Chapter 3

Decision Analysis

Decision Theory

• Decision theory is the analytic and systematic approach for making the best decision.

Features of Decision Making

• Decision making is for __________.a. past b. future c. both past and future

• A decision is about a (an) _________.a. status b. action c. condition

• The process of making decision is a process of __________.a.producing b. manufacturing c. creating

d. cooking e. selecting f. fabricating

Components in Decision Making (1 of 2)

• Alternatives of a decision– A list of choices, one of which will be selected

as the decision by the decision maker.

• States of Nature– Possible conditions that may actually occur

in the future, which will affect the outcome of your decision but are beyond your control.

Components in Decision Making (2 of 2)

• Payoffs– a payoff is the outcome of a decision

alternative under a state of nature. The larger the payoff the better.

• The decision alternatives, states of nature and payoffs are organized in a decision table.

Decision Table for the Thompson Lumber Example

Decision Alternatives

States of Nature

Favorable Market

Unfavorable Market

Build a large plant $200,000 -$180,000

Build a small plant $100,000 -$20,000

Doing nothing $0 $0

Types of Decision Making• Decision making under certainty

– The outcome of a decision alternative is known (i.e., there is only one state of nature.)

• Decision making under risk– The outcome of a decision alternative is not

known, but its probability is known.

• Decision making under uncertainty– The outcome of a decision alternative is not

known, and even its probability is not known.

Decision Making under Uncertainty

• The outcome of a decision alternative is not known, and even its probability is not known.

• A few criteria (approaches) are available for the decision makers to select according to their preferences and personalities.

Criterion 1: Maximax (Optimistic)

• Step 1. Pick maximum payoff of each alternative.

• Step 2. Pick maximum of those maximums in Step 1; its corresponding alternative is the decision.

• “Best of bests”.

Maximax Decision for Thompson Lumber

States of Nature Row

Decision Favorable Unfavorable Maximum

Alternatives market market

Large plant 200,000 –180,000 200,000

Small plant 100,000 –20,000 100,000

Do nothing 0 0 0

Max(Row max’s) = Max(200,000, 100,000, 0) = 200,000.

So, the decision is ‘Large plant’.

For Whom?

• MaxiMax is an approach for:– Risk taker who tends not to give up

attractive opportunities regardless of possible failures, or

– Optimistic decision maker in whose eyes future is bright.

Criterion 2: Maximin (Pessimistic)

• Step 1. Pick minimum payoff of each alternative

• Step 2. Pick the maximum of those minimums in Step 1, its corresponding alternative is the decision

• “Best of worsts”

Maximin Decision for Thompson Lumber

States of Nature Row

Decision Favorable Unfavorable Minimum

Alternatives market market payoffs

Large plant 200,000 –180,000 –180,000

Small plant 100,000 –20,000 –20,000

Do nothing 0 0 0

Max(Row Min’s) = Max(–180,000, –20,000, 0) = 0.

So, the decision is ‘do nothing’.

For Whom?

• MaxiMin is an approach for:– Risk averter who tends to avoid bad

outcomes despite of some possible attractive outcomes; or

– Pessimistic decision maker in whose eyes future is obscure.

Criterion 3: Hurwicz (Realism)

• Step 1. Calculate Hurwicz value for each alternative

• Step 2. Pick the alternative of largest Hurwicz value as the decision.

Hurwicz Value

• Hurwicz value of an alternative

= (row max)() + (row min)(1-)

where (01) is called coefficient of realism.

Decision by Hurwicz Valuefor Thompson Lumber

=0.8States of Nature

Decision Favorable Unfavorable Hurwicz Alternatives market market values

Large plant 200,000 –180,000 124,000Small plant 100,000 –20,000 76,000Do nothing 0 0 0

Max(Hurwicz values) = Max(124,000,76,000,0) = 124,000.So, the decision is ‘large plant’.

For Whom?

• Hurwicz method can be used by decision makers with different preferences on risks. – For a person who tends to take risk, a larger

is used;– For a person who tends to be conservative, a

smaller is used.

• What if = 1?

• What if = 0?

Criterion 4: Equally Likely

• Step 1. Calculate the average payoff for each alternative.

• Step 2. The alternative with highest average if the decision.

Decision by Equally Likelyfor Thompson Lumber

States of Nature Row Decision Favorable Unfavorable Average Alternatives market market

Large plant 200,000 –180,000 10,000Small plant 100,000 –20,000 40,000Do nothing 0 0 0

Max(Row avg’s) = Max(10,000, 40,000, 0) = 40,000.So, the decision is ‘small plant’.

For Whom?

• Equally Likely method is for the decision maker who does not have particular preference on taking or avoiding risks.

Criterion 5: Minimax Regret

• Step 1. Construct a ‘regret table’,

• Step 2. Pick maximum regret of each row in regret table,

• Step 3. Pick minimum of those maximums in Step 2, its corresponding alternative is the decision.

Regret

• Regret is amount you give up due to not picking the best alternative in a given state of nature.

• Regret = Opportunity cost = Opportunity loss

Payoff Table for Thompson Lumber and Column Maximums States of Nature

Decision Favorable Unfavorable

Alternatives market market

Large plant $200,000 -$180,000

Small plant $100,000 -$20,000

Doing nothing $0 $0

Column Max$200,000 $0

Regret Table for Thompson Lumber

States of Nature

Decision Favorable Unfavorable

Alternatives market market

Large plant $0 $180,000

Small plant $100,000 $20,000

Doing nothing $200,000 $0

Minimax Regret Decision for Thompson Lumber

Regret TableStates of Nature

Decision Favorable Unfavorable Row Alternatives market market Maximum

Large plant 0 180,000 180,000Small plant 100,000 20,000 100,000Do nothing 200,000 0 200,000

Min(Row max’s) = Min{180,000, 100,000, 200,000} = 100,000.

