Decision Analysis

Decision Analysis

A set of alternative actions We may chose whichever we please

A set of possible states of nature Only one will be correct, but we don’t know in

advance A set of outcomes and a value for each

Each is a combination of an alternative action and a state of nature

Value can be monetary or otherwise

Decision Analysis

Certainty Decision Maker knows with certainty what the state of

nature will be - only one possible state of nature Uncertainty / Ignorance

Decision Maker knows all possible states of nature, but does not know probability of occurrence

Risk Decision Maker knows all possible states of nature,

and can assign probability of occurrence for each state

Decision Making Under CertaintyDecision VariableUnits to build 150

Parameter EstimatesCost to build (/unit) 6,000$ Revenue (/unit) 14,000$ Demand (units) 250

Consequence VariablesTotal Revenue 2,100,000$ Total Cost 900,000$

Performance MeasureNet Revenue 1,200,000$

Decision Making Under Ignorance – Payoff Table

Kelly Construction Payoff Table (Prob. 8-17)

Low (50 units) Medium (100 units) High (150 units)

Build 50 400,000 400,000 400,000

Build 100 100,000 800,000 800,000

Build 150 (200,000) 500,000 1,200,000

State of Nature

DemandAlternative Actions

Decision Making Under Ignorance

MaximaxSelect the strategy with the highest possible

return Maximin

Select the strategy with the smallest possible loss

LaPlace-BayesAll states of nature are equally likely to occur. Select alternative with best average payoff

Maximax: The Optimistic Point of View Select the “best of the best” strategy

Evaluates each decision by the maximum possible return associated with that decision (Note: if cost data is used, the minimum return is “best”)

The decision that yields the maximum of these maximum returns (maximax) is then selected

For “risk takers” Doesn’t consider the “down side” risk Ignores the possible losses from the selected

alternative

Maximax Example

Low (50 units) Medium (100 units) High (150 units) Max

Build 50 400,000 400,000 400,000 400,000

Build 100 100,000 800,000 800,000 800,000

Build 150 (200,000) 500,000 1,200,000 1,200,000

State of NatureMaximax CriterionDemand

Alternative Actions

Kelly Construction

Maximin: The Pessimistic Point of View

Select the “best of the worst” strategyEvaluates each decision by the minimum

possible return associated with the decisionThe decision that yields the maximum value

of the minimum returns (maximin) is selected For “risk averse” decision makers

A “protect” strategyWorst case scenario the focus

Maximin

Low (50 units) Medium (100 units) High (150 units) Min

Build 50 400,000 400,000 400,000 400,000

Build 100 100,000 800,000 800,000 100,000

Build 150 (200,000) 500,000 1,200,000 (200,000)

State of NatureMaximin CriterionDemand

Alternative Actions

Kelly Construction

Decision Making Under Risk Expected Return (ER)*

Select the alternative with the highest (long term) expected return

A weighted average of the possible returns for each alternative, with the probabilities used as weights

* Also referred to as Expected Value (EV) or Expected Monetary Value (EMV) **Note that this amount will not be obtained in the short term, or if the decision is a one-time event!

Expected Return

Low (50 units) Medium (100 units) High (150 units) ER

Build 50 400,000 400,000 400,000 400,000

Build 100 100,000 800,000 800,000 660,000

Build 150 (200,000) 500,000 1,200,000 570,000

Probability 0.2 0.5 0.3 1.0

State of NatureExpected

ReturnDemandAlternative

Actions

Expected Value of Perfect Information EVPI measures how much better you could do on

this decision if you could always know when each state of nature would occur, where: EVUPI = Expected Value Under Perfect Information

(also called EVwPI, the EV with perfect information, or EVC, the EV “under certainty”)

EVUII = Expected Value of the best action with imperfect information (also called EVBest )

EVPI = EVUPI – EVUII EVPI tells you how much you are willing to pay for

perfect information (or is the upper limit for what you would pay for additional “imperfect” information!)

Expected Value of Perfect Information

Low (50 units) Medium (100 units) High (150 units) ER

Build 50 400,000 400,000 400,000 400,000

Build 100 100,000 800,000 800,000 660,000

Build 150 (200,000) 500,000 1,200,000 570,000

Probability 0.2 0.5 0.3 1.0

Best Decision 400,000 800,000 1,200,000 840,000

EVPI 180,000

State of NatureExpected

ReturnDemandAlternative

Actions

Using Excel to Calculate EVPI: Formulas View

A B C D E123 Payoffs States of Nature Expected Return4 Alternatives Low (50 units) Medium (100 units) High (150 units) ER5 Build 50 400000 400000 400000 =SUMPRODUCT(B5:D5,B$8:D$8)6 Build 100 100000 800000 800000 =SUMPRODUCT(B6:D6,B$8:D$8)7 Build 150 -200000 500000 1200000 =SUMPRODUCT(B7:D7,B$8:D$8)8 Probability 0.2 0.5 0.39 Best Decision =MAX(B5:B7) =MAX(C5:C7) =MAX(D5:D7)1011 EVwPI = =SUMPRODUCT(B9:D9,B8:D8)12 EVBest = =MAX(E5:E7)13 EVPI = =E11-E1214

Kelly Construction

A newsvendor can buy the Wall Street Journal newspapers for 40 cents each and sell them for 75 cents.

