356. EMV
Expected Monetary Value (EMV) — the standard decision criterion under decision analysis: pick the act with the highest expected payoff.
For act with payoff in state (probability ):
Optimal act: .
356.1. Example
| State (high, ) | State (low, ) | EMV | |
|---|---|---|---|
| Act (build big) | |||
| Act (build small) | |||
| Act (don’t build) |
(All values in millions of dollars.) Optimal by EMV: , EMV .
356.2. 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
356.3. 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
356.4. The St. Petersburg paradox
A coin is flipped until tails. If flips, payoff . Expected payoff: .
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.
356.5. See also
- Decision Trees
- Decision Analysis
- EVPI / EVSI
- Utility Theory — when EMV isn’t enough