1. Decision Analysis (Decision Tables, Utility)
Y. İlker TOPCU, Ph.D.
twitter.com/yitopcu

2. Introduction One dimensional (single criterion) decision making
Single stage vs. multi stage decision making
Decision analysis is an analytical and systematic way to tackle problems
A good decision is based on logic: rational decision maker (DM)

3. Components of Decision Analysis
A state of nature is an actual event that may occur in the future.
A payoff matrix (decision table) is a means of organizing a decision situation, presenting the payoffs from different decisions/alternatives given the various states of nature.

4. Basic Steps in Decision Analysis
Clearly define the problem at hand
List the possible alternatives
Identify the possible state of natures
List the payoff (profit/cost) of alternatives with respect to state of natures
Select one of the mathematical decision analysis methods (models)
Apply the method and make your decision

5. Types of Decision-Making Environments
Type 1: Decision-making under certainty
DM knows with certainty the payoffs of every alternative .
Type 2: Decision-making under uncertainty
DM does not know the probabilities of the various states of nature. Actually s/he knows nothing!
Type 3: Decision-making under risk
DM does know the probabilities of the various states of nature.

6. Decision Making Under Certainty
Instead of state of natures, a true state is known to the decision maker before s/he has to make decision
The optimal choice is to pick an alternative with the highest payoff

7. Decision Making Under Uncertainty
Maximax
Maximin
Criterion of Realism
Equally likelihood
Minimax

8. Decision Table / Payoff Matrix

9. Maximax Choose the alternative with the maximum optimistic level
ok = {oi} = { {vij}}

10. Maximin Choose the alternative with the maximum security level
sk = {si} = { {vij}}

11. Criterion of Realism
Hurwicz suggested to use the optimism-pessimism index (a)
Choose the alternative with the maximum weighted average of optimistic and security levels
{a oi + (1 – a) si} where 0≤a≤1

12. 0.1667 ≤ a ≤ 0.6154  “Construct small plant”
Criterion of Realism
120a-20 = 0  a =
380a-180 = 120a-20  a =
0 ≤ a ≤  “Do nothing”
≤ a ≤  “Construct small plant”
≤ a ≤ 1  “Construct large plant”

13. Equally Likelihood
Laplace argued that “knowing nothing at all about the true state of nature” is equivalent to “all states having equal probability”
Choose the alternative with the maximum row average (expected value)

14. Minimax Choose the alternative with the minimum worst (maximum) regret
Savage defined the regret (opportunity loss) as the difference between
the value resulting from the best action given that state of nature j is the true state and
the value resulting from alternative i with respect to state of nature j
Choose the alternative with the minimum worst (maximum) regret

15. Summary of Example Results
METHOD DECISION
Maximax “Construct large plant”
Maximin “Do nothing”
Criterion of Realism depends on a
Equally likelihood “Construct small plant”
Minimax “Construct small plant”
The appropriate method is dependent on the personality and philosophy of the DM.

16. Solutions with QM for Windows

17. Decision Making Under Risk
Probability
Objective
Subjective
Expected (Monetary) Value
Expected Value of Perfect Information
Expected Opportunity Loss
Utility Theory
Certainty Equivalence
Risk Premium

18. Probability
A probability is a numerical statement about the likelihood that an event will occur
The probability, P, of any event occurring is greater than or equal to 0 and less than or equal to 1:
0  P(event)  1
The sum of the simple probabilities for all possible outcomes of an activity must equal 1

19. Objective Probability
Determined by experiment or observation:
Probability of heads on coin flip
Probability of spades on drawing card from deck
P(A): probability of occurrence of event A
n(A): number of times that event A occurs
n: number of independent and identical repetitions of the experiments or observations
P(A) = n(A) / n

20. Subjective Probability
Determined by an estimate based on the expert’s
personal belief,
judgment,
experience,
and
existing knowledge of a situation

