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Assume the following problem: You're deciding whether to invest into an opportunity with uncertain cost $c$ and value $v$. The cost has been estimated to be normally distributed with 90% CI between 1 and 5 million. Similarly, the value has been estimated to be normally distributed with 90% CI between 1 and 20 million. The default decision is to invest. However, you're interested in finding the value of reducing the uncertainty about both the cost and value in order to be certain about your decision.

I intuitively understand that here we're after the Expected Value of Perfect Information or $EVPI$ which is equal to the potential loss in case we take the wrong decision.

Questions:

  1. How do we represent the problem above mathematically? Intuitively, I would write something like: $EVPI=\int_c\int_v(v-c)p(c)p(v)$
  2. How do we solve the problem for a generic case? Is MCMC or a simple Monte Carlo a good choice here?

I would prefer a layman-level answer as I'm not very strong with stats.

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  • $\begingroup$ Instead of "CI" might you mean "standard deviation"? CI (confidence interval) does not seem applicable. $\endgroup$
    – whuber ♦
    Jun 18, 2013 at 20:07
  • $\begingroup$ I agree it seems unusually phrased, but if you take it as meaning that 90% of the density is within those bounds, then you can uniquely identify both parameters of the distribution. $\endgroup$
    – Nameless
    Jun 18, 2013 at 20:25
  • $\begingroup$ @Nameless That's not actually a confidence interval, though. $\endgroup$
    – Glen_b
    Jun 19, 2013 at 1:48
  • $\begingroup$ Thanks for pointing this out - I really meant that 90% of the density is within those bounds. That should suffice to compute the mean and variance. $\endgroup$
    – David
    Jun 19, 2013 at 9:25

1 Answer 1

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I am assuming we have an expected value maximizing decision maker (DM). Without additional information, the decision maker would invest, because the expected return is positive. However, for certain realizations of the two random variables (whenever $v<c$), this decision will result in a loss.

It seems reasonable that the value of perfect information is exactly equal to those expected losses, because the DM would exactly avoid investment in these cases. A rational expected value maximizing DM would therefore be willing to pay up to this amount for the information. But your mathematical representation doesn't cover the fact that the decision is only changed if $(v-c)$ is negative. Hence, we should get something like $$EVPI=-\int_{-\infty}^{\infty} \int_{-\infty}^{c} (v-c) f_c f_v dv dc,$$ that is, for every realization $c$, we only consider the cases where $v<c$. Since the DM avoids these losses with perfect information, we multiply by $-1$. I doubt, though, that this can be solved analytically, because integrals and normal distributions don't go well together (uniform distributions, on the other hand, should generally have closed form solutions).

Since I didn't even know that MCMC was until I looked it up just now, I can't say more than you probably could regarding question 2.

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  • $\begingroup$ Thanks Nameless, your answer is making sense. I understand that in general, the equation cannot be solved analytically. How could one go about solving this though? What kind of simulation would be appropriate? $\endgroup$
    – David
    Jun 19, 2013 at 9:29
  • $\begingroup$ I guess I would try some symbolic math tool, like mathematica or Matlab with symbolic toolbox. Again, I don't really know how to do it numerically. $\endgroup$
    – Nameless
    Jun 19, 2013 at 9:46

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