Chapter 14 Decision Making
Applying Probabilities to the
Decision-Making Process in
the face of uncertainty.
In order to make the best
decision, with the information
available, the decision maker
utilizes certain decision strategies
to evaluate the possible benefits
and losses of each alternative.
When making a decision in the
face of uncertainty, ask:
1) What are my possible
Alternatives or Courses of
Action?
2) How can the future affect each
action?
What are my possible Alternatives
or Courses of Action?
Before selecting a course of action, the decision
maker must have at least two possible
alternatives to evaluate before making his
choice.
I want to invest $1 million for 1 year.
I narrow my choices to three alternatives
(actions):
Example:
Alternative 1: Invest in guaranteed income
certificate paying 10%.
Alternative 2: Invest in a bond with a coupon
value of 8%.
Alternative 3: Invest in a well-diversified
portfolio of stocks.
The Alternatives (Actions) are
under the decision maker’s control.
How can the future affect each
alternative (action)?
Unless you’ve got a crystal
ball
Future uncertainties may derail
the most perfect of plans.
These future events are also
referred to as States of Nature
Example: Economic conditions, foremost among
which is interest rates.
Interest rates increase.
Interest rates stay the same.
Interest rates decrease.
To account for future
uncertainties (events)
We assign probabilities to measure
the likelihood of a future event
occurring.
Example:
Interest rates
Increase
Stay same
Decrease
Probabilities
Probabilities
.2
.5
.3
Future events (states of nature or
outcomes), are out of the
decision maker’s control and
often strictly a matter of
chance.
Yet, the impact of these
events
Affect the payoffs/losses
which determine the decision
making process.
Important Distinction!
The action (alternative)
is under the decision
maker’s control.
The event (state of
nature) that ultimately
occurs is strictly a
matter of chance.
Associated with each alternative (action)
and event (state of nature) is a
corresponding payoff or profit.
Interest rates
Increase
Stay same
Decrease
GIC
$100,000
$100,000
$100,000
PAYOFF TABLE
Bond
-$50,000
$80,000
$180,000
Stocks
$150,000
$ 90,000
$ 40,000
Interest rates
Increase
Stay same
Decrease
GIC
$100,000
$100,000
$100,000
PAYOFF TABLE
Bond
-$50,000
$80,000
$180,000
Stocks
$150,000
$ 90,000
$ 40,000
If I could predict the future with certainty,
I would choose the alternative with the
highest payoff (profit).
Instead of focusing on profits, I could look at the
Opportunity Loss associated with each
combination of an alternative and the economic
condition affecting that alternative’s profitability.
Opportunity Loss
The difference between
the profit I made on the
alternative I chose and the
profit I could have made
had the best decision
been made.
PAYOFF TABLE
Bond
-$50,000
$80,000
$180,000
Interest rates
Increase
Stay same
Decrease
GIC
$100,000
$100,000
$100,000
Interest rates
Increase
Stay same
Decrease
OPPORTUNITY LOSS TABLE
GIC
Bond
$50,000
$200,000
0
20,000
80,000
0
Stocks
$150,000
$ 90,000
$ 40,000
Stocks
0
10,000
140,000
NOTE:
Since Opportunity Loss is the difference
between two decisions, it can not be
expressed as a negative number.
If I could predict the future with certainty,
I would choose the alternative (action)
with the highest payoff or lowest loss.
Interest rates
Increase
Stay same
Decrease
GIC
$100,000
$100,000
$100,000
Interest rates
Increase
Stay same
Decrease
PAYOFF TABLE
Bond
-$50,000
$80,000
$180,000
Stocks
$150,000
$ 90,000
$ 40,000
OPPORTUNITY LOSS TABLE
GIC
Bond
$50,000
$200,000
0
20,000
80,000
0
Stocks
0
10,000
140,000
In many decision problems, it is impossible
to assign Empirical probabilities to the
economic events, or states of nature, that
affect profits and losses. In many cases,
probabilities are assigned Subjectively.
Interest rates
Increase
Stay same
Decrease
Probabilities
.2
.5
.3
Using probabilities, we calculate
the Expected Monetary Value for
each alternative or action.
