Decision Analysis. (to p2). What is it? What is the objective? More example Tutorial: 8 th ed:: 5, 18, 26, 37 9 th ed: 3, 12, 17, 24. (to p5). (to p50). What is it?. It concerns with making a decision or an action based on a series of possible outcomes
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What is it?
What is the objective?
8th ed:: 5, 18, 26, 37
9th ed: 3, 12, 17, 24
It concerns with making a decision or an action based on a series of possible outcomes
The possible outcomes are normally presented in a table format called
“ Payoff Table”
A payoff table consists of
Columns: state of nature (ie events)
Rows: choice of decision
It corresponding entries are “payoff values”
Our objective here is to determine which decision should be chosen based on the payoff table
How to achieve it?
Simple two-dimensional cases
Posterior analysis with Bayesian rules
3) minimax regret,
5) equal likelihood
Known as “optimistic” person, he/she will pick the decision that has the maximum of maximum payoffs .
How to derive this decision?
Step1: For each row (or decision), select its max values
Step 2: Choose the decision that has the max value of
Known as the “pessimistic” person, she/he will select the decision that has the maximum of the minimum from the payoff table
Select this action
How to obtain it?
Step 1: For each row (or decision), obtain its min values
Step 2: Choose the decision that has the max value from step 1
Regret is the difference between the payoff from the best decision and all other decision payoffs.
The decision maker attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret.
How to obtain it?
Step 1: Find out the max value for each column
Step 2: Subtract value of step 1 to all entry of that column
Step 3: Find out the max value for each row
Step 4: Choose the decision that has smallest value of step 3
Max 100,000 30,000
(100,000 - 50,000)
(30,000 - 30,000)
Note: this is a regret table.
Step 4: Choose the decision that has smallest value of step 3
Select this decision
- The Hurwicz criterion is a compromise between the maximax and maximin criterion.
- A coefficient of optimism, , is a measure of the decision maker’s optimism.
- The Hurwicz criterion multiplies the best payoff by and the worst payoff by 1- ., for each decision, and the best result is selected.
Apartment building $50,000(.4) + 30,000(.6) = 38,000
Office building $100,000(.4) - 40,000(.6) = 16,000
Warehouse $30,000(.4) + 10,000(.6) = 18,000
- The equal likelihood ( or Laplace) criterion multiplies the decision payoff for each state of nature by an equal weight, thus assuming that the states of nature are equally likely to occur.
Apartment building $50,000(.5) + 30,000(.5) = 40,000
Office building $100,000(.5) - 40,000(.5) = 30,000
Warehouse $30,000(.5) + 10,000(.5) = 20,000
Select this, Max
A dominant decision is one that has a better payoff than another decision under each state of nature.
The appropriate criterion is dependent on the “risk” personality and philosophy of the decision maker.
Criterion Decision (Purchase)
Maximax Office building
Maximin Apartment building
Minimax regret Apartment building
Hurwicz Apartment building
Equal liklihood Apartment building
Here, we need to understand the following concepts:
1) Expected values
2) Expected Opportunity loss
3) Expected values of Perfect Information
(Ex. 18 and 26)
4) Decision Tree
a) basic tree
b) sequential tree
5) Posterior analysis with Bayesian rules
(Ex. 33, 35 and 37)
Table 12.7 Payoff table with Probabilities for States of Nature
Select this, Max
The expected opportunity loss is the expected value of the regret for each decision (Minimax)
EOL(Apartment) = $50,000(.6) + 0(.4) = 30,000
EOL(Office) = $0(.6) + 70,000(.4) = 28,000
EOL(Warehouse) = $70,000(.6) + 20,000(.4) = 50,000
Table 12.8 Regret (Opportunity Loss) Table with Probabilities for States of Nature
Select this, Min
The expected value of perfect information (EVPI) is the maximum amount a decision maker would pay for additional information.
EVPI = (expected value with perfect information) –(expected value without perfect information)
EVPI = the expected opportunity loss (EOL) for the best decision.
How to compute them?
Table 12.9 Payoff Table with Decisions, Given Perfect Information
Best outcome Worst outcome
Best outcome for each column
Decision with perfect information: $100,000(.60) + 30,000(.40) = $72,000 Decision without perfect information: EV(office) = $100,000(.60) - 40,000(.40) = $44,000 EVPI = $72,000 - 44,000 = $28,000 EOL(office) = $0(.60) + 70,000(.4) = $28,000
A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches).Decision Trees (1 of 2)
Payoff Table for Real Estate Investment Example
Decision tree for real estate investment example
The expected value is computed at each probability node:
EV(node 2) = .60($50,000) + .40(30,000) = $42,000
EV(node 3) = .60($100,000) + .40(-40,000) = $44,000
EV(node 4) = .60($30,000) + .40(10,000) = $22,000
Choose branch with max value:
Use to denote not chosen path(s)
A sequential decision tree uses to illustrate a series of decisions.
How to make use of them?
