Choice model estimates

We estimated the model separately for the four different treatments, with the same priors on the hyper parametersp(φ), ensuring that any difference in the posteriors derive from the data alone.

Our model is highly non-linear in parameters and this makes it necessary to formulate weakly informative and regularizing priors to ensure that NUTS will converge. We use symmetric normal priors on theµ_k’s, and positive half-Cauchy for the standard deviations:

µ_k∼N(0,1), fork∈ {α,β,ρ}, (13)

µ_λ ∼N(0.01,1), (14)

σ_k∼Cauchy₊(0,1), fork∈ {α,β,ρ,λ}. (15) With 3 chains of 3000 warmup iterations and 3000 sampling iterations, we evaluated con-vergence with visual inspection of the sampling trace, the number of effective samples and the R-statistic of Gelman and Rubin (1992). For one treatment (now), the chains did not seem tob converge properly, and for this treatment we re-ran estimation with 6 chains, 6000 warmup iter-ations and 6000 sampling iteriter-ations and then the chains passed our tests. In Figure 5 we present the marginal distribution of the samples of the hyper parameter posteriors for the probability weighting functions and the implied expectations forα andβ.⁸

Looking across the columns of Figure 5, there are no striking treatment differences, and most parameters seem reasonably precisely estimated. One exception is that the σ_β values in thenowtreatment are higher than those in the other treatments (andP(σ_β^now>σ_β^short) =0.99).

More disturbing is the high level of all the σ_β’s. While it is not straightforward to interpret what the implications are of the estimated parameters, we can sample(α,β)using the median estimates of (µ_α,µ_β,σ_α,σ_β) and then examine the implied probability weighting functions.

Figure 6 shows 500 samples of the weighting functions (and the weighting function of a par-ticipant with median parameter values). There is a strikingly large heterogeneity in weighting functions. A substantial proportion seems to put almost no positive weight on probabilities at all, which implies that they evaluate lotteries by their worst possible outcome. This is one way that the model can account for the proportion of people that never chose the lottery in any of their 28 situations. The median parameter values imply a moderate under-weighting of all probabilities large enough to be a feature of the lotteries we exposed our participants to (in our lotteries, probabilities started at 1/6).

Turning to Figure 7 and the marginal posterior distributions of(µ_ρ,µ_λ,σ_ρ,σ_λ), one thing we can note about the distribution of α is that the median participant in treatments now and shortseem to be approximately risk neutral (with posteriorsµ_α centered around 0), with point estimates of median ρ at 1.03 in both thenowand the short treatment. However, the median participant in thelongandnevertreatments seem to be somewhat risk seeking, with medianρ’s of 1.15 and 1.16 in thelongandnevertreatments respectively. Considering the narrowness of the posteriors, we have some confidence in this conclusion, the posteriors imply thatP(µ_ρ^never>

µ_ρ^now) =0.95,P(µ_ρ^never>µ_ρ^short) =0.97,P(µ_ρ^long>µ_ρ^now) =0.94, andP(µ_ρ^long>µ_ρ^now) =0.96.

The differences are not quite as stark in the meanρ’s, reflecting the role of heterogeneities.

The estimates of µ_λ, reflecting the impact of random utility on the median participant are for the most part centered around−1.75 (somewhat higher in theshort treatment). This esti-mate implies that if two alternatives have monetary worth 10 NOK apart (aboutAC1), the most

7See e.g Gelman, Carlin, Stern, Dunson, Vehtari, and Rubin (2013) for a text book treatment or Nilsson, Rieskamp, and Wagenmakers (2011) for a more complete discussion in the context of choices under risk.

8With logz∼N(µ,σ²), the expectation is calculated asE[z] =exp(µ+σ²/2).

