2.7 Proof of Theorem 2
2.7.3 Proof of Lemma 10
We linearize the equation in system (2) of Lemma 8. The result is:
logv(xkt)−logλk−logpkt =0 if t=0, (2.12) logv(xkt) +logβ+tlogδ−logλk−logpkt =0 if t>0, (2.13) x> x0 =⇒logv(x0)≥logv(x), (2.14) logβ≥0, (2.15) logδ ≤0. (2.16) In the system comprised by (2.12), (2.13), (2.14), (2.15), and (2.16), the unknowns are the real numbers logβ, logδ, logλk, and logvkt for all k = 1, . . . ,K and t = 1, . . . ,T.
First, we are going to write the system of inequalities from (2.12) to (2.16) in matrix form.
We shall define a matrix Asuch that there are positive numbers vkt, λk, β, and δ the logs of which satisfy equations (2.12) and (2.13) if and only if there is a solutionu ∈RK×(T+1)+2+K+1 to the system of equations
A·u =0,
and for which the last component of uis strictly positive.
Let Abe a matrix with K×(T+1)rows andK×(T+1) +2+K+1 columns, defined as follows: we have one row for every pair (k,t), one column for every pair (k,t), two columns for each k, and two additional columns. Organize the columns so that we first have theK×(T+1)columns for the pairs (k,t), then two columns, which we shall refer to as theβ-column andδ-column, respectively, then Kcolumns (one for eachk), and finally one last column. In the row corresponding to(k,t)the matrix has zeroes everywhere with the following exceptions: it has a 1 in the column for (k,t), it has a 1 ift >0 and it has a 0 if t =0 in the β-column, it has tin the δ-column, it has a −1 in the column fork, and−logpkt in the very last column.
Thus, matrix A looks as follows:
(1,1) ··· (k,t) (k,t0) ··· (K,T) β δ 1 ··· k ··· K p
... ... ... ... ... ... ... ... ... ... ...
(k,t=0) 0 · · · 1 0 · · · 0 0 t 0 · · · −1 · · · 0 −logpkt (k,t0>0) 0 · · · 0 1 · · · 0 1 t0 0 · · · −1 · · · 0 −logpkt0
... ... ... ... ... ... ... ... ... ... ...
.
Consider the system A·u = 0. If there are numbers solving equations (2.12) and (2.13), then these define a solution u ∈ RK×(T+1)+2+K+1 for which the last component is 1. If, on the other hand, there is a solution u ∈ RK×(T+1)+2+K+1 to the system A·u = 0 in which the last component is strictly positive, then by dividing through by the last component of u we obtain numbers that solve equation (2.12) and (2.13).
In second place, we write the system of inequalities (2.14), (2.15), and (2.16) in matrix form. Let Bbe a matrix with K×(T+1) +2+K+1 columns. Define B as follows: one row for every pair (k,t) and (k0,t0) with xkt > xkt00; in the row corresponding to(k,t) and (k0,t0) we have zeroes everywhere with the exception of a −1 in the column for (k,t)and a 1 in the column for (k0,t0). Finally, we have the last two rows, where we have zeroes everywhere with one exception. In the first row, we have a−1 at(K×(T+1) +1)-th column; in the second row, we have a −1 at (K×(T+1) +2)-th column. We shall refer to the first row as the β-row, which captures (2.15). We also shall refer to the second row as the δ-row, which captures (2.16). For (general) QHD, we do not have a β-row.
In third place, we have a matrix E that captures the requirement that the last component of a solution be strictly positive. The matrix E has a single row and K×(T+1) +2+K+1 columns. It has zeroes everywhere except for 1 in the last column.
To sum up, there is a solution to system (2.12), (2.13), (2.14), (2.15), and (2.16) if and only if there is a vector u ∈ RK×(T+1)+2+K+1 that solves the system of equations and linear inequalities:
(S1): A·u =0, B·u≥0, E·u0.
