2.1.1 Decision task

The design was based on that of Abdellaoui et al. (Reference Abdellaoui, Baillon, Placido and Wakker2011). In each round, the participants were asked to evaluate lotteries involving drawing from virtual, Ellsberg-like urns. Two types of urns were used: the known and the unknown. The known urn always contained one ball of each of eight colors. In the unknown urn there were also eight colored balls, but participants did not know the particular composition. For example, there could be three blue balls, zero green balls, etc. In each case each ball had the same probability to be drawn. The same was true of each color in the known urns only.

At the beginning of each round a clock would start counting down from 60 s. Participants saw the graphical representation of the urn (known or unknown) and the information how much money they could win when particular color was drawn, see Fig. 1. They were asked to type in the amount they considered just as good as the lottery. They did so twice in each round. Upon confirming their initial (and typically rapid) choice, participants saw their selected amount displayed below the picture and were invited to rethink it and amend it or type in the same one again if they were sure it correctly represented their preference. Thus, each participant was allowed to change his or her mind at most once per round. Once the second amount (the same as the first one or a different one) was typed in and confirmed or when the time was gone, the participant could move on to the second round.

Fig. 1

Decision screen

At the end of the experiment the computer randomly selected: one round, one second of this round (1–60) and an amount of money X that could be obtained for sure, drawn from the range of possible payoffs in this round (e.g., from 40 to 100 in Fig. 1) to determine participants’ payoffs. If the participant in the selected second of the round had been indicating that her certainty equivalent of the lottery was lower than X, she received X. If she indicated it was higher than X, the lottery was played out and she received one of the possible outcomes (a Becker-DeGroot-Marschak procedure). In the case a second was selected in which no decision had been made yet, the lottery or X would be assigned randomly. Thereby participants were motivated to make their first decision as quick as possible and then to indicate if they changed their mind after consideration.

For example, assume that the computer drew round 1, in which the participant could win 100 PLN with probability 4/8, or 40 PLN otherwise, as shown in Fig. 1. Furthermore, the 25th second of this round was selected and sure amount X equal to 70. Suppose that in the fifth second of Round 1 the participant had evaluated the lottery at 64 and then she amended it to 68 in the 21st second. Because 68 is smaller than X = 70, in the 25th second of the round she was revealing that she liked X better than the lottery. She would thus receive 70 for sure. Now suppose she had typed in 72 in the third second and only changed it to 65 in the 30th second. Then, in the 25th second she was indicating that she liked the lottery better, so the lottery would be played out. In such a case, she would win 100 PLN with probability 4/8 and 40 PLN otherwise.

This double-response method has previously been applied by Krawczyk and Sylwestrzak (Reference Krawczyk and Sylwestrzak2018) to binary choices aimed at eliciting social preference. A closely related design was also pursued by Agranov et al. (Reference Agranov, Caplin and Tergiman2015) investigating a guessing game and Kessler et al. (Reference Kessler, Kivimaki and Niederle2017), who looked at modified binary dictator games and prisoner’s dilemmas. Beyond the domain studied, there are two main differences. First, in these studies, participants could change their opinion many times, not just once. In the study by Agranov et al. (Reference Agranov, Caplin and Tergiman2015) this in some cases led to seemingly nonsensically large number of changes; further, Kessler et al. (Reference Kessler, Kivimaki and Niederle2017) ended up reporting the initial and the final choices only. This suggests that allowing just dual response, naturally corresponding to the dual-system theories, may be a reasonable design choice. Second, compared to these studies, we only mildly punished indecision: as mentioned before, one of the options was implemented randomly, whereas in the case of Agranov et al. (Reference Agranov, Caplin and Tergiman2015) and Kessler et al. (Reference Kessler, Kivimaki and Niederle2017), participants earned zero if a second was randomly selected in which there was no decision yet. This design choice is related to the first difference: the possibility that some participants make an instant; essentially random initial choice out of fear of receiving nothing has more bearing on the data if only one decision change is allowed.

