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Prospect theory and the “forgotten” fourfold pattern of risk preferences

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Abstract

Markowitz (Journal of Political Economy 60:151–158, 1952) identified a fourfold pattern of risk preferences in outcome magnitude: When outcomes are large, people are risk averse in gains and risk seeking in losses, but risk preferences reverse when the outcomes are small, with people exhibiting risk seeking in gains and risk aversion in losses. This fourfold pattern was not addressed by either version of prospect theory (Kahneman and Tversky Econometrica 47:363–391, 1979; Tversky and Kahneman Journal of Risk and Uncertainty 5:297–323, 1992). We show how prospect theory can accommodate the pattern by combining an overweighting of low probabilities with a decreasingly elastic value function. We then examine the performance of prospect theory with two decreasingly elastic value functions: Prospect theory performs better, both quantitatively and qualitatively, with a normalized logarithmic value function than with a normalized exponential value function. We discuss several issues, and speculate about why Tversky and Kahneman did not address Markowitz’s fourfold pattern.

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Notes

  1. For choices between a sure thing x and a gamble (x, p), original prospect theory (Kahneman and Tversky 1979) is equivalent to cumulative prospect theory (Tversky and Kahneman 1992).

  2. The normalized logarithmic value function over gains can be derived from Bernoulli’s (1738/1954; see also Schoemaker 1982) utility function u(x) = blog((a + x)/a), which accommodates both decreasing absolute risk aversion and increasing relative risk aversion over states of wealth. The normalized logarithmic utility function is obtained when a = b = 1/α, thus covering both attitudes toward risk with a single parameter.

  3. An alternative would be Prelec’s (1998) one-parameter probability-weighting function in the respective domains, w(p) = exp(−(−log(p)γ) if x ≥ 0 and w(p) = exp(−(−log(p)δ) if x ≤ 0, but this function performed worse on our data than Tversky and Kahneman’s (1992).

  4. An alternative would be a Fechnerian choice rule, in which choice probability derives from the difference between the values of the options, i.e., P = F((V(R) – V(S)/ε). Under the difference interpretation of Luce’s (1959) choice axiom, F is the logistic distribution function. This choice rule performed worse on our data than the ratio choice rule.

  5. When incorporating the power value function, prospect theory could not be estimated reliably.

  6. Mullet (1992) confirmed the multiplicative rule for participants rating the cash equivalent of gambles, but observed an additive rule for those rating the attractiveness of gambles.

  7. This stake-dependence of a power value function is only a crude measure of “decreasing elasticity” (not designated as such by the authors), amounting to a discontinuous value function, with constant elasticity at either side of the discontinuity.

  8. It should be recognized, however, that the stake-dependent probability-weighting function and the crude approximation of a decreasingly elastic value function addressed the same regularities in the data, and that they competed with one another in the same analysis. Alternatively, one might specify two alternative models, one with a constant-elasticity value function and a stake-dependent probability-weighting function, the other with a stake-independent probability-weighting function and a value function exhibiting continuously decreasing elasticity, and then compare the models on predictive accuracy.

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Acknowledgments

This work was supported by the Fundação para a Ciência e Tecnologia [project POCI 2010; grant numbers PTDC/PSI-PCO/101447/2008, PTDC/MHC-PCN/3805/2012], by the Economic and Social Research Council [grant number ES/K002201/1], and by the Leverhulme Trust [grant number RP2012-V-022]. Our article has greatly benefited from comments given by Peter Wakker.

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  1. European University, Quinta do Bom Nome, Estrada da Correia 53, 1500-210, Lisbon, Portugal

    Marc Scholten

  2. Warwick Business School, Scarman Road, Coventry, CV4 7AL, UK

    Daniel Read

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Correspondence to Marc Scholten.

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Scholten, M., Read, D. Prospect theory and the “forgotten” fourfold pattern of risk preferences. J Risk Uncertain 48, 67–83 (2014). https://doi.org/10.1007/s11166-014-9183-2

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