Sleepiness, choice consistency, and risk preferences

Abstract

We investigate the consistency and stability of individual risk preferences by manipulating cognitive resources. Participants are randomly assigned to an experiment session at a preferred time of day relative to their diurnal preference (circadian matched) or at a non-preferred time (circadian mismatched) and choose allocations between two risky assets [using the Choi et al. (Am Econ Rev 27(5):1921–1938, 2007), design]. We find that choices of circadian matched and mismatched subject are statistically similar in terms of satisfying basic requirements for preference consistency. However, mismatched subjects tend to choose riskier asset bundles.

Appendices

Appendix 1: Subject Instruction

1.1 Instructions

This is an experiment in decision making. Your payoffs will depend partly on your decisions and partly on chance. Your payoffs will not depend on the decisions of the other participants in the experiment. Please pay careful attention to the instructions as a considerable amount of money is at stake.

The entire experiment should be completed within an hour and a half. At the end of the experiment, you will be paid privately. Your total payoff in this experiment will consist of $5 as a participation fee (simply for showing up on time), plus whatever payoff you receive from the decision experiment. Details of how your payoff will depend on your decisions will be provided below.

During the experiment, we will speak in terms of experimental tokens instead of dollars. Your payoffs will be calculated in terms of tokens and then translated at the end of the experiment into dollars at the following rate:

$$\begin{aligned} 2\; \hbox {Tokens} = 1\; \hbox {Dollar} \end{aligned}$$

1.2 The decision problem

In this experiment, you will participate in 50 independent decision problems that share a common form. This section describes in detail the process that will be repeated in all decision problems and the computer program that you will use to make your decisions.

In each decision problem, you will be asked to allocate tokens between two accounts, labeled x and y. The x account corresponds to the x-axis and the y account corresponds to the y-axis in a two-dimensional graph. Each choice will involve choosing a point on a line representing possible token allocations. Examples of lines that you might face appear in the graph below. Many lines are shown on the same graph to highlight that there will be a variety of different lines you could face, but each decision you make will involve only one line on the graph, as you will see further in these instructions.

In each choice, you may choose any x and y pair that is on the line. For example, as illustrated in the next graph below, choice a represents a decision to allocate 14 tokens in the x account and 70 tokens in the y account. Another possible allocation is b, in which you allocate 40 tokens in the x account and 30 tokens in the y account.

Each decision problem will start by having the computer select such a line randomly from the set of lines that intersect with at least one of the axes at 50 or more tokens but with no intercept exceeding 100 tokens. The lines selected for you in different decision problems are independent of each other and independent of the lines selected for any of the other participants in their decision problems.

To choose an allocation in each decision problem, use the mouse to move the pointer on the computer screen to the allocation that you desire. When you are ready to make your decision, left-click to enter your chosen allocation. After that, confirm your decision by clicking on the Submit button that will appear after your decision is made. Note that you can choose only x and y combinations that are on the line (you may also choose either endpoint on any line if you so desire). The next graph shows a picture of the actual decision screen you will see in the experiment. Notice that where you position the pointer on the line will highlight exactly what combination of x and y is at that location on the line. This same information is also shown in the information area to the right of the graph (the example graph shows additional information that will be discussed next in these instructions. It also indicates a 20 round experiment, although today’s experiment will be 50 rounds in length).

Once you have confirmed your choice for that decision round, press the OK button. Your payoff in each decision round is determined by the number of tokens in your x account and the number of tokens in your y account. At the end of the round, the computer will randomly select one of the accounts, x or y. There is an equal chance that either account will be selected, and this random selection occurs separately and independently of each participant. You will only receive as payment the number of tokens you allocated to the account that was chosen. (In the example graph directly above, if account x is selected you would receive 22.7 tokens, and if account y is selected you would receive 56 tokens). The random selection of account x or y in a decision round will not be shown to you until the very end of the experiment.

Once a decision round is finished, you will be asked to make an allocation in another independent decision. This process will be repeated until all 50 decision rounds are completed. At the end of the last round, you will be informed that the experiment has ended.