However, he must buy the papers before he knows how many he can actually sell. If he buys more papers than he can sell, he disposes of the excess at no additional cost. If he does not buy enough papers, he loses potential sales now and possibly in the future.

Suppose that the loss of future sales is captured by a loss of goodwill cost of 50 cents per unsatisfied customer.

The Newsvendor Model

The demand distribution is as follows:

P0 = Prob{demand = 0} = 0.1

P1 = Prob{demand = 1} = 0.3

P2 = Prob{demand = 2} = 0.4

P3 = Prob{demand = 3} = 0.2

Each of these four values represent the states of nature. The number of papers ordered is the decision. The returns or payoffs are as follows:

State of Nature (Demand)

0 1 2 3

Decision

0 0 -50 -100 -1501 -40 35 -15 -652 -80 -5 70 203 -120 -45 30 105

Payoff = 75(# papers sold) – 40(# papers ordered) – 50(unmet demand)

Where 75¢ = selling price 40¢ = cost of buying a paper 50¢ = cost of loss of goodwill

Now, the ER is calculated for each decision:

State of Nature (Demand)

0 1 2 3

Decision

0 0 -50 -100 -150 -85 1 -40 35 -15 -65 -12.52 -80 -5 70 20 22.53 -120 -45 30 105 7.5

ER

Prob. 0.1 0.3 0.4 0.2

ER1 = -40(0.1) + 35(0.3) – 15(0.4) – 65(0.2) = -12.5

ER2 = -80(0.1) – 5(0.3) + 70(0.4) + 20(0.2) = 22.5

ER3 = -120(0.1) – 45(0.3) + 30(0.4) – 105(0.2) = 7.5

ER0 = 0(0.1) – 50(0.3) – 100(0.4) – 150(0.2) = -85

Of these four ER’s, choose the maximum,and order 2 papers

ER(new) = 0(0.1) + 35(0.3) + 70(0.4) + 105(0.2)

State of Nature

0 1 2 3

Decision

0 0 -50 -100 -1501 -40 35 -15 -652 -80 -5 70 203 -120 -45 30 105

Prob. 0.1 0.3 0.4 0.2

= 59.5ER(current) = 22.5

EVPI = 59.5 – 22.5 = 37.0 cents

The decision that yields the maximum of these maximum returns (maximax) is then selected.

This method evaluates each decision by the maximum possible return associated with that decision.

Maximax Criterion: The Maximax criterion is an optimistic decision making criterion.

Then, the decision that yields the maximum value of the minimum returns (maximin) is selected.

Maximin Criterion: The Maximin criterion is an extremely conservative, or pessimistic, approach to making decisions.

Maximin evaluates each decision by the minimum possible return associated with the decision.

So, using the 3 criteria, we made the following decisions regarding the newsvendor data:

Criteria DecisionMaximin Cash Flow Order 1 paper

Expected Return Order 2 papers

Maximax Cash Flow Order 3 papers

Most people are risk-averse, which means they would feel that the loss of a certain amount of money would be more painful than the gain ofthe same amount of money. Utility functions in decision analysis measure the “attractiveness” of money.

Utility can be thought of as a measure of “satisfaction.”

THE RATIONALE FOR UTILITY

Utility

1.00.9100.8500.775

0.680

0.524

100 200 300 400 500 600 Dollars

Typical risk-averse utility function:

Go from $400 to $500 results

in

A gain in

utility of

0.06

To illustrate, first suppose you have $100 and someone gives you an additional $100. Note that your utility increases by

U(200) – U(100) = 0.680 – 0.524 = 0.156

Now suppose you start with $400 and someone gives you an additional $100. Now your utility increases by

U(500) – U(400) = 0.910 – 0.850 = 0.060

This illustrates that an additional $100 is less attractive if you have $400 on hand than it is if you start with $100.

Utilities and Decisions under RiskSummary:

Utility is a way to incorporate risk aversion into the expected return calculation.

Calculating a utility function is out of the scope of this course, but it can be calculated by a series of lottery questions (e.g., Would you prefer one million dollars or a 50% chance of earning five million?).