1 1 Slide © 2009 South-Western, a part of Cengage Learning Slides by John Loucks St. Edward’s University

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© 2009 South-Western, a part of Cengage Learning© 2009 South-Western, a part of Cengage Learning

Slides by

JohnLoucks

St. Edward’sUniversity

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Chapter 5Chapter 5Utility and Game TheoryUtility and Game Theory

The Meaning of UtilityThe Meaning of Utility Utility and Decision MakingUtility and Decision Making Utility: Other ConsiderationsUtility: Other Considerations Introduction to Game TheoryIntroduction to Game Theory Mixed Strategy GamesMixed Strategy Games

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For the upcoming year, Swofford has three For the upcoming year, Swofford has three real estatereal estateinvestment alternatives, and future real estate investment alternatives, and future real estate prices areprices areuncertain. The possible investment payoffs are uncertain. The possible investment payoffs are below.below.

States of NatureStates of Nature Real Estate Prices:Real Estate Prices:

Go Up Remain Same Go Go Up Remain Same Go DownDown

Decision AlternativeDecision Alternative ss11 ss22 ss33

Make Investment A, Make Investment A, dd11 30,000 20,000 30,000 20,000 -50,000-50,000

Make Investment B, Make Investment B, dd22 50,000 -20,000 50,000 -20,000 -30,000-30,000

Do Not Invest, Do Not Invest, dd33 0 0 0 0 0 0

Probability .3 .5 Probability .3 .5 .2 .2

PAYOFF TABLEPAYOFF TABLE

Example: Swofford, Inc.Example: Swofford, Inc.

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Expected Value (EV) ApproachExpected Value (EV) Approach

If the decision maker is risk neutral the If the decision maker is risk neutral the expected value approach is applicable.expected value approach is applicable.

EV(EV(dd11) = .3(30,000) + .5( 20,000) + .2(-50,000) = ) = .3(30,000) + .5( 20,000) + .2(-50,000) = $9,000$9,000

EV(EV(dd22) = .3(50,000) + .5(-20,000) + .2(-30,000) = -) = .3(50,000) + .5(-20,000) + .2(-30,000) = -$1,000$1,000

EV(EV(dd33) = .3( 0 ) + .5( 0 ) + .2( 0 ) = ) = .3( 0 ) + .5( 0 ) + .2( 0 ) = $0$0

Considering no other factors, the optimal Considering no other factors, the optimal decision appears to decision appears to dd11 with an expected with an expected monetary value of $9,000……. but is it?monetary value of $9,000……. but is it?


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Other considerations:Other considerations:• Swofford’s current financial position is weakSwofford’s current financial position is weak..• The firm’s president believes that, if the next The firm’s president believes that, if the next

investment results in a substantial loss, investment results in a substantial loss, Swofford’s future