What threshold best describes aggregate behavior?

We use the estimation results from the mixed logit regressions done for the case where there is no CS to predict choice when there is a CS. If the consumer is Naive, his choice will be predicted by applying parameter estimates from the model with no CS to the data with CS, which obtains estimates(p^{N a}_{LP CS}, p^{N a}_{HP CS}, p^{N a}_IS)for the probabilities to choose the LPCS, HPCS

Table 8: Regressions with no CS, 3 and 6-menus.

(1) (2) (3) (4) (5) (6)

Logit 3-menus Probit 3-menus MixLogit 3-menus Logit 6-menus Probit 6-menus MixLogit 6-menus main

unit price (up) −18.7200∗ ∗∗ −16.2710∗ ∗∗ −20.2468∗ ∗∗ −16.2815∗ ∗∗ −6.6052∗ ∗∗ −17.4617∗ ∗∗

(−6.89) (−6.87) (−6.91) (−8.49) (−6.19) (−7.67)

up×hard menu −9.7361∗ ∗ −11.7958∗ ∗∗ −10.2537∗ ∗ −24.3972∗ ∗∗ −9.6439∗ ∗∗ −26.9113∗ ∗∗

(−2.86) (−3.50) (−3.14) (−6.63) (−4.77) (−7.19)

up×score shape task 20.7627∗ 14.2704+ 20.9114∗ 10.8997 4.1976∗ 13.4579

(2.24) (1.77) (2.23) (1.53) (1.96) (1.61)

position 0.0656∗ −0.0916+ 0.0671∗ ∗ 0.0053 0.0240 0.0046

(2.56) (−1.95) (2.63) (0.53) (1.22) (0.44)

shape −0.3621∗ ∗∗ −0.3705∗ ∗∗ −0.3961∗ ∗∗ −0.3339∗ ∗∗ −0.1509∗ ∗∗ −0.3958∗ ∗∗

(−12.05) (−11.55) (−9.21) (−14.52) (−6.58) (−9.54)

size −0.0121∗ ∗∗ −0.0108∗ ∗∗ −0.0137∗ ∗∗ −0.0019 −0.0002 −0.0019

(−5.28) (−4.19) (−4.03) (−0.92) (−0.23) (−0.41)

SD

shape 0.3836∗ ∗∗ 0.4549∗ ∗∗

(9.81) (9.48)

size 0.0352∗ ∗∗ 0.0537∗ ∗∗

(8.39) (11.81)

N 10854 10854 10854 21708 21708 21708

ll −3757.4104 −3747.6265 −3689.0559 −6103.5141 −6042.2092 −5881.8136

tstatistics in parentheses

+p <0.10, *p <0.05, **p <0.01, ***p <0.001

Note: One subject did not perform the shape comparison task, so the regressions are based on 201 subjects choosing among 18 menus with no CS.

26

and IS respectively. If he follows the CS rule, he will choose the LPCS. If he follows the Thresh-old rule then the probability he chooses the LPCS isp^{T h}_{LP CS} = p^{N a}_{LP CS}(LP CS, IS×v), which is to be interpreted as the probability a Naive consumer would choose the LPCS if his choice was restricted to either the LPCS or the IS and the price of the IS was multiplied by a factor v. We computed for each consumer the thresholdvj that maximizes their maximum like-lihood. Behavior of subjects with a high value ofvj is close to following the CS rule, while that of those with lowv_jis close to Dominance Editing, that is, eliminating dominated offers from one’s consideration set and comparing remaining offers based on their signals.¹⁴ One can similarly predict the choices made by a consumer following the Signal First heuristic.

Compared to predictions based on the Naive rule, the Threshold rule makes use of two additional degrees of freedom as it requires information about what is the CS and requires estimating the threshold used by the subjects. This will be taken into account by comparing rules using the Akaike Information Criterion.

In mathematical terms, the likelihood function isf(y, θ) = QN t=1

QM j=1

p^y_tj^tj withtdenoting the menu,N the total number of menus presented to consumers,jdenoting the option,M the number of options, andy_tj = 1iffy_t =j,0otherwise, wherebyy_tis the consumer’s choice.

p_tj =Pr(y_t =j)is the predicted probability, which depends on the rule we assume for con-sumers’ choice, so for examplep_tj = 1iffjis the LPCS and the consumer is assumed to follow the CS rule.yis the vector of choices andθare the parameters determining the choice among options.

Table 9 reports the log-likelihood, the values of the AIC and of the Bayesian information criterion (“BIC”) for each rule, for 3 and 6-menus.¹⁵ The last column contains the value of thresholdvthat maximizes the log-likelihood for the Threshold rule. The numbervreported there is to be interpreted as “consumers appear to consider IS options asvtimes more ex-pensive when they are presented next to CS options than when they are presented next to other IS options”. This measures the price penalty applied to IS options when compared to the LPCS. For more interpretation of this number, see the detailed explanation in section 2.

14The rules above predict that the HPCS will never be chosen. However, as we saw, this is not the case in our data. One therefore has to take account that some consumers choose the HPCS. We therefore do a separate regression so as to determine the probabilitypLP CSwith which the LPCS is chosen among CS offers. Note that in this case, only the offer’s position and its price may determine the choice, along with some case-specific variables, since both shape and area are the same in a CS. One then modifies the formulas above as follows: In the case of the CS rule: p^CS_{LP CS} = pLP CS andp^CS_{HP CS} = 1−pLP CS and in the case of the Threshold rule: p^{T h}_{LP CS} = pLP CSp^{N a}_{LP CS}(LP CS, IS×v),p^{T h}_{HP CS} = (1−pLP CS)p^{N a}_{HP CS}(HP CS, IS×v)andp^{T h}_IS = 1−p^{T h}_{LP CS} −p^{T h}_{HP CS}. Formulas are slightly longer in the case of 6-menus but can be inferred from the above.

15We only study 6 menus with one CS.

Table 9: Rules scores, aggregate behavior.

Naive Signal Dominance Threshold v First Editing Heuristic

3-menus LL −3 484 −2 994 −3 073 −2 984 1.05

df 8 9 9 10

AIC 6 984 6 006 6 164 5 988

BIC 7 034 6 062 6 220 6 050

N 3 618 3 618 3 618 3 618

6-menus LL −5 954 −5 769 −5 788 −5 762 1.04

df 8 9 9 10

AIC 11 924 11 556 11 594 11 544 BIC 11 974 11 612 11 650 11 606

N 3 618 3 618 3 618 3 618

The Threshold rule gives the best predictions for both menu lengths. In terms of thresh-old, an IS offer suffers a 4 to 5% price penalty compared to the LPCS offer, which is a con-siderable amount. Assuming consumers follow the Signal First heuristic or do Dominance Editing does not either attain better values in the AIC or BIC criteria. The Naive rule is clearly rejected in all cases so consumers clearly do take CS information into account.

When mapping payoffs by menu length and difficulty in the case with no CS (table 4) to the predictions from our simulations (Graph 3), the standard error of the consumers’ er-ror term when assessing unit prices appears to have been be about 0.15, in which case the optimal thresholdv_j would be between 1.2 and 1.4, which is a lot more than the 1.04-1.05 threshold determined above. This indicates perhaps that they were over-confident in their ability to make accurate choices. We check this in the following by determining thresholds individual by individual and comparing this to optimal thresholds given individual accuracy determined from choice among menus with no CS.

3.3.3 What threshold did consumers use individually, and did they choose their