Basic Business Statistics (8 th Edition)

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Basic Business Statistics (8 th Edition). Chapter 17 Decision Making. Chapter Topics. The payoff table and decision trees Opportunity loss Criteria for decision making Expected monetary value Expected opportunity loss Return to risk ratio Expected profit under certainty

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Chapter 17

Decision Making

Chapter Topics
• The payoff table and decision trees
• Opportunity loss
• Criteria for decision making
• Expected monetary value
• Expected opportunity loss
• Expected profit under certainty
• Decision making with sample information
• Utility

Features of Decision Making
• List alternative courses of action
• List possible events or outcomes or states of nature
• Determine “payoffs”
• (Associate a payoff with each course of action and each event pair)
• (Evaluate criteria for selecting the best course of action)

List Possible Actions or Events

Two Methods of Listing

Payoff Table

Decision Tree

Payoff Table (Step 1)

Consider a food vendor determining whether to sell soft drinks or hot dogs.

Course of Action (Aj)

Sell Soft Drinks (A1)

Sell Hot Dogs (A2)

Event (Ei)

Cool Weather (E1) x11 =\$50x12 = \$100

Warm Weather (E2) x21 = \$200 x22 = \$125

xij = payoff (profit) for event i and action j

Payoff Table (Step 2):Do Some Actions Dominate?
• Action A “dominates” action B if the payoff of action A is at least as high as that of action B under any event and is higher under at least one event.
• Action A is “inadmissible” if it is dominated by any other action(s).
• Inadmissible actions do not need to be considered.
• Non-dominated actions are called “admissible.”

Action C “dominates” Action D

Payoff Table (Step 2):Do Some Actions Dominate?

(continued)

Course of Action (Aj)Production Process

Event (Ei)

Level of Demand

A B C D

70 80 100 100

120 120 125 120

200 180 160 150

Low

Moderate

High

Decision Tree:Example

Food Vendor Profit Tree Diagram

x11 = \$50

Cool Weather

Warm Weather

Soft Drinks

x21 = \$200

Hot Dogs

x12 = \$100

Cool Weather

Warm Weather

x22 =\$125

Opportunity Loss: Example

Highest possible profit for an event Ei - Actual profit obtained for an action Aj

Opportunity Loss (lij)

Event: Cool Weather

Action: Soft Drinks Profit x11 : \$50

Alternative Action: Hot Dogs Profit x12 : \$100

Opportunity Loss l11 = \$100 - \$50 = \$50

Opportunity Loss l12 = \$100 - \$100 = \$0

Opportunity Loss: Table

Alternative Course of Action

Event Optimal Profit of Sell Soft Drinks Sell Hot Dogs Action Optimal Action

Cool Hot 100 100 - 50 = 50 100 - 100 = 0 Weather Dogs

Warm Soft 200 200 - 200 = 0 200 - 125 = 75 Weather Drinks

Decision Criteria
• Expected monetary value (EMV)
• The expected profit for taking an action Aj
• Expected opportunity loss (EOL)
• The expected loss for taking action Aj
• Expected value of perfect information (EVPI)
• The expected opportunity loss from the best decision

Decision Criteria -- EMV

Expected Monetary Value (EMV) =

Sum(monetary payoffs of events)  (probabilities of the events)

Number of events

N

Vj

Xij

Pi

i = 1

EMVj = expected monetary value of action j

Xi,j = payoff for action j and event i

Pi = probability of event i occurring

Decision Criteria -- EMV Table Example: Food Vendor

PiEvent MV xijPi MV xijPi

Soft Hot

Drinks Dogs

.50 Cool \$50 \$50 .5 = \$25 \$100 \$100.50 = \$50

.50 Warm \$200 \$200 .5 = 100 \$125 \$125.50 = 62.50

EMV Soft Drink = \$125

EMV Hot Dog = \$112.50

Highest EMV = Better alternative

Decision Criteria -- EOL

Expected Opportunity Loss (EOL)

Sum (opportunity losses of events)  (probabilities of events)

N

Lj

lij

Pi

i =1

EOLj= expected opportunity loss of action j

li,j = opportunity loss for action j and event i

Pi = probability of event i occurring

Decision Criteria -- EOL Table Example: Food Vendor

PiEvent Op Loss lijPi Op Loss lijPi

Soft Drinks Hot Dogs

.50 Cool \$50 \$50.50 = \$25 \$0 \$0.50 = \$0

.50 Warm 0 \$0 .50 = \$0 \$75 \$75 .50 = \$37.50

EOL Soft Drinks = \$25

EOL Hot Dogs = \$37.50

Lowest EOL = Better Choice

EVPI
• Expected value of perfect information (EVPI)
• The expected opportunity loss from the best decision
• Represents the maximum amount you are willing to pay to obtain perfect information

