# Swayed by Friends or by the Crowd?

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4 Age age <20 20 =< age <30 30 =< age <40 40 =< age <50 50 =< age <60 age > 60 3% 0% 8% 6% Education High School Graduate Associates Degree Bachelors Degree Graduate Degree 19% 24% Table 1. Positive Predictor Estimated Coefficients z-value α f (0.027) α s (0.050) Note: Standard errors are shown in parentheses. Signif. codes: 0 *** ** 0.01 * Pseudo-R 2 = % 59% Figure 2. Demographics questions were discarded and not considered in the analysis. We ran the experiments until we got 350 valid responses for each study. Overall, we rejected 33% of the responses across all 3 studies because they were invalid. The average completion time for each valid HIT was seconds while the average completion time for the invalid HITS was 153.3; this suggests that the workers that were rejected had not taken the task as seriously as the rest of the workers. In addition to the validity check questions, each study consisted of 8 questions (with the format of Figure 1) which we use in our analysis and 3 demographics questions asking about the gender, the age and the education level of the respondent. Overall, 36% of the approved respondents, were female. Other demographic information for the approved respondents according to the self-reporting of the workers are plotted in Figure 2. Model Specification For each question, there are two options that the worker can select form, which we refer to as option 1 and option 2. We denote the number of stars S i and the number of friend recommendations by F i, for i = 1, 2. To predict the probability that option 1 is selected, we conduct a logistic regression on our dataset of choices with the difference in the number of stars (i.e., S 1 S 2 ) and friends (i.e., F 1 F 2 ) for each question as the predictor variables. In particular, P r[choose option 1] = logit(α s (S 1 S 2 )+α f (F 1 F 2 )). We note that a number of other empirical studies also use the logit choice function to model social influence [20, 10, 23]. RESULTS In this section, we report the results from each survey separately. 53% 4% Study 1: Positive Opinions for Hotels Model 1 As mentioned in the previous section, we examine the results by fitting a logistic regression model. We first only considered the difference in the number of stars and friends as predictor variables. The estimated coefficients along with other parameters are depicted in Table 1, and as can be seen both are statistically significant. Observe that both coefficients are positive; this is intuitive, since more stars (resp. more positive recommendations) indicate that the option is better and thus the worker is more likely to select it. Finally, the pseudo -R 2 for this model 1 is 0.95, indicating that the fit is very good. Interpretation of the coefficients We interpret the coefficients for our model in terms of marginal effects on the odds ratio. The odds ratio measures the probability that the dependent variable is equal to 1 relative to the probability that it is equal to zero. For the logit model, the log odds of the outcome is modeled as a linear combination of the predictor variables; therefore, the odds ratio of a coefficient is equal to exp(coefficient). Since α s = 0.735, we conclude that a unit increase in S 1 S 2, multiplies the initial odds ratio by exp(0.735) = In other words, the relative probability of choosing option 1 increases by 107%. For the friends predictor variable, the odds ratio is equal to exp(0.204) = 1.22, meaning that, the relative probability of selecting option 1 increase by 22% if F 1 F 2 increases by one unit. To further assess the predictive power of the model, we performed cross validation. We left out one question at a time and estimated the coefficients using the remaining questions. Then, we predicted the probabilities for the question that was left out. The predicted values were very close in all cases with absolute mean difference of The actual values and their differences can be found in Table 2. Finally, we used one of the questions of this survey twice in Amazon s Mechanical Turk (in two separate HITS) in order to see whether workers would react to the question 1 We computed Efron s pseudo -R 2 which is defined as follows: R 2 = 1 P N i=1 (yi ˆπi)2 P N i=1 (yi ȳ)2 where, N is the number of observations in the model, y is the dependent variable, ȳ is the mean of the y values, and ˆπ is the probabilities predicted by the logit model. The numerator of the ratio is the sum of the squared differences between the actual y values and the predicted π probabilities. The denominator of the ratio is the sum of squared differences between the actual y values and their mean [13]. 4

