EVPI vs VSS

334. EVPI vs VSS

Two complementary metrics for how much uncertainty modeling adds to a decision problem:

EVPI (Expected Value of Perfect Information) — how much you’d gain by knowing $𝜉$ exactly before deciding
VSS (Value of Stochastic Solution) — how much you gain by modeling uncertainty at all

334.1. Three problem versions

For a stochastic program with random parameter $𝜉$ :

	Problem	Decision
SP	${min}_{𝑥} 𝐸_{𝜉} [𝑓 (𝑥, 𝜉)]$	accounts for uncertainty
WS (wait-and-see)	$𝐸_{𝜉} [{min}_{𝑥} 𝑓 (𝑥, 𝜉)]$	decide after observing $𝜉$
EV (expected value)	${min}_{𝑥} 𝑓 (𝑥, 𝐸 [𝜉])$	decide using average $𝜉$ , ignore variability

Their optimal values: $𝑧^{SP}, 𝑧^{WS}, 𝑧^{EV}$ .

334.2. EVPI

EVPI = 𝑧^{SP} - 𝑧^{WS}

(For minimization: lower is better, so EVPI is positive when WS < SP, meaning perfect info would help.)

Interpretation: upper bound on the value of information. Knowing $𝜉$ in advance enables tailored optimal decisions per scenario — usually does better than committing to one decision before observing.

334.3. VSS

VSS = EEV - 𝑧^{SP}

where EEV (Expected result of the Expected-value solution) is the expected cost of using the EV-optimal $𝑥$ :

EEV = 𝐸_{𝜉} [𝑓 (𝑥^{EV}, 𝜉)]

Interpretation: value of modeling uncertainty. If you ignore variability and just use mean parameters, your decision can be poor when actual $𝜉$ deviates. VSS measures this loss.

334.4. Why both matter

Situation	EVPI	VSS
Worth investing in better forecasts?	yes if EVPI is large	—
Worth doing stochastic programming instead of EV?	—	yes if VSS is large
EVPI small, VSS small	decision is robust to uncertainty
EVPI large, VSS small	forecasting is helpful, but you’d make the same decision anyway
EVPI small, VSS large	can’t improve via forecasting, but must explicitly model variability
Both large	strongest case for both better data and stochastic modeling

334.5. Worked example

From two-stage recourse example:

$𝑧^{SP}$ : solve stochastic program → optimal $𝑥^{SP}$ and expected cost
$𝑧^{WS}$ : best $𝑥$ in each scenario separately, average over scenarios
$𝑧^{EV}$ : solve with $𝐸 [𝐷]$ replacing $𝐷$ ; then compute actual expected cost = $EEV$

A small calculation gives concrete EVPI and VSS. The relative magnitudes depend on:

Spread of scenarios (larger spread → larger EVPI/VSS)
Cost asymmetry between scenarios (asymmetric → larger VSS)
Convexity of cost in $𝜉$ (concave → larger EVPI, by Jensen)

334.6. Sanity check identities

EVPI $\geq 0$ always (perfect info can’t hurt)
VSS $\geq 0$ always (stochastic modeling can’t hurt)
VSS $\leq EEV - 𝑧^{WS}$ (stochastic is no better than perfect info)

334.7. See also

Stochastic Programming
Two-stage Recourse
EVPI (Decision Analysis) — discrete analog
EVSI — value of sample info