Download data and study materials from OSF
University College London
Sample size: 1322
Field period: 10/31/2005-11/09/2005
n experiment by Tversky and Kahneman (1981) illustrates that people's tendency to evaluate risky decisions separately can lead them to choose combinations of choices that are first-order stochastically dominated by other available combinations. We investigate the generality of this effect both theoretically and experimentally. We show that for any decisionmaker who does not have constant-absolute-risk-averse preferences and who evaluates her decisions one by one, there exists a simple pair of independent binary decisions where the decisionmaker will make a dominated combination of choices. We also characterize, as a function of a person's preferences, the amount of money that she can lose due to a single mistake of this kind. The theory is accompanied by both a real-stakes laboratory experiment and a large-sample survey from the general U.S. population. Replicating Tversky and Kahneman's original experiment where decisionmakers with prototypical prospect-theory preferences will choose a dominated combination, we find that 28% of the participants do so. In the survey we ask the respondents about several hypothetical large-stakes choices, and find higher proportions of dominated choice combinations. A statistical model that estimates preferences from the survey results is best fit by assuming people have utility functions that are close to prospect-theory value functions and that about 83% of people bracket narrowly. None of these results varies strongly with the personal characteristics of participants. We also demonstrate directly that dominated choices are driven by narrow bracketing: when we eliminate the possibility of narrow bracketing by using a combined presentation of the decisions, the dominated choices are eliminated in the laboratory experiment and are greatly reduced in the survey.
- Do laboratory subjects give up money (in the first-oder-stochastic dominance sense) because they do not combine multiple choices into a single (global) choice problem?
- Do TESS respondents give up money due to this effect?
- We show theoretically that decisionmakers can incur losses if two conditions are met: first, their risk preferences are different from constant absolute risk aversion (CARA), and second, they do not integrate multiple choices into one global choice problem (although they should). This raise the question of 'whose preferences differ from CARA?' and 'who fails to integrate choices?' The possible answer categories are given by the personal characteristics of TESS respondents, as well as additional questions that we ask about their mathematical background.
- Do the losses go away if the questions are presented in a combined fashion, so that the integration is done automatically?
- The respondents were presented with a series of (hypothetical-large-scale-payoff) choices under risk. Pairs of choices could potentially yield a violation of first-order stochastic dominance, if the respondents chose one particular choice combination.
- One treatment variation was to present some pairs of choices as two choices to some respondents, and as one combined choice to other respondents
- Another variation was to include/not include other, additional choices, to see whether this inclusion had an effect on the violation rate in the original pair of choices.
The frequency of making dominated combinations of choices
In the laboratory experiment that replicates Tversky and Kahneman, 28% of subjects chose the dominated combination (out of 4) if payments were real and small scale. 34% chose it when payments were hypothetical and small scale, and 54% when they were hypothetical and large scale. Of the TESS respondents, 66% chose the dominated combination, with hypothetical and large-scale payments. In the three new examples, 40-50% chose the dominated combinations.
A statistical model estimation suggests that the degree to which respondents integrate their choices is small and indistinguishable for almost all subgroups. Men and women have somewhat different propensities to combine choices, as do white versus non-white respondents. (Men and non-whites integrate more.) But the differences either do not carry over to reliable differences in choices (men and women have the same violation rates overall) or are not robust to a different specification of the model. The estimated preferences are somewhat different between some subgroups: younger respondents, nonwhite respondents, high-income respondents and math-skilled respondents are more risk neutral. But except for white/nonwhite, this does not protect them from dominance violations.
When choices are presented as combined choices, the violation rates are strongly reduced in two cases, and less strongly reduced in one case.
The paper establishes that, under a very wide set of preferences, the failure to combine several decisions can lead a decisionmaker to make systematic mistakes in the form of dominated choices. The experiments show systematic choice patterns that confirm this prediction, and estimates of our statistical model suggest that narrow bracketing can organize the data far better then any broad-bracketing model where the choices are combined into global problems.
A methodological note concerns the question of how to devise empirical estimates of risk preferences. Narrow bracketing implies that empirical estimates of risk attitudes will vary widely with the assumptions about the scope of the decision problem that the agents face, and how well those assumptions match the way agents themselves isolate choices in their minds. The currently prevalent approach of researchers reporting estimates that can be varied by several orders of magnitudes by dint of imposing varied assumptions about the scope of decisionmakers' choices needs a substitute. Our statistical analysis demonstrates that it may be possible to include a simultaneous estimation of the agents' degrees of bracketing so as to add more discipline to the measurement of risk attitudes.
The original experiment by Tversky and Kahneman involved one risky choice where all payoffs were weakly positive and one where all payoffs weew weakly negative. It exploits that people are often risk seeking in the domain of losses, and risk averse in the domain of gains, as predicted by Prospect Theory. We introduce three additional examples of pairs of gambles, where other properties of Prospect Theory (and related models) are used or tested. Several of these gambles have payoffs both in the losses and in the gains. In each example, the decisionmaker has 2x2=4 combined choices available.
Rabin, M., and G. Weizsacker. 2009. "Narrow bracketing and dominated choices." The American Economic Review 99:1508-1543.