Decision Making Under Uncertainty:

Pay Off Table and Decision Tree

Decision Making Under Uncertainty

A set of quantitative decision-making

techniques for decision situations

where uncertainty exists

Decision Making

States of nature– events that may occur in the future– decision maker is uncertain which state of

nature will occur– decision maker has no control over the states

of nature

Payoff Table

A method of organizing & illustrating the payoffs from different decisions given various states of nature

A payoff is the outcome of the decision

Payoff Table

States Of Nature

Decision a b

1 Payoff 1a Payoff 1b

2 Payoff 2a Payoff 2b

Decision-Making Models Under Uncertainty

Maximax choose decision with the maximum of the

maximum payoffs Maximin

choose decision with the maximum of the minimum payoffs

Minimax regretchoose decision with the minimum of the

maximum regrets for each alternative

Hurwicz – choose decision in which decision payoffs are

weighted by a coefficient of optimism, – coefficient of optimism () is a measure of a

decision maker’s optimism, from 0 (completely pessimistic) to 1 (completely optimistic)

Equal likelihood (La Place) – choose decision in which each state of nature

is weighted equally

Decision Making Under Uncertainty Example

Expand $ 800,000 $ 500,000

Maintain status quo 1,300,000 -150,000

Sell now 320,000 320,000

States Of Nature

Good Foreign Poor Foreign

Decision Competitive Conditions Competitive Conditions

Maximax Solution

Expand: $ 800,000

Status quo: 1,300,000 Maximum

Sell: 320,000

Decision: Maintain status quo

Maximin Solution

Expand: $ 500,000 Maximum

Status quo: -150,000

Sell: 320,000

Decision: Expand

Minimax Regret Solution

$ 1,300,000 - 800,000 = 500,000 $ 500,000 - $500,000 = 0

1,300,000 - 1,300,000 = 0 500,000 - (-150,000) = 650,000

1,300,000 - 320,000 = 980,000 500,000 - 320,000 = 180,000

Good Foreign Poor Foreign

Competitive Conditions Competitive Conditions

Expand: $ 500,000 MinimumStatus quo: 650,000Sell: 980,000Decision: Expand

Regret Value

Hurwicz Solution

= 0.3, 1- = 0.7

Expand: $ 800,000 (0.3) + 500,000 (0.7) = $590,000 **

Status quo: 1,300,000 (0.3) -150,000 (0.7) = 285,000

Sell: 320,000 (0.3) + 320,000 (0.7) = 320,000

Decision: Expand** Maximum

Equal Likelihood Solution

Two decisions, weight = 0.50 for each state of nature

Expand: $ 800,000 (0.50) + 500,000 (0.50) = $650,000 **

Status quo: 1,300,000 (0.50) -150,000 (0.50) = 575,000

Sell: 320,000 (0.50) + 320,000 (0.50) = 320,000

Decision: Expand

**Maximum

Decision Making With Probabilities

Risk involves assigning probabilities to states of nature

Expected value is a weighted average of decision outcomes in which each future state of nature is assigned a probability of occurrence

Expected Value

EV x p ix ixi

n

where

ix outcome i

p ix probability of outco

( )

1

me i

Expected Value Example

70% probability of good foreign competition

30% probability of poor foreign competition

EV(expand) $ 800,000 (0.70) + 500,000 (0.30)

= $710,000

EV(status quo) $1,300,000 (0.70) - 150,000 (0.30)

= 865,000 Maximum

EV(sell) $ 320,000 (0.70) + 320,000 (0.30)

= 320,000

Decision: Maintain status quo

Case of Pay off Table application

An ICT (Information and communication technology) company wants to analyze the future of its business. There are 4 decision alternatives: expand the company, maintain status quo, decrease the business size up to 50% of the current size and sell the company. From the business analysis there will be two possibilities: good economic condition and bad economic condition. If the economic condition is good the profit of the expansion will be Rp. 900 million and only Rp. 400 million when the economic condition is bad. If the economic condition is good the profit of maintain status quo will be Rp. 1.000 million and only Rp. 50 million when the economic condition is bad. If the economic condition is good the profit of decrease the business will be Rp. 600 million and only Rp. 300 million when the economic condition is bad. When the company is sold the current price is Rp. 350 million. Solve this decision problem by using maximax, maximin, minimax, hurwicz (with alpha = 0.3) and Equal likelihood. Based on the analysis provide your best suggestion.

Sequential Decision Trees

A graphical method for analyzing decision situations that require a sequence of decisions over time

Decision tree consists ofSquare nodes - indicating decision points

Circles nodes - indicating states of nature

Arcs - connecting nodes

Decision tree basics: begin with no uncertainty

Basic setup:Trees run left to right chronologically.Decision nodes are represented as squares.Possible choices are represented as lines (also called branches).The value associated with each choice is at the end of the branch.

