Question

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Answer 1

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.A common discipline that uses the EVPI concept is health economics. In that context and when looking at a decision of whether to adopt a new treatment technology, there is always some degree of uncertainty surrounding the decision, because there is always a chance that the decision turns out to be wrong. The expected value of perfect information analysis tries to measure the expected cost of that uncertainty, which “can be interpreted as the expected value of perfect information (EVPI), since perfect information can eliminate the possibility of making the wrong decision” at least from a theoretical perspective.

Equation:

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 player, while the column index j describes a random variable that the player does not yet have knowledge of, that has probability p_j of being in state j. If the player is to choose i without knowing the value of j, the best choice is the one that maximizes the expected monetary value:

EMV = max i ∑ j p j R i j {\displaystyle {\mbox{EMV}}=\max _{i}\sum _{j}p_{j}R_{ij}} ${\displaystyle {\mbox{EMV}}=\max _{i}\sum _{j}p_{j}R_{ij}}$

where

∑ j p j R i j {\displaystyle \sum _{j}p_{j}R_{ij}} ${\displaystyle \sum _{j}p_{j}R_{ij}}$

is the expected payoff for action i i.e. the expectation value, and

EMV = max i {\displaystyle {\mbox{EMV}}=\max _{i}} ${\displaystyle {\mbox{EMV}}=\max _{i}}$

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

EV | PI = ∑ j p j ( max i R i j ) , {\displaystyle {\mbox{EV}}|{\mbox{PI}}=\sum _{j}p_{j}(\max _{i}R_{ij}),} ${\displaystyle {\mbox{EV}}|{\mbox{PI}}=\sum _{j}p_{j}(\max _{i}R_{ij}),}$

where p j {\displaystyle p_{j}} $p_{j}$ is the probability that the system is in state j, and R i j {\displaystyle R_{ij}} $R_{ij}$ is the pay-off if one follows action i while the system is in state j. Here ( max i R i j ) , {\displaystyle (\max _{i}R_{ij}),} ${\displaystyle (\max _{i}R_{ij}),}$ indicates the best choice of action i for each state j.

The expected value of perfect information is the difference between these two quantities,

EVPI = EV | PI − EMV . {\displaystyle {\mbox{EVPI}}={\mbox{EV}}|{\mbox{PI}}-{\mbox{EMV}}.} ${\displaystyle {\mbox{EVPI}}={\mbox{EV}}|{\mbox{PI}}-{\mbox{EMV}}.}$

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. Since EV|PI is necessarily greater than or equal to EMV, EVPI is always non-negative.

EVPI provides a criterion by which to judge ordinary imperfectly informed 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.

•EVPI measures how much better you could do on this decision if you could always know when each state of nature would occur, where:

–EVUPI = Expected Value Under Perfect Information (also called EVwPI, the EV with perfect information, or EVC, the EV “under certainty”)

–EVUII = Expected Value of the best action with imperfect information (also called EVBest )

–EVPI = EVUPI – EVUII

•EVPI tells you how much you are willing to pay for perfect information (or is the upper limit for what you would pay for additional “imperfect” information!)

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