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Theory and Decision

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01.08.2014

Risk behavior for gain, loss, and mixed prospects

verfasst von: Peter Brooks, Simon Peters, Horst Zank

Erschienen in: Theory and Decision | Ausgabe 2/2014

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Abstract

This study extends experimental tests of (cumulative) prospect theory (PT) over prospects with more than three outcomes and tests second-order stochastic dominance principles (Levy and Levy, Management Science 48:1334–1349, 2002; Baucells and Heukamp, Management Science 52:1409–1423, 2006). It considers choice behavior of people facing prospects of three different types: gain prospects (losing is not possible), loss prospects (gaining is not possible), and mixed prospects (both gaining and losing are possible). The data supports the distinction of risk behavior into these three categories of prospects, Further, probability weighting and diminishing sensitivity of utility as predicted by PT are observed. Loss aversion is, however, less pronounced, except for choices where one prospect is degenerate. The data suggests that the probability of losing may be relevant for loss aversion.

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Experimental analyses for dominance criteria of higher order are discussed elsewhere (e.g., Deck and Schlesinger 2010). The independence axiom of expected utility entails itself a dominance principle that is appealing: first-order stochastic dominance (FSD) requires that the prospect should be preferred that has higher decumulative probability among all outcomes than a second prospect. FSD is a simple criterion, people agree with this principle and apply it in simple situations. However, in nontransparent choice situations many violate this dominance criterion; see Birnbaum and Navarrete (1998) and Birnbaum (2005) for experimental evidence. The aforementioned biases have little influence on FSD.

See also Hogarth and Einhorn (1990), Tversky and Kahneman (1992), Wu and Gonzalez (1996), Gonzalez and Wu (1999), Abdellaoui (2000), Bleichrodt and Pinto (2000), Bleichrodt et al. (2001), Etchart-Vincent (2004), Abdellaoui et al. (2005), and Abdellaoui et al. (2011).

As degenerate lotteries are identified with the corresponding outcome, the prospect \((p_{1:}0,\ldots ,p_{n}:0)\), that has no gains and no losses, is identified with the reference outcome zero.

See Tversky and Kahneman (1992), Goldstein and Einhorn (1987), Prelec (1998), Diecidue et al. (2009), or Abdellaoui et al. (2010) for parametric specifications incorporating optimism, insensitivity, and pessimism.

In the experiment we use prospects in which each outcome has probability \( 1/5 \); for prospects with equal outcomes the latter are displayed with the coalesced probabilities, e.g., two 1/5 chances of obtaining £5 are presented as a single 2/5 chance for £5.

As we restrict attention to prospects that have the same mean this definition makes sense. SSD is, in general, also defined for prospects without equal means. The more general definition implies our definition used here but the reverse implication does not hold.

The case \(c=d\) occurs naturally in the parametric probability weighting functions of Goldstein and Einhorn (1987), Tversky and Kahneman (1992), Lattimore et al. (1992), Prelec (1998), and Diecidue et al. (2009).

Based on the aggregate data of Abdellaoui (2000) the decision weights of 0.33 (0.25) are approximately 0.33 for gains and 0.35 for losses (0.29 for gains and 0.29 for losses) for the one-parameter Tversky and Kahneman (1992) probability weighting function. For the two-parameter Goldstein and Einhorn (1987) probability weighting function we have \( w^{+}(0.33)\approx 0.30\) and \(w^{-}(0.33)\approx 0.35\) (\(w^{+}(0.25)\approx 0.3\) and \(w^{-}(0.25)\approx 0.29\))

See Appendix 1.

See Cohen et al. (1985), Camerer (1989), Battalio et al. (1990), Harless (1992), Harless and Camerer (1994), Myagkov and Plott (1997), Di Mauro and Maffioletti (2002), Smith et al. (2002), Mason et al. (2005), and Brooks and Zank (2005).

The descriptions “safer” or “riskier” are short for “safer in the SSD-sense” or “riskier in the SSD-sense,” respectively.

We have also used the parameters estimates of Abdellaoui (2000) for the two-parameter probability weighting function of Goldstein and Einhorn (1987). These give similar predictions.

Adding the data for the mixed condition and a parameter, \(\lambda \), for LA gives similar estimates. \(\lambda =0.93\) (\(\mathrm{SE}=0.059\)), is found insignificantly different from one.

The estimated value for the slope of the regression line, \(\hat{\beta }=1.3\), has a standard error of 0.1732. This results in a test statistic value of 7.506 while the upper \(5\,\%\) critical value of the \(t_{1}\) distribution is 6.314. The intercept for this regression line takes the value of 37 with a standard error of 1.8707.

Because very few subjects were MWSD for gains and for losses we skip the corresponding table.

We observe that this test is equivalent to verifying if common outcomes being gains or being losses matters for choice behavior

Another explanation could be that the high number tasks involving mixed prospects and loss prospects may have generated pessimism about gaining any amount of money out of this experiment. This would explain why most subjects were not willing to pay a large proportion of their earnings from the experiment to participate again in the same study (a finding which can be interpretted as a form of loss aversion). 82 subjects provided us with such information: on average, those who lost from their fixed payment (31 subjects earned £6.64 on average) were willing to pay \(44, 66\,\%\) of their ernings; those who gained (27 subjects earned £24.22 on average) were willing to pay \(23.85\,\%\) of their earnings, and those who neither gained nor lost (24 subjects received £17) were willing to pay \(24.5\,\%\) of their earnings to repeat the experiment.

Recall that the expected pay from the experiment was £17, while the minimum one can ensure is £2. Thus, it may well be, that the frequent reoccurence of tasks with potential losses has induced many subjects to exhibit more risk neutral behavior in the SSD-sense.

However, this result is obtained using a valuation task (i.e., assessing the value of an experiment where losing is likely) which may trigger additional LA.

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Titel: Risk behavior for gain, loss, and mixed prospects
verfasst von: Peter Brooks
Simon Peters
Horst Zank
Publikationsdatum: 01.08.2014
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 2/2014
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-013-9396-x

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