Expected value of perfect information

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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.^[1]

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 $R i j$ 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.