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Slides 8a: Introduction

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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

- 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 Ignorance – Payoff Table

Kelly Construction Payoff Table (Prob. 8-17)

Decision Making Under Ignorance

- Maximax
- Select the strategy with the highest possible return

- Maximin
- Select the strategy with the smallest possible loss

- LaPlace-Bayes
- All 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

Kelly Construction

Maximin: The Pessimistic Point of View

- Select the “best of the worst” strategy
- Evaluates each decision by the minimum possible return associated with the decision
- The decision that yields the maximum value of the minimum returns (maximin) is selected

- For “risk averse” decision makers
- A “protect” strategy
- Worst case scenario the focus

Maximin

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 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!)

Using Excel to Calculate EVPI: Formulas View

KellyConstruction

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 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:

Decision

0 1 2 3

0 0 -50 -100 -150

1 -40 35 -15 -65

2 -80 -5 70 20

3 -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

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

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

State of Nature (Demand)

Decision

0 1 2 3

ER

0 0 -50 -100 -150 -85

1 -40 35 -15 -65 -12.5

2 -80 -5 70 20 22.5

and order 2 papers

3 -120 -45 30 105 7.5

Prob. 0.1 0.3 0.4 0.2

Now, the ER is calculated for each decision:

Of these four ER’s, choose the maximum,

Decision

0 1 2 3

0 0 -50 -100 -150

1 -40 35 -15 -65

2 -80 -5 70 20

3 -120 -45 30 105

Prob. 0.1 0.3 0.4 0.2

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

= 59.5

ER(current) = 22.5

EVPI = 59.5 – 22.5 = 37.0 cents

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

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

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

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

CriteriaDecision

Maximin Cash Flow Order 1 paper

Expected Return Order 2 papers

Maximax Cash Flow Order 3 papers

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

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 of

the same amount of money.

Utility functions in decision analysis measure the “attractiveness” of money.

Utilitycan be thought of as a measure of “satisfaction.”

1.0

0.910

0.850

0.775

0.680

0.524

A gain in utility of 0.06

100 200 300 400 500 600 Dollars

Go from $400 to $500 results in

Typical risk-averse utility function:

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 Risk you an additional $100. Note that your utility increases by

Summary:

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?).

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