Expected Utility

See how to score a lottery. Give each outcome a happiness score, weight it by probability, and add up.

The key: 💰 money does not pile up happiness at a steady rate. The richer you are, the less one more coin delights you. So the average prize, paid for sure, feels better than the gamble itself. That difference is the Jensen gap.

For example, flip a coin: heads pays 100 won, tails pays nothing. The average prize is 50 won. But offer a sure 40 won instead, and many friends pick the 40. The sure thing is easier on the heart.

U(L)=∑npn u(xn)U(L) = \sum_n p_n\, u(x_n)
u concave ⇒U(L)≤u(E[L])(Jensen)u \text{ concave } \Rightarrow U(L) \le u(\mathbb{E}[L]) \quad \text{(Jensen)}

Part of these learning paths

Examples & Exercises

A consumer with Bernoulli utility u(x)=xu(x) = \sqrt{x} evaluates the lottery LL paying 150150 with probability 1/21/2 and 5050 with probability 1/21/2.

Compute the expected value, the expected utility, and the certainty equivalent directly, and verify that the Jensen inequality below holds strictly.

E[L]=12⋅150+12⋅50=100\mathbb{E}[L] = \tfrac{1}{2} \cdot 150 + \tfrac{1}{2} \cdot 50 = 100
U(L)=12150+1250≈9.659U(L) = \tfrac{1}{2}\sqrt{150} + \tfrac{1}{2}\sqrt{50} \approx 9.659
U(L)≤u(E[L])U(L) \le u(\mathbb{E}[L])
A 50-50 lottery over 150150 and 5050, u(x)=xu(x) = \sqrt{x}.
Show proof

  1. The expected value is E[L]=100\mathbb{E}[L] = 100. Expected utility is U(L)=12150+1250≈12(12.247+7.071)≈9.659U(L) = \frac{1}{2}\sqrt{150} + \frac{1}{2}\sqrt{50} \approx \frac{1}{2}(12.247 + 7.071) \approx 9.659.

  2. The certainty equivalent cc solves u(c)=U(L)u(c) = U(L): c≈9.659\sqrt{c} \approx 9.659, so c≈93.30c \approx 93.30. The utility of the expected value is u(E[L])=100=10>9.659=U(L)u(\mathbb{E}[L]) = \sqrt{100} = 10 > 9.659 = U(L) — strict Jensen for concave uu.

    Jensen gap=u(E[L])−U(L)≈10−9.659=0.341>0\text{Jensen gap} = u(\mathbb{E}[L]) - U(L) \approx 10 - 9.659 = 0.341 > 0

  3. The risk premium is E[L]−c≈100−93.30=6.70\mathbb{E}[L] - c \approx 100 - 93.30 = 6.70. In the figure this reads off as the vertical segment above the expected value (the Jensen gap) and the certainty-equivalent marker sitting left of the expected value.

The same lottery LL is evaluated by a risk-neutral consumer with Bernoulli utility u(x)=xu(x) = x.

Find the expected utility, the certainty equivalent, and the Jensen gap, and show the risk premium is zero.

U(L)=12⋅150+12⋅50=100=E[L]U(L) = \tfrac{1}{2} \cdot 150 + \tfrac{1}{2} \cdot 50 = 100 = \mathbb{E}[L]
The same lottery with linear utility u(x)=xu(x) = x — risk neutrality.
Show proof

  1. With linear utility, U(L)=12⋅150+12⋅50=100=E[L]U(L) = \frac{1}{2} \cdot 150 + \frac{1}{2} \cdot 50 = 100 = \mathbb{E}[L]. The expectation operator is linear, so a linear uu passes it straight through.

  2. The certainty equivalent is c=U(L)=100=E[L]c = U(L) = 100 = \mathbb{E}[L]. The Jensen gap is u(E[L])−U(L)=100−100=0u(\mathbb{E}[L]) - U(L) = 100 - 100 = 0, and the risk premium E[L]−c=0\mathbb{E}[L] - c = 0 — the risk-neutral consumer is indifferent between the lottery and its expected value.

    In the figure the chord lies exactly on the curve.

7. Choice under Uncertainty