Expected Utility

Question

I really don't know how to interpret the graph. Can someone help me?

I thought of doing 0.6253+0.3751 to find the expected value of the lottery but where is the sure bet?

Answer 1

the first thing to start with is to realize how the graph works. Any point inside the graph gives you three probabilities: $Prob_1$, $Prob_2$ and $Prob_3$. Remember that while you only see the first and the last, the $Prob_2$ is given as the probability remaining: $Prob_2 = 1 - Prob_1 - Prob_3$ and in the graph you can compute it as the distance from the hypotheneuse (see attached graph below):

What you can do in your situation then is to count the expected income for extreme cases. You know that in first extreme case you get with $Prob_3 = 1/2$ the income $I = 3$ and with $Prob_2 = 1/2$ the income $I = 2$.

This way you can count both extreme cases:

$$ E[I_{right}] = \frac{5}{8} * 3 + \frac{3}{8} * 1 = 2,25 $$

$$ E[I_{left}] = \frac{1}{2} * 3 + \frac{1}{2} * 2 = 2,5 $$

What you can see is that the right case gives you less expected income than the left one but they are both prefered the same way. And here's the thing: The right one is more dramatic (either you get the best prize or the worst), whereas the left one is more modest (lower probability of getting the best prize but on the other hand you do not ger the worst one either). Since the right one has smaller expected income AND it is more dramatic AND the consumer is indifferent between this and the other one, it means that he is RISK LOVER. In other words, he is willing to pay something for the adrenaline coming from more risky bet. One short remark: You can compare the slope between expected-iso income line and utility function to come to the same conclusion.

What concerns the second question, it tells you the same story, John prefers the right case to obtaining $2.50$ for certain (which is the expected income from the left case btw). Based on the first information given, I tried to fit the coefficient $k$ of his utility function which I assumed to be $U(I) = I^k$, while the coefficient I obtained was around $k = 3.2$. Here it is important to note that this might be biased solution because I do not know exactly his utility function $U(I)$ but for the information given it might be all right...

Then what you do is to compare:

$$ A = (2)^{3.2} \approx 9.2 $$

$$ B = 0.35 * (2.5)^{3.2} + 0.65*1 \approx 7.2 $$

Which results in a lottery A to be prefered.