gXt ∈ w b , where Xt is the seg￾ment share of w just before purchase t. The recommender’s inputs are segment shares (market shares within a segment of similar users), and its output is a product. The system modeled rec￾ommends the product with higher segment share. This choice of g has a parallel with collaborative fil￾ters, which identify similar customer segments and recommend the most popular item within them (e.g., “people who bought X also bought Y ”). This recommender can be represented by the step function gXt=P w recommendedXt= ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ 0 Xt < 1 2  1 2 Xt = 1 2  1 Xt > 1 2  (1) where Xt ∈ 0 1. Figure 2 plots this. When Xt = 1 2 and the products have equal shares, the recommen￾dation is determined by a Bernoulli( 1 2 ) trial. To start, the recommender does not favor either product, and we assume X1 = 1 2 . Assumption 6. The segment of consumers constitut￾ing Xt is preselected and does not change over time. Figure 2 Recommender gXt 0 0.5 1.0 0 0.5 1.0 Xt g(Xt ) Figure 3 A Two-Urn Model for Recommender Systems p (1–r) r g(Xt ) Urn 1 Urn 2 This segment of similar consumers is identified based on past behavior, possibly from purchases of products in other categories. The assumption that the group does not evolve is for tractability, but it does have a parallel with business practice. In indus￾try, real-time updating of segments is often computa￾tionally prohibitive, so many firms update segments periodically.2 The process defined by these assumptions can be illustrated by an urn model. Consider the two-urn system of Figure 3. Urn 1 contains balls represent￾ing products w and b. A fraction p of the balls in urn 1 are white; this fraction is the consumer’s pur￾chase probability for w in the absence of recommen￾dations. Urn 2 is the recommender: its contents reflect the sales history within the segment, and it produces recommendations according to gXt, where Xt is the fraction of w in urn 2 just before t. To start, urn 2 contains one w and one b. At time t = 1, a ball is drawn with replacement from urn 1 representing the consumer’s choice before seeing the recommenda￾tion. Next, a ball is drawn with replacement from urn 2 according to gXt, representing the recommended product. With probability r, the consumer accepts the recommendation, and with probability (1−r) the con￾sumer retains the original choice. The ball chosen represents the actual product purchased; a copy of it is added to urn 2, which is equivalent to updat￾ing the recommender’s sales history (e.g., the firm’s database). Consumer 2 then arrives, and the process repeats (p and r are the same, but X2 is used instead of X1), and so on for other customers. From these assumptions, the probability that w is purchased at time t is f Xt = P w chosen on occasion tXt = p1−r +gXtr 2 Section 5 presents an alternate approach where we relax these assumptions. Specifically, we model the consumer’s decision pro￾cess, consider multiple products with a no-purchase option, and allow segments to evolve over time.