Decision Making Models

Prof. Yongwon Seo

(seoyw@cau.ac.kr)

College of Business Administration, CAU

Decision Theory

• Decision theory problems are characterized by the following:

– A list of alternatives.

– A list of possible future states of nature.

– Payoffs associated with each alternative/state of nature combination.

– Two categories of decision situations:

• Probabilities can be assigned to future occurrences

• Probabilities cannot be assigned to future occurrences

– Decision making criteria

2

Components of Decision Making

• A state of nature is an actual event that may occur in the future.

• A payoff table is a means of organizing a decision situation, presenting the payoffs from different decisions given the various states of nature.

3

DECISION MAKING WITHOUT PROBABILITIES

Decision Making Models

4

[Ex1] Real Estate Investments: Decision Making Without Probabilities

5

Alternatives and States of Nature

• List of Alternatives : Sometimes “Do-nothing” should be considered

– Suppose that a real estate investor must decide on a plan for purchasing a certain piece of property. After careful consideration, the investor has ruled out “do nothing” and is left with the following list of acceptable alternatives:

• 1. Apartment building

• 2. Office building

• 3. Warehouse

• States of nature : possible future conditions (events)

– Suppose that the profitability of the investment is influenced by the future economic conditions. The investor views the possibilities as

• 1. Good economic conditions

• 2. Poor economic conditions

6

Payoff table

• Payoff Table

• Decision Making Criteria

maximax, maximin, minimaxminimax regret, Hurwicz, equal likelihood

7

Maximax Criterion

• In the maximax criterion the decision maker selects the decision that will result in the maximum of maximum payoffs ; an optimistic criterion.

8

Maximin Criterion

• In the maximin criterion the decision maker selects the decision that will reflect the maximum of the minimum payoffs; a pessimistic criterion.

9

Minimax Regret Criterion

• Regret is the difference between the payoff from the best decision and all other decision payoffs.

• An approach that takes all payoffs into account.

10

Original Payoff Table

Regret Table

Minimax Regret Criterion (cont)

• The manager calculates regrets for all alternatives under each state of nature. Then the manager identifies the maximum regret for each alternative.

• Finally, the manager attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret.

11

Hurwicz Criterion

• The approach offers the decision maker a compromise between the maximax and the maximin criteria.

– Requires the decision maker to specify a degree of optimism, in the form of a coefficient of optimism α, with possible values of α ranging from 0 to 1.00.

– The closer the selected value of α is to 1.00, the more optimistic the decision maker is, and the closer the value of α is to 0, the more pessimistic the decision maker is.

– Criteria: α(Best Payoff) + (1-α)(Worst Payoff)

• α=0 : equivalent to maximin

• α=1 : equivalent to maximax

12

Hurwicz Criterion (cont)

The Hurwicz criterion is a compromise between the maximax and maximincriteria.

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. Here, = 0.4.

13

Decision Values

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

Equal Likelihood Criterion

• 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.

14

Decision Values

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

Summary of Criteria Results

■ 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 likelihood Apartment building

15

DECISION MAKING WITH PROBABILITIES

Decision Making Models

16

Decision Making with Probabilities

• Decision making under partial uncertainty

– Distinguished by the present of probabilities for the occurrence of the various states of nature under partial uncertainty.

– The term risk is often used in conjunction with partial uncertainty.

• Sources of probabilities

– Subjective estimates

– Expert opinions

– Historical frequencies

17

Expected Value (EV)

• Expected value (EV) is computed by multiplying each decision outcome under each state of nature by the probability of its occurrence.

18

Expected Opportunity Loss (EOL)

• The Expected Opportunity Loss (EOL) is the expected value of the regret for each decision.

• The alternative with the smallest expected loss is selected as the best choice.

19

Expected Value of Perfect Information

• The expected value of perfect information (EVPI) is the maximum amount a decision maker would pay for additional information.

– upper bound of money to spend to obtain perfect information

• EVPI equals “the expected value given perfect information(EPC: Expected Payoff under Certainty)” minus “the expected value without perfect information(EV)” for the best decision.

– EPC = 0.6 * 100,000 + 0.4 * 30,000 = 72,000

– EVPI = EPC – EV(best) = 72,000 – 44,000 = 28,000

• EVPI always equals the expected opportunity loss (EOL) for the best decision.

