# Towards a Theory Model for Product Search

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3 generates a money utility U m(i). The decision to purchase X j generates a product utility U p(x j) and, simultaneously, paying the price p j decreases the money utility to U m(i p j). Assuming that the consumer strives to optimize its own utility, the purchased product X j is the one that gives the highest increase in utility. This approach naturally generates a ranking order for the products: The products that generate the highest increase in utility should be ranked on top. Thus, to compute the increase in utility, we need the gained utility of product U p(x j) and the lost utility of money U m(i) U m(i p j) Utility of Product Modeling the utility of a product can be traced back to Lancaster s characteristics theory [15] and Rosen s hedonic price model [24]. The hedonic price model assumes that differentiated products are described by vectors of objectively measured characteristics. In addition, the utility that a consumer has for a product can be decomposed into a set of utilities for each product characteristic. According to this model, a product X with K features can be represented by a K-dimensional vector X = x 1,...,x K, where x k represents the amount or quality of the k-th characteristic of the product. The overall utility of product X is then modeled by the function U p(x 1,..., x K ). One of the critical issues in this model is how to estimate the aggregated utility from the individual product characteristics. Based on the hedonic price model, we assume that each product characteristic is associated with a weight representing consumers desirability towards that characteristic. Under this assumption, we further refine the definition of overall utility to be the aggregation of weighted utilities from the observed individual characteristics and an unobserved characteristic ξ: U p(x) =U p(x 1,...,x K )= K β k x k + ξ, (1) k=1 where β k represents the corresponding weight that the consumer assign to the k-th characteristic x k. Notice that with ξ we capture the influence of all product characteristics that are not explicitly accounted in our model. So, a product that consumers perceive as high-quality due to a characteristic not explicitly captured in our measurements (e.g. brand name), will end up having a high value of ξ Utility of Money Given the utility of a product, to analyze consumers motivation to trade money for the product, it is also necessary to understand the utility of money. This concept is defined as consumers happiness for owning monetary capital. Based on Alfred Marshall s well-established principles [17], utility of money has two basic properties: increasing and concave. Increasing: An increase in the amount of money will cause an increase in the utility of money. In other words, the more money someone has, the better. Concave: The increase in utility, or marginal utility of money, is diminishing as the amount of money increases. Based on these properties, an example of the utility function for money is shown in Figure 1. Note that with the concave form of the utility function, the slope is decreasing hence the marginal utility of money is diminishing. So, \$100 is more important for someone with \$1000 than for someone with \$100,000. This also implies that consumers are risk-averse under normal circumstances. This is because with the same probability to winorlose,losingn dollars in the assets will cause a drop in the utility larger than the boost while winning N dollars. Figure 1: A concave, bounded, increasing function for 1utility of money, approximated with a linear function for small changes The concave assumption can be relaxed when the changes in money are small. For most transactions, we often assume that the marginal utility of money is approximately constant. More formally, we assume that a consumer with income I receives a money utility U m(i). Paying the price p decreases the money utility to U m(i p). Assuming that p is relatively small compared to the disposable income I, the marginal utility of money remains mostly constant in the interval [I p, I] [17]. Under this assumption, the utility of money that the consumer will lose by paying the price p for product X, canbethereby represented in a quasi-linear form as follows: U m(i) U m(i p) =α I α (I p) =α(i) p, (2) where α(i) denotes the marginal utility for money for someone with disposable income I Challenges Given the utility of product and utility of money, we can now derive the utility surplus as the increase in utility, or excess utility, after the purchase. More formally, the mathematical definition for utility surplus is provided as follows. Definition 1.: The utility surplus (US), for a consumer with disposable income I, when buying a product X priced at p, isthegain in the utility of product U p minus the loss in the utility of money U m. US = U p(x) [U m(i) U m(i p)] + ε i j = β k x k + ξ α p + }{{} }{{} ε (3) k } {{ } Utility of money Stochastic error Utility of product Note that ξ is a product-specific disturbance scalar summarizing unobserved characteristics of product X, andε is a stochastic error term that is assumed to be i.i.d. across products and consumers in the selection process and is usually assumed to follow a Type I extreme-value distribution. In this theory model, the key challenge is to estimate the corresponding weights assigned by consumers towards money and product dimensions. We discuss this next. 3. ESTIMATION OF MODEL PARAMETERS In the previous section, we have introduced the background of utility theory, characteristics-based theory, and surplus. Recall that our main goal is to identify the best product (with the highest surplus) for a consumer. This is complicated by the fact that utility, and therefore surplus, of consumers is private

