Expected utility and risk aversion

Expected monetary value (EMV) treats a certain $50 and a coin-flip between $0 and $100 as identical, because both have expectation $50. Almost nobody is indifferent between them — most people prefer the certain $50. This is risk aversion, and expected value alone cannot represent it.

The von Neumann–Morgenstern (vNM) axioms. vNM (1944) proved that if a decision-maker's preferences over lotteries (probability distributions over outcomes) satisfy four axioms — completeness, transitivity, continuity, and independence (if $A\succ B$ then a mixture of $A$ with $C$ is preferred to the same mixture of $B$ with $C$, for any $C$ and mixing weight) — then there exists a utility function $u$, unique up to positive affine transformation, such that the decision-maker ranks lotteries by their expected utility $E[u(X)]$. This is the axiomatic bedrock of the "EU" special case of ADE: set $\lambda=0$ (no distortion), $\varepsilon=0$ (no ambiguity aversion), $\omega=0$ (no regret), and ADE's score collapses exactly to $E[u(X_a)]$.

Risk aversion and its measures. A concave $u$ ($u''<0$) represents risk aversion: by Jensen's inequality, $E[u(X)] \le u(E[X])$, so the decision-maker values a risky prospect below its expected value. Arrow–Pratt quantify local risk aversion with the absolute risk aversion coefficient $\text{ARA}(x) = -u''(x)/u'(x)$ and relative risk aversion $\text{RRA}(x)=x\cdot\text{ARA}(x)$. ADE uses the exponential ("CARA") utility $u(x)=\rho(1-e^{-x/\rho})$ because $\text{ARA}(x)=1/\rho$ is constant — risk tolerance $\rho$ does not depend on wealth level, which makes the parameter directly interpretable ("I am indifferent between a certain loss of $$\rho$ and a coin flip between losing nothing and losing roughly $$2\rho$") and lets $\rho\to\infty$ recover risk-neutral EMV smoothly, a convenient property for a single unifying engine. (Users who need wealth-dependent risk aversion — e.g. CRRA power utility $u(x)=x^{1-\gamma}/(1-\gamma)$ — can supply pre-transformed payoffs; ADE's distortion and ambiguity layers still apply on top.)

Certainty equivalent and risk premium. The certainty equivalent $CE_a = u^{-1}(E[u(X_a)])$ is the guaranteed amount as good as the risky prospect $X_a$. The risk premium $\pi_a = E[X_a] - CE_a$ is exactly what a risk-averse decision-maker would pay to eliminate the risk — the actuarial rationale for insurance premium loadings above pure expected loss. ADE reports both for every action.

Worked intuition. Take a $100,000 exposure with a 10% chance of a total loss and 90% chance of no loss. $E[X] = -10{,}000$. With CARA utility and $\rho = 50{,}000$: $E[u(X)] = 0.1\cdot u(-100{,}000) + 0.9\cdot u(0)$. Since $u(-100{,}000) = 50{,}000(1-e^{2}) \approx -319{,}453$, $E[u(X)]\approx -31{,}945$, giving $CE = -50{,}000\ln(1-(-31{,}945)/50{,}000) \approx -50{,}000\ln(1.639)\approx -24{,}540$. The risk premium is $-10{,}000-(-24{,}540)=14{,}540$: this decision-maker would rationally pay up to $14,540 above the $10,000 expected loss to transfer the risk — the essence of why insurance exists as a mutually beneficial trade even though it has a negative expected value for the buyer.

Limits. EU is silent on ambiguity (unknown probabilities, not just unknown outcomes — see Ellsberg) and on regret; ADE layers $\varepsilon$ and $\omega$ on top precisely to address what EU alone cannot.