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Heterogeneity in preferences towards complexity

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Abstract

We analyze lottery-choice data in a way that separately estimates the effects of risk aversion and complexity aversion. Complexity is represented by the number of different outcomes in the lottery. A finite mixture random effects model is estimated which assumes that a proportion of the population are complexity-neutral. We find that around 33% of the population are complexity-neutral, around 50% complexity-averse, and the remaining 17% are complexity-loving. Subjects who do react to complexity appear to have a bias towards complexity aversion at the start of the experiment, but complexity aversion reduces with experience, to the extent that the average subject is (almost) complexity-neutral by the end of the experiment. Complexity aversion is found to increase with age and to be higher for non-UK students than for UK students. We also find some evidence that, when evaluating complex lotteries, subjects perceive probabilities in accordance with Prospective Reference Theory.

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Notes

  1. The experiment was programmed and conducted with the experiment software z-Tree (Fischbacher 2007).

  2. The instructions and questionnaire are provided in Appendix A.

  3. Note that task types 1 and 5 have only 440 observations, while all other task types have 480. The reason for this difference is a minor programming bug that led to the loss of data on 40 of the 80 subjects’ choices on each of the two choice problems: VC3-S3 and SW-S3.

  4. We thank Kip Viscusi for suggesting this extension to our model.

  5. It is important to realise that (6) does not apply when J = 1. In this case, \( {\tilde{p}}_j={p}_j=1 \); that is, the probability of the certain outcome is correctly interpreted as 1.

  6. The STATA code used for estimation is available from the authors on request.

  7. When τ is added as a determinant of risk aversion to Model 4, its asymptotic t-statistic is −0.40 and the p-value is 0.69.

  8. When the three explanatory variables male, (age-18), and non_UK are all added to the risk-aversion equation in Model 5, the LogL rises very slightly to −2410.25, but the associated AIC is 4848.5, which is vastly inferior to Model 5.

  9. Changing an energy tariff, a mobile phone tariff, or a bank account are decisions that can in principle be made frequently. However, in practice such decisions are typically seen to be quite infrequent (Sitzia et al. 2012). In addition, frequent changes in tariffs are likely to impede learning. Hence we expect complexity aversion to play an important role in these situations.

  10. The STATA code used for estimation is available from the authors on request.

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Acknowledgments

Funding from the University of East Anglia, advice from Bob Sugden and from participants at the Asia Pacific meeting of the Economic Science Association in Auckland in February 2014, and research assistance from Axel Sonntag are gratefully acknowledged. The usual disclaimer applies. The experimental instructions can be found in the appendix.

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Correspondence to Peter G. Moffatt.

Appendices

Appendix A

1.1 Experimental instructions

In the course of this experiment, over a number of periods you will be asked to choose between lotteries that pay returns in experimental points with given probabilities.

In the table below you see an example of a lottery of the kind a unit of which you can choose each period:

Table A Example of product

In this example, the lottery will give you the chance to earn the following returns at the end of the experiment: 50 points with a probability of 35% (Outcome 1); 15 points with a probability of 15% (Outcome 2); 80 points with a probability of 18% (Outcome 3); and so on.

If you see a lottery which provides a given return with a probability of 100%, this means that, if you choose this lottery, you will get that return for sure.

1.2 Earnings

At the end of the experiment the computer will randomly select one of the periods that will be used to determine your earnings. The computer will then randomly select one outcome of the lottery you have chosen in that period based on the probabilities. This outcome determines the return of that lottery and the points you have earned in the experiment. Every 10 points you own are converted into 1 pound, and so for example 80 points are worth 8 pounds.

Before starting to take decisions, we ask you to fill the enclosed questionnaire, with the only purpose of checking whether you have understood these instructions. Raise your hand when you have completed the questionnaire.

1.3 Questionnaire

1) Your earnings in the experiment are the sum of your earnings each period?

Yes ______No ______

2) Consider the example lottery in Table A. What is the probability of obtaining a return of 80?

_________

PLEASE RAISE YOUR HAND WHEN YOU HAVE FINISHED.

THANK YOU FOR ANSWERING THE QUESTIONNAIRE

Appendix B

2.1 Construction of log-likelihood function

For a given subject (i), who is of type j, facing a given choice problem (t), we define:

$$ \nabla {U}_{it,j}\left({\alpha}_i,{\gamma}_i\right)={U}_{i,j}\left({\mathrm{p}}_{\mathrm{t}},\ {\mathrm{x}}_{\mathrm{t}}\right)-{U}_{i,j}\left({\mathrm{q}}_{\mathrm{t}},\ {\mathrm{y}}_{\mathrm{t}}\right) $$
(A1)

where the two terms on the right are defined according to one of (4a)-(4d), depending on the type of subject i.