Compute the expected values for each stage
Then choose action that produces the highest value.
Max (1.29M-0.8M, 1.36M-0.2M)
Bayesian analysis deals with posterior information, and it can be computed based on their conditional probabilities
Example: Consider the following example. Using expected value criterion, best decision was to purchase office building with expected value of $444,000, and EVPI of $28,000.
can be obtained here?
Let consider the following information:
g = good economic conditions
p = poor economic conditions
P = positive economic report
N = negative economic report
From Economic analyst,we can obtain the following “conditional probability”, the probability that an event will occur given that another event has already occurred.
P(Pg) = .80
P(Ng) = .20
P(Pp) = .10
P(Np) = .90
From about information, we can further compute the posterior information such as:
How it works?
Total = 0.8+0.2 = 1.0
(A posterior probability is the altered marginal probability of an event based on additional
Example for its application:
Let P(g) = .60; P(p) = .40
Then, the posterior probabilities is:
P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)] = (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923
Systematic way to compute it?
Other posterior (revised) probabilities are:
P(gN) = .250
P(pP) = .077
P(pN) = .750
Where are these values located/appeared in our tree diagram?
Computation of Posterior Probabilities
P(P) = P(P/g)P(g) +P(P/p) P(p) =0.8* 0.6 +0.1* 0.4 = 0.52
- Decision tree below differs from earlier versions in that :
1. Two new branches at beginning of tree represent report outcomes;
2. Newly introduced
Decision tree with posterior probabilities
- EV (apartment building) = $50,000(.923) + 30,000(.077) = $48,460
- EV (strategy) = $89,220(.52) + 35,000(.48) = $63,194
The derivative is obtained
What good are these values?
how much should we pay for, say, sampling a survey
From the decision tree
Table 12.13 Payoff Table for Auto Insurance Example
Expected Cost (insurance) = .992($500) + .008(500) = $500
Expected Cost (no insurance) = .992($0) + .008(10,000) = $80
- Decision should be do not purchase insurance, but people almost always do purchase insurance.
- Utility is a measure of personal satisfaction derived from money.
- Utiles are units of subjective measures of utility.
- Risk averters forgo a high expected value to avoid a low-probability disaster.
- Risk takers take a chance for a bonanza on a very low-probability event in lieu of a sure thing.
a. Determine the best decision without probabilities using the 5 criteria of the chapter.
b. Determine best decision with probabilities assuming .70 probability of good conditions, .30 of poor conditions. Use expected value and expected opportunity loss criteria.
c. Compute expected value of perfect information.
d. Develop a decision tree with expected value at the nodes.
e. Given following, P(Pg) = .70, P(Ng) = .30, P(Pp) = 20, P(Np) = .80, determine posterior probabilities using Bayesians rule.
f. Perform a decision tree analysis using the posterior probability obtained in part e.
Step 1 (part a): Determine Decisions Without Probabilities
Maximax Decision: Maintain status quo
Status quo 1,300,000 (maximum)
Maximin Decision: Expand
Expand $500,000 (maximum)
Status quo -150,000
Minimax Regret Decision: Expand
Expand $500,000 (minimum)
Status quo 650,000
Hurwicz ( = .3) Decision: Expand
Expand $800,000(.3) + 500,000(.7) = $590,000
Status quo $1,300,000(.3) - 150,000(.7) = $285,000
Sell $320,000(.3) + 320,000(.7) = $320,000
Equal Liklihood Decision: Expand
Expand $800,000(.5) + 500,000(.5) = $650,000
Status quo $1,300,000(.5) - 150,000(.5) = $575,000
Sell $320,000(.5) + 320,000(.5) = $320,000
Step 2 (part b): Determine Decisions with EV and EOL
Expected value decision: Maintain status quo
Expand $800,000(.7) + 500,000(.3) = $710,000
Status quo $1,300,000(.7) - 150,000(.3) = $865,000
Sell $320,000(.7) + 320,000(.3) = $320,000
Expected opportunity loss decision: Maintain status quo
Expand $500,000(.7) + 0(.3) = $350,000
Status quo 0(.7) + 650,000(.3) = $195,000
Sell $980,000(.7) + 180,000(.3) = $740,000
Step 3 (part c): Compute EVPI
EV given perfect information = 1,300,000(.7) + 500,000(.3) = $1,060,000
EV without perfect information = $1,300,000(.7) - 150,000(.3) = $865,000
EVPI = $1.060,000 - 865,000 = $195,000
Step 4 (part d): Develop a Decision Tree
Step 5 (part e): Determine Posterior Probabilities
P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)] = (.70)(.70)/[(.70)(.70) + (.20)(.30)] = .891 P(p P) = .109
P(gN) = P(Ng)P(g)/[P(Ng)P(g) + P(Np)P(p)] = (.30)(.70)/[(.30)(.70) + (.80)(.30)] = .467 P(pN) = .533 Step 6 (part f): Perform Decision tree Analysis with Posterior Probabilities