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: Read what 'same instant and further utility' means for Babu.
Saying every extra unit of gain gives Babu the same utility as the one before it means his marginal utility of money is constant, so his utility function is a straight line, \(u(x) = kx\) for some constant \(k\), not the curved, diminishing-sensitivity shape the passage describes for most people.

Step 2: See what a linear utility function does to expected utility.
With \(u(x)=kx\), the expected utility of any gamble is just \(k\) times its expected value. Multiplying every option's expected value by the same positive constant \(k\) never changes which option is larger, so for Babu, ranking options by expected utility and ranking them by plain expected value always give the identical answer. This means option A (decide by Expected Value) and option B (decide by Expected Utility) are both correct advice at the same time, they are not rival theories for Babu, they agree completely.

Step 3: Bring in how many times Babu will actually play.
Babu wants to play as many times as possible over 50 years, with each play lasting at most ten seconds, so the number of plays runs into millions. By the law of large numbers, when an independent gamble is repeated this many times, the realised long-run average result converges to the theoretical expected value regardless of how the decision is framed. In that setting, all the sophisticated machinery of utility theory (modelling curvature, risk premiums, and so on) does not add anything beyond the simple rule 'always take the higher expected value,' so a specialised consultant genuinely has nothing extra to contribute here.

Step 4: Combine the three pieces of advice.
Since A, B and C are all individually true and useful things to tell Babu given his linear utility and his huge number of repeated plays, the complete and most defensible advice includes all three together, not just any two of them.

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