Subject i (of type j) chooses lottery (qt, yt) over lottery (pt, xt) if the following inequality holds:

$$ \nabla {U}_{it,j}\left({\alpha}_i,{\gamma}_i\right)+{\varepsilon}_{it}<0 $$
(A2)

where ε it is a stochastic term with distribution:

$$ {\varepsilon}_{it}\sim N\left(0,\ \exp \left(\varphi {\tau}_{it}\right)\right) $$
(A3)

The stochastic term ε may be interpreted as computational error, and its variance is assumed to depend on τit, which is the position of task t in subject i’s sequence of tasks (τ = 1,…,54). According to (A3), this variance is 1 at the start of the experiment—a normalization that is required for identification of the model. A negative sign of the parameter φ will indicate that the magnitude of errors tends to decrease as the experiment progresses (an experience effect), while a positive sign will indicate an increase in errors over the course of the experiment (a boredom effect).

We define the binary variable yit as follows. yit = +1 if subject i chooses (qt, yt) over (pt, xt); yit = −1 if subject i chooses (pt, xt) over (qt, yt). By (A2) and (A3):

$$ P\left({y}_{it}=1\Big|i\in j\right)=P\left[\nabla {U}_{it,j}\left({\alpha}_i,{\gamma}_i\right)+{\varepsilon}_{it}<0\right]=\varPhi \left(-\frac{\nabla {U}_{it,j}\left({\alpha}_i,{\gamma}_i\right)}{\sqrt{ \exp \left(\varphi {\tau}_{it}\right)}}\right)\kern1em i=1,\cdots, n\kern1.25em t=1,\cdots, T $$
(A4)

Where “i ∈ j” denotes that subject i is of type j. Note that we are assuming that there are in total n subjects each facing T choice problems.

As noted in (5) in the main text, we are assuming between-subject heterogeneity both in risk aversion (α) and in complexity aversion (γ), and their joint distribution will be assumed to be:

$$ \left(\begin{array}{c}\hfill \alpha \hfill \\ {}\hfill \gamma \hfill \end{array}\right)\sim N\left[\left(\begin{array}{c}\hfill {\theta}_1\hfill \\ {}\hfill {\theta}_2\hfill \end{array}\right),\left(\begin{array}{cc}\hfill {\eta}_1^2\hfill & \hfill \rho {\eta}_1{\eta}_2\hfill \\ {}\hfill \rho {\eta}_1{\eta}_2\hfill & \hfill {\eta}_2^2\hfill \end{array}\right)\right] $$
(A5)

ρ is the correlation coefficient between risk aversion and complexity aversion.

The likelihood contribution associated with subject i is:

$$ {L}_i={\displaystyle \underset{\alpha }{\int }{\displaystyle \underset{\gamma }{\int }{\displaystyle \sum_{j=1}^4{\pi}_j}{\displaystyle \prod_{t=1}^T\varPhi \left(-{y}_{it}\times \frac{\nabla {\cup}_{it,j}\left({\alpha}_i,{\gamma}_i\right)}{\sqrt{ \exp \left(\varphi {\tau}_{it}\right)}}\right)f\left(\alpha, \gamma; {\theta}_1,{\eta}_1,{\theta}_2,{\eta}_2,\rho \right) d\gamma d\alpha}}} $$
(A6)

where f(α, γ; θ 1, η 1, θ 2, η 2, ρ) is the joint density function of α and γ, and the parameters πj, j = 1…4 are mixing proportions, representing the proportion of the population who are of each type.

A further extension to the model allows risk aversion and complexity aversion to depend on experience within the experiment, and also on demographics. In Models 5 and 6, we assume that:

$$ \begin{array}{l}\hfill {\theta}_{1, it}={\theta}_{10}+{\theta}_{11}{\tau}_{it}\hfill \\ {}\hfill {\theta}_{2, it}={\theta}_{20}+{\theta}_{21}{\tau}_{it}+{\theta}_{22}mal{e}_i+{\theta}_{23}{\left( age-18\right)}_i+{\theta}_{24}non\_u{k}_i\hfill \end{array} $$
(A7)

where τit is, as in (A3), the position of task t in subject i’s sequence of tasks (τ = 1,…,54). (A7) allows the means (over the population) of the coefficients of risk aversion and complexity aversion, to change with experience and demographics. For example, the mean of the coefficient of complexity aversion (for an 18-year-old, female, UK subject) is θ 20 at the start of the sequence, and changes by an amount θ 21 with each task undertaken.