The argument now follow along the lines of the proof of Theorem 1. Suppose that there is no solutionu and let(θ,η,π) be an integer vector solving system:
(S2) : θ·A+η·B+π·E=0, η ≥0, π >0.
The following has the same proof as Claim 1.
Claim 9. (i) θ·A1+η·B1 =0; (ii) θ·A2+η·B2 = 0; (iii) θ·A3+η·B3 = 0; (iv) θ·A4 =0; and (v) θ·A5+π·E5=0.
We transform the matrices A and B based on the values of θ and η, as we did in the proof of Theorem 1. Let us define a matrix A∗ from A and B∗ from B, as we did in the proof of Theorem 1. We can prove the same claims (i.e., Claims 2, 3, 4, 5, and 6) as in the proof of Theorem 1. The proofs are the same and omitted. In particular, we can show that there exists a sequence of pairs (xkti
i,xkt00i
i
)ni=∗1 that satisfies (1) in SAR-PQHD. We shall use the sequence of pairs (xkti
i,xkt00i
i
)ni=∗1 as our candidate violation of SAR-PQHD.
Claim 10. The sequence(xktii,xtk00i
i)ni=∗1satisfies (2), (3), and (4) in SAR-PQHD.
Proof. We first establish (2). Note that A∗3 is a vector, and in row r the entry of A∗3 is as follows. There must be a(k,t) of whichr is a copy. Then the component at row r of A∗3 is t if r is original and −t if r is converted. Now, when r appears as original there is some i for which t =ti, when r appears as converted there is somei for whicht =t0i. So for eachrthere is i such that(A∗3)r is either ti or−t0i.
By Claim 9 (iii),θ·A3+η·B3=0. Recall thatθ·A3equals the sum of the rows of A∗3. Moreover, B3 is a vector that has zeroes everywhere except a −1 in the δ row (i.e., K×(T+1) +2th row). Therefore, the sum of the rows of A∗3 equals ηK×(T+1)+2, whereηK×(T+1)+2 is theK×(T+1) +2th element of η. Since η ≥0, therefore, ∑i:ti>0ti−∑i:t0i>0t0i =ηK×(T+1)+2 ≥0, and condition (2) in the axiom is satisfied.
Next, we show (3). By Claim 9 (ii), θ·A2+η·B2=0. Recall that θ·A2 equals the sum of the rows of A∗2. Moreover, B2 is a vector that has zeroes everywhere except a −1 in the β-row (i.e., K×(T+1) +1th row). Therefore, the sum of the rows of A∗2 equals ηK×(T+1)+1, whereηK×(T+1)+1 is theK×(T+1) +1th element of η. Since η ≥ 0, therefore, #{i : ti > 0} −#{i : t0i > 0} = ηK×(T+1)+1 ≥ 0, and condition (3) in the axiom is satisfied. (For (general) QHD, B2 is a zero vector in the β-row (i.e.,K×(T+1) +1th row). Therefore, #{i: ti >0} −#{i : t0i >0} =0, and condition (3) in SAR-QHD is satisfied.)
Now we turn to (4). By Claim 9 (iv), the rows of A∗4 add up to zero. Therefore, the number of times thatkappears in an original row equals the number of times that it appears in a converted row. By Claim 6, then, the number of times k appears as ki equals the number of times it appears ask0i. Therefore, condition (4) in the axiom is satisfied.
Finally, we can show that ∏ni=∗1pktii/pkt00i
i > 1, which completes the proof of Lemma 5 as the sequence (xktii,xkt00i
i)ni=∗1 would then exhibit a violation of SAR- PQHD. The proof is the same as that of the corresponding lemma in the proof of Theorem 1.
