Chapter 12 & Module E

Decision Analysis

&

Game Theory

Components of Decision Making (D.M.)

Decision alternatives - for managers to choose from.

States of nature - that may actually occur in the future regardless of the decision.

Payoffs - payoff of a decision alternative in a state of nature.The components are given in Payoff Tables.

A Payoff Table (It shows payoffs of different decisions at different states of nature)

Investment States of Nature

decision Economy Economy

alternatives good bad

Apartment $ 50,000 $ 30,000

Office 100,000 - 40,000

Warehouse 30,000 10,000

Types of Decision Making (D.M.) - 1

Deterministic D.M. (D.M. under certainty):– Only one “state of nature”,– Payoff of an alternative is known,– Examples:

Problems for LP, IP, transportation, and network flows.

Types of Decision Making (D.M.) - 2

D.M. without probabilities (D.M. under uncertainty):– More than one states of nature;– Payoff of an alternative is not known at the

time of making decision;– Probabilities of states of nature are not

known.

Types of Decision Making (D.M.) - 3

D.M. with probabilities (D.M. under risk)– More than one states of nature;– Payoff of an alternative is not known at the

time of making decision;– Probabilities of states of nature are known

or given.

Types of Decision Making (D.M.) - 4

D.M. in competition (Game theory)– Making decision against a human

competitor.

Decision Making without Probabilities

No information about possibilities of states of nature.

Five criteria (approaches) for a decision maker to choose from, depending on his/her preference.

Criterion 1: Maximax Pick the maximum of the maximums of

payoffs of decision alternatives. (Best of the bests)Investment States of Nature max

decision Economy Economy payoffs

alternatives good bad (bests)

Apartment $ 50,000 $ 30,000 $50,000

Office 100,000 - 40,000 100,000

Warehouse 30,000 10,000 30,000

Decision:

Whom Is MaxiMax for?

MaxiMax method is for optimistic decision makers who tend to grasp every chance of making money, who tend to take risk, who tend to focus on the most fortunate outcome of an alternative and overlook the possible catastrophic outcomes of an alternative.

Criterion 2: Maximin Pick the maximum of the minimums of

payoffs of decision alternatives. (Best of the worsts)Investment States of Nature min

decision Economy Economy payoff

alternatives good bad (worsts)

Apartment $ 50,000 $ 30,000 $30,000

Office 100,000 - 40,000 - 40,000

Warehouse 30,000 10,000 10,000

Decision:

Whom Is MaxiMin for?

MaxiMin method is for pessimistic decision makers who tend to be conservative, who tend to avoid risks, who tend to be more concerned about being hurt by the most unfortunate outcome than the opportunity of being fortunate.

Criterion 3: Minimax Regret

Pick the minimum of the maximums of regrets of decision alternatives. (Best of the worst regrets)

Need to construct a regret table first.

Regret of a decision under a state of nature= (the best payoff under the state of nature)

– (payoff of the decision under the state of nature)

Investment States of Nature decision Economy Economy alternatives good bad

Apartment $ 50,000 $ 30,000Office 100,000 - 40,000Warehouse 30,000 10,000

Payoffs

Investment States of Nature max decision Economy Economy regret alternatives good bad

Apartment $ 50,000 $ 0 $50,000Office 0 70,000 70,000Warehouse 70,000 20,000 70,000

Regrets

Decision:

Whom Is MiniMax Regret for?

MiniMax regret method is for a decision maker who is afraid of being hurt by the feeling of regret and tries to reduce the future regret on his/her current decision to minimum. “I concern more about the regret I’ll have than how much I’ll make or lose.”

Criterion 4: Hurwicz

Pick the maximum of Hurwicz values of decision alternatives. (Best of the weighted averages of the best and the worst)

Hurwicz value of a decision alternative

= (its max payoff)() + (its min payoff)(1-)

where (01) is called coefficient of optimism.

Investment States of Nature decision Economy Economy alternatives good bad

Apartment $ 50,000 $ 30,000Office 100,000 - 40,000Warehouse 30,000 10,000

Payoffs

Investment decision Hurwicz Values alternatives

Apartment 50,000(0.4)+30,000(0.6) = 38,000Office 100,000(0.4)40,000(0.6) = 16,000Warehouse 30,000(0.4)+10,000(0.6) = 18,000

Hurwicz Values with =0.4

Decision:

Whom Is Hurwicz Method for?

