ECON30019 Chap.11 Expected Value and Expected Utility under Risk
Expected Value and Expected Utility under Risk
From this chapter the objects being ranked are no longer alternatives but lotteries, and the preference relation of the earlier chapters is asked to order them. A prospect is a set of possible outcomes with their probabilities, and expected value weights each outcome by its chance, a principle that arrived in the seventeenth century from the problem of dividing an interrupted game's stake.
The chapter shows why that is not enough, using the game whose expected value has no finite limit and which nobody would pay much to play, and follows the resolution that puts utility inside the expectation. Curvature then carries risk attitude: concave means risk averse, linear risk neutral, convex risk seeking, with the risk premium's sign reporting which.
It closes with the two axioms beyond rationality that expected utility requires, and the representation theorem that ties them to the formula.
What this chapter covers
- 01
Prospects and lotteries, and what changed about the objects being ranked
- 02
Expected utility theory as both a normative and a descriptive claim
- 03
Expected value, and the division problem it came from
- 04
The game with an unbounded expected value, and what it refutes
- 05
Putting utility inside the expectation, and why the series then converges
- 06
Diminishing marginal utility, concavity, and the plain-language version
- 07
One gamble, three curvatures, three recommendations
- 08
The risk-attitude trichotomy and the sign of the risk premium
- 09
Certainty equivalent, computed by inverting the utility function
- 10
The independence and Archimedean axioms, and the representation theorem
Certainty equivalent and risk premium
- 1Expected value: half of 100 plus half of 36, which is 50 plus 18, giving 68.
- 1Expected utility: half of the square root of 100 plus half of the square root of 36, which is half of ten plus half of six, giving 8.
- 1Certainty equivalent: solve the square root of the certainty equivalent equals 8, so it is 64.
- 1Risk premium: expected value minus certainty equivalent, 68 minus 64, giving 4.
- 1State the attitude: the premium is positive, so the grower is risk averse.
- 1Justify it against curvature: the square root has diminishing marginal utility and is therefore concave, so utility lost by falling to 36 outweighs utility gained by rising to 100.
Key terms
- Prospect
- A list of outcomes that could occur together with the chance of each, which is the object a preference relation ranks once risk enters.
- Lottery
- A probability distribution over outcomes, with non-negative components summing to one. Mixtures of lotteries are what the independence axiom is about.
- Expected utility
- The probability-weighted sum of the utilities of the outcomes, as distinct from the utility of the probability-weighted sum of the outcomes.
- Certainty equivalent
- The sure sum a person would trade a given gamble for without regret, obtained by inverting the utility function at the gamble's expected utility.
- Risk premium
- The expected value minus the certainty equivalent. Its sign reports the risk attitude: positive for risk averse, negative for risk seeking, zero for risk neutral.
- Independence axiom
- The requirement that mixing two ranked lotteries with the same third lottery at the same probability preserves the ranking, which makes preferences linear in probabilities.
Expected Value and Expected Utility under Risk FAQ
Is a risk-averse person who refuses a favourable gamble being irrational?
No. Expected utility theory leaves your appetite for risk entirely open. Refusing a gamble whose expected value exceeds the certain alternative is exactly what the theory predicts for someone with a sufficiently concave utility function, and it remains fully consistent with every axiom. What the theory forbids is the choice depending on how the options are described.
What is the difference between expected utility and the utility of the expected value?
They are different objects. Expected utility averages the utilities of the outcomes; the other takes the utility of the average outcome. For a concave function the utility of the average is at least as large, so confusing them inflates the figure, and the gap between them is exactly what makes the risk premium positive.
Why does the theory need two axioms beyond rationality?
Because a utility representation over lotteries is not enough on its own. Independence delivers linearity in probabilities, which is what allows the expectation to be taken at all. The Archimedean axiom delivers continuity, ensuring no lottery is infinitely good or bad, without which a single catastrophic outcome would dominate every comparison however small its probability.
Exam move
Fix an order of operations and use it every time: expected value of each option on one line each, expected utility of each option on the next two, then the verdict. Questions in this family almost always ask for both comparisons and a comment on why they differ, so the ordered layout is the answer structure, and it makes a reversal impossible to miss.
Convert every cost to a negative outcome as you read it, then always prefer the larger number. Practise inverting three utility functions for the certainty equivalent, and finish every answer with the sign check against curvature.
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