$(1)$ It has been shown that more people choose $B$ when presented with problem $1$ and when presented with problem $2,$ most people choose $C.$ Do these choices violate expected utility theory? Why? $($Assume $u(0)=0)(5$ marks$)$

Now you are considering using prospect theory to justify the decisions presented in $(1).$ You proposed the following value function

$v(z)=\{\begin{array}{ccc}{z}^{\mathrm{.}85}& \text{w}\text{h}\text{e}\text{n}& z0\\ -3(-z{)}^{\mathrm{.}85}& \text{w}\text{h}\text{e}\text{n}& z<mn\; id="EditorElement\_16611446">0</mn>\end{array}$

and the following probability decision weighting function where $$ has been set at $0\mathrm{.}6$

$(p)=\frac{p^{}}{(p^{}+(1-p{)}^{}{)}^{\frac{1}{}}}$

Answer question $(2)-(4).$

$(2)$ Predict individual's decision with the value function and simple probabilities. $(5$ marks$)$

$(3)$ Predict individual's decision with the value function and weighting function. $(5$ marks$)$

$(4)$ Compare your answers to $(2)$ and $(3).$ Explain which setting better predicts

Prospect Theory $(20$ marks$)$

Consider the following prospect choices:

$\backslash$table$[[$Question $1],[$Prospect $A,$Prospect $B$ individual's decision and why. $(5$ marks$)$