Hurwicz method is for an extreme risk taker (=1), an extreme risk averter (=0), and a person between the two extremes ( is somewhere between 1 and 0) .

Criterion 5: Equal Likelihood

Pick the maximum of the average payoffs of decision alternatives. (Best of the plain averages)

Average payoff of a decision alternative

natureofstatesofnumber

ealternativthatofpayoffsallofsum

Investment States of Nature decision Economy Economy alternatives good bad

Apartment $ 50,000 $ 30,000Office 100,000 - 40,000Warehouse 30,000 10,000

Payoffs

Investment decision Average Payoffs alternatives

Apartment (50,000+30,000) / 2 = 40,000Office (100,00040,000) / 2 = 30,000Warehouse (30,000+10,000) / 2 = 20,000

Average Payoffs

Decision:

Whom Is Equally Likelihood for?

Equally likelihood method is for a decision maker who tends to simply use the average payoff to judge an alternative.

Dominated Alternative If alternative A’s payoffs are lower than

alternative B’s payoffs under all states of nature, then A is called a dominated alternative by B.

A dominated alternative can be removed from the payoff table to simplify the problem.

Investment States of Nature decision Economy Economy alternatives good bad

Apartment $ 50,000 $ 30,000Office 100,000 - 40,000Warehouse 30,000 10,000

Decision Making with Probabilities

The probability that each state of nature will actually occur is known.

States of Nature

Investment Economy Economy

decision good bad

alternatives 0.6 0.4

Apartment $ 50,000 $ 30,000

Office 100,000 - 40,000

Warehouse 30,000 10,000

Criterion:Expected Payoff

Select the alternative that has the largest expected payoff.

Expected payoff of an alternative:

n=number of states of nature

Pi=probability of the i-th state of nature

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

n

i

ii PV1

*

Example

Decision Alt’s

Econ

Good

0.6

Econ

Bad

0.4 Expected payoff

Apartment 50,000 30,000

Office 100,000 -40,000

Warehouse 30,000 10,000

Expected Opportunity Loss (EOL)

Each decision alternative has an EOL which is the expected value of the opportunity costs (regrets).

The alternative with minimum EOL has the highest expected payoff.

Investment States of Nature decision Economy Economy alternatives good bad

Apartment $ 50,000 $ 30,000Office 100,000 - 40,000Warehouse 30,000 10,000

Payoffs 0.6 0.4

Investment States of Nature decision Economy Economy alternatives good bad

Apartment $ 50,000 $ 0Office 0 70,000Warehouse 70,000 20,000

Opp Loss Table 0.6 0.4

Example (cont.)

EOL (apartment)

= 50,000*0.6 +0*0.4 = 30,000 EOL (office)

=0*0.6+70,000*0.4 = 28,000 EOL (warehouse)

= 70,000*0.6+20,000*0.4 = 50,000

Minimum EOL = 28,000 that is associated with Office.

(Max Exp. Payoff) vs. (Min EOL)

The alternative with minimum EOL has the highest expected payoff.

The alternative selected by (Max expected payoff) and by (Min EOL) are always same.

Expected Value of Perfect Information (EVPI)

It is a measure of the value of additional information on states of nature.

It tells up to how much you would pay for additional information.

An ExampleIf a consulting firm offers to provide “perfect information

about the future with $5,000, would you take the offer?

States of Nature

Investment Economy Economy

decision good bad

alternatives 0.6 0.4

Apartment $ 50,000 $ 30,000

Office 100,000 - 40,000

Warehouse 30,000 10,000

Another Example You can play the game for many times. What is your rational strategy of “guessing”? Someone offers you perfect information about

“landing” at $65 per time. Do you take it? If not, how much you would pay?

