Expected Utility Decision Theory Problem

Question

Consider a person who choose among lotteries. Each lottery is of the form (p1, p2, p3) where p1 is the probabilty of getting Rs.5, p2 is the probabilty of getting Rs.1 and p3 is the probabilty of getting Rs.0. This person prefer lottery (0, 1, 0) to lottery (0. 1, 0. 89,0. 01). If this,person maximum expected utility and is faced with lotteries (0, 0. 11, 0. 89) and (0. 1, 0, 0. 9), which,lottery should he prefer?

Answer 1

Let the Utility of Rs.0 be $0$, Rs.1 be $1$ and of Rs.5 be $u_5$. the preference of $(0,1,0)$ over $(0.1,0.89,0.01)$ tells us that: $$ \begin{aligned} U(0,1,0)&>U(0.1,0.89,0.01)\\ 1&>0.1u_1+0.89\\ u_5&<1.1 \end{aligned} $$ Where $U(a,b,c)=a\times u_5+b$ denotes the expected utility of lottery $(a,b,c)$.

Now look at the expected utility of lotteries $(0,0.11,0.89)$ and $(0.1,0,0.9)$: $$ \begin{aligned} U(0,0.11,0.89)&=0.11\\ U(0.1,0,0.9)&<0.1\times 1.1=0.11 \end{aligned} $$ That is the second lottery has less expected utility that the first, so the first $(0,0.11,0.89)$ is prefered.