North Side

South Side

Japanese

Greek

Vietnam

Thai

Example: deciding where to eat dinner

Assigning values to the nodes involves defining goals.

Example: deciding where to eat dinner

Taste versus Speed

4

3

1

2

1

2

4

3

North Side

South Side

Japanese

Greek

Vietnam

Thai

To solve a tree, work backwards, i.e. right to left.

Example: deciding where to eatdinner

Speed

1

2

4

3

North Side

South Side

Japanese

Greek

Vietnam

Thai

Value =4

Value =4

Value =2

Decision making under uncertainty

Chance nodes are represented by circles.

Probabilities along each branch of a chance node must sum to 1.

Example: a company deciding whetherto go to trial or settle a lawsuit

Go to trial

Settle

Win [p=0.6]

Lose [p= ]

Solving a tree with uncertainty:

The expected value (EV) is the probability-weighted sum of the possible outcomes:

pwinx win payoff + plosex lose payoff

In this tree, “Go to trial” has a cost associated with it that “Settle” does not.

We’re assuming the decision-maker is maximizing expected values.

Go to trial

Settle

Win [p=0.6]

Lose [p=0.4]

-$4M

-$8M

$0

-$.5MEV= -$3.2M

EV= -$3.7M

-$3.7M

Decision tree notation

Go to trial

Settle

Win [p=0.6]

Lose [p=0.4]

-$4M

-$8M

$0

-$.5M

-$4m

-$8.5M

-$.5M

EV= -$3.2M

EV= -$3.7M

Value of optimal decision

Chance nodes(circles)

Terminal valuescorresponding toeach branch (thesum of payoffsalong the branch).

Probabilities(above the branch)

Payoffs(below the branch)

Decision nodes(squares)

-$3.7M

-$4M

Running totalof net expectedpayoffs(below the branch)

Expected value of chance node (or certainty equivalent)

Example of a Decision Tree Problem

A glass factory specializing in crystal is experiencing a substantial backlog, and the firm's management is considering three courses of action:

A) Arrange for subcontractingB) Construct new facilitiesC) Do nothing (no change)

The correct choice depends largely upon demand, which may be low, medium, or high. By consensus, management estimates the respective demand probabilities as 0.1, 0.5, and 0.4.

A glass factory specializing in crystal is experiencing a substantial backlog, and the firm's management is considering three courses of action:

A) Arrange for subcontractingB) Construct new facilitiesC) Do nothing (no change)

The correct choice depends largely upon demand, which may be low, medium, or high. By consensus, management estimates the respective demand probabilities as 0.1, 0.5, and 0.4.

Example of a Decision Tree Problem (Continued): The Payoff Table

0.1 0.5 0.4Low Medium High

A 10 50 90B -120 25 200C 20 40 60

The management also estimates the profits when choosing from the three alternatives (A, B, and C) under the differing probable levels of demand. These profits, in thousands of dollars are presented in the table below:

The management also estimates the profits when choosing from the three alternatives (A, B, and C) under the differing probable levels of demand. These profits, in thousands of dollars are presented in the table below:

Step 1. We start by drawing the three decisions

A

B

C

Step 2. Add our possible states of nature, probabilities, and payoffs

A

B

C

High demand (0.4)

Medium demand (0.5)

Low demand (0.1)

$90$50

$10

High demand (0.4)

Medium demand (0.5)

Low demand (0.1)

$200$25

-$120

High demand (0.4)

Medium demand (0.5)

Low demand (0.1)

$60$40

$20

Step 3. Determine the expected value of each decision

High demand (0.4)High demand (0.4)

Medium demand (0.5)Medium demand (0.5)

Low demand (0.1)Low demand (0.1)

AA

$90$90

$50$50

$10$10

EVA=0.4(90)+0.5(50)+0.1(10)=$62EVA=0.4(90)+0.5(50)+0.1(10)=$62

$62$62

Step 4. Make decision

High demand (0.4)

Medium demand (0.5)

Low demand (0.1)

High demand (0.4)

Medium demand (0.5)

Low demand (0.1)

A

B

CHigh demand (0.4)

Medium demand (0.5)

Low demand (0.1)

$90$50

$10

$200$25

-$120

$60$40

$20

$62

$80.5

$46

Alternative B generates the greatest expected profit, so our choice is B or to construct a new facility

Alternative B generates the greatest expected profit, so our choice is B or to construct a new facility

Format of a Decision Tree

State of nature 1

B

Payoff 1

State of nature 2

Payoff 2

Payoff 3

2

Choose A’1

Choose A’2

Payoff 6State of nature 2

2

Payoff 4

Payoff 5

Choose A’3

Choose A’4

State of nature 1

Choose A

’

Choose A’2

1

Decision PointChance Event

Case of Decision Tree application

See Attached Problem