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Theory and Decision

Erschienen in:

01.12.2014

Probabilistic risk attitudes and local risk aversion: a paradox

verfasst von: Vjollca Sadiraj

Erschienen in: Theory and Decision | Ausgabe 4/2014

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Abstract

Prominent theories of decision under risk that challenge expected utility theory model risk attitudes at least partly with transformation of probabilities. This paper shows how attributing local risk aversion to attitudes towards probabilities can produce extreme probability distortions that imply paradoxical risk aversion.

Vorheriger Artikel Introduction to FUR XV special issue

Nächster Artikel The role of intuition and reasoning in driving aversion to risk and ambiguity

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Formally, urn $R_{g}$ is prospect $\{\$ 900,0;p,1-p\}, p=(g+5)/100$ and urn ${S}_{g}$ is prospect $\{\$ 900,\$ 200,0;p-1/2n,1/n,1-p-1/2n\}$, $n = $10. All prospects R and S considered in this paper are of these types.

The supposition that the safer prospect is weakly preferred to the riskier prospect for all $g$ from {0, 5,..., 90} is made here for simplicity of exposition. Section 3 provides general results for cases when the weak preference for the safer urn is observed only for some subset of {0,5,..., 90} and strict preference for the riskier urn is observed outside the subset. Note that the latter preferences cannot be rationalized by expected utility.

For a power utility specification, $x^{r}$ the supposition $q>1$ is equivalent to $r>0.461,$ which is plausible (Harrison and Rutström 2008). For smaller values of $r$, one can construct an example with other prizes. For example, for $r= 0.3$, the high, middle, and the low prizes that satisfy condition $q>1$ are $900, $300, and $100.

To be able to construct numerical illustrations for implications of inequality (1), I need to choose a value for $q$. Prizes used in the example, 900 and 200, are within the range of payoffs in Abdellaoui et al. (2008) study. In my numerical illustrations I use 2 as a lower bound of $q$ since the value of 900 being at least three times the value of 200 is satisfied for the power estimates in their study. Their reported estimated (mean) power exponent on the gain domain is 0.86.

The supposition $q>1$ requires that the high prize of €40 is valued at least twice as much as €10, which is arguably plausible; for a power utility specification, $x^{r}$ this supposition is equivalent with $r>0.5.$

For $q>1, \kappa (q,k^{*}-k,k-k_*)$ approaches infinity as $k^{*}-k$ (see lemma 5.1 in the appendix). For given probabilities $p$ and $p^{*}$, verify that $k^{*}-k\ge 2n(p^{*}-p)-1;$ so the larger the value of $n$, the larger the value of $\kappa (.).$

Let 0.5M denote $500,000 and use subadditivity of $v(.)$ and $f(0.75)-f(0.5)\le 0.00004(1-f(0.5))$ to verify that $v(0.5M)(f(0.75)-f(0.5))\le 25000v(20)\left( {f(0.75)-f(0.5)} \right) <v(20)(1-f(0.5)).$ Rearrange terms in the last inequality to get $v(0.5M)f(0.75)<v(0.5M)f(0.5)+v(20)(1-f(0.5));$ hence, $0.5 million or $20 with equal probability is preferred over $0.5 million or $0 with probabilities 3/4 and 1/4.

Abdellaoui, M., Bleichrodt, H., & L’Haridon, O. (2008). A tractable method to measure utility and loss aversion under prospect theory. Journal of Risk and Uncertainty, 36, 245–266.CrossRef

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Harrison, G. W., & Rutström, E. E. (2008). Risk aversion in the laboratory. In J. C. Cox & G. W. Harrison (Eds.), Risk aversion in experiments. Research in experimental economics (Vol. 12). Greenwich, CT: JAI Press.

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Titel: Probabilistic risk attitudes and local risk aversion: a paradox
verfasst von: Vjollca Sadiraj
Publikationsdatum: 01.12.2014
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 4/2014
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-013-9410-3

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