Some implications of common consequences in lotteries

Abstract

This work studies the implications of some aspects of preferences toward risk in the choice between two binary lotteries exhibiting a common consequence. The results obtained are then applied to two different problems: the choice between two risky challenges characterized by different rewards in the case of success and different probabilities of success and the choice between self-protection and self-insurance in the presence of the risk of incurring financial loss.

Appendices

Appendix A

Lemma 1

Given the two non-negative random variables Xand Ysatisfying Y_FSD ≥ Xand$E[X]<E[Y]<+\infty$, let Zbe the random variable with probability density function

$$ f_{Z}(t)=\frac{\bar{F}_{Y}(t)-\bar{F}_{X}(t)}{E[Y]-E[X]} $$

with t ≥ 0 and where$\bar {F}_{X}$and$\bar {F}_{Y}$are the survival functions of Xand Y. If g(.) is a measurable and differentiable function such that E[g(X)] and E[g(Y )] are finite then E[g^′(Z)] is finite and

$$ E[g(Y)]-E[g(X)]=E[g'(Z)][E[Y]-E[X]]$$

Proof

See Di Crescenzo (1999), Proposition 3.1 and Theorem 4.1 . □

Appendix B

We prove here that $\tilde {z}$ dominates $\tilde {s}$ via first order stochastic dominance. Given the functions $F_{\tilde {z}}(.)$ and $F_{\tilde {s}}(.)$ which are the cumulative distribution functions of $\tilde {s}$ and $\tilde {z}$, $\tilde {z}$ dominates $\tilde {s}$ via first order stochastic dominance if $G(j)=F_{\tilde {s}}(j)-F_{\tilde {z}}(j)\geq 0$ for every j with a strict inequality for some j. Also note that $\frac {dG}{dj}=f_{\tilde {s}}(j)-f_{\tilde {z}}(j)$ where $f_{\tilde {s}}(j)$ and $f_{\tilde {z}}(j)$ are the density functions of $\tilde {s}$ and $\tilde {z}$ respectively. By applying Proposition 3.1 in Di Crescenzo 1999 (see again Appendix A), we obtain:

$$ f_{\tilde{z}}(j)= \left\{\begin{array}{ll} 0 & k_{0}>j\\ \frac{1-\bar{F}_{U(\tilde{x}_{1})}(j)}{E[U(\tilde{y}_{1})]-E[U(\tilde{x}_{1}])} & k_{1}>j\geq k_{0} \\ \frac{1}{E[U(\tilde{y}_{1})]-E[U(\tilde{x}_{1}])} & k_{2}>j\geq k_{1}\\ \frac{\bar{F}_{U(\tilde{y}_{1})}(j)}{E[U(\tilde{y}_{1})]-E[U(\tilde{x}_{1}])}& k_{3}>j\geq k_{2}\\ 0 & j\geq k_{3} \end{array}\right. $$

(12)

where $\bar {F}_{U(\tilde {x}_{1})}(j)$ and $\bar {F}_{U(\tilde {y}_{1})}(j)$ denote the survival function of the random variables $U(\tilde {x}_{1})$ and $U(\tilde {y}_{1})$ respectively, and

$$ f_{\tilde{s}}(j)= \left\{\begin{array}{ll} 0 & x_{0}>j \\ \frac{1}{E[U(\tilde{y}_{1})]-U(x_{0})} & k_{2}>j\geq x_{0}\\ \frac{\bar{F}_{U(\tilde{y}_{1})}(j)}{E[U(\tilde{y}_{1})]-U(x_{0})} & k_{3}>j\geq k_{2}\\ 0 & j\geq k_{3} \end{array}\right. $$

(13)

This implies that:

$$ f_{\tilde{s}}(j)-f_{\tilde{z}}(j)= \left\{\begin{array}{ll} 0 & x_{0}>j\\ \frac{1}{E[U(\tilde{y}_{1})]-U(x_{0})}& k_{0}>j\geq x_{0}\\ \frac{1}{E[U(\tilde{y}_{1})]-U(x_{0})}- \frac{1-\bar{F}_{U(\tilde{x}_{1})}(j)}{E[U(\tilde{y}_{1})]-E[U(\tilde{x}_{1}])} & k_{1}>j\geq k_{0} \\ \frac{1}{E[U(\tilde{y}_{1})]-U(x_{0})}-\frac{1}{E[U(\tilde{y}_{1})]-E[U(\tilde{x}_{1}])} & k_{2}>j\geq k_{1}\\ \frac{\bar{F}_{U(\tilde{y}_{1})}(j)}{E[U(\tilde{y}_{1})]-U(x_{0})}- \frac{\bar{F}_{U(\tilde{y}_{1})}(j)}{E[U(\tilde{y}_{1})]-E[U(\tilde{x}_{1}])}& k_{3}>j\geq k_{2}\\ 0 & j\geq k_{3} \end{array}\right. $$

(14)

It is clear that $f_{\tilde {s}}(j)<f_{\tilde {z}}(j)$ for j ≥ k₁ and that $f_{\tilde {s}}(j)>f_{\tilde {z}}(j)$ for j < x₀. Moreover, since $\bar {F}_{U(\tilde {x}_{1})}(k_{1})=0$ and $\bar {F}_{U(\tilde {x}_{1})}(k_{0})=1$ and $\bar {F}_{U(\tilde {x}_{1})}(.)$ is a decreasing function in the interval [k₀, k₁] there exist a value j₀ ∈ [k₀, k₁] such that $f_{\tilde {s}}(j)<f_{\tilde {z}}(j)$ for j > j₀ and that $f_{\tilde {s}}(j)>f_{\tilde {z}}(j)$ for j < j₀. All this implies that G(j) = 0 for j ≥ k₃, G(j) = 0 for j < x₀, G(j) increasing for j < j₀ and G(j) decreasing for j > j₀. This implies, in turn, that G(j) ≥ 0 ∀j ∈ [x₀, k₃] with a strict inequality for some j, which, as shown above, means that $\tilde {z}$ dominates $\tilde {s}$ via first order stochastic dominance.