Land on ‘Head’ Land on ‘Tail’

Guess ‘Head’ $100 - $60

Guess ‘Tail’ - $80 $150

Calculating EVPI

EVPI

= EVwPI – EVw/oPI

= (Exp. payoff with perfect information) –

(Exp. payoff without perfect information)

Expected payoff with Perfect InformationEVwPI

where n=number of states of nature hi=highest payoff of i-th state of nature

Pi=probability of i-th state of nature

n

i

ii Ph1

Example for Expected payoff with Perfect Information

States of Nature

Investment Economy Economy

decision good bad

alternatives 0.6 0.4

Apartment $ 50,000 $ 30,000

Office 100,000 - 40,000

Warehouse 30,000 10,000

hi 100,000 30,000

Expected payoff with perfect information

= 100,000*0.6+30,000*0.4 = 72,000

Expected payoff without Perfect Information

Expected payoff of the best alternative selected without using additional information. i.e.,

EVw/oPI = Max Exp. Payoff

Example for Expected payoff without Perfect Information

Decision Alt’s

Econ

Good

0.6

Econ

Bad

0.4 Expected payoff

Apartment 50,000 30,000 42,000

*Office 100,000 -40,000 *44,000

Warehouse 30,000 10,000 22,000

Expected Value of Perfect Information (EVPI) in above Example

EVPI

= EVwPI – EVw/oPI

= 72,000 - 44,000

= $28,000

Example Revisit

0.5 0.5

Land on “Head” Land on “Tail”

Guess “Head” $100 -$60

Guess “Tail” -$80 $150

Up to how much would you pay for a piece of information about result of “landing”?

EVPI is equal to (Min EOL)

EVPI is the expected opportunity loss (EOL) for the selected decision alternative.

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

EVPI is a Benchmark in Bargain

EVPI is the maximum $ amount the decision maker would pay to purchase perfect information.

Value of Imperfect Information

Expected value of imperfect information

= (discounted EVwPI) – EVw/oPI

= (EVwPI * (% of perfection)) – EVw/oPI

Decision Tree Decision tree is used to help make a series of

decisions. A decision tree is composed of decision nodes

(square), chance nodes (circle), and payoff nodes (final or tip nodes).

A decision tree reflects the decision making process and the possible payoffs with different decisions under different states of nature.

Making Decision on a Decision Tree

It is actually a process of marking numbers on nodes.

Mark numbers from right to left. For a chance (circle) node, mark it with its

expected value. For a decision (square) node, select a

decision and mark the node with the number associated with the decision.

Example p.551

Example p.558-559

Game Theory

Game theory is for decision making under competition.

Two or more decision makers are involved, who have conflicting interests.

Two-Person Zero-Sum Game

Two decision makers’ benefits are completely oppositei.e., one person’s gain is another person’s loss

Payoff/penalty table (zero-sum table):– shows “offensive” strategies (in rows) versus

“defensive” strategies (in columns);– gives the gain of row player (loss of column

player), of each possible strategy encounter.

Example 1 (payoff/penalty table)

Athlete Manager’s Strategies

Strategies (Column Strategies)

(row strat.) A B C

1 $50,000 $35,000 $30,000

2 $60,000 $40,000 $20,000

Two-Person Constant-Sum Game

For any strategy encounter, the row player’s payoff and the column player’s payoff add up to a constant C.

It can be converted to a two-person zero-sum game by subtracting half of the constant (i.e. 0.5C) from each payoff.

Example 2 (2-person, constant-sum)

During the 8-9pm time slot, two broadcasting networks are vying for an audience of 100 million viewers, who would watch either of the two networks.

Payoffs of nw1 for the constant-sum of 100(million)

Network 2

Network 1 western Soap Comedy

western 35 15 60

soap 45 58 50

comedy 38 14 70

An equivalent zero-sum table

Network 2

Network 1 western Soap Comedy

western -15 -35 10

soap - 5 8 0

comedy -12 -36 20

Equilibrium Point

In a two-person zero-sum game, if there is a payoff value P such that

P = max{row minimums} = min{column

maximums}

then P is called the equilibrium point, or saddle point, of the game.

Example 3 (equilibrium point)

Athlete Manager’s Strategies

Strategies (Column Strategies)

(row strat.) A B C

1 $50,000 $35,000 $30,000

2 $60,000 $40,000 $20,000

Game with an Equilibrium Point: Pure Strategy

The equilibrium point is the only rational outcome of this game; and its corresponding strategies for the two sides are their best choices, called pure strategy.

The value at the equilibrium point is called the value of the game.

At the equilibrium point, neither side can benefit from a unilateral change in strategy.

Pure Strategy of Example 3

Athlete Manager’s Strategies

Strategies (Column Strategies)

(row strat.) A B C

1 $50,000 $35,000 $30,000

2 $60,000 $40,000 $20,000

Example 4 (2-person, 0-sum)

Row

Players Column Player Strategies

Strategies 1 2 3

1 4 4 10

2 2 3 1

3 6 5 7

Mixed Strategy

If a game does not have an equilibrium, the best strategy would be a mixed strategy.