So, the minimax regret decision is ‘small plant’.

For Whom?

• MiniMax Regret is an approach for the decision maker who hates the feeling of having regrets.

Decision Making under Risk

• The outcome of a decision alternative is not known, but its probability is known.

Max EMV Approach

• Step 1. Calculate EMV for each alternative.

• Step 2. Pick the alternative with highest EMV as the decision.

EMV – expected monetary value

• EMV of an alternative is the expected value of possible payoffs of that alternative.

• EMV

n=number of states of nature

P(Xi)=probability of the i-th state of nature

Xi=payoff of the alternative under the i-th state of nature

n

iii XPX

1

* )(

Example of Thompson Lumber

States of Nature

Decision Favorable Unfavorable

Alternatives market market EMV

0.5 0.5

Large plant $200,000 -$180,000 10,000

Small plant $100,000 -$20,000 40,000

Doing nothing $0 $0 0

Minimum EOL Approach

Step 1. Generate the opportunity loss table.

Step 2. Calculate the expected value (EOL) for each alternative in the opportunity loss table.

Step 3. Pick up the alternative with the minimum EOL.

Opportunity Loss Table

• Opportunity loss = Regret = Opp. cost

• Opportunity loss table = Regret Table

Payoff Table for the Thompson Lumber Example

Decision Alternatives

States of Nature

Favorable Market

Unfavorable Market

Build a large plant $200,000 -$180,000

Build a small plant $100,000 -$20,000

Doing nothing $0 $0

Opportunity Loss table and EOLfor Thompson Lumber

States of Nature

Decision Favorable Unfavorable

Alternatives market market EOL

0.5 0.5

Large plant $0 $180,000

Small plant $100,000 $20,000

Doing nothing $200,000 $0

Expected Value of Perfect Information (EVPI)

• It is value of additional information for better decision making.

• It is an upper bound on how much to pay for the additional information.

Calculating EVPI

• EVPI = (Exp. payoff with perfect information) –

(Exp. payoff without perfect information)

= EVwPI – EVw/oPI

EVw/oPI

• EVw/oPI is the average payoff you expect to get based only on the information given in the decision table without the help of additional information.

• EVw/oPI = Max (EMV)

EVw/oPI = Maximum EMV

States of Nature

Decision Favorable Unfavorable

Alternatives market market EMV

0.5 0.5

Large plant $200,000 -$180,000 10,000

Small plant $100,000 -$20,000 40,000

Doing nothing $0 $0 0

Since Max EMV = 40,000,

EVw/oPI = 40,000.

EVwPI

• EVwPI is the average payoff you can get if following the perfect information about the state of nature in the future.

• EVwPI

where n=number of states of nature

bi=best payoff of i-th state of nature

Pi=probability of i-th state of nature

n

i

ii Pb1

EVwPI for the Example of Thompson

States of Nature

Decision Favorable Unfavorable

Alternatives market market EMV

0.5 0.5

Large plant $200,000 -$180,000 10,000

Small plant $100,000 -$20,000 40,000

Doing nothing $0 $0 0

bi $200,000 $0

EVwPI = 200,000*0.5+0*0.5 = 100,000

EVPI for Thompson Lumber

• EVwPI = 200,000*0.5 + 0*0.50 = $100,000

• EVw/oPI = Maximum EMV = $40,000

• EVPI = EVwPI – EVw/oPI

= $100,000 – $40,000

= $60,000

EVPI is a Benchmark in Purchasing Additional Information

• EVPI is the maximum $ amount the decision maker would pay to purchase the additional information about the states of nature (from a consulting firm, for example).

What if Information Is Not Perfect?

• In most cases, information about future is not “perfect”. We need to discount EVwPI properly in those cases.

• If you have 80% of confidence on the information, then

Expected Value of Additional Information = EVAI

= EVwPI * 80% - EVw/oPI

Maximum EMV, Minimum EOL, and EVPI

• The decision selected by the Maximum EMV approach is always the same as the decision selected by the Minimum EOL approach. (why?)

• The value of EVPI is equal to the value of minimum EOL. (why?)

An Example

• You can play the game for many times.• Someone offers you perfect information about “landing” at the price of $65 per time. Do you

take it? If not, how much would you pay?• (See the handout of class work)

Land on ‘Head’ Land on ‘Tail’

Guess ‘Head’ $100 - $60

Guess ‘Tail’ - $80 $150

In the Tossing Coin Example

• EMV for “guess Head” = $20.

• EMV for “guess Tail” = $35* (Max EMV).

• EOL for “guess Head” = $105

• EOL for “guess Tail” = $90* (Min EOL)

• EVwPI = $125, EVw/oPI=$35

• EVPI = $90

Maximum average payoff per game

Alt. 1, Guess “Head”

Alt. 2, Guess “Tail”

EMV

EMV

regr

et regr

et

average payoff

average payoff

EO

L EO

LAlternatives

$

125

20

35

How to Set Up a Decision Table

• A decision table is set up by the decision maker.

• Determine decision alternatives and states of nature.

• Determine the payoffs of each alternative under the states of nature.

• See case “Garden Salad” in class work.