Decision Analysis A method for determining optimal strategies

when faced with several decision alternatives and an uncertain pattern of future events.

The Decision Analysis Approach

Identify the decision alternatives - di

Identify possible future events - sj

mutually exclusive - only one state can occur exhaustive - one of the states must occur

Determine the payoff associated with each decision and each state of nature - Vij

Apply a decision criterion

Types of Decision Making Situations

Decision making under certainty state of nature is known decision is to choose the alternative with the best

payoff

Types of Decision Making Situations

Decision making under uncertainty The decision maker is unable or unwilling to

estimate probabilities Apply a common sense criterion

Decision Making Under Uncertainty

Maximin Criterion (for profits) - pessimistic list minimum payoff for each alternative choose alternative with the largest minimum

payoff

Decision Making Under Uncertainty

Maximax Criterion (for profits) - optimistic list maximum payoff for each alternative choose alternative with the largest maximum

payoff

Decision Making Under Uncertainty

Minimax Regret Criterion calculate the regret for each alternative and each

state list the maximum regret for each alternative choose the alternative with the smallest maximum

regret

Decision Making Under Uncertainty

Minimax Regret Criterion Regret - amount of loss due to making an incorrect

decision - opportunity cost

|| * ijjij VVR

Types of Decision Making Situations

Decision making under risk Expected Value Criterion

compute expected value for each decision alternative

select alternative with “best” expected value

Computing Expected Value Let:

P(sj)=probability of occurrence for state sj

and N=the total number of states

Computing Expected Value

Since the states are mutually exclusive and exhaustive

jsP

sPsPsPsP

j

N

j

Nj

allfor 0)(

and

1)()()()(1

21

Types of Decision Making Situations

Then the expected value of any decision di is

ij

N

j

ji VsPdEV )()(1

Decision Trees A graphical representation of a decision

situation Most useful for sequential decisions

Decision Making Under Risk:Another Criterion

Expected Regret Criterion Compute the regret table Compute the expected regret for each alternative Choose the alternative with the smallest expected

regret The expected regret criterion will always yield

the same decision as the expected value criterion.

Expected Regret Criterion The expected regret for the preferred decision

is equal to the Expected Value of Perfect Information - EVPI

EVPI is the expected value of knowing which state will occur.

EVPI – Alternative to Expected Regret

EVPI – Expected Value of Perfect Information EVwPI – Expected Value with Perfect

Information about the States of Nature EVwoPI – Expected Value without Perfect

Information about the States of Nature EVPI=|EVwPI-EVwoPI|

Bayes Law

In this equation, P(B) is called the prior probability of B and P(B|A) is called the posterior, or sometimes the revised probability of B. The idea here is that we have some initial estimate of the probability of B, we get some additional information about whether A happens or not, and then we use Bayes Law to compute this revised probability of B.

)()|()()|()(

)()|()()|(

)()|()|(

BPBAPBPBAPAP

BPBAPBPBAP

BPBAPABP

Expected Value of Sample Information – EVSI

EVSI – Expected Value of Sample Information EVwSI – Expected Value with Sample

Information about the States of Nature EVwoSI – Expected Value without Sample

Information about the States of Nature EVSI=|EVwSI-EVwoSI|

Efficiency of Sample Information – E

Perfect Information has an efficiency rating of 100%, the efficiency rating E for sample information is computed as follows:

Note: Low efficiency ratings for sample information might lead the decision maker to look for other types of information

100EVPI

EVSIE

Accounting for Risk in Decision Analysis

Mean-Variance

j

m

j

iiji pERrVar

1

)(2

Accounting for Risk in Decision Analysis

Utility Theory replacing the payoffs with a unitless scale that

accounts for both the value of the payoff and the decision makers risk attitude

Risk Aversion A decision maker is risk averse if he/she would

prefer a certain x dollars to a risky alternative with ER=x dollars.

Accounting for Risk in Decision Analysis

Direct assessment of utility Utility functions

parameter.

aversionrisk theis and payoff

theofamount theis where

1)(

exampleFor /

r

x

xU erx