Game without an Equilibrium Point

A player may benefit from unilateral change for any pure strategy. Therefore, the game would get into a loop.

To break loop, a mixed strategy is applied.

Example:

Company I Company II Strategies

Strategies B C

2 8 4

3 1 7

Mixed Strategy

A mixed strategy for a player is a set of probabilities each for an alternative of the player.

The expected payoff of row player (or the expected loss of column player) is called the value of the game.

Example:

Company I Company II Strategies

Strategies B C

2 8 4

3 1 7

Let mixed strategy for company I be

{0.6, 0.4}; and for Company II be

{0.3, 0.7}.

Equilibrium Mixed StrategyAn equilibrium mixed strategy

makes expected values of any player’s individual strategies identical.

Every game contains one equilibrium mixed strategy.

The equilibrium mixed strategy is the best strategy.

How to Find Equilibrium Mixed Strategy

By linear programming (as introduced in book)

By QM for Windows, – we use this approach.

Both Are Better Off at Equilibrium

At equilibrium, both players are better off, compared to maximin strategy for row player and minimax strategy for column player.

No player would benefit from unilaterally changing the strategy.

A Care-Free Strategy

The row player’s expected gain remains constant as far as he stays with his mixed strategy (no matter what strategy the column player uses).

The column player’s expected loss remains constant as far as he stays with his mixed strategy (no matter what strategy the row player uses).

Unilateral Change from Equilibrium by Column Playerprobability 0.1 0.9

B C

0.6 Strat 2 8 4

0.4 Strat 3 1 7

Unilateral Change from Equilibrium by Column Playerprobability 1.0 0

B C

0.6 Strat 2 8 4

0.4 Strat 3 1 7

Unilateral Change from Equilibrium by Row Player

probability 0.3 0.7

B C

0.2 Strat 2 8 4

0.8 Strat 3 1 7

A Double-Secure Strategy

At the equilibrium, the expected gain or loss will not change unless both players give up their equilibrium strategies.

– Note: Expected gain of row player is always equal to expected loss of column player, even not at the equilibrium, since 0-sum)

Both Leave Their Equilibrium Strategies

probability 0.8 0.2

B C

0.5 Strat 2 8 4

0.5 Strat 3 1 7

Both Leave Their Equilibrium Strategies

probability 0 1

B C

0.2 Strat 2 8 4

0.8 Strat 3 1 7

Penalty for Leaving Equilibrium

It is equilibrium because it discourages any unilateral change.

If a player unilaterally leaves the equilibrium strategy, then– his expected gain or loss would not change,

and– once the change is identified by the competitor,

the competitor can easily beat the non-equilibrium strategy.

Find the Equilibrium Mixed Strategy

Method 1: As on p.573-574 of our text book. The method is limited to 2X2 payoff tables.

Method 2: Linear programming. A general method.

Method we use: Software QM.

Implementation of a Mixed Strategy Applied in the situations where the mixed

strategy would be used many times. Randomly select a strategy each time

according to the probabilities in the strategy.

If you had good information about the payoff table, you could figure out not only your best strategy, but also the best strategy of your competitor (!).

Dominating Strategy vs. Dominated Strategy

For row strategies A and B: If A has a better (larger) payoff than B for any column strategy, then B is dominated by A.

For column strategies X and Y: if X has a better (smaller) payoff than Y for any row strategy, then Y is dominated by X.

A dominated decision can be removed from the payoff table to simplify the problem.

Example:

Company I Company II Strategies

Strategies A B C

1 9 7 2

2 11 8 4

3 4 1 7

Find the Optimal Mixed Strategy in 2X2 Table

Suppose row player has two strategies, 1 and 2, and column player has two strategies, A and B.

For row player:

Let p be probability of selecting row strategy 1. Then the probability of selecting row strategy 2 is (1-p).

Represent EA and EB by p, where EA (EB) is the expected payoff of the row player if the column player chose column strategy A (B).

Set EA = EB , and solve p from the equation.

For column player:

Let p be probability of selecting column strategy A. Then the probability of selecting column strategy B is (1-p).

Represent E1 and E2 by p, where E1 (E2) is the expected payoff of the row player if the column player chose column strategy A (B).

Set E1 = E2 , and solve p from the equation.