1.3 Your earnings

Your earnings in the experiment are determined as follows. At the end of the experiment, the computer will randomly select one decision round to carry out (that is, 1 out of 50). The round selected depends solely upon chance, and it is equally likely that any round will be chosen. Once a round is chosen, you will receive the number of tokens you allocated to the account (x or y) that was randomly selected for that round. Keep in mind that there is an equal chance that account x or y will be chosen for your token payoff in any given round.

The round selected, your choice and your payment (in terms of tokens) will be shown in the large window that appears at the center of the program dialog window. At the end of the experiment, the tokens will be converted into money. Each token will be worth 0.5 dollars (in other words, your “tokens” payoff will be divided by 2 to get your payoff in dollars). Your final cash earnings in the experiment will be your earnings in the round selected plus the $5 show-up fee. You will receive your payment as you leave the experiment.

1.4 Rules

Please do not share your decisions with anyone else in today’s experiment, please do not talk with anyone during the experiment, and please remain silent until everyone is finished. If there are no further questions, you are ready to start, and an experimenter will start your experiment program.

Appendix 2A: Pre-experiment online survey (answered prior to recruitment)

(some responses elicited using slider bars, matrix response boxes, or other standard online survey features)

——————————

What is your gender?

• Female

• Male

Please enter your valid email address (e.g., johndoe@emailserver.com)

This is required if you wish to be entered into the database for eligibility and recruitment consideration for future cash-compensation research experiments

What is your ethnicity?

• Hispanic or Latino

• Not Hispanic or Latino

What is your racial category?

• American Indian/Alaska Native

• Asian

• Native Hawaiian or Other Pacific Islander

• Black or African American

• White (Caucasian)

• Mixed

• Other (please specify in text box)

What is your age?

Are you a student, faculty, or staff?

Over the last 2 weeks, how often have you been bothered by the following problems?

(Options for each are “Not at all”, “Several Days”, “Over Half of the Days”, or “Nearly Every Day”)

• Feeling nervous, anxious or on edge

• Not being able to stop or control worrying

• Worrying too much about different things

• Little interest or pleasure in doing things

• Trouble relaxing

• Being so restless that it is hard to sit still

• Becoming easily annoyed or irritable

• Feeling down, depressed, or hopeless

• Felling afraid as if something awful might happen.

Considering only your own “feeling best” rhythm, at what time would you get up if you were entirely free to plan your day?

• Between 5:00 and 6:30 a.m.

• Between 6:31 and 8:00 a.m.

• Between 8:01 and 9:30 a.m.

• Between 9:31 and 11:00 a.m.

• Between 11:01 a.m. and noon

During the first half-hour after waking up in the morning, how tired do you feel?

• Very tired

• Fairly tired

• Fairly refreshed

• Very refreshed

At what time in the evening do you feel tired and, as a result, in need of sleep?

• Between 8:00 pm and 9:00 pm

• Between 9:01 pm and 10:30 pm

• Between 10:31 pm and 12:30 am

• Between 12:31 am and 2:00 am

• Between 2:01 am and 3:00 am

At what time of the day do you think you reach your “feeling best” peak?

• Between midnight and 4:30 a.m.

• Between 4:31 a.m. and 7:30 a.m.

• Between 7:31 a.m. and 9:30 a.m.

• Between 9:31 a.m. and 4:30 p.m.

• Between 4:31 p.m. and 9:30 p.m.

• Between 9:31 p.m. and midnight.

One hears about “morning” and “evening” types of people. Which ONE of these types do you consider yourself to be?

• Definitely a ‘morning-type’

• More likely a ‘morning-type’ than an ‘evening-type’

• More likely an ‘evening-type’ than a ‘morning-type’

• Definitely an ‘evening-type’

Over the last 7 nights, what is the average amount of sleep you obtained each night?

Last night, how much sleep did you get?

What do you feel is the optimal amount of sleep for you personally to get each night? (optimal in terms of next day alertness, performance, and functionality for you personally.)

How likely are you to doze off or fall asleep in the following situations, in contrast to just feeling tired? This refers to your usual way of life in recent times. Even if you have not done some of these things recently, try to work out how they would have affected you.

