decision analysis chapter 15: hillier and lieberman dr. hurley’s agb 328 course
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Decision AnalysisChapter 15: Hillier and LiebermanDr. Hurley’s AGB 328 Course
Terms to KnowAlternative, State of Nature, Payoff,
Payoff Table, Prior Distribution, Prior Probabilities, Maximin Payoff Criterion, Maximum Likelihood Criterion, Bayes’ Decision Rule, Crossover Point, Posterior Probabilities, Probability Tree Diagram, Expected Value of Perfect Information, Expected Value of Experimentation, Nodes, Branches, Decision Node, Event Node
Terms to Know Cont.Backward Induction Procedure,
Spider Chart, Tornado Chart, Utility Function for Money, Decreasing Marginal Utility for Money, Risk Averse, Increasing Marginal Utility of Money, Risk Neutral, Risk Seekers, Exponential Utility Function
Goferbroke Company ExampleTrying to maximize payoff from land that
may have oil given The company has two options: drill or sell
the landIf the company drills for oil and oil exists,
they expect a payoff of $700KIf the company drills for oil and oil does
not exist, they expect a payoff of -$100KIf the company sells the land it receives
$90K whether the oil exists or notThere is a 1 in 4 chance that oil exists
Payoff Table for Goferbroke
Nature
Alternatives Oil Exists Oil Does Not Exist
Drill $700K -$100K
Sell $90K $90K
Prior Probability 25% 75%
Maximin Payoff CriterionThis criterion identifies the worst
payoff for each decision that you could make and maximizes the highest of these amounts◦For Goferberoke this would be to sell
the landThis criterion is for the very
cautious
Maximum Likelihood CriterionThis criterion requires you to
select the best payoff from the highest likelihood state of nature
For Goferbroke, the best decision based on this criterion is to sell the land
Bayes’ Decision RuleThis criterion calculates the expected value
of each decision and then chooses the maximum of these expected values
For Goferbroke, the expected payoff for drilling is 100K while for selling it is 90K
A nice attribute about Bayes decision rule is that you can conduct a sensitivity analysis to find what probability would cause you to change your decision from the given prior probabilities◦ You can do this by finding the probability that will
cause one decisions expected payoff to equal another decisions expected payoff
Bayes TheoremLet Ai represent the true state is i
where i = 1,2,…,nLet Bj represent the finding/event
j occurring where j = 1,2,…,mLet P(•) represent the probability
operator and P(•|•) represent the conditional probability operator
Then Bayes Theorem states:
Bayes Theorem Using a Tree Diagram
P(A1)
P(A2)
P(B2 |A1)
P(B1 |A1)
P(B1 |A2)
P(B2 |A2)
P(B1|A1)P(A1)
P(B2|A1)P(A1)
P(B1|A2)P(A2)
P(B2|A2)P(A2)
𝑃 (𝐴1|𝐵1 )=𝑃 (𝐵1|𝐴1 )𝑃 (𝐴1)
𝑃 (𝐵1|𝐴1 )𝑃 ( 𝐴1 )+𝑃 (𝐵1|𝐴2 )𝑃 (𝐴2)
Using Bayes Theorem Using a Tree Diagram for GoferrbrokeLet the probability of finding oil
be 25% and 75% for not finding oil given no prior information
Let the probability of finding oil be 60% given information that is favorable to finding oil
Let the probability of finding oil be 40% given information that is not favorable to finding oil
Using Bayes Theorem Using a Tree Diagram for Goferrbroke Cont.Let the probability of not finding
oil be 20% given information that is favorable to finding oil
Let the probability of not finding oil be 80% given information that is not favorable to finding oil
Using Bayes Theorem Using a Tree Diagram for Goferrbroke
P(Oil)=0.25
P(No Oil)=0.75
P(Unfavorable |Oil)=0.4
P(Favorable |Oil)=0.6
P(Favorable |No Oil)= 0.2
P(Unfavorable |No Oil)=0.8
P(Favorable|Oil)P(Oil) = 0.25*0.6 = 0.15
P(Unfavorable|Oil)P(Oil)= 0.25*0.4 = 0.1
P(Favorable|No Oil)P(No Oil)= 0.75*0.2 = 0.15
P(Unfavorable |No Oil)P(No Oil)= 0.75*0.8 = 0.6
In-Class Activity (Not Graded)What are:
◦P(Oil|Unfavorable)◦P(No Oil|Favorable)◦P(No Oil|Unfavorable)
Calculating Expected Payoffs of the Alternatives Given Information from Seismic Study Expected payoff if you drill given that
the findings were unfavorable:
Expected payoff if you sell given that the findings were unfavorable:
Expected payoff if you drill given that the findings were favorable:
Expected payoff if you sell given that the findings were favorable:
Expected Payoff with Perfect Information (EPPI)
◦EPPI calculates the expected value of the decisions made given perfect information This measures assumes that you will have
chosen the best alternative given the state of nature that occurs
Hence you will multiply the probability of the state of nature by the best payoff achievable in that state For the Goferbroke example, if oil exists you would
choose to drill receiving 700 and if oil does not exists you would choose to sell receiving 90
◦Goferbroke’s EPPI = 0.25*700+0.75*90 = 242.5
Expected Value of Perfect Information (EVPI)EVPI = Expected Payoff with Perfect
Information – Expected Payoff without Perfect Information◦Expected Payoff without Perfect Information
is just the value you get by using Bayes Decision Rule of maximizing expected payoff
Goferbroke’s EVPI = 242.5-100=142.5If the seismic survey was a perfect
indicator, you would choose to do it because the EVPI is greater than the cost of the survey
Expected Payoff with Experimentation (EPE)EPE =
◦Where: P(Bj) is the probability that finding j occurs
E(payoff|Bj) represents the expected payoff that you get if finding j occurs Note that this payoff does not factor in the cost of
collecting the needed information
Goferbroke’s EPE = P(Favorable)*E(payoff|Favorable) + P(Unfavorable)*E(payoff|Unfavorable) = 0.3*300 + 0.7*90 = 153
Expected Value of Experimentation (EVE)EVE = expected payoff with
information – expected payoff without experimentation
Goferbroke’s EVE = 153 – 100 = 53◦Since this exceeds the cost of the
information, Goferbroke would proceed with undergoing the survey
Decision TreesDecision trees can be a useful tool when
examining how to make the optimal decisions when there is multiple alternatives to choose from◦ In the trees, you have decision nodes which are
represented as squares and event/chance nodes that are represented by circles
◦You also have the payoffs that occur due to a sequence of decision and event nodes occurring
To solve these decision trees you work your way from the end of the tree to the beginning of the tree
Goferbroke’s Decision Tree ExampleDiscussed in class
In-Class Activity (Not Graded)Do problem 15.2-7Do Problem 15.4-3
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