A yet further extension allows misinterpretation of stated probabilities in accordance with Prospective Reference Theory. If the transformed probabilities defined in (6) are used in place of true probabilities, we may rewrite (A1) as:

$$ \tilde{\nabla}{\operatorname{U}}_{it,j}\left({\alpha}_i,{\gamma}_i\right)={\operatorname{U}}_{i,j}\left({\tilde{\mathrm{p}}}_{\mathrm{t}},\ {\mathrm{x}}_{\mathrm{t}}\right)-{\operatorname{U}}_{i,j}\left({\tilde{\mathrm{q}}}_{\mathrm{t}},\ {\mathrm{y}}_{\mathrm{t}}\right) $$
(A8)

where \( {\tilde{\mathrm{p}}}_{\mathrm{t}} \) and \( {\tilde{\mathrm{q}}}_{\mathrm{t}} \) are vectors of transformed probabilities. Then the likelihood function is constructed using \( \tilde{\nabla}{U}_{it,j} \) defined in (A8) in place of ∇ U  it,j in (A6).

If Prospective Reference Theory were assumed, and specification (A7) were used, there would be a total of 15 parameters to estimate: θ10, θ11, η1, θ20, θ21, θ22, θ23, θ24, η2, ρ, φ, δ, and three of the four mixing proportions: π1, π2, π3. The models we in fact estimate contain somewhat fewer parameters since they are restricted versions of the general model.

Estimation is performed using the method of Maximum Simulated Likelihood (MSL) (Train 2003). This requires the use of two sets of Halton draws, which, when converted to normality, represent simulated realizations of the random parameters α and γ. Maximization of the simulated likelihood function is performed using the ml routine in STATA.Footnote 10

Having estimated the model, posterior type probabilities can be obtained, and also posterior subject-specific estimates of risk aversion and complexity aversion. To obtain the posterior type probabilities, we use a version of Bayes’ rule:

$$ P\left(i\in j\mid {y}_{i1}\cdots {y}_{it}\right)=\frac{{\widehat{\pi}}_j{\displaystyle \underset{\alpha }{\int}\underset{\gamma }{\int}\prod_{t=1}^T\varPhi \left(-{y}_{it}\times \frac{\nabla {U}_{it,j}\left(\alpha, \gamma \right)}{\sqrt{ \exp \left(\varphi {\tau}_{it}\right)}}\right)f\left(\alpha, \gamma; {\widehat{\theta}}_1,{\widehat{\eta}}_1,{\widehat{\theta}}_2,{\widehat{\eta}}_2,\widehat{\rho}\right) d\gamma d\alpha}}{{\widehat{L}}_i} $$
(A9)

Where hats indicate MLEs, and \( {\widehat{L}}_i \) is the likelihood contribution for subject i as defined in (A6), with parameters replaced by MLEs.

To obtain the posterior subject-specific estimates of risk aversion, we use:

$$ {\widehat{\alpha}}_i=E\left({\alpha}_i\mid {y}_{i1}\cdots {y}_{iT}\right)=\frac{{\displaystyle \underset{\alpha }{\int}\underset{\gamma }{\int}\alpha \prod_{t=1}^T\varPhi \left(-{y}_{it}\times \frac{\nabla {U}_{it,4}\left(\alpha, \gamma \right)}{\sqrt{ \exp \left(\varphi {\tau}_{it}\right)}}\right)f\left(\alpha, \gamma; {\widehat{\theta}}_1,{\widehat{\eta}}_1,{\widehat{\theta}}_2,{\widehat{\eta}}_2\right) d\gamma d\alpha}}{{\displaystyle \underset{\alpha }{\int}\underset{\gamma }{\int}\prod_{t=1}^T\varPhi \left(-{y}_{it}\times \frac{\nabla {U}_{it,4}\left(\alpha, \gamma \right)}{\sqrt{ \exp \left(\varphi {\tau}_{it}\right)}}\right)f\left(\alpha, \gamma; {\widehat{\theta}}_1,{\widehat{\eta}}_1,{\widehat{\theta}}_2,{\widehat{\eta}}_2\right) d\gamma d\alpha}} $$
(A10)

Note that we are conditioning on the subject being of type 4 (responsive to both variance and complexity) when computing their posterior risk aversion. Posterior subject-specific estimates of complexity aversion, denoted \( {\widehat{\gamma}}_i \), are computed using a formula similar to (A10).

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Moffatt, P.G., Sitzia, S. & Zizzo, D.J. Heterogeneity in preferences towards complexity. J Risk Uncertain 51, 147–170 (2015). https://doi.org/10.1007/s11166-015-9226-3

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