The following table represents a decision analysis problem (in units of a thousand dollars)
Suppose you can obtain information which will tell you, with certainty, whether or not state 1 (S1) will occur. What is the maximum amount you would be ready to pay to obtain the information about S1? (Using Bayes' Rule)
I already calculated the expected payoffs of each Alternative w/o the perfect information which is E[A1] = $33,000 E[A2] = $29,000 E[A3] = $39,000.
Not really sure how to calculate the expected total payoff with or without the S1 information, or how to determine it's worth? Thank you.
You have access to information that can discriminate for sure whether nature $S$ is in state $s_1$ or not. You can think of this as a conditional probability mass function $q$ over a piece of data $D$ that assumes either value 0 (if $S\neq s_1$) or 1 (if $S=s_1$), e.g. $$ q(D=1 \mid S=s) = \begin{cases} 1 & \text{if } s=s_1 \\ 0 & \text{if } s\neq s_1 \end{cases}. $$ and similarly for $q(D=0 \mid S=s)$.
You may now apply Bayes' rule to determine what is your updated information regarding the state of nature given either $D=0$ or $D=1$. Both of these will have some expected value $U_0$ or $U_1$ computed similarly to the prior information.
Note that both $D=0$ and $D=1$ have a prior probability $P(D=d)$, so the expected utility before actually obtaining the information is $$ P(D=0) U_0 + P(D=1) U_1. $$
The price you should be willing to pay for this information should not exceed the improvement in the expected value above compared to the expected value of the prior information.