Expected Monetary Value (EMV)
Interest rates
Increase
Stay same
Decrease
GIC
$100,000
$100,000
$100,000
Interest rates
Increase
Stay same
Decrease
PAYOFF TABLE
Bond
-$50,000
$80,000
$180,000
Stocks
$150,000
$ 90,000
$ 40,000
Probabilities
.2
.5
.3
EXPECTED MONETARY VALUE
EMV (GIC) = .2(100,000) + .5(100,000) + .3(100,000) = $100,000
EMV (Bond) = .2(-50,000) + .5(80,000) + .3(180,000) = $84,000
EMV (Stocks) = .2(150,000) + .5(90,000) + .3(40,000) = $87,000
To maximize profits, choose the
Alternative with the highest EMV.
EXPECTED MONETARY VALUE
EMV (GIC) = .2(100,000) + .5(100,000) + .3(100,000) = $100,000
EMV (Bond) = .2(-50,000) + .5(80,000) + .3(180,000) = $84,000
EMV (Stocks) = .2(150,000) + .5(90,000) + .3(40,000) = $87,000
What does the EMV represent?
If the investment is made a large
number of times (infinite)
with bonds,
* 20% of the investments will result in a $50,000
loss,
* 50% will result in an $80,000 profit, and
* 30 % will result in $180,000 profit.
The average of all these investments is the
EMV of $84,000.
Expected Opportunity Loss
Decision (EOL)
Expected Opportunity Loss (EOL)
Interest rates
Increase
Stay same
Decrease
Interest rates
Increase
Stay same
Decrease
OPPORTUNITY LOSS TABLE
GIC
Bond
$50,000
$200,000
0
20,000
80,000
0
Stocks
0
10,000
140,000
Probabilities
.2
.5
.3
EXPECTED OPPORTUNITY LOSS (EOL) DECISION
EOL (GIC) = .2(50,000) + .5(0) + .3(80,000) = $34,000
EOL (Bond) = .2(200,000) + .5(20,000) + .3(0) = $50,000
EOL (Stocks) = .2(0) + .5(10,000) + .3(140,000) = $47,000
To minimize losses, choose the
Alternative with the lowest EOL.
EXPECTED OPPORTUNITY LOSS (EOL) DECISION
EOL (GIC) = .2(50,000) + .5(0) + .3(80,000) = $34,000
EOL (Bond) = .2(200,000) + .5(20,000) + .3(0) = $50,000
EOL (Stocks) = .2(0) + .5(10,000) + .3(140,000) = $47,000
Example
A vendor at a baseball
game must determine
whether to sell ice cream
or soft drinks at today’s
game. The vendor
believes that the profit
made will depend on
the weather.
Based on past experience at this time
of year, the vendor estimates the
probability of warm weather as 60%.
Event
Cool Weather
Warm Weather
ACTION
Sell Soft Drinks ($)
50
60
Sell Ice Cream ($)
30
90
Event
Cool Weather
Warm Weather
ACTION
Sell Soft Drinks ($)
50
60
Sell Ice Cream ($)
30
90
1) Compute the EMV for selling soft drinks
and selling ice cream.
2) Compute the EOL for selling soft drinks
and ice cream.
3) Based on the previous results, which
should the vendor sell, ice cream or soft
drinks? Why?
EMV
EMV (soft drinks) = .4(50) + .6(60)
= 20 + 36
= $56
EMV (ice cream) = .4(30) + .6(90)
= 12 + 54
= $66
Sell ice cream
Stay tuned
The Value of Additional Information
If we knew in advance which future event
or state of nature would occur, we
would capitalize on this knowledge and
maximize our profits/minimize our
losses.
But our “knowledge” of future events or states of
nature is sometimes tenuous at best.
Leaving us to ask ourselves, “Am I making the
best decision?”
To determine which course of
action (alternative)
to select, we assign
probabilities to the
likelihood of each
future event occurring.
Probabilities are assigned based on :
• past data,
• the subjective opinion of the decision
maker,
• Or knowledge about the probability
distribution that the event may follow.
Better information makes better decisions.
But what are you willing to pay?
Data is costly to acquire:
* Money & time
• Cognitive energy
• Staff effort
• Opportunity costs of failing to do other
things with the money or time.