5 Table 2. Cross validation for study 1 Left out question Actual Predicted Difference Q Q Q Q Q Q Q Q Table 3. Stduies 1 and 2: demographic information included Predictor Study 1 Study 2 α f (0.19) 0.35 (0.07) α s (0.08) 0.66 (0.12) gender f (0.07) (0.06) gender s (0.16) (0.11) edu1 s 0.11 (0.16) (0.11) edu1 f 0.04 (0.08) (0.07) edu2 s (0.46) 0.16 (0.40) edu2 f (0.23) 0.18 (0.63) edu3 s (0.16) (-3.1) edu3 f (0.08) (-0.31) age1 s (0.39) (0.15) age1 f 0.02 (0.21) (0.11) age2 s 0.06 (0.16) 0.17 (0.12) age2 f 0.11 (0.08) 0.16 (0.08) age3 s (0.34) (0.16) age3 f (0.18) (0.10) age4 s (0.16) 0.10 (0.11) age4 f (0.08) 0.04 (0.07) age5 s 0.16 (0.40) 0.05 (0.24) age5 f 0.03 (0.22) 0.15 (0.17) Note: Standard errors are shown in parentheses. Signif. codes: 0 *** ** 0.01 * in similar ways. We found that the percentage of workers that chose the first option of the question was similar in both cases (26% versus 24%), further validating our approach. Model 1 In Model 1, we included all self reported demographic information as predictor variables in addition to the stars and friend recommendation variables. This information includes: gender, age (in five 10-year brackets from 20 to 60 years old: called age1-5), and education level (high school, associates degree, bachelors degree, and graduate degree: called edu1-3). More specifically, we coded the following variables as dummy variables. The estimated coefficients and other information is shown in the second column of Table 3. As can be seen in Table 3, these extra coefficients are not statistically significant. This suggests that people in different demographics trade off ratings from the public and friend recommendations similarly. Table 4. Negative Predictor Estimated Coefficients z-value α f (0.030) α s (0.050) Note: Standard errors are shown in parentheses. Signif. codes: 0 *** ** 0.01 * Pseudo-R 2 = 0.95 Table 5. Cross validation for study 2 Left out question Actual Predicted Difference Q Q Q Q Q Q Q Q Study 2: Negative Opinions for Hotels Model 2 In the previous subsection, we examined how positive recommendations from friends and the number of stars influence users choices. In this section we look at negative opinions from friends instead of positive recommendations. In particular, each option is characterized by the number of stars (based on information from the general public) as well as the number of friends who have negative opinions about it. We examine the data by fitting a logit model. The estimated coefficients are reported in Table 4. As can be seen in the table both variables are statistically significant and the pseudo-r 2 measure for this model is 0.95 which implies that the model is a good fit. Moreover, as we would expect, the friends coefficient is negative in this case, as more negative opinions from friends decrease the probability that the worker selects an option. Interpretation of the coefficients Similarly to Study 1, we interpret the coefficients for our model in terms of marginal effects on the odds ratio. For the present model (negative recommendations), the fact that α s = means that one unit increase in S 1 S 2, multiplies the initial odds ratio by exp(0.503) = In other words, the relative probability of choosing option 1 increases by 65%. For the friends predictor variable, the odds ratio is equal to exp( 0.281) = 0.75, which means that the relative probability of selecting option 1 decreasing by 25%. Equivalently, the relative probability of selecting option 1 when F 1 F 2 decreases by one unit is (exp(0.281) 1) 32%. For this study we did cross validation as well to test the predictive power of our model. The results are shown in Table 5. The predicted and actual values are very close (mean absolute difference = 0.231). Model 2 Similarly to Model 1, in this model we include all variables: stars, friends opinions, and demographics information in the model. The results are shown in the last column 5

8 Makret Share 1 Standard deviation = Standard deviation = Frequency Market Share for A Number of users Figure 3. 3 sample paths of market shares for the product with higher initial rating (S 0 1 = 4 and S0 2 = 2). Frequency Standard deviation = 1.0 runs usually converge to different limits. For instance, Figure 3 shows three sample paths for the market share of option 1 when σ = 1 and θ = 0 and initial star ratings of S 0 1 = 4 and S 0 2 = 2 stars respectively. We ran same simulations for other numbers of initial stars which were qualitatively the same Market Share for A Figure 4. Market share value frequencies over 500 runs (θ = 0, and S 0 1 = 4 and S0 2 = 2). The multiplicity of limit points has been previously observed in settings with one type of social influence. In particular, [7] shows that when objective evidence is weak (and thus decisions are only affected by social influence), then the limit point is not predetermined, or in other words, there are many potential limit points [7]. We observe that this also holds in a setting with two types of social influence (friends and the general public). However, in contrast to the setting of Bendor et al., in our setting certain limit points are more likely than others. Frequency threshold = Market Share for A In general, there is a range of likely limit points, which depends on the standard deviation σ (of the normal distribution from which the ratings come from) and the threshold θ. We study the effect of σ and θ on the distribution of limit points. Frequency threshold = 2 Figure 4 shows the histograms for the frequency of different limit points when θ = 0 for σ = 0.5 and σ = 1, S 0 1 = 4 and S 0 2 = 2 (based on 500 runs). These histograms approximate the corresponding distributions of limit points. We observe that the range of likely outcomes becomes narrower when the standard deviation is smaller. In other words, the variability of limit points increases when the variability of ratings increases. Frequency Market Share for A threshold = We next consider the effect of the threshold θ for a fixed σ. The histograms in Figure 5 show the market share for option 1 for θ = 0, 2, 4 when σ = 1 (based on 500 runs) with initial star ratings of S 0 1 = 3 and S 0 2 = 2. We observe that as θ increases, the distribution of limit points increases (in Market Share for A Figure 5. Frequencies of market shares for the product with higher initial ratings over 500 runs (σ = 1, and S 0 1 = 3 and S0 2 = 2). 8

10 20. Páez, A., Scott, D., and Volz, E. A discrete-choice approach to modeling social influence on individual decision making. Environment and Planning B: Planning and Design 35, 6 (2008), Peeters, G., and Czapinski, J. Positive-negative asymmetry in evaluations: The distinction between affective and informational negativity effects. European review of social psychology 1 (1990), Salganik, M., Dodds, P., and Watts, D. Experimental study of inequality and unpredictability in an artificial cultural market. Science 311, 5762 (2006), Sorensen, A. Social learning and health plan choice. The RAND Journal of Economics 37, 4 (2006), Taylor, S. Asymmetrical effects of positive and negative events: The mobilization-minimization hypothesis. Psychological Bulletin 110, 1 (1991), Tucker, C., and Zhang, J. How does popularity information affect choices? a field experiment. Management Science, forthcoming (2011). 26. Welch, I. Sequential sales, learning, and cascades. Journal of Finance (1992),

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