The Perfect-Info Ceiling (EVPI)

In the last lesson you learned to ask would it change my action? — and to bin any test that fails it. Now we put a number on the tests that pass. We start at the top, with the best possible test: a clairvoyant who answers with perfect certainty. Whatever a real test could tell you, it can’t beat knowing for sure. So the value of perfect information — EVPI — is the ceiling: the absolute most any test, survey, or expert could ever be worth on this decision. Nail the ceiling first and every real test slots in beneath it.

The set-up: should you launch?

Let’s make it concrete with a decision we’ll carry through the next three lessons. You can launch a new product or walk away. The world is either good (the market wants it) or bad (it doesn’t). The payoffs, in thousands of dollars:

Your honest prior is that the market is good with probability 40% (so bad with 60%). That’s the whole problem: a payoff table and a prior. Everything else is arithmetic.

Step 1 — what’s your best move with no extra info?

Weigh each action by the prior. This is just expected value — each outcome times its probability, summed:

$EV(\text{Launch}) = 0.40 \times 400 + 0.60 \times (-300) = 160 - 180 = -20$

$EV(\text{Walk away}) = 0.40 \times 0 + 0.60 \times 0 = 0$

Walking away (0) beats launching (−20), if only barely. So with nothing but your prior, your best move is to walk away, and the value of that best prior action — call it $V_0$ — is 0. Hold onto that: right now you launch nothing and pocket nothing.

Step 2 — what would a crystal ball be worth?

Now imagine a clairvoyant who tells you the true state before you decide, with perfect certainty. You’d use it perfectly: in every good world you’d launch and bank 400; in every bad world you’d walk away and take 0 instead of eating −300.

You don’t know in advance which message you’ll get, so weigh the clairvoyant-guided outcomes by the same prior:

$EV(\text{with crystal ball}) = 0.40 \times \underbrace{400}_{\text{launch, good}} + 0.60 \times \underbrace{0}_{\text{walk, bad}} = 160$

With perfect foresight you’d expect to make 160. Without it you make 0. The value of the perfect information is the difference:

$\text{EVPI} = 160 - 0 = \mathbf{160}$

EVPI is the expected cost of the mistake you’d otherwise make

There’s a second way to see the same number, and it’s the one to remember. Acting on your prior, you walk away from every world — including the 40% good ones where launching would have made 400. That’s the mistake: walking away from a winner. It costs you 400 each time it happens, and it happens 40% of the time:

$\text{expected cost of the mistake} = 0.40 \times 400 = 160$

Same 160. EVPI is exactly the expected cost of the error your prior-based choice walks you into — because perfect information is precisely what lets you stop making that error. This is why value can never exceed the stakes: you can’t save more than the mistake was going to cost.

When the ceiling is the floor: EVPI = 0

Run the same machine on a lopsided prior and watch the value vanish. Suppose you were 95% sure the market is bad:

$EV(\text{Launch}) = 0.05 \times 400 + 0.95 \times (-300) = 20 - 285 = -265$

You walk away — emphatically. Now the crystal ball: it’s worth $0.05 \times 400 = 20$ with perfect play, and your prior action already banks 0, so EVPI $= 20 - 0 = 20$. Push to 99% bad and it drops to 4. The more lopsided you are, the less a perfect answer can do, because you’re already taking the action you’d take in almost every world. At the extreme — 100% sure — EVPI is exactly 0: nothing left to learn, because nothing could change your mind.

Why start at the ceiling?

Because it’s the cheapest sanity check in all of decision analysis. Before you haggle over which survey to buy or how accurate it is, compute EVPI. If the perfect answer is worth less than the price of the survey, you can stop right there — no real test can be worth buying, because every real test is worth less than perfect information. EVPI turns a fuzzy “should we research this?” into a hard upper bound you can compare against a price tag in thirty seconds.

You now have the ceiling: on the launch decision, perfect information is worth 160, and that caps everything. But nobody sells perfect information. Next we drop from the crystal ball to a real, noisy test — a market survey that’s right most of the time but not always — and work out what that is worth. Spoiler baked into the maths: it’ll come in under 160, every time.