The Troika paradox

doi:10.1016/j.econlet.2011.12.042

Economics Letters

Volume 115, Issue 2, May 2012, Pages 236-239

https://doi.org/10.1016/j.econlet.2011.12.042 Get rights and content

Abstract

In three binary choice problems, people reveal a choice pattern which falsifies expected utility theory and many generalized non-expected utility theories. This new paradox challenges popular non-expected utility models analogously to how the Allais paradox challenged neoclassical expected utility theory.

Highlights

► We present a new paradox for decision making under risk. ► The new paradox challenges many popular non-expected utility models. ► Experimental evidence on the new paradox is presented.

Section snippets

Falsified theories

Consider a binary lottery $L$ which yields outcome $x$ with probability $p \in [0, 1]$ and outcome $y$ with probability 1- $p$ . Suppose that a decision maker finds outcome $x$ to be at least as good as outcome $y$ . Let $u (x)$ and $u (y)$ denote the utility of outcome $x$ and outcome $y$ respectively. Many theories of decision under risk postulate that the utility of lottery $L$ is given by formula (1). $U (L) = u (x) \cdot w (p) + u (y) \cdot [1 - w (p)] .$

A non-decreasing function $w : [0, 1] \to [0, 1]$ satisfies $w (0) = 0$ , $w (1) = 1$ . This function differs across

Experiment

In the experiment, subjects faced three binary choice problems described in the introduction. The probability information was conveyed through a composition of red and black cards. Fig. 1 shows the first binary choice problem as it was displayed in the experiment (translated from German).

The experiment was conducted at the University of Innsbruck (Austria). Altogether, 35 undergraduate students took part in two experimental sessions, which were conducted on the same afternoon (October 14,

Conclusion

The paradox presented in this paper combines old empirical facts into a new puzzle. People simultaneously exhibit risk aversion when facing probable gains and risk seeking when facing small-probability gains. Friedman and Savage (1948) previously tried to explain this fact. It was also popularized by Tversky and Kahneman (1992) as an ingredient of the fourfold pattern of risk attitudes.

Equally well-known is another empirical fact. When facing lotteries of a similar expected value, people often

References (28)

Peter Fishburn
Nontransitive measurable utility
Journal of Mathematical Psychology
(1982)
Uday Karmarkar
Subjectively weighted utility: a descriptive extension of the expected utility model
Organizational Behavior and Human Performance
(1978)
David Bell
Regret in decision making under uncertainty
Operations Research
(1982)
Pavlo Blavatskyy
Reverse common ratio effect
Journal of Risk and Uncertainty
(2010)
Pavlo Blavatskyy
Modifying the mean-variance approach to avoid violations of stochastic dominance
Management Science
(2010)
David Butler et al.
Imprecision as an account of the preference reversal phenomenon
American Economic Review
(2007)
Soo Hong Chew
A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox
Econometrica
(1983)
Soo Hong Chew et al.
Mixture symmetry and quadratic utility
Econometrica
(1991)
Ward Edwards
The prediction of decisions among bets
Journal of Experimental Psychology
(1955)
Gustav Fechner
Elements of Psychophysics
(1860)

Milton Friedman et al.

The utility analysis of choices involving risk

Journal of Political Economy

(1948)

Faruk Gul

A theory of disappointment aversion

Econometrica

(1991)

Jagdish Handa

Risk, probability, and a new theory of cardinal utility

Journal of Political Economy

(1977)

David Harless et al.

The predictive utility of generalized expected utility theories

Econometrica

(1994)

Cited by (8)

Probability weighting and L-moments
2016, European Journal of Operational Research
Several popular generalizations of expected utility theory—cumulative prospect theory, rank-dependent utility and Yaari's dual model—allow for non-linear transformation of (de-)cumulative probabilities. This paper shows an unexpected connection between probability weighting and the statistical theory of L-moments. Specifically, cubic probability weighting results in a linear tradeoff between the expected value (the first L-moment), Gini (1912) mean difference statistic (the second L-moment, also known as L-scale) and the third L-moment (measuring skewness). Inverse S-shaped probability weighting function crossing the 45° line at a probability ≤0.5 reflects an aversion to the dispersion of outcomes and an attraction to positively skewed distributions.
Axiomatization of weighted (separable) utility
2014, Journal of Mathematical Economics
Citation Excerpt :
An improved understanding of the descriptive properties of weighted utility then calls for empirical tests of bi-separable transitivity, which are rather scarce in the literature (see, however Blavatskyy, 2012).
Nontrivial decision problems typically involve a trade-off among multiple attributes of choice options. One simple way of resolving such trade-offs is to aggregate multiple attributes into one real-valued index, known as weighted or separable utility. Applications of weighted utility can be found in choice under risk (expected utility) and uncertainty (subjective expected utility), intertemporal choice (discounted utility) and welfare economics (utilitarian social welfare function). This paper presents an alternative behavioral characterization (preference axiomatization) of weighted utility. It is shown that necessary and sufficient conditions for weighted utility are completeness, continuity, bi-separable transitivity (and transitivity if none of the attributes is null, or inessential).
Compound invariance implies prospect theory for simple prospects
2013, Journal of Mathematical Psychology
Citation Excerpt :
For example, Prelec’s (1998) characterization of compound invariance probability weighting under prospect theory becomes the same mathematical theorem as the classical characterizations of CARA (constant absolute risk averse, or linear–exponential) and CRRA (constant relative risk averse, or log-power) utility under expected utility. Other “identical” theorems concern conditionally invariant probability weighting (Prelec, 1998) and the constant relative decreasing impatient (renamed unit invariant in this paper) time discount functions of Bleichrodt, Rohde, and Wakker (2009) and Ebert and Prelec (2007). Thus we provide the simplest proofs of these results presently available in the literature, and we extend them to domains more general than considered before.
Behavioral conditions such as compound invariance for risky choice and constant decreasing relative impatience for intertemporal choice have surprising implications for the underlying decision model. They imply a multiplicative separability of outcomes and either probability or time. Hence the underlying model must be prospect theory or discounted utility on the domain of prospects with one nonzero outcome. We indicate implications for richer domains with multiple outcomes, and with both risk and time involved.
Basic geometric dispersion theory of decision making under risk: Asymmetric risk relativity, new predictions of empirical behaviors, and risk triad
2021, Decision Analysis
Out-of-Sample Predictions of CPT and Geometric Dispersion Theory of Descriptive Choice under Risk, Empirical Testbeds, Paradoxes, and Risk Triad
2018, SSRN
A synopsis of mean-multiplicative deviation and risk dispersion theory
2017, 67th Annual Conference and Expo of the Institute of Industrial Engineers 2017

View all citing articles on Scopus

View full text

The Troika paradox

Abstract

Highlights

Section snippets

Falsified theories

Experiment

Conclusion

Journal of Mathematical Psychology

Organizational Behavior and Human Performance

Regret in decision making under uncertainty

Operations Research

Reverse common ratio effect

Journal of Risk and Uncertainty

Modifying the mean-variance approach to avoid violations of stochastic dominance

Management Science

Imprecision as an account of the preference reversal phenomenon

American Economic Review

A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox

Econometrica

Mixture symmetry and quadratic utility

Econometrica

The prediction of decisions among bets

Journal of Experimental Psychology

Elements of Psychophysics

The utility analysis of choices involving risk

Journal of Political Economy

A theory of disappointment aversion

Econometrica

Risk, probability, and a new theory of cardinal utility

Journal of Political Economy

The predictive utility of generalized expected utility theories

Econometrica