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: Work out the expected value of each option.
Expected value is probability multiplied by the gain. First option: \(0.4 \times 100 = 40\). Second option: \(0.5 \times 80 = 40\). Third option: \(0.8 \times 50 = 40\). All three options come out to exactly the same expected value of $40, so plain expected value cannot possibly separate them, this question is really asking us to bring in expected utility instead.

Step 2: Recall what 'risk taking' means for a person choosing between equal expected value options.
A risk-seeking person has a convex utility function. When two prospects share the same expected value, a convex utility function always places a strictly higher expected utility on the more spread out, extreme option (lower probability, higher single payoff) compared to the more 'sure-like' option (higher probability, smaller payoff), which is the exact opposite of the concave, risk-averse case described in the passage.

Step 3: Rank the three options by how 'spread out' they are.
The first option has the lowest probability (0.4) paired with the highest single payoff ($100), so it is the most extreme and spread out of the three. The third option has the highest probability (0.8) paired with the smallest payoff ($50), so it behaves the most like a near-sure outcome. The second option sits between the two.

Step 4: Apply the risk-seeking preference.
Since Babitha is confirmed to be risk taking, and a convex utility function favours the most spread-out option among equal-expected-value alternatives, she is most likely to favour the option with the lowest probability and highest payoff, which is the first option. This also rules out the second and third options, and rules out indifference, since a convex utility function does NOT treat equal-expected-value options as equally good, unlike a linear one.

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