Which bets would a rational person definitely want to take, assuming they maximize expected utility:

I. Win $1.5 if a coin comes up heads and lose $1 if a coin comes up tails

II. Win $1.5 million if a coin comes up heads and lose $1 million if a coin comes up tails

III. Win $1.5 if the S&P 500 rises by more than 10% next year; lose $1 if the S&P falls by more than 10% next year

Question

Which  bets would a rational person definitely want to take, assuming they  maximize expected utility:

I. Win $1.5 if a coin comes up heads and lose $1 if a coin comes up tails

II. Win $1.5 million if a coin comes up heads and lose $1 million if a coin comes up tails

III. Win $1.5 if the S&P 500 rises by more than 10% next year; lose $1 if the S&P falls by more than 10% next year

BlackTom AI · Accepted Answer

We start by restating the question and listing the options to ensure clarity.
Question: Which bets would a rational person definitely want to take, assuming they maximize expected utility:
I. Win $1.5 if a coin comes up heads and lose $1 if a coin comes up tails
II. Win $1.5 million if a coin comes up heads and lose $1 million if a coin comes up tails
III. Win $1.5 if the S&P 500 rises by more than 10% next year; lose $1 if the S&P falls by more than 10% next year

Option I: The gamble pays +1.5 with probability 0.5 and -1 with probability 0.5. Its expected monetary value is 0.5×1.5 + 0.5×(−1) = 0.75 − 0.5 = +0.25 dollars.
Option II: The gamble pays +1.5 million with probability 0.5 and −1 million with probability 0.5. Its expected monetary value is 0.5×1.5,000,000 + 0.5×(−1,000,000) = 750,000 − 500,000 = +250,000 dollars.
Option III: The gamble pays +1.5 with probability p for a >10% rise in the S&P and −1 with probability q for a >10% fall, where p and q depend on market dynamics. Without explicit probabilities, the expected value cannot be computed, and the statement hinges on those probabilities and the decision maker’s utility over those outcomes.

Now, analyze each option individually:
Option I – Why this is or isn’t attractive under expected utility: The net payoff distribution is symmetric around a small positive expected value of +0.25 dollars. If the decision maker has linear or modestly concave utility around this wealth level, this positive expected value tends to make the gamble attractive, though the small magnitude means the benefit is uncertain in practice. Importantly, there is no catastrophic downside here beyond the fixed −1 when tails occur, and the positive EV suggests a tendency to accept under many utility specifications. However, whether it is “definitely” desired depends on wealth and risk aversion; for some risk-averse utilities, even a positive EV may not guarantee acceptance, but with such a small stake the downside is limited.
Option II – Why this is less likely to be “definitely” desirable: The bet offers a large potential gain (+$1.5 million) but also a large potential loss (−$1 million), and its expected value is +$250,000. While that is positive under linear or mildly concave utility, the large variance and the asymmetric loss (a big loss relative to the gain) make the decision highly sensitive to the agent’s absolute wealth and risk preferences. A strictly risk-averse individual or one with a concave utility that sharply discriminates against large losses may reject this bet even though the EV is positive. Hence, II is not guaranteed to be taken by all rational agents maximizing expected utility.
Option III – Why this is unclear or not definitely attractive: This bet’s attractiveness hinges entirely on the probabilities of the S&P rising more than 10% or falling more than 10% in the next year. Without specified probabilities, we cannot compute an expected utility, and different plausible probability assessments (e.g., near-0.5, or highly skewed toward one outcome) will yield different judgments under any reasonable utility function. Therefore, III cannot be declared definitively attractive without more information about p and q (and the investor’s utility over those monetary outcomes).

Summarizing the comparative judgments:
- I has a positive expected value with small stakes, making it reasonably attractive under many utility forms, but claiming it is definitively desirable depends on the exact utility function and wealth level; the reasoning here shows it as a plausible positive EV gamble.
- II has a positive EV but involves a much larger exposure to risk, so it is not guaranteed to be chosen by all rational agents maximizing expected utility, due to risk aversion and large potential losses relative to gains.
- III lacks explicit probabilities, so its expected utility cannot be determined, rendering it not definitively desirable.

Taking all of this into account, the option that stands out as most consistently favorable across typical utility specifications and wealth levels is Option I, while II and III do not meet the same universal justification.

IFEPIA7022_001_2025_3 - Economics of Finance EOF in-class Quiz 10-7-25

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