21. Payoff Matrix with Probabilities

22. Expected (Monetary) Value
Choose the alternative with the maximum weighted row average
EV(ai) = vij P(qj)

23. Sensitivity Analysis EV (Large plant) = $200,000P – $180,000 (1 – P)
EV (Small plant) = $100,000P – $20,000(1 – P)
EV (Do nothing) = $0P + $0(1 – P)

24. Sensitivity Analysis Large Plant Small Plant EV P -200000 -150000
-50000
50000
100000
150000
200000
250000
0.2
0.4
0.6
0.8 1 P EV
Point 1
Small Plant
Large Plant
Point 2

25. Expected Value of Perfect Information
A consultant or a further analysis can aid the decision maker by giving exact (perfect) information about the true state: the decision problem is no longer under risk; it will be under certainty.
Is it worthwhile for obtaining perfectly reliable information: is EVPI greater than the fee of the consultant (the cost of the analysis)?
EVPI is the maximum amount a decision maker would pay for additional information

26. Expected Value of Perfect Information
EVPI = EV with perfect information
– Maximum EV under risk
EV with perfect information: 200*.6+0*.4=120
Maximum EV under risk: 52
EVPI = 120 – 52 = 68

27. Expected Opportunity Loss
Choose the alternative with the minimum weighted row average of the regret matrix
EOL(ai) = rij P(qj)

28. Utility Theory
Utility assessment may assign the worst payoff a utility of 0 and the best payoff a utility of 1.
A standard gamble is used to determine utility values: When DM is indifferent between two alternatives, the utility values of them are equal.
Choose the alternative with the maximum expected utility
EU(ai) = u(ai) = u(vij) P(qj)

29. Standard Gamble for Utility Assessment
Best payoff (v*)
u(v*) = 1
Worst payoff (v–)
u(v–) = 0
Certain payoff (v)
u(v) = 1*p+0*(1–p)
(p)
(1–p)
Lottery ticket
Certain money

30. Utility Assessment (1st approach)
Best payoff (v*)
u(v*) = 1
Worst payoff (v–)
u(v–) = 0
Certain payoff (x1)
u(x1) = 0.5
(0.5)
Lottery ticket
Certain money I v*
u(v*) = 1 x1
u(x1) = 0.5 x2
u(x2) = 0.75
(0.5)
Lottery ticket
Certain money II x1
u(x1) = 0.5 v–
u(v–) = 0 x3
u(x3) = 0.25
(0.5)
Lottery ticket
Certain money
In the example:
u(-180) = 0 and u(200) = 1
x1= 100  u(100) = 0.5
x2 = 175  u(175) = 0.75
x3 = 5  u(5) = 0.25
III

31. Expected Utility (Example 1)

32. Utility Assessment (2nd approach)
Best payoff (v*)
u(v*) = 1
Worst payoff (v–)
u(v–) = 0
Certain payoff (vij)
u(vij) = p
(p)
(1–p)
Lottery ticket
Certain money
In the example:
u(-180) = 0 and u(200) = 1
For vij=–20, p=%70  u(–20) = 0.7
For vij=0, p=%75  u(0) = 0.75
For vij=100, p=%90  u(100) = 0.9

33. Expected Utility (Example 2)

34. Preferences for Risk Risk aversion (avoidance)
Risk neutrality (indifference)
Utility
Risk proness (seeking)
Monetary outcome

35. Certainty Equivalence
If a DM is indifferent between accepting a lottery ticket and accepting a sum of certain money, the monetary sum is the Certainty Equivalent (CE) of the lottery
Z is CE of the lottery (Y>Z>X). X Y Z p
1–p
Lottery ticket
Certain money

36. Risk Premium
Risk Premium (RP) of a lottery is the difference between the EV of the lottery and the CE of the lottery
If the DM is risk averse (avoids risk), RP > 0
S/he prefers to receive a sum of money equal to expected value of a lottery than to enter the lottery itself
If the DM is risk prone (seeks risk), RP < 0
S/he prefers to enter a lottery than to receive a sum of money equal to its expected value
If the DM is risk neutral, RP = 0
S/he is indifferent between entering any lottery and receiving a sum of money equal to its expected value