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. (Expected value is the sum of possible outcomes weighted by their probability of occurrence.) Preference for a sure outcome over a risky prospect of equal expected value is called risk averse; indeed, people tend to be risk averse when choosing between prospects with positive outcomes. The tendency towards risk aversion can be explained by the notion of diminishing sensitivity, first formalized by Daniel Bernoulli in 1738. Just as the impact of a candle is greater when it is brought into a dark room than into a room that is well lit, so, suggested Bernoulli, the utility resulting from a small increase in wealth will be inversely proportional to the amount of wealth already in one's possession. It has since been assumed that people have a subjective utility function, and that preferences should be described using expected utility instead of expected value. According to expected utility, the worth of a gamble offering a 50 percent chance to win $200 (otherwise nothing) is \(0.50 \times u(200)\), where \(u\) is the person's concave utility function. (A function is concave or convex if a line joining two points on the curve lies entirely below or above the curve, respectively.) It follows from a concave function that the subjective value attached to a gain of $100 is more than 50 percent of the value attached to a gain of $200, which gives a preference for the sure $100 gain and, hence, risk aversion.

Consider now a choice between losses. When asked to choose between a prospect that offers a 50 percent chance to lose $200 (otherwise nothing) and the alternative of losing $100 for sure, most people prefer to take an even chance at losing $200 or nothing over a sure $100 loss. This is because diminishing sensitivity applies to negative as well as to positive outcomes: the impact of an initial $100 loss is greater than that of the next $100. This gives a convex function for losses and a preference for risky prospects over sure outcomes of equal expected value, called risk seeking. Except for prospects that involve very small probabilities, risk aversion is generally seen in choices involving gains, while risk seeking tends to hold in choices involving losses.

Based on the above passage, look at the decision situations faced by three persons: Babu, Babitha and Bablu.

Suppose the instant and further utility of each unit of gain is the same for Babu. Babu has decided to play as many times as possible before he dies. He expects to live for another 50 years. A game does not last more than ten seconds. Babu is confused about which theory to trust for making his decision and seeks the help of a renowned decision making consultant, Roy Associates. What should Roy Associates' advice to Babu be?

• A. Babu can decide on the basis of the Expected Value hypothesis.
• B. Babu should decide on the basis of the Expected Utility hypothesis.
• C. 'Mr. Babu, I'm redundant.'
• D. A, B and C

Hint:

A constant marginal utility means Expected Value and Expected Utility always agree for Babu, and playing millions of times means the law of large numbers already does the consultant's job for him.

Answer 1

The Correct Option is D

Step 1: Translate 'same instant and further utility' into a formula.
If every additional unit of gain feels exactly as valuable as the previous unit, Babu's utility function has constant slope, so $u(x) = kx$ for some fixed $k > 0$. There is no diminishing sensitivity here, unlike the concave/convex shapes the passage uses to explain other people's risk attitudes.

Step 2: Show Expected Value and Expected Utility never disagree for this $u$.
For any gamble with outcomes $x_1, x_2, ...$ and probabilities $p_1, p_2, ...$, expected utility is $\sum p_i \cdot k x_i = k \sum p_i x_i = k \cdot (\text{expected value})$. Since $k$ is a fixed positive number, whichever option has the bigger expected value also has the bigger expected utility, always. So options A and B are both valid, simultaneously correct guidance for Babu, they are the same ranking in disguise.

Step 3: Bring the repetition count into the picture.
Babu plans to squeeze in as many ten-second games as he can across roughly 50 years, which works out to a huge number of independent repetitions. The law of large numbers says that over that many repetitions, the average result he actually experiences will settle very close to the expected value of a single play, no matter which theoretical lens (value or utility) is used to describe the decision. In a setting like this, an expert consultant's specialised modelling does not change the practical advice at all, the simple 'always take the option with the higher expected value' rule already captures everything that matters.

Step 4: Put all three pieces together.
Because A and B coincide exactly (Step 2) and the repetition count makes the consultant's extra expertise unnecessary (Step 3), all three statements, A, B and C, hold true for Babu's specific situation, so the fullest and most accurate advice is all three together.

Final Answer:
Roy Associates should tell Babu A, B and C. \[ \boxed{\text{A, B and C}} \]