Expected Value Models
—EMV & EOL—
 
Once a probability distribution has been assessed for each set of uncertain states of natureand this can always be done, subjectively it is straightforward to apply the next step: compute the expected value for each action alternative. Since there are two ways to look at the same problem (actual monetary values and opportunity losses), we can compute the expected values on either one of the payoff tables.
 

Expected Monetary Value
Referring to the original payoff matrix, the formula for expected monetary value (EMV) is:
EMV (Ai  )  =  E (Ai  )  =  Σ p ( Rij  )
where i refers to the matrix's rows and j refers to the columns.

Thus, using the probability distribution derived previously, we obtain:


Alternatively, we can make use of the decision tree:




Expected Opportunity Loss
Recall from the Savage criterion that an opportunity loss is the payoff difference between the best possible outcome under S and the actual outcome resulting from choosing Ai  given that Sj  occurs. Referring now to the opportunity loss matrix, the formula for expected opportunity loss (EOL) is:
EOL (Ai  )  =  E (Ai  )  =  Σj  pj  OLij  )
Obviously, the same probability distribution applies (since the states of nature are the same):


EOL can also be depicted with a decision tree, of course. (Exercise left to the reader.)




Note that for a given probability distribution, the expected payoffs (EMV and EOL) for every action alternative Ai   always add up to a constant. In our case, they always add up to 4.2. Thus, the Max EMV corresponds with the Min EOL.





The 4.2 value represents the Expected Value given Perfect Information (EVgPI), and is obtained as follows:



We obtain the expected value of the above lottery ( the EVgPI ) thusly:


     EVgPI  =  Σj  pj  (Rij*)



The Expected Value of Perfect Information ( EVPI ) Is then:


   EVPI = EVgPI - EMV*


If the above lottery is solved using opportunity losses instead of monetary values, we get:


   EVPI = EOL*

 

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