Evaluating Change in Objective Ambiguous Mortality Probability: Valuing Reduction in Ambiguity Size and Risk Level

Abstract

This article develops a valuation model to evaluate mortality probability changes under objective ambiguity, where multiple mortality probabilities are exogenously given. We construct the valuation model based on $\upalpha$-maxmin expected utility to evaluate the reduction in ambiguity size and risk level as well as estimate the subjects’ ambiguity attitude. Our model can bring an interesting policy implication. If the subjects are ambiguity-averse, a reduction in ambiguity size with a constant risk level can increases welfare. Thus, even if risk level cannot be lowered, reduction in ambiguity size can also be used as a measure toward increasing welfare. Our model can empirically estimate this welfare change resulting from the reduction in ambiguity size. To demonstrate how our model works, we apply it to survey data on a public program that reduces mortality probability in accidents caused by wildlife.

Appendices

Appendix 1

The ambiguity attitude parameter, ${\alpha }$, can be intuitively interpreted by considering the familiar Ellsberg’s (1961) urn example. An urn contains 30 red balls and 60 green and blue balls in unspecified proportions. The subjects are asked to compare between bets on a red and a green ball. If they win (or lose), they get $100 (or $0). According to many experimental results, most people choose to bet on red, because they dislike an ambiguous situation but favor certain probability. Let the subjects be $\upalpha$-MEU maximizers. Then, the expected utility of a bet on a red ball is $\frac{1}{3}{u}\left( \$ 100 \right) +\frac{2}{3}{u}(\$ 0)$, while that on a green ball is

$$\begin{aligned}&{\alpha }\; {\min }_{{p}\in \left[ {0,\frac{2}{3}} \right] } \left\{ {{pu}\left( {\$ 100} \right) +\left( {1-{p}} \right) {u}\left( {\$ 0} \right) } \right\} \nonumber \\&\qquad +\left( {1-{\alpha }} \right) {\max }_{{p}\in \left[ {0,\frac{2}{3}} \right] } \left\{ {{pu}\left( {\$ 100} \right) +\left( {1-{p}} \right) {u}\left( {\$ 0} \right) } \right\} , \end{aligned}$$

which is equivalent to $\frac{2}{3}\left( {1-{\alpha }} \right) {u}\left( {\$ 100} \right) +\left( {\frac{1}{3}+\frac{2}{3}{\alpha }} \right) {u}\left( {\$ 0} \right)$. Thus, in this example, for them to bet on a red ball (i.e., have an ambiguity-averse attitude), ${\alpha }$ has to be greater than $\frac{1}{2}$.

Appendix 2

Here, we show the detailed derivation of Eq. (9).

$$\begin{aligned} \Pr \left( {{A}_{ i} } \right)&= \Pr \left( {\textit{EU}^{\alpha }\left( {{p}_{1{i}}^{ h} ,{p}_{1{i}}^{ l}, {y}_{ i} -{t}_{ i} ,{\alpha }} \right) >\textit{EU}^{\alpha }\left( {{p}_{0{ i}}^{ h}, {p}_{0{ i}}^{ l} ,{y}_{ i}, {\alpha }} \right) } \right) \\&= \Pr \left( {\textit{EU}\left( {{p}_{1{i}}^{\alpha } ,{y}_{ i} -{t}_{ i} } \right) >\textit{EU}\left( {{p}_{0{ i}}^{\alpha }, {y}_{ i} } \right) } \right) \\&= \Pr \left( \beta {\ln }\left( {{y}_{ i} -{t}_{ i} } \right) +\theta {p}_{1{i}}^{\alpha } {\ln }\left( {{y}_{ i} -{t}_{ i} } \right) +{c}_1 +{\epsilon }_{1{i}} >\beta \ln \left( {{y}_{ i} } \right) \right. \nonumber \\&\left. \quad +\theta {p}_{0{ i}}^{\alpha } {\ln }\left( {{y}_{ i} } \right) +{c}_0 +{\epsilon }_{0{ i}} \right) \\&= \Pr \left( {\epsilon }_{0{ i}} -{\epsilon }_{1{i}} <\beta \ln \left( {\frac{{y}_{ i} -{t}_{ i} }{{y}_{ i} }} \right) +{\theta }\left( {\alpha {p}}_{1{i}}^{ h} +\left( {1-{\alpha }} \right) {p}_{1{i}}^{ l} \right) {\ln }\left( {{y}_{ i} -{t}_{ i} } \right) \right. \\&\quad \left. -{\theta }\left( {{\alpha {p}}_{0{ i}}^{ h} +\left( {1-{\alpha }} \right) {p}_{0{ i}}^{ l} } \right) {\ln }\left( {{y}_{ i} } \right) +{c}_1 -{c}_0 \right) \\&= \Pr \left( {\epsilon }_{0{ i}} -{\epsilon }_{1{i}} <\beta \ln \left( {\frac{{y}_{ i} -{t}_{ i} }{{y}_{ i} }} \right) +\theta \alpha \ln \left( {\frac{\left( {{y}_{ i} -{t}_{ i} } \right) ^{{ p}_{1{i}}^{ h} }}{{y}_{ i}^{{ p}_{0{ i}}^{ h} } }} \right) \right. \\&\quad \left. +{\theta }\left( {1-{\alpha }} \right) \ln \left( {\frac{\left( {{y}_{ i} -{t}_{ i} } \right) ^{{ p}_{1{i}}^{ l} }}{{y}_{ i}^{{ p}_{0{ i}}^{ l} } }} \right) +{c}_1 -{c}_0 \right) \\&= \Pr \left( {\epsilon }_{ i} <\beta \ln \left( {\frac{{y}_{ i} -{t}_{ i} }{{y}_{ i} }} \right) +\theta \alpha \ln \left( {\frac{\left( {{y}_{ i} -{t}_{ i} } \right) ^{{ p}_{1{i}}^{ h} }}{{y}_{ i}^{{ p}_{0{ i}}^{ h} } }} \right) \right. \\&\quad \left. +{\theta }\left( {1-{\alpha }} \right) \ln \left( {\frac{\left( {{y}_{ i} -{t}_{ i} } \right) ^{{ p}_{1{i}}^{ l} }}{{y}_{ i}^{{ p}_{0{ i}}^{ l} } }} \right) +{c} \right) , \end{aligned}$$

where ${\epsilon }_{i} \equiv {\epsilon }_{0{i}} -{\epsilon }_{1{i}} ,{c}\equiv {c}_1 -{c}_0$. The second, third, and fourth equalities hold—the second because of Eq. (2), the third because of Eq. (8), and the fourth because of the definition of ${p}_{j}^{\alpha }$.