Expected Profit Under Certainty - Expected Monetary Value of the Best Alternative

EVPI (should be a positive number)

EVPI Computation

Expected Profit Under Certainty

= .50(\$100) + .50(\$200)

= \$150

Expected Monetary Value of the Best Alternative

= \$125

EVPI = \$150 - \$125 = \$25

= Lowest EOL

= The maximum you would be willing to spend to obtain perfect information

Taking Account of VariabilityExample: Food Vendor

2 for Soft Drink

= (50 -125)2 .5 + (200 -125)2 .5 = 5625

 for Soft Drink = 75

CVfor Soft Drinks = (75/125)  100% = 60%

2 for Hot Dogs = 156.25  for Hot dogs = 12.5

CVfor Hot dogs = (12.5/112.5)  100% = 11.11%

Expresses the relationship between the return (expected payoff) and the risk (standard deviation)

You might want to choose hot dogs. Although soft drinks have the higher Expected Monetary Value, hot dogs have a much larger return to risk ratio and a much smaller CV.

Decision Making in PHStat
• PHStat | decision-making | expected monetary value
• Check the “expected opportunity loss” and “measures of valuation” boxes
• Excel spreadsheet for the food vendor example

Permits revising old probabilities based on new informationDecision Making with Sample Information

Prior

Probability

New

Information

Revised

Probability

Revised Probabilities Example: Food Vendor

Additional Information: Weather forecast is COOL.

When the weather is cool, the forecaster was correct 80% of the time.

When it has been warm, the forecaster was correct 70% of the time.

F1 = Cool forecast

F2 = Warm forecast

E1 = Cool Weather = 0.50

E2 = Warm Weather = 0.50

P(F1 | E1) = 0.80 P(F1 | E2) = 0.30

Prior Probability

Revising Probabilities Example:Food Vendor
• Revised Probability (Bayes’s Theorem)

Revised EMV Table Example: Food Vendor

PiEvent Soft xijPi Hot xijPi

Drinks Dogs

.73 Cool \$50 \$36.50 \$100 \$73

.27 Warm \$200 54 125 33.73

EMV Soft Drink = \$90.50

EMV Hot Dog = \$106.75

Revised probabilities

Highest EMV = Better alternative

Revised EOL Table Example: Food Vendor

PiEvent Op Loss lijPi OP Loss lijPi

Soft Drink Hot Dogs

.73 Cool \$50 \$36.50 \$0 0

.27 Warm 0 \$0 75 20.25

EOL Soft Drinks = 36.50

EOL Hot Dogs = \$20.25

Lowest EOL = Better Choice

Revised EVPI Computation

Expected Profit Under Certainty

= .73(\$100) + .27(\$200)

= \$127

Expected Monetary Value of the Best Alternative

= \$106.75

EPVI = \$127 - \$106.75 = \$20.25

= The maximum you would be willing to spend to obtain perfect information

Taking Account of Variability: Revised Computation

2 for Soft Drinks

= (50 -90.5)2 .73 + (200 -90.5)2 .27 = 4434.75

 for Soft Drinks = 66.59

CVfor Soft Drinks = (66.59/90.5)  100% = 73.6%

2 for Hot Dogs = 123.1875  for Hot dogs = 11.10

CVfor Hot dogs = (11.10/106.75)  100% = 10.4%

You might want to choose Hot Dogs. Hot Dogs have a much larger return to risk ratio.

Revised Decision Makingin PHStat
• PHStat | decision-making | expected monetary value
• Check the “expected opportunity loss” and “measures of valuation” boxes
• Use the revised probabilities
• Excel spreadsheet for the food vendor example

Utility
• Utility is the idea that each incremental \$1 of profit does not have the same value to every individual
• A risk averse person, once reaching a goal, assigns less value to each incremental \$1.
• A risk seeker assigns more value to each incremental \$1.
• A risk neutral person assigns the same value to each incremental \$1.

Three Types of Utility Curves

Utility

Utility

Utility

\$

\$

\$

Risk Averter: Utility rises slower than payoff

Risk Seeker:Utility rises faster than payoff

Risk-Neutral: Maximizes Expected payoff and ignores risk