Question

Decisions are often 'risky' in the sense that their outcomes are not known with certainty. Presented with a choice between a risky prospect that offers a 50 percent chance to win $200 (otherwise nothing) and an alternative of receiving $100 for sure, most people prefer the sure gain over the gamble, although the two prospects have the same expected value. Preference for a sure outcome over a risky prospect of equal expected value is called risk averse; people tend to be risk averse when choosing between prospects with positive outcomes, a pattern explained by diminishing sensitivity: the utility from a small increase in wealth falls as the wealth already held rises, so the person's utility function \(u\) is concave for gains. A convex utility function is the mirror image of this: it gives MORE weight to the extreme, spread-out outcome of a gamble than to a 'safer,' more sure-like outcome of the same expected value, and a person with such a function is called risk taking or risk seeking.

Based on the above, look at the decision situation faced by Babitha.

Babitha played a game in which she had three options, with probabilities 0.4, 0.5 and 0.8 respectively. The gains from the three outcomes are likely to be $100, $80 and $50 respectively. An expert has pointed out that Babitha is a risk taking person. According to the expected utility hypothesis, which option is Babitha most likely to favour?

• A. First
• B. Second
• C. Third
• D. Babitha would be indifferent to all three options.

Hint:

All three options tie at an expected value of $40, so the tie has to be broken by Babitha's convex, risk-seeking utility function, which favours the most extreme, lowest-probability, highest-payoff option.

Answer 1

The Correct Option is A

Step 1: Notice the trap: check expected value first.
$0.4 \times 100 = 40$, $0.5 \times 80 = 40$, $0.8 \times 50 = 40$. Every option gives exactly $40 in expected value, so a person deciding purely on expected value would truly be indifferent between them, this is the trap the fourth option is designed to catch. But the question specifically asks us to use the expected utility hypothesis for a risk-taking person, not plain expected value.

Step 2: Translate 'risk taking' into a shape of utility function.
A risk-seeking (risk-taking) person has a convex utility curve, meaning the marginal utility of an extra dollar of gain INCREASES with the amount at stake, rather than decreasing the way it does for a risk-averse, concave utility holder. A convex curve rewards the more extreme, higher-variance outcome more than the safer one, even when both have identical expected value.

Step 3: Compare the three options on variance, not just expected value.
The first option (0.4 probability, $100 gain) has the widest gap between its 'win' and 'no win' outcomes, so it is the highest-variance, most extreme of the three. The third option (0.8 probability, $50 gain) is the closest to a near-certain small gain, so it is the lowest-variance, most 'sure-like' choice.

Step 4: Match Babitha's risk-taking profile to the ranking.
A convex utility function assigns the first (highest-variance) option the highest expected utility among the three, even though all three tie on plain expected value. Since Babitha is risk taking, she is most likely to pick the option that a convex utility function ranks highest, which is the first option, not the second, third, or a tie between all three.

Final Answer:
Babitha is most likely to favour the First option. \[ \boxed{\text{First}} \]