Chapter 3
When the Eyes Say Buy: Visual Fixations during Hypothetical
Consumer Choice Improve Prediction of Actual Purchases
3.1 Introduction
Real choices are binding consequential commitments to a course of action, like ac- cepting a job or voting in an election. However, scientists and policy makers who are interested in real choices often rely on hypothetical statements about what people would choose, rather than what they do actually choose. Measurement of hypothetical choice is common in many fields, and is usually done for practical reasons. Examples include pre-election polling in politics (e.g., Gallup presiden- tial election poll), marketing surveys of potential new products to forecast sales (Chandon et al., 2004; Green and Srinivasan, 1990; Infosino, 1986; Jamieson and Bass, 1989; Raghubir and Greenleaf, 2006; Schlosser et al., 2006; Silk and Urban, 1978; Urban et al., 1983), artificial choices about moral dilemmas or measurement of “sacred” values, which cannot be actually enforced for ethical reasons (Berns et al., 2012; FeldmanHall et al., 2012b; Greene et al., 2004, 2001; Hariri et al., 2006; Kühberger et al., 2002; Monterosso et al., 2007), eliciting quality-adjusted- life-years (QALY) to choose medical procedures (Cutler et al., 1997; Garber and Phelps, 1997; Gold et al., 1996; Zeckhauser and Shepard, 1976), and surveys used to estimate dollar value of goods that are not traded in markets (such as clean air or the prevention of oil spills) for cost-benefit analysis (Carson, 2012; Carson and Hanemann, 2005; Shogren, 2005, 2006).
The maintained assumption in all these research areas is that hypothetical
choices offer some useful relation to real choice. However, many comparisons show that hypothetical and real choices can differ systematically. The differ- ences are collectively called “hypothetical bias.” Typically, it is an upward “Yes bias”: people overstate their intentions to buy new products and vote, compared to actual rates of purchase and voting (Blumenschein et al., 2008; Bohm, 1972;
Cummings et al., 1995; Johannesson et al., 1998; List and Gallet, 2001; Little and Berrens, 2004; Murphy et al., 2005). A small number of brain imaging studies have found common valuation regions (Kang et al., 2011) or emotion regions (Feldman- Hall et al., 2012a) for both types of choice, as well as distinct regions which are more strongly activated during real choice (Kang and Camerer, 2013).
Given the possibility of hypothetical bias, an important practical challenge is how to accurately forecast real choices from data on hypothetical choices. A good forecasting correction method is also likely to be scientifically valuable, if it can create knowledge about the detailed mechanism that produces bias (and how it varies across types of choices and people).
Different forecasting methods have been tried. Methods can be sorted into two categories: procedures and enhanced pre-choice measurement. Procedural approaches change how questions are asked or choice data are processed. Mea- surement methods collect more data and use them to improve forecasting.
Procedures Many studies have explored different experimental or statistical pro- cedures that might reduce the bias. Statistical procedures (“calibration”) search for a predictable measurable relations between the hypothetical and real choices, and then test how well that relation can be used to forecast actual choices from hy- pothetical ones within-sample, or in a new case (Blackburn et al., 1994; Fox et al., 1998; Kurz, 1974; List and Shogren, 1998, 2002; Shogren, 1993). For example, in our study we observe that about 55% of subjects choose to purchase consumer goods hypothetically, but only 23% do when choices are real. So one could take a hypothetical purchase rate in a new sample, and multiply it by 0.23/0.55 =0.40, to crudely estimate a real purchase rate.
Calibration methods such as these have been extended to account for socio- demographics variables in hypothetical bias. They are useful for many purposes.
However, calibration has not been well-tuned to adjust for likely vagaries of spe- cific goods and choice contexts, as flagged by List and Shogren (1998, 2002).
A more ambitious procedure is to search for a way of asking hypothetical questions that gives answers which are closer to real-choice answers. Champ et al.
(1997) ask respondents how “certain” they are (on a 10-point scale) about whether
they would actually donate the stated amount to a project if asked to do so. Cum- mings and Taylor (1999) use the “cheap talk” protocol: the design includes an explicit discussion of the hypothetical bias problem (what it is and why it might occur) at the beginning of the experiment. Following findings in social psychol- ogy, Jacquemet et al. (2013) use a “solemn oath,” asking participants to swear on their honor to give honest answers, as a truth-telling commitment device. Finally, the “dissonance-minimizing format” of Blamey et al. (1999) and Loomis et al.