(options are “would NEVER doze or fall asleep”, “SLIGHT chance of dozing or falling asleep”, “MODERATE chance of dozing or falling asleep”, or “HIGH chance or dozing or falling asleep”)

• Sitting and reading

• Watching TV

• Sitting, inactive in a public place (e.g., a theater or a meeting)

• As a passenger in a car for an hour

• Lying down to rest in the afternoon when circumstances permit

• Sitting and talking to someone

• Sitting quietly after lunch without alcohol

• In a car, while stopped for a few minutes in traffic.

Do you have a diagnosed sleep disorder?

• Yes

• No

• Not diagnosed, but I believe I may have a sleep disorder

If you responded “Yes” (i.e., you have a diagnosed sleep disorder), what is it?

Appendix 2B: Survey given during experiment

Appendix 2C: Technical details of preference function tests

1. Testing expected utility theory.

A test of expected utility in asset markets has been proposed by Varian (1983), and a maintained assumption of Varian’s test is that the Bernoulli utility function over money is concave, i.e., individuals are risk averse. Recently, Polisson et al. (2015) have proposed an alternative test of expected utility that does not rely on assuming risk aversion.

The test by Polisson et al. consists of finding a set of numbers, i.e., utilities, that are consistent with the observed revealed preference relation and with an extension of the observed revealed preference relation over the lattice of points generated by the experiment.²² Polisson et al. also propose a way to calculate a number similar to Afriat’s critical cost to efficiency corresponding to the test of expected utility theory.

We use the approach by Polisson et al. to test for adherence to expected utility theory.

2. Testing rank-dependent utility.

To test for rank-dependent utility in asset markets, we propose a straightforward generalization of Varian’s (1983) test of expected utility to allow for the possibility of rank-dependent valuation of outcomes.

It should be noted that rank-dependent utility can be tested in asset market experiments even if the probability of states of nature does not vary. A simple example will illustrate this point. Consider we observe the following three revealed preference relations: $(\hbox {x}_{1},\hbox {y}_{1})\hbox {R}(\hbox {x}_{2},\hbox {y}_{2})$, $(\hbox {x}_{3},\hbox {y}_{2})\hbox {R}(\hbox {x}_{1},\hbox {y}_{3})$ and $(\hbox {x}_{2},\hbox {y}_{3})\hbox {R}(\hbox {x}_{3},\hbox {y}_{1})$ where aRb means a is revealed preferred to b. We also assume that $\hbox {x}_{1}<\hbox {y}_{1}$, $\hbox {x}_{2}>\hbox {y}_{2}$, $\hbox {y}_{1}>\hbox {x}_{3}$, $x_{3}>\hbox {y}_{2}$ and $\hbox {x}_{2}>\hbox {y}_{3}$.

These allocations are not consistent with expected utility theory. To see this, suppose u rationalizes the choices above when the probability of obtaining asset x is p ($0<p<1$). We must have that:

• i. ${\mathrm {u({x}_{1})p+u(y_{1})(1-p) > u(x_{2})p+u(y_{2})(1-p)}}$

• ii. ${\mathrm {u({x}_{3})p+u(y_{2})(1-p) > u(x_{1})p+u(y_{3})(1-p)}}$

• iii. ${\mathrm {u({x}_{2})p+u(y_{3})(1-p) > u(x_{3})p+u(y_{1})(1-p)}}$

Adding inequalities (i) and (ii), we obtain that $\hbox {u}(\hbox {x}_{3})\hbox {p}+\hbox {u}(\hbox {y}_{1})(1-\hbox {p}) > \hbox {u}(\hbox {x}_{2})\hbox {p}+\hbox {u}(\hbox {y}_{3})(1-\hbox {p})$, which contradicts inequality (iii).

