Expected Value of Perfect Information (EVPI) and Expected Value of Sample Information (EVSI) quantify how much a decision maker should rationally pay for additional information. EVPI measures the value of eliminating uncertainty entirely, while EVSI measures the value of reducing uncertainty through imperfect signals or samples. This page explains both measures with diagrams and numerical examples.

1. EVPI: Expected Value of Perfect Information

EVPI answers: “How much would I pay to know the future with certainty?”

\[ EVPI = EV_{with\ PI} – EV_{with\ current\ information} \]

2. EVPI Example with Diagram

Decision: Launch a new product.

           (Launch)
             /   \
         Strong  Weak
         0.4     0.6
         +500    -200

           (Don't Launch)
                0

Step 1: Expected value with current information

\[ EV_{launch} = 0.4(500) + 0.6(-200) = 80 \] \[ EV_{no\ launch} = 0 \] Best decision = Launch → EVwCI = 80

Step 2: Expected value with perfect information

If Strong → Launch → +500 If Weak → Don’t Launch → 0

\[ EV_{PI} = 0.4(500) + 0.6(0) = 200 \]

Step 3: EVPI

\[ EVPI = 200 – 80 = 120 \]

Interpretation: Perfect information is worth up to 120 units.

3. EVSI: Expected Value of Sample Information

EVSI measures the value of imperfect information (e.g., surveys, tests, pilot studies).

\[ EVSI = EV_{with\ sample\ information} – EV_{with\ current\ information} \]

EVSI is always ≤ EVPI.

4. EVSI Example with Imperfect Test

Same product launch problem, but now a market survey predicts Strong/Weak with 80% accuracy.

Step 1: Compute posterior probabilities

Let S = Strong, W = Weak, T = Test predicts Strong.

\[ P(T|S)=0.8,\quad P(T|W)=0.2 \] \[ P(S)=0.4,\quad P(W)=0.6 \]

\[ P(T)=0.8(0.4)+0.2(0.6)=0.44 \]

\[ P(S|T)=\frac{0.8(0.4)}{0.44}=0.727 \] \[ P(W|T)=0.273 \]

Step 2: Compute EV given test result

If test predicts Strong: \[ EV = 0.727(500)+0.273(-200)= 291.4 \] → Launch

If test predicts Weak: \[ P(S|T^c)=0.133,\quad P(W|T^c)=0.867 \] \[ EV = 0.133(500)+0.867(-200)= -113.1 \] → Don’t Launch

Step 3: Expected value with sample information

\[ EV_{SI} = P(T)(291.4) + P(T^c)(0) \] \[ = 0.44(291.4)=128.2 \]

Step 4: EVSI

\[ EVSI = 128.2 – 80 = 48.2 \]

Interpretation: The survey is worth up to 48.2 units.

5. Summary Table