Expected value of perfect information

In decision theory, the expected value of perfect information (EVPI) is the price that one would be willing to pay in order to gain access to perfect information. [Douglas Hubbard "How to Measure Anything: Finding the Value of Intangibles in Business" pg. 46, John Wiley & Sons, 2007]

The problem is modeled with a payoff matrix "R_ij" in which the row index "i" describes a choice that must be made by the payer, while the column index "j" describes a random variable that the payer does not yet have knowledge of, that has probability "p_j" of being in state "j". If the payer is to choose "i" without knowing the value of "j", the best choice is the one that maximizes the expected monetary value::$mbox\{EMV\}\; =\; max\_i\; sum\_j\; p\_j\; R\_\{ij\}.\; ,$where:$sum\_j\; p\_j\; R\_\{ij\}.\; ,$is the expected payoff for action "i" i.e. the expectation value, and:$mbox\{EMV\}\; =\; max\_i\; ,$is choosing the maximum of these expectations for all available actions.On the other hand, with perfect knowledge of "j", the player may choose a value of "i" that optimizes the expectation for that specific "j". Therefore, the expected value given perfect information is:$mbox\{EV\}|mbox\{PI\}\; =\; sum\_j\; p\_j\; (max\_i\; R\_\{ij\}),\; ,$where $p\_j$ is the probability that the system is in state "j", and $R\_\{ij\}$ is the pay-off if one follows action "i" while the system is in state "j".Here $(max\_i\; R\_\{ij\}),\; ,$ indicates the best choice of action "i" for each state "j".

The expected value of perfect information is the difference between these two quantities,:$mbox\{EVPI\}\; =\; mbox\{EV\}|mbox\{PI\}\; -\; mbox\{EMV\}.\; ,$This difference describes, in expectation, how much larger a value the player can hope to obtain by knowing "j" and picking the best "i" for that "j", as compared to picking a value of "i" before "j" is known.

EVPI provides a criterion by which to judge ordinary mortal forecasters. EVPI can be used to reject costly proposals: if one is offered knowledge for a price larger than EVPI, it would be better to refuse the offer. However, it is less helpful when deciding whether to accept a forecasting offer, because one needs to know the quality of the information one is acquiring.

References

ee also

*Expected value of sample information