331. 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 exactly before deciding
- VSS (Value of Stochastic Solution) — how much you gain by modeling uncertainty at all
331.1. Three problem versions
For a stochastic program with random parameter :
| Problem | Decision | |
|---|---|---|
| SP | accounts for uncertainty | |
| WS (wait-and-see) | decide after observing | |
| EV (expected value) | decide using average , ignore variability |
Their optimal values: .
331.2. EVPI
(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 in advance enables tailored optimal decisions per scenario — usually does better than committing to one decision before observing.
331.3. VSS
where EEV (Expected result of the Expected-value solution) is the expected cost of using the EV-optimal :
Interpretation: value of modeling uncertainty. If you ignore variability and just use mean parameters, your decision can be poor when actual deviates. VSS measures this loss.
331.4. 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 |
331.5. Worked example
From two-stage recourse example:
- : solve stochastic program → optimal and expected cost
- : best in each scenario separately, average over scenarios
- : solve with replacing ; then compute actual expected cost =
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 (concave → larger EVPI, by Jensen)
331.6. Sanity check identities
- EVPI always (perfect info can’t hurt)
- VSS always (stochastic modeling can’t hurt)
- VSS (stochastic is no better than perfect info)
331.7. See also
- Stochastic Programming
- Two-stage Recourse
- EVPI (Decision Analysis) — discrete analog
- EVSI — value of sample info