5 becomes: How can we find the preferences of various consumer types by simply observing the aggregate product demand? Model Specification We solve this issue by monitoring demand for similar products in different markets, for which we know the distribution of consumers. Since the same product will have the same demand from a given demographic group, any differences in demand across markets can be attributed to the different demographics. The following simplified example illustrates the intuition behind this approach. Example 3. Consider an example where we have two cities, A and B and two types of consumers: business trip travelers and family trip travelers. City A is a business destination with 80% of the travelers being business travelers and 20% families. City B is mainly a family destination with 10% business travelers and 90% family travelers. In city A, we have two hotels: Hilton (A 1) and Doubletree (A 2). In city B, we have again two hotels: Hilton (B 1) and Doubletree (B 2). Hilton hotels (A 1, B 1)have a conference center but no pool, and Doubletree hotels (A 2, B 2) have a pool but no conference center. To keep the example simple, we assume that preferences of consumers do not change when they travel in different cities and that prices are the same. By observing demand, we see that demand in city A (business destination) is 820 bookings per day for Hilton and 120 bookings for Doubletree. In city B (family destination) the demand is 540 bookings per day for Hilton and 460 bookings for Doubletree. Since the hotels are identical in the two cities, the changes in demand must be the result of different traveler demographics, hinting that a conference center is desirable for business travelers. For this paper, to extract consumer preferences, we use the Random-Coefficient Model [6], introduced by Berry, Levinsohn, and Pakes, and commonly referred to as the BLP model. This model extends the basic Logit model by assuming the coefficients β and α inequation6tobedemographic-specific. LetT i be a vector representing consumer type, which can specify a particular purchase context, age group, and so on. In the simplest case, we can have a binary variable for each consumer group. With the preferences being now demographic-specific, we write the utility surplus for consumer i, oftypet i, when buying product j, with features x 1 j,...,x k j, at price p j to be: US i j = k β k (T i ) x k j α(i i ) p j + ξ j + ε i j. (9) For the Logit model, in Equation 4, we used V (α, β) tostylistically separate the population preferences from the idiosyncratic behavior of the consumer. We now do the same for the BLP model, separating the mean population preferences from the demographic-specific preferences. So, we write β k (T i )= ( βk + βt k T i), where β k is the mean of the preference distribution, and βt k is a vector capturing the variation in the preferences from different consumer types. Similarly, we model α i as a function of income I i : α(i i )= ( ᾱ + α II i). Notice that we assume α I and β T to be independent. We rewrite USj i as: US i j = k ( βk + β k T T i) x k j + ξ j ( ᾱ + α I I i) p j + ε i j. (10) We use δ j = ᾱ p j + k β k x k j + ξ j to represent the mean utility of product j. Then, as in the Logit model, we derive the choice probability for j, by integrating over the population demographic and income distributions P (T )andp (I): ( exp δ j + α I I i p j + ) k βk T T i x k j P (choice j)= 1+ l exp ( δ l + α I I i p l + k βk T T ) i x k dp (T ) dp (I) l (11) We explain next how we compute this integral and how we extract the parameters that capture the population preferences Estimation Methodology With the model in hand, now we discuss how we identify the unknown parameters δ j, α I and β T. We apply methods similar to those used in [6, 7] and [25]. In general, we estimated the parameters by searching the parameter space in an iterative manner, using the following steps: 1. Initialize the parameters δ (0) j and θ (0) =(α (0) I,β (0) T )using a random choice of values. 2. Estimate market shares s j given θ and δ. 3. Estimate most likely mean utility δ j given the market shares. 4. Find the best parameters ᾱ and β k that minimize the unexplained remaining error in δ j and evaluate the generalized method of moments (GMM) objective function. 5. Use Nelder-Mead Simplex algorithm to update the parameter values for θ =(α I,β T ) and go to Step 2, until minimizing the GMM objective function. We describe the steps in more detail below. Calculating market share s j: To form the market equations (i.e., model predicted market share = observed market share), we need two things: the right-hand side s obs j that can be observed from our transaction data, and the left-hand side s j, derived from Equation 11. Unfortunately, the integral in Equation 11 is not analytic. To approximate this integral, we proceed as follows: Given the demographic distribution, we generate a consumer randomly, with a known demographic and income and, therefore, known preferences. Then, using the standard Logit model (Equation 6), we generate the choice of the product for this consumer. For example, assume that we have the following joint demographic distribution of travel purpose and age group: [ Age 45 Age > 45 Business 15% 15% Family 30% 40% ]. In this case, we have a 40% probability of generating a sample consumer with family travel purpose and age above 45. By repeating the process and obtaining N T samples of demographics T i and N I samples of income I i,wecancomputeanunbiased estimator of the Equation 11 integral: 5 s j(δ j θ) 1 N I 1 N T ( N I N T exp δ j + α I I i p j + ) k βk T T ik x k j I i T i 1+ l exp ( δ l + α I I i p l + k βk T T ) ik x k. l (12) Estimate mean utility δ j: Since we know how to compute market shares from the parameters, we can now find a value of δ j that best fits the observed market shares. (Notice that, conditional on θ =(α I,β T ), market share s j canbeviewedas a function of the mean utility δ j.) We apply the contraction mapping method recommended by Berry [6], which suggests computing the value for δ using an iterative approach: δ (t+1) j = δ (t) j +(ln(s obs j ) ln(s j(δ (t) j θ))). (13) The procedure is guaranteed to converge [6] and find δ j that satisfies s j(δ j θ) =s obs j. 5 We use NT = N I = 100 in our study.

