EMV

359. EMV

Expected Monetary Value (EMV) — the standard decision criterion under decision analysis: pick the act with the highest expected payoff.

For act $𝑎_{𝑖}$ with payoff $𝑉_{𝑖 𝑗}$ in state $𝑠_{𝑗}$ (probability $𝑝_{𝑗}$ ):

EMV (𝑎_{𝑖}) = \sum_{𝑗} 𝑝_{𝑗} 𝑉_{𝑖 𝑗}

Optimal act: $𝑎^{*} = arg {max}_{𝑖} EMV (𝑎_{𝑖})$ .

359.1. Example

	State $𝑠_{1}$ (high, $𝑝 = 0.6$ )	State $𝑠_{2}$ (low, $𝑝 = 0.4$ )	EMV
Act $𝑎_{1}$ (build big)	$80$	$- 10$	$0.6 \cdot 80 + 0.4 \cdot (- 10) = 44$
Act $𝑎_{2}$ (build small)	$40$	$20$	$0.6 \cdot 40 + 0.4 \cdot 20 = 32$
Act $𝑎_{3}$ (don’t build)	$0$	$0$	$0$

(All values in millions of dollars.) Optimal by EMV: $𝑎_{1}$ , EMV $= 44$ .

359.2. When EMV is the right criterion

Repeated decisions: long-run average is the right measure; gambler’s-ruin issues smoothed out
Risk-neutral decision-maker: dollars are interchangeable
Decisions small relative to wealth: linear approximation valid
Probabilities are reasonably well-known: estimation error is small relative to payoffs

359.3. When EMV is not the right criterion

One-shot decision with large downside: someone facing potential ruin will rationally avoid the high-EMV gamble (insurance, hedging)
Risk-averse decision-maker: use utility theory — payoffs run through a concave utility function before averaging — see Utility Theory
Deep uncertainty: probabilities not just unknown but unknowable — use non-probabilistic criteria
Multiple objectives: not just money — see multi-criteria decision analysis

359.4. The St. Petersburg paradox

A coin is flipped until tails. If $𝑛$ flips, payoff $= 2^{𝑛}$ . Expected payoff: $\sum_{𝑛} {(\frac{1}{2})}^{𝑛} \cdot 2^{𝑛} = \sum_{𝑛} 1 = \infty$ .

But no rational person pays infinite money to play. Resolves: utility is concave (log-utility famously), and expected utility is finite. Bernoulli’s solution from 1738 — first hint that EMV alone isn’t enough.

359.5. See also

Decision Trees
Decision Analysis
EVPI / EVSI
Utility Theory — when EMV isn’t enough