(1999) include additional response categories that permit respondents to express support for a project or policy without having to commit dollars. 1
Several meta-analyses have been conducted to evaluate effects of diverse ex- perimental methods on hypothetical bias to find variables that account for the variation in bias across goods and contexts (Carson et al., 1996; List and Gal- let, 2001; Little and Berrens, 2004; Murphy et al., 2005). Hypothetical bias is in- fluenced by the distinction between willingness-to-pay or willingness-to-accept, public goods and private goods, and elicitation methods.
All calibration methods also rely on extrapolating from a past hypothetical- actual relation to the future. An example of where this can backfire is politics.
Historically, polls asking people whom they intend to vote for overestimated the actual vote for black candidates on election day (Keeter and Samaranayake, 2007).
However, this so-called “Bradley effect” (also known as “Wilder effect”) has grad- ually eroded over time (Hopkins, 2009).
Further search for ideal procedures to pose hypothetical questions that yield responses that predict real answers is surely worthwhile. However, there is no current consensus on a single method that works effectively across choice con- texts. We therefore turn to measuring more variables.
Pre-choice measures Another approach that has been explored more tentatively is to measure psychological or neural variables that are recorded during the pro- cess of hypothetical choice, and use those measures to forecast actual choice. 2 These measures will often precede choice, so we generally call them “pre-choice”
measures. We report new evidence from this approach using measures of vi-
1Other procedures, such as asking respondents to consider budget constraints and budgetary substitutes, are shown to be ineffective (Loomis et al., 1994; Neill, 1995).
2Recent study by Bernheim et al. (2015) has features of both procedural and measurement methods approaches. Their proposed method, called “non-choice revealed preference,” involves estimation of statistical relationships between choices and non-choice variables and prediction of choices using those relationships together with non-choice data under new environment. They propose that non-choice reactions could include from simple ratings (liking, familiarity, certainty, happiness, etc.) to physiological reactions including brain activities.
sual attention—both mouse-based lookup of information (mousetracking), and eyetracking recordings.3
We record visual attention as people make hypothetical choices (about con- sumer products, for example). On some trials people choose to (hypothetically) buy the product, and on others they don’t buy. On later trials they are surprised by the opportunity to actually buy some of those same products.
The motivating hypothesis is that what people looked at during the initial hypothetical choice will help forecast whether they will stick with their original hypothetical choice, or will change their minds when making a subsequent real choice. 4 A quick preview of the main result is the following: during hypotheti- cal choice, the more people look at prices, and the longer they take to transition from looking to making a choice, the more likely they are to switch a hypothet- ical “Buy” to a real “Don’t buy.” The improvement in prediction is not large in magnitude. However, it provides initial evidence that some improvement is pos- sible using pre-choice measures, and further efforts designed at maximizing the improvement in prediction could do much better.
Note that a few recent studies have measured functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) signals and used them to fore- cast actual choices, (e.g., Levy et al., 2011; Smith et al., 2014; Tusche et al., 2010).
While interesting and promising, none of these studies are specifically designed to make the leap from hypothetical pre-choice thinking to lateractual choices. We discuss their methods and compare them to ours in the concluding discussion.
The remaining of the paper is organized as follows. We describe our experi- mental design in Section 3.2, report the main results in Section 3.3, and discuss our results and conclude in Section 3.4. Additional results and experimental de- tails are reported in Sections B.1 to B.4.
3There is only one study directly comparing results from both measures on a common task (Lohse and Johnson, 1996). As in that study, we find that the main regularities are common across both visual fixation measures.
4This is partly motivated by the “mind eye hypothesis” which assumes that what a person is looking at indicates what they are currently thinking about or attending to (Just and Carpenter, 1980). Studies have shown a connection between eyetracking patterns and users’ decision making processes (Goldberg et al., 2002). Remarkably, eye movements can be even more accurate than con- scious recall in predicting whether people have seen a visual stimulus before (and eye movements are associated with hippocampal activity measured by fMRI; Hannula and Ranganath, 2009).