However, these inequalities do not contradict rank-dependent utility theory. Let w(p) be the probability weight of p. This theory requires that

• i. ${\mathrm {(1-w(1-p))u(x_{1})+w(1-p)u(y_{1}) > (1-w(p))u(y_{2})+w(p)u(x_{2})}}$

• ii. ${\mathrm {(1-w(p))u(y_{2})+w(p)u(x_{3})> (1-w(1-p))u(x_{1})+w(1-p)u(y_{3})}}$

• iii. ${\mathrm {(1-w(p))u(y_{3})+w(p)u(y_{2}) > (1-w(1-p))u(x_{3})+w(1-p)u(y_{1})}}$

Consider inequality (i). By assumption, $\hbox {x}_{1}<\hbox {y}_{1}$ and $\hbox {x}_{1}$ obtains with probability p and $\hbox {y}_{1}$ obtains with probability 1-p. Rank-dependent utility theory weighs the lower ranked outcome, $\hbox {x}_{1}$, with weight $1-\hbox {w}(1-\hbox {p})$, with the probability that outcomes are larger than $\hbox {x}_{1}$ (w(1) $=$ 1) minus the probability that outcomes are strictly larger than $\hbox {x}_{1}$ (w(1–p)). Similarly, rank-dependent utility theory weighs the largest outcome, $\hbox {y}_{1}$, with the probability that outcomes are larger than $\hbox {y}_{1}$ (w(1–p)) minus the probability that outcomes are strictly larger than $\hbox {y}_{1}$ (w(0) $=$ 0). The remaining expressions are obtained in a similar manner.

Adding up inequalities (i) and (ii), we obtain that $\mathrm {w(1-p)u(y_{1})+w(p)u(x_{3}) >} \mathrm {w(1-p)u(y_{3})+w(p)u(x_{2})}$. This inequality would violate rank-dependent utility theory if w(1-p) $=$ 1-w(p) which is not true in general. It is easy to show that if either $\hbox {x}_{\mathrm{i}}>\hbox {y}_{\mathrm{i}}$ or $\hbox {x}_{\mathrm{i}}<\hbox {y}_{\mathrm{i}}$ for = 1, 2, 3, this pattern of behavior would also violate rank-dependent utility.

In sum, it is possible to test for rank-dependent utility theory separated from expected utility even if probabilities are fixed. This follows from the fact that probability weighting depends on the rank of outcomes.

We now show how to test for rank-dependent utility theory in the context of two essential Arrow–Debreu securities. Let the consumption data be ${\mathrm {(x_{ik},p_{ik},\pi _{ik})}}$, i $=$ 1, 2, k = 1,..., N, where ${\mathrm {x_{ik},p_{ik},\pi _{ik}}}$ are consumption bundles of state contingent assets, their prices and the probability of each state i.

In the context of two state-contingent assets, rank-dependent utility theory is equivalent to the following representation of preferences:

$$\begin{aligned}&{\mathrm {u(x_{1k},x_{2k}) = (1-w(\pi _{2k}))u(x_{1k}) + w(\pi _{2k})u(x_{2k})\, if\, x_{1k }\le x_{2k}}}\hbox { and}\\&{\mathrm {u(x_{1k},x_{2k}) = w(\pi _{1k})u(x_{1k}) + (1-w(\pi _{1k}))u(x_{2k})\, if\, x_{1k }> x_{2k}}} \end{aligned}$$

where u is a monotone increasing value function and w is a monotone increasing probability weighting function with w(0) $=$ 0 and w(1) $=$ 1.

Because rank-dependent utility theory evaluates lottery prizes according to their ranks, the utility representation of preferences is not differentiable everywhere. In particular, the corresponding indifference curves will have a kink whenever $\hbox {x}_{1\mathrm{k}}=\hbox {x}_{2\mathrm{k}}$.

To derive the testable hypotheses of rank-dependent utility with a concave value function, we note that if u is differentiable a consumer choosing $(\hbox {x}_{1\mathrm{k}},\hbox {x}_{2\mathrm{k}})$, where $\hbox {x}_{1\mathrm{k} }>\hbox {x}_{2\mathrm{k}}$ solves the following problem:

$$\begin{aligned} {\mathrm {max_{x1k,x2k} \{w(\pi _{1k})u(x_{1k}) + (1-w(\pi _{1k}))u(x_{2k}): p_{1k}x_{1k }+ p_{2k}x_{2k }\le 1\}}} \end{aligned}$$

with a symmetric representation for the case in which $\hbox {x}_{1\mathrm{k}}< \hbox {x}_{2\mathrm{k}}$.