0.00 0.25 0.50 0.75 1.00

0.00 0.17 0.33 0.50 0.67 0.83 1.00

Probability

Probability weight (treatment "short")

Figure 6: Probability weighting functions

Note: The graph shows 500 different probability weighting functions. For illustration, parameters are drawn from theshortprior of(α,β)given hyper parameters that are medians from the posterior of(µα,µ_β,σα,σ_β). Also shown is the 45-degree line and the weighting function implied by the median parameter values (in bold). The probabilities of the lotteries used in the experiment are indicated on the first axis.

valuable alternative will be chosen with probability Λ(exp(−1.75)·10) =0.85, which seems like a reasonable amount of random utility. If we instead look at the posterior formeanλ, we see some disturbingly large values. This reflects the influence of a long tail of the log normal distribution, which drives the choice probabilities to 1 for a segment of the population. Work-ing together with how some participants underweight probabilities in the extreme (as shown in Figure 6), this allows the model to predict the fairly large proportion of the participants who never chose to take the lottery (as shown in Figure 4).

5.4 Discussion

In our pre-analysis plan we registered five main hypotheses.⁹ In this section we relate our pre-specified hypotheses to the evidence from Section 5.1 and Section 5.2. There are few papers that provide relevant information about risk taking on behalf of others, so our hypothesis were to a large degree based on what others have found studying risk taking on behalf of self.

Our first hypothesis was that decision makers do not distinguish between no delay (the nowtreatment) and a short delay (theshorttreatment). We intended this hypothesis to mainly concern the mode informing participants about the outcome. A one-week delay is about as

9Our study was registered at the American Economic Association’s registry for randomized controlled trials (RCT ID: AEARCTR-0004403).

rho_murho_sigmamean_rholambda_mulambda_sigmamean_lambda

now short long never

−0.20.00.20.60.70.80.91.01.251.501.752.00−2.0−1.51.52.02.53.005101520

0.0

0.2

0.4

0.6 0.00.20.40.6 0.00.20.40.6 0.00.20.40.6 Parameter value

Fraction of sim ulated posterior

Figure7:Posteriordistributionofhyper-parametersforriskaversionandrandomutility Note:Marginaldistributionsofthe(µρ,µλ,σρ,σλ)parametersforeachofthefourtreatments(fromthesameestimationrunastheposteriorsinFigure5)Column3and6 presenttheimpliedposteriorforE[ρ]andE[λ].

short as practically possible while working with a survey provider. It is, however, contrary to Zimmermann (2015) who found that even a two-day delay is sufficient to encourage more risk taking for self. The reduced form evidence of Section 4 certainly do not provide any evidence against this hypothesis. The choice model also do not provide evidence of any difference in risk aversion, but we do find that in thenowtreatment there is more heterogeneity inβ, one of the weighting function parameters.

Our second hypothesis, based on previous evidence on risk-taking on behalf of self, was that there is more risk taking with a long than with a short delay. This would be in line with evidence on risk taking for self (Shelley, 1994; ¨Onc¸¨uler, 2000; Noussair and Wu, 2006; Abdellaoui et al., 2011; Onay, La-Ornual, and ¨Onc¸¨uler, 2013; Savadori and Mittone, 2015), but most of these also conflate the timing of information with the timing of payment. Our reduced form evidence provides evidence against this, with a precisely estimated null effect. The choice model estimates do suggest that the median participant in thelongtreatment is somewhat risk seeking, in contrast to the risk neutrality of the median participant we find in thenowandshort treatments, but this difference is not sufficiently strong to have effects on average risk taking (it is counteracted by a number of smaller differences in other parameters). We also formulated an hypothesis to follow up, that differences in risk taking between short and long delays are due primarily to differences in probability weighting rather than differences in risk aversion (as found for decision taking on own behalf by Abdellaoui et al. (2011)). The choice model estimates clearly contradict this hypothesis.

Our final two hypotheses were competing predictions about how decisions with no revela-tion of uncertainty would be closer to either no delay at all (hypothesis four), or to an imaginary extension of delay into eternity which would be closer to a long delay (hypothesis five). These hypotheses concern specific features of decision taking on behalf of others; previous studies examining decision taking for self obviously cannot hide the information completely since ex-perimenters eventually have to pay participants. Our reduced form evidence does not speak to this hypothesis, since we did not find any differences in average risk taking between treatments.

To the extent that we find any systematic differences in the choice model estimates, with re-spect to theρ parameter for risk aversion we find that thenevertreatment is closer to thelong treatment, providing some evidence for hypothesis five and against hypothesis four.