On top of the double-response treatment described above, a no time pressure treatment was additionally run, in which participants took as much time as they wanted to make a decision. Only the final decision mattered (and was incentivized using an analogous Becker-DeGroot-Marschak procedure). This treatment obviously provided less useful evidence on the effect of deliberation and was, therefore, run with a smaller group of participants only. Its main purpose was to verify if final decisions of the double-response treatment are similar in terms of timing and contents to decisions taken under no time pressure at all.

At the end of the experiment participants received information on the round, second of the round and amount of money X that would determine their payoff. Then they completed Frederick’s (Reference Frederick2005) cognitive reflection test (CRT) and a short post-experimental questionnaire (sex, age and field and year of study). These are very briefly discussed in the online appendix, which can also be consulted for the detailed description of the procedure.

2.1.2 Decision task parameters; estimation of probability weights; hypotheses

In total, participants made their decisions in 32 paid rounds, preceded by two practice rounds, with possible payoffs ranging from 0 PLN to 100 PLN (ca. 24 euro), see table C1 in the online appendix for the parameters of all lotteries. Some of the rounds were identical up to the coloring of the winning balls, allowing assessment of consistency in participants’ choices. Half of the participants played the 13 rounds with known urns first and then 19 rounds with unknown urns (we will refer to this condition as “Known First”), whereas the order was reversed for the other half (“Unknown First”).

It can be noted that some of the tasks involved 50/50 chances to get different non-zero rewards. Following the semi-parametric method of Abdellaoui et al. (Reference Abdellaoui, Baillon, Placido and Wakker2011), the certainty equivalents from these rounds were used to estimate the parameter of power value function, with the weight of 0.5 as an additional parameter. Non-linear least squares method was applied. Resulting values of each possible reward, together with certainty equivalents provided in remaining rounds (involving probabilities other than 50/50 for 100 PLN and zero otherwise) were subsequently used to calculate weights for these probabilities. Indeed, if for some $j\in \left(1,2,3,5,6,7,8\right)$ we denote the amount reported to be as good as 100 PLN with probability j/8 by CE (standing for the certainty equivalent), we conclude that w(j/8) = (CE/100)^ρ, where ρ is the individual parameter of the power value function. The seven probability weights can then be used to estimate parameters of Prelec’s (Reference Prelec1998) two-parameter probability weighting function,

$w\left(p\right)=exp\left(-\beta \left(-ln\left(p\right){)}^{\alpha }\right)\right).$

Objective probabilities, w(p) = p results from $\alpha =\beta =1$ . Lower values of α (often reported in previous studies) correspond to low sensitivity to changes in probability away from the reference points of absolute impossibility and absolute certainty (inverse-S). Higher values of α would signify lower sensitivity near these thresholds. Low values of β characterize elevated probability weighting curves (optimism) while high β means pessimism. This procedure was applied separately to each participant’s choices under specific conditions (initial choices for known urns, final choices for known urns, initial choices for unknown urns, final choices for unknown urns). As a result we have four separate sets of estimates of r, w(p)’s (for each $j\in \left(1,\dots ,8\right)$ ), α, and β for each participant making decisions under double-response and two for each participant making decisions under no time pressure.

Analogously, we consider an alternative specification, under which the probability weights are subjected to simple linear regression on the unit interval, $w\left(p\right)=c+sp$ . Following Abdellaoui et al., we focus on $a=1-s$ as an index of likelihood insensitivity and $b=1-s-2c$ (which is the difference between the “dual” intercept arising when we flip the picture 180° and the standard intercept) as an index of pessimism.

In view of existing literature, we hypothesize that subjects will initially focus predominantly on the (most salient) lowest possible outcome (the lower outcome of the lottery), especially when probabilities are not explicitly given, only considering other variables later. In accordance with the attentional drift–diffusion model (which has been shown to correctly explain choices and decision times in other domains, see Krajbich et al. Reference Krajbich, Lu, Camerer and Rangel2012) subjects will tend to accumulate evidence in favor of the option currently being the focus of their attention. In the context of our experiment this means that the initial choices will be more risk-avoiding (high b and β) and show less sensitivity to the probability (low a/α) than final choices, especially in the case of unknown urns.