Problem $3.(20$ points$)$ Recall the Allais paradox from lecture.

Question $1$: $($$$1,1)$ or $($$$1,0\mathrm{.}89$;$$5,0\mathrm{.}10$;$$0,0\mathrm{.}01)$

Question $2$: $($$$1,0\mathrm{.}11$;$$0,0\mathrm{.}89)$ or $($$$5,0\mathrm{.}10$;$$0,0\mathrm{.}90)$

Most people prefer the first lottery in Question $1$ but the second lottery in Question $2.$ Suppose someone chooses according to prospect theory with value function

$v(x)=\{\begin{array}{ccc}{x}^{0\mathrm{.}1}& if& x0\\ 2x& if& x<mn\; id="EditorElement\_3792648">0</mn>\end{array}$

where all units are in $billions. The probability weighing function they use is

$(p)=\frac{\sqrt[\phantom{2}]{p}}{(\sqrt[\phantom{2}]{p}+\sqrt[\phantom{2}]{1-p}{)}^2}$

Show that even without any editing, this can explain the Allais paradox $($set reference point to $$0$ in both questions$).$