1. Is this number how much I want to pay for / sell for this lottery?
“This is a valuation task, and you are asked to fill in how much is this lottery worth to you. Our payment procedure is designed to guarantee that it is for your best interest to fill in the exact valuation in your mind, which dominates both overstating and understating this value.” (Specifically avoid mentioning “buy” or “sell” in the explanation.)
2. In case the subject gives a valuation lower than the lower outcome in the lottery. “Sorry to interrupt. You can surely put whatever amount you see proper as your valuation. This is just a reminder, because here you put a valuation lower than the lower possible outcome in the lottery, and I want to clarify the rules in case there is any misunderstanding. Since we will only randomly draw a number from the lower outcome and the higher outcome of a given lottery, in this case X and Y (X<Y are the two outcomes of the lottery this subject is valuating), therefore giving a valuation lower than the lower outcome of the lottery means that all the random number we draw would be higher than your valuation and therefore you will be paid that amount. In the extreme case, if we draw X, your valuation indicates that you prefer to be paid X, rather than receiving this lottery that gives you at least X. Is this what you prefer?”
3. In case the subject gives a valuation higher than the higher outcome in the lottery. “Sorry to interrupt. You can surely put whatever amount you see proper as your valuation. This is just a reminder, because here you put a valuation higher than the higher possible outcome in the lottery, and I want to clarify the rules in case there is any misunderstanding. Since we will only randomly draw a number from the lower outcome and the higher outcome of a given lottery, in this case X and Y (X<Y are the two outcomes of the lottery this subject is valuating), therefore giving a valuation higher than the higher outcome of the lottery means that all the random number we draw would be lower than your valuation and therefore
you will receive the lottery. In the extreme case, if we draw Y, your valuation indicates that rather than receiving Y, you prefer to receive the lottery that gives you at most Y. Is this what you prefer?”
4. In the cash treatment, make sure the subject put all the notes and coins for valuation back to the box after finishing each valuation.
Chapter 6
Are Black Swans Really Ignored?
Re-examining Decisions from
Experience
6.1
Introduction
Studies of decisions from experience (henceforth, DFE) investigate decision situa- tions in which people rely on personal experiences when facing uncertainty. Decision makers often have no access to possible choice outcomes, let alone to the correspond- ing probabilities. Instead, they make decisions based on the past observations in their memory. DFE better captures real life decisions than traditional "Decisions from De- scription" (henceforth, DFD) where payoffs and probabilities are fully specified, which rarely happens the case in real life. In the usual sampling paradigm of DFE (Hertwig et al., 2004), subjects learn about unknown payoff distributions by drawing samples with replacement. With merely these cases in memory, they make their final decisions. Since Hertwig et al. (2004), an intriguing discrepancy between the two decision paradigms, which is called the DFE-DFD gap, has received plenty of attention. The common view in the DFE literature is that rare and extreme events, so called "black
swans", are underweighted under the DFE paradigm whereas they are overweighted under the DFD paradigm (for a review, see Hertwig and Erev (2009)). This implies a complete reversal of the inverse S-shaped probability weighting that has been docu- mented by many empirical studies under DFD (Abdellaoui, 2000; Bleichrodt and Pinto, 2000; Bruhin et al., 2010; Booij et al., 2010; Fehr-Duda et al., 2006; Gonzalez and Wu, 1999; Tversky and Kahneman, 1992; Wu and Gonzalez, 1996).
The DFE literature has indicated that the DFE-DFD gap is a robust empirical phenomenon. Although the under-sampling of rare events due to reliance on small samples mostly explains the early findings on the gap (Hadar and Fox, 2009; Fox and Hadar, 2006; Hertwig et al., 2004), later studies have shown that it does not provide a complete account (Barron and Ursino, 2013; Camilleri and Newell, 2009; Hau et al., 2010, 2008; Ungemach et al., 2009). Moreover, different attitudes towards risk (known probabilities in DFD) and ambiguity (partially unknown probabilities in DFE) are another cause of the gap (Abdellaoui et al., 2011; Kemel and Travers, 2015). Despite the robustness of the DFE-DFD gap, whether it can actually amount to a reversal - or only an attenuation - of the inverse S-shaped probability weighting is still unclear in the literature.
In addition to the sampling error and ambiguity, there are two extra confounds that render the inferences about probability weighting problematic in DFE studies. The first concerns an aggregation problem when there is a lack of control over the sampling experience of subjects. Because of the random nature of the sampling process - where the sampling is made with replacement and subjects decide when to stop sampling - each subject relies on her own distinct subjective experiences while making her choices in the sampling paradigm. Importantly, this heterogeneity at the individual level causes potential distortions at the aggregate level due to averaging artifacts (see Estes,1956; Estes,2002; Sidman,1952). We elaborate on this issue in the section of the DFE-DFD Gap.
The second confound concerns is regarding the role of utilities in the investigation of probability weighting. In proceeding studies of DFE, the underweighting of rare outcomes is typically inferred from the preference of sure gains over EV-equivalent lotteries with rare probability (for example, a preference of $1 for sure over a lottery
with 10% chance of winning $10 and $0 otherwise). It seems that the prevalent risk seeking for unlikely gains under DFD turns into an aversion under DFE (see also the review of the DFE literature by Rakow and Newell 2010). However, it is important to recognize that the aversion to unlikely gains may as well be due to concave utility (possibly coupled with an unbiased probability weighting) as it may be due to an underweighting of unlikely events.
This paper provides a reliable measurement of probability weighting under DFE by resolving the aforementioned problems. First, we used Barron and Ursino’s (2013) adjustment of the sampling paradigm to obtain a control over the sampling experience of each individual subject. Specifically, all of our subjects were required to carry out complete sampling from finite outcome distributions without replacement. Hence, they acquired the precise sampling information that matched with the objective probabilities without any sampling error or ambiguity1.
Next, rather than relying on indirect inferences, we measured probability weighting by a rigorous two-stage methodology (Abdellaoui, 2000; Bleichrodt and Pinto, 2000; Etchart-Vincent, 2004; Qiu and Steiger, 2010). In particular, controlling for the utility curvature in the first stage, each choice in the second stage exactly revealed over- weighting or underweighting of probabilities. Thus, our experimental setup enabled us to identify the direction and the magnitude of the deviations from expected utility (henceforth, EU), and hence find out what the exact DFE-DFD gap is.