Whenever $\hbox {x}_{1\mathrm{k}}= \hbox {x}_{2\mathrm{k}}$, the problem is equivalent to the simultaneous solution of the following problems:

If the function u is concave, first-order conditions are necessary and sufficient for the existence of a maximum. For decisions with $\hbox {x}_{1\mathrm{k} }> \hbox {x}_{2\mathrm{k}}$, we will have that

${\mathrm {u'(x_{1k})=\lambda _{k}p_{1k}/w(\pi _{1k})}}$ and ${\mathrm {u'(x_{2k})=\lambda _{k}p_{2k}/(1-w(\pi _{1k}))}}$, and for decisions with $\hbox {x}_{1\mathrm{k}}< \hbox {x}_{2\mathrm{k}}$, we will have that ${\mathrm {u'(x_{1k})=\lambda _{k}p_{1k}/(1-w(\pi _{2k}))}}$ and $\mathrm {u'(x_{2k})=}\mathrm {\lambda _{k}p_{2k}/w(\pi _{2k})}$. Where $\lambda _{\mathrm{k}}$ denotes the Lagrange multiplier associated with the problem. Whenever $\hbox {x}_{1\mathrm{k} }= \hbox {x}_{2\mathrm{k}}$ all these conditions hold simultaneously. In such a case, it is possible for $\lambda _{\mathrm{ik}}$, i $=$ 1, 2, to vary for each separate case. Diewert (2012) discusses the conditions under which it is possible to find numbers that satisfy the first-order conditions of the problem.

Below we establish the system of linear inequalities associated with rank-dependent utility theory with a concave value function. Since, function u is concave, we know that

$$\begin{aligned} {\mathrm {u(x_{ik}) \le u(x_{jm}) + u'(x_{jm})(x_{ik}- x_{jm})}} \end{aligned}$$

Using the first-order conditions of the associated problem, we have that

$$\begin{aligned}&{\mathrm {u(x_{ik}) \le u(x_{im}) + (\lambda _{m}p_{jm}/w(\pi _{jm}))(x_{ik}- x_{jm})}}\hbox { if } {\mathrm {x_{jm }> x_{-j,m}}}\\&{\mathrm {u(x_{ik}) \le u(x_{im}) + (\lambda _{m}p_{jm}/(1-w(\pi _{-j,m})))(x_{ik}- x_{jm}) if x_{jm }< x_{-j,m}}} \end{aligned}$$

In the case $\hbox {x}_{\mathrm{jm}}= \mathrm{x}_{-\mathrm{j,m}}$, both equations above hold for potentially different values of $\lambda _{\mathrm{j,m}}$, $j = 1, 2$.

A concave rank-dependent utility representation of preferences exists if there are positive numbers $\{\hbox {u}_{1\mathrm{k}}$, $\hbox {u}_{2\mathrm{k}}$, $\lambda _{1k}$, $\lambda _{2k}\}$ ²³ and numbers $\{\hbox {w}_{1\mathrm{k}}$, $\hbox {w}_{2\mathrm{k}}\}$ ²⁴ between 0 and 1 satisfying the above equations. The proof of the existence of a rank-dependent utility representation of preferences if the above inequalities are satisfied follows standard arguments.

