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