# Expected value of perfect information

- 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

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