– EVPI = EOL(best)

– EOL : loss due to imperfect info.

20

DECISION TREE

Decision Making Models

21

Decision Tree Format

• Decision trees are used by decision makers to obtain a visual portrayal of decision alternatives and their possible consequences.

– : Decision Node

• Branches from : alternatives

– : Event (Chance, Probability) Node

• Branches from : possible futures with corresponding probabilities

Decision Tree: Example

23

Decision Tree: Example (cont)

24

EV in Decision Trees

• 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

• Branches with the greatest expected value are selected.

– Office building is selected as the best decision

– EV=$44,000

25

EV2 = $42,000

EV3 = $44,000

EV4 = $22,000

EV2 = $42,000

EV3 = $44,000

EV4 = $22,000

Decision Tree: Sequential Decision Example

■ A sequential decision tree is used to illustrate a situation requiring a series of decisions.

■ Used where a payoff table, limited to a single decision, cannot be used.

■ The next slide shows the real estate investment example modified to encompass a ten-year period in which several decisions must be made.

26

Sequential Decision Tree Example

27

Sequential Decision Tree Example

28

$2,540,000

$1,390,000

$1,290,000

Sequential Decision Tree Example

29

$2,540,000

$1,390,000

$1,290,000

$1,740,000

$790,000

Sequential Decision Tree Example

30

$1,290,000

$1,360,000

$1,290,000

$1,360,000

Sequential Decision Tree Example

31

$1,160,000

Best Strategy

32

Decision Analysis with Additional Information : Bayesian Analysis

• Bayesian analysis uses additional information to alter the marginal probability of the occurrence of an event.

• In the real estate investment example, using the expected value criterion, the best decision was to purchase the office building with an expected value of $44,000, and EVPI of $28,000.

33

Decision Analysis with Additional Information : Bayesian Analysis (cont)

• A conditional probability is the probability that an event will occur given that another event has already occurred.

• An economic analysis may provide additional information for future economic conditions. The conditional probabilities are known as follows:

g = good economic conditions, p = poor economic conditionsP = positive economic report, N = negative economic report

P(Pg) = .80 P(Ng) = .20

P(Pp) = .10 P(Np) = .90

• Prior probabilities for good or poor economic conditions in the real estate decision:

P(g) = .60; P(p) = .40

34

Decision Trees with Posterior Probabilities

■ A posterior probability is the altered marginal probability of an event based on additional information.

■ Posterior probabilities by Bayes’ rule:

P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)]

= (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923

■ Posterior (revised) probabilities for decision:

P(gP) = .923 P(pP) = .077

P(gN) = .250 P(pN) = .750

■ P 𝑃 = 𝑃 𝑃∩𝑔 +𝑃 𝑃∩𝑝 = 𝑃 𝑃 𝑔 𝑃 𝑔 +𝑃 𝑃 𝑝 𝑃 𝑝 = 0.8∗0.6+0.1∗0.4= 0.52

■ P 𝑁 = 𝑃 𝑁∩𝑔 +𝑃 𝑁∩𝑝 = 𝑃 𝑁 𝑔 𝑃 𝑔 +𝑃 𝑁 𝑝 𝑃 𝑝 = 0.2∗0.6+0.9∗0.4= 0.48

35

Decision Tree with Posterior Probabilities

36

Good Econ.

Poor Econ.

Good Econ.

Poor Econ.

Good Econ.

Poor Econ.

Good Econ.

Poor Econ.

Good Econ.

Poor Econ.

Good Econ.

Poor Econ.

Decision Trees with Posterior Probabilities

37

• What is your best strategy?

EV for simple option vs. strategic decision

• EV for simple option vs. strategic decision

– Without sample information, EV(best) is achieved with office

• EV(office) = $100,000(.6) - 40,000(.4) = $44,000

– With sample information,

• EV(if P office, if N apartment) = $89,220(.52) + 35,000(.48) = $63,194

• The expected value of sample information (EVSI) is the difference between the expected value with and without information:

– EVSI = $63,194 - 44,000 = $19,194

• The efficiency of sample information is the ratio of the expected value of sample information to the expected value of perfect information:

– efficiency = EVSI /EVPI = $19,194/ 28,000 = .68

38