8 Figure 2: Diagonal: Histograms for the continuous variables in the data set (market share, price, ext amenities, value, room, location, checkin, service, business service, rooms, class, crime, int amenities, review count, cleaningness, tripadvisor rating). Top: correlations of variable pairs. Bottom: scatterplots with ellipses for joint distributions of variable pairs. Most Price Price Hotel # of # of # of Diversity: City TripAdvisor Travelocity Booked Low to High to Class Reviews Rooms Amenities Price with High Low Class New York 77% 63% 61% 57% 71% 88% 76% 89% 60% 80% Los Angeles 72% 58% 71% 59% 84% 89% 87% 86% 69% 76% San Francisco 79% 57% 65% 62% 70% 82% 68% 79% 79% 72% Orlando 83% 81% 62% 63% 73% 79% 73% 79% 61% 79% New Orleans 61% 69% 60% 78% 69% 80% 72% 91% 58% 85% Salt Lake City 61% 80% 69% 66% 79% 83% 73% 70% 76% 79% Significance p =0.05 p = 0.01 p = (Sign Test Level 56% 59 % 61% N = 200) Table 2: Consumer-surplus-based ranking vs. existing baselines. The percentage shows the number of users than indicated preference for the ranking of our surplus-based ranking, compared to the corresponding baseline ranking. The comparison was a pairwise blind comparison of rankings of 10 hotels. Each cell corresponds to a separate test using N = 200 users, for a total of comparisons. Across all comparisons, our ranking fares better in a statistically significant manner. Given the noise inherent in the Mechanical Turk ratings (some users may give random answers in such tests and vote equally for both approaches), the results show a strong superiority of the surplus-based rankings. are times more sensitive to online reviews compared to the age demographic of 65+ year old. 6.2 User Study: Ranking Comparison With the estimated weights, we can derive the consumer surplus for each product (hotel), which can then be used to generate rankings. In our experiments, we used six metropolitan areas for which we generated hotel rankings (we used big cities, so that we have a meaningful number of hotels to rank). Rankings based on the average consumer surplus vs. existing ranking baselines.: We used 10 different, existing rankings as a baseline for comparison, and we compared each baseline against our own surplus-based ranking. For each city and each ranking baseline, we performed pair-wise blind tests, asking 200 anonymous users on Amazon Mechanical Turk 11 to compare pairs of rankings and tell us which of the hotel ranking lists they prefer better. In all 60 comparisons, each using We restricted participation to US-based users. Mechanical Turk users are representative of the general US population. users, the majority of users preferred our surplus-based rankings, in a statistically significant manner. Given that some MTurk users may be giving us random results, we expect the real performance of our algorithm to be even better than indicated in the numbers in Table 2. Qualitative analysis: We also asked consumers why they chose a particular ranking, to better understand how users interpret the surplus-based ranking. Many users that liked our ranking indicated that our ranking promotes the idea that price is not the only main factor in rating the quality of products. Moreover, users strongly preferred the diversity provided by our ranking across both price and quality. In contrast, the other ranking approaches tend to list products of only one type (e.g., hotels with high review ratings are often very expensive hotels and the most booked are often 3-star mediocre hotels). We should emphasize at this point that our algorithm does not try explicitly to introduce diversity in the results. This is a direct outcome of our economic-based approach: If a segment of the market is systematically underpriced (hence making the best deals a homogeneous list), then we expect the market forces to fix this

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