We follow Diewert (2012) in introducing a new variable S (S$\ge$0) to be added to all the Afriat inequalities above which equals 0 only if the system of equations has a solution. We also follow Diewert (2012) in restricting $\lambda$ to be larger than one²⁵ and normalizing income to be equal to one. This slack variable can be used to assess the fitness of the data with respect to rank-dependent expected utility. Consider that a subject is an expected utility maximizer. Adding up the inequalities, we find

$$\begin{aligned}&{\mathrm {\pi _{1m}u(x_{1k})+\pi _{2m}u(x_{2k}) \le \pi _{1m}u(x_{1m})+\pi _{2m}u(x_{2m})+\lambda _{m}(p_{1m}x_{1k} + p_{2k}x_{2k }}}\\&\quad {\mathrm {- p_{1m}x_{1m} - p_{2m}x_{2m})}} \end{aligned}$$

and because income is assumed to be equal to one, we have that

$$\begin{aligned} {\mathrm {\pi _{1m}u(x_{1k})+\pi _{2m}u(x_{2k}) \le \pi _{1m}u(x_{1m})+\pi _{2m}u(x_{2m})+\lambda _{m}(p_{1m}x_{1k} + p_{2k}x_{2k }- 1)}} \end{aligned}$$

A violation of expected utility would occur if the utility of allocation $(\hbox {x}_{1\mathrm{k}}$, $\hbox {x}_{2\mathrm{k}})$ is preferred to allocation $(\hbox {x}_{1\mathrm{m}},\hbox {x}_{2\mathrm{m}})$ and it was affordable when $(\hbox {x}_{1\mathrm{m}},\hbox {x}_{2\mathrm{m}})$ was chosen. The equation above would then be violated. Suppose that income is adjusted to level $1-\varepsilon$ such that allocation $(\hbox {x}_{1\mathrm{k}},\hbox {x}_{2k})$ is not longer affordable. The minimum value that variable $\varepsilon$ takes gives a measure of how much the model departs from the theoretical assumptions. Note that variable $\varepsilon$ holds a relationship with the slack variable S since $\hbox {max}_{\mathrm{k}} \{\lambda _{\mathrm{k}}\varepsilon \}$ will solve the equations above. Since $\lambda$ is always larger than one, the variable S gives us an upper bound of the value of $\varepsilon$. In the analysis below, we use 1-S as an estimate of the lower bound of the critical cost to efficiency for the test of rank-dependent utility.

3. Results.

Figure 5 shows the distribution of Afriat’s critical cost to efficiency (CCEI) corresponding to the test of expected utility developed by Polisson et al. (2015). The mean CCEI of circadian matched subjects is 0.891 and mean CCEI of the circadian mismatch subjects is 0.876. These means are not statistically significantly different (t test $=$ 0.7697, p value $=$ 0.4424, rank sum test p value $=$ 0.5033). Only two subjects, both of them circadian mismatched, have a CCEI equal to 1. This means that only two subjects satisfy conditions for expected utility without any violations of the theory.

Fig. 5

Distribution of critical cost to efficiency (CCEI) for test of expected utility (Polisson et al. 2015)

Fig. 6

Distribution of Critical Cost to Efficiency (CCEI) for test of rank-dependent utility with concave utility function and w(1/2)$\le$1/2

Figure 6 shows the distribution of Afriat’s critical cost to efficiency (CCEI) corresponding to the test of rank-dependent expected utility with a concave utility function and w(1/2) $\le$ 1/2. While the test presented in Sect. 2 does not require w(1/2) $\le$ 1/2, we consider it to be a natural restriction given the clustering of decisions around the 50-50 allocation. The mean CCEI of circadian matched subjects is 0.797 and mean CCEI of the circadian mismatch subjects is 0.817. These means are not statistically significantly different (t test $=$ 0.7844, p value $=$ 0.4337, rank sum test p value $=$ 0.8434). Only two subjects, both of them circadian mismatched, pass the rank-dependent utility test without any violations (CCEI $=$ 1).

The test of rank-dependent utility implicitly finds a value for a subject’s weight for probability 1/2. We find that these estimates are not statistically different according to a t test (0.6869, p value $=$ 0.4930), but they are different according to the rank sum test (p value $=$ 0.0480). To check if there is a difference in estimated probability weight of 1/2, we test if the proportion having weights below 1/2 is different across groups. Thirty-three percent of circadian matched subjects have estimated weights below 1/2 and 21 % of circadian mismatched subjects have weights below 1/2. These proportions are statistically different (t test $=$ −1.9662, p value $=$ 0.0507). This provides additional support that circadian matched subjects are more likely to choose safe options than circadian mismatched subjects.