Posterior Analysis
Determining a Decision Strategy
 
The posterior-analysis tree is made much the same as the prior tree, depicting the chronological sequence of events. ACME's first decision is now whether or not to consult BBA. If ACME chooses not to consult, then they should base their manufacturing-plant decision on the prior analysis, which recommended action alternative JR  with an EMV* of $2.7 million. This is shown on the lower branch of the first (leftmost) decision node. In fact, the prior-analysis tree shown on the previous page logically belongs attached to this lower branch.
 
If ACME decides to consult BBA (branch C ), they will receive one of three possible forecasts: FH, FM or FW. (Interestingly, they know the probabilities of getting those forecasts, courtesy of Bayes' Theorem.) Now, once they receive a forecast, they would revise their prior distribution accordingly, proceed to the posterior analysis and determine the optimal decision in light of the given forecast. But since they know beforehand the probability of getting each forecast (the preposteriors), they can do even better: they can determine what their best decision would be under each possible forecast and then take the expected value of the forecast node itself and compare it to the prior EMV*. This will tell them if it's worthwhile to conduct the market research before committing their money. That is to say, Bayes' Theorem allows them to evaluate if the additional information they are thinking of acquiring is in fact worth acquiring.


 
Note that after each forecast branch, ACME must evaluate a mini decision tree almost identical to the one in the prior analysis, except for the revised probabilities. The revised probabilities are conditional on the forecast, so they are different for every mini-tree. The payoffs in each mini-tree have not changed, however. This is because this analysis uses gross payoffspostponing deduction of the cost of consultation until the rollback is completed. Another approach is to use net payoffsdeduct the $0.4 million consulting cost at each end branch of the mini-trees. Either approach is valid. If there is only one deduction in the model, gross analysis is easier. But if the problem involves more than one deduction, especially if incurred costs vary in the different branches, then net analysis is recommended in order to keep accounts clear.
 
The uncertainty node at the end of the C branch (the forecast uncertainty) shows the gross expected value (yellow) and net expected value (aquamarine) given ACME consults BBA. These quantities are called the Gross Expected Value given Imperfect Information (GEV | II) and the Net Expected Value given Imperfect Information (NEV | II), respectively. The term "imperfect information" is used to distinguish real-world information from the idealized "perfect information" of our hypothetical prophet.
 
Finally, notice that since EMV* = 2.7 is greater than NEV | II = 2.6901, ACME should not consult with BBA: the decision is "DO NOT CONSULT BBA". This is because the additional information provides no added expected benefit. In fact, it results in a decrease of expected value.

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The decision turned out to be "DO NOT CONSULT BBA" because the cost of of consulting was too high: BBA's consulting fee was $400,000 (or $0.4 million). But knowing this, ACME could make a counteroffer: Would BBA accept $200,000? Let's suppose that BBA says they could do the job for $300,000. Should ACME accept BBA's new proposition?

To answer this question, take the GEV | II and deduct the new cost of consulting. This gives a new NEV | II = 2.7901, which is greater than than EMV* = 2.7. So now ACME should consult. They also obtain a complete DECISION STRATEGY:

ACME's DECISION STRATEGY (if cost of consulting is $0.3 million)
ACME should consult BBA. Then:
•  If BBA forecasts a HIGH DEMAND, then ACME should build a LARGE plant. Their EMV will be $4.5786 million.
•  If BBA forecasts MEDIUM DEMAND, ACME should build a JUST RIGHT plant. Their EMV will be $3.2545 million.
•  If BBA forecasts LOW DEMAND, then ACME should build a SMALL plant. Their EMV will be $1.3637 million.

ACME does not know what BBA's forecast will be. But no matter what it turns out to be, ACME now knows exactly what it should do. In other words, ACME has a complete decision strategy.

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