Goal is to gather data as long as:
the MARGINAL COST is
NO MORE than the
MARGINAL BENEFITS of
the additional data.
At some point, data gathering must stop
and the decision must be made.
The Value of Additional Information
Expected Payoff with Perfect
Information (EPPI)
EPPI is the maximum price
that a decision maker
should be willing to pay
for perfect information.
With perfect information,
I would know what to expect
so I would select the
optimum course of action
for each future event.
Expected Payoff With Perfect
Information (EPPI)
EPPI is also known as Expected Profit under Certainty.
Interest rates
Increase
Stay same
Decrease
Interest rates
Increase
Stay same
Decrease
GIC
$100,000
$100,000
$100,000
PAYOFF TABLE
Bond
-$50,000
$80,000
$180,000
Stocks
$150,000
$ 90,000
$ 40,000
Probabilities
.2
.5
.3
EPPI = .2(150,000) + .5(100,000) + .3(180,000) = $134,000
EXPECTED PROFIT UNDER CERTAINTY is the expected profit you could make if
you have perfect information about WHICH event will occur.
If the Expected Profit Under Certainty (EPPI)
is the profit I expect to make if have perfect information
about which event will occur, how much should I be
willing to pay for this “perfect” information?
EPPI = .2(150,000) + .5(100,000) + .3(180,000) = $134,000
The $134,000 does NOT represent the MAXIMUM
amount I’d be willing to pay for perfect
information because I could have made an
expected profit of EMV= $100,000 WITHOUT
perfect information.
Expected Value of Perfect
Information
EVPI = EPPI – EMV*
= $134,000 - $100,000
= $34,000
If perfect information were available, the
decision maker should be willing to pay up
to $34,000 to acquire it.
Besides profits & losses, is there
something else we should
consider?
Variability!
When comparing two or more actions,
especially with vastly different means,
evaluate the relative risk associated with
each action.
Coefficient of Variation (CV)
Return to Risk Ratio (RRR)
Coefficient of Variation (CV)
Measures the relative size of the variation
compared with the arithmetic mean (EMV).
CV = σ ÷ EMV
σ = √Σ (x- µ)2 · P (X)
Where µ = EMV*
Interest rates
Increase
Stay same
Decrease
Interest rates
Increase
Stay same
Decrease
GIC
$100,000
$100,000
$100,000
PAYOFF TABLE
Bond
-$50,000
$80,000
$180,000
Stocks
$150,000
$ 90,000
$ 40,000
Probabilities
.2
.5
.3
EXPECTED MONETARY VALUE
EMV (GIC) = .2(100,000) + .5(100,000) + .3(100,000) = $100,000
EMV (Bond) = .2(-50,000) + .5(80,000) + .3(180,000) = $84,000
EMV (Stocks) = .2(150,000) + .5(90,000) + .3(40,000) = $87,000
CV = σ ÷ EMV
σ = √Σ (x- µ)2 · P (X)
Where µ = EMV*
Interest rates
Increase
Stay same
Decrease
GIC
$100,000
$100,000
$100,000
PAYOFF TABLE
Bond
-$50,000
$80,000
$180,000
Stocks
$150,000
$ 90,000
$ 40,000
EMV* = $100,000
σstocks = √Σ (x- µ)2 · P (X)
= √(150,000 – 100,000) 2
(40,000 – 100,000) 2 (.3)
= 40,373
(.2) + (90,000 – 100,000) 2 (.5) +
Interest rates
Increase
Stay same
Decrease
GIC
$100,000
$100,000
$100,000
PAYOFF TABLE
Bond
-$50,000
$80,000
$180,000
Stocks
$150,000
$ 90,000
$ 40,000
EMV* = $100,000
σbonds = √Σ (x- µ)2 · P (X)
= √(-50,000 – 100,000) 2
(180,000 – 100,000) 2 (.3)
= 81,363
(.2) + (80,000 – 100,000) 2 (.5) +
Coefficient of Variation
CVstocks = (σ ÷ EMV) 100%
= (40,373 ÷ 100,000) 100%
= 40.4%
CVbonds = (σ ÷ EMV) 100%
= (81,363 ÷ 100,000) 100%
= 81.4%
Return-to-Risk Ratio (RRR)
RRR = EMV ÷ σ
RRR stocks = 100,000 ÷ 40,373 = 2.48
RRR bonds = 100,000 ÷ 81,363 = 1.23
Although Bonds & stocks have a comparable EMV, the
RRR for stocks is substantially higher than bonds & the
stocks’ CV much smaller than that of bonds.
EXPECTED MONETARY VALUE
EMV (GIC) = .2(100,000) + .5(100,000) + .3(100,000) = $100,000
EMV (Bond) = .2(-50,000) + .5(80,000) + .3(180,000) = $84,000
EMV (Stocks) = .2(150,000) + .5(90,000) + .3(40,000) = $87,000
RRR stocks = 100,000 ÷ 40,373 = 2.48
RRR bonds = 100,000 ÷ 81,363 = 1.23
Homework
A baker must decide how many specialty
cakes to bake each morning. From past
experience, she knows that the daily
demand for cakes ranges from 0 to 3.
Each cake costs $3.00 to produce and
sells for $8.00, and any unsold cakes
are thrown in the garbage at the end of
the day.
Set up a payoff table to help the baker
decide how many cakes to bake.
Payoff Table
Produce
Demand
Bakeo
Bake1
Bake2
Bake3
Sello
0
-3.00
-6.00
-9.00
Sell1
0
5.00
2.00
-1.00
Sell2
0
5.00
10.00
7.00
Sell3
0
5.00
10.00
15.00
Opportunity Loss Table
Produce
Demand
Bakeo
Bake1
Bake2
Bake3
Sello
0
3.00
6.00
9.00
Sell1
5.00
0
3.00
6.00
Sell2
10.00
5.00
0
3.00
Sell3
15.00
10.00
5.00
0
Assuming probability of each event is equal:
Sell = .25
EMV(0) = $ 0
EMV(1) = .25(-3) + .25(5) + .25(5) + .25(5)
= $3.00
EMV(2) = .25(-6) + .25(2) +.25(10) + .25(10)
= $4.00
EMV(3) = .25(-9) + .25(-1) +.25(7) + .25(15)
= $3.00
EMV* decision is to bake 2 cakes.
Assuming probability of each event is equal:
Sell = .25
EOL(0) = .25(0) + .25(5) + .25(10) =.25(15)
= $7.50
EOL(1) = .25(3) + .25(0) + .25(5) =.25(10)
= $4.50
EOL(2) = .25(6) + .25(3) + .25(0) =.25(5)
= $3.50
EOL(3) = .25(9) + .25(6) + .25(3) =.25(0)
= $4.50
EOL* decision is to bake 2 cakes.
EVPI = EPPI – EMV*
EPPI= .25(-3) + .25(5) + .25(10) + .25(15)
= $6.75
EMV = $4.00
EVPI = $6.75 – 4.00 = $2.75
Homework
The manager of a large shopping center in
Buffalo is in the process of deciding on
the type of snow clearing service to hire
for his parking lot. Two services are
available. The White Christmas
Company will clear all snowfalls for a
flat fee of $40,000 for the entire winter
season. The Weplowmen Company
charges $18,000 for each snowfall it
clears.
Set up the payoff table to help the manager decide,
assuming that the number of snowfalls per winter
season ranges from 0 to 4.
Payoff Table
Demand Flat Fee
Pay per
snowfall
# of
Snowo
-40,000
0
# of
Snow1
-40,000
-18,000
# of
Snow2
-40,000
-36,000
# of
Snow3
-40,000
-54,000
# of
Snow4
-40,000
-72,000
Using subjective assessments, the manager
has assigned the following probabilities
to the number of snowfalls. Determine
the optimal decision.
•
•
•
•
•
p(0) = .05
p(1) = .15
p(2) = .30
p(3) = .40
p(4) = .10
EMV (flat fee) = - $40,000
EMV (pay per snowfall) = .5(0) + .15(-18,000) +
.3(-36,000) + .4( -54,000) + .1(-72,000)
= -$42,000
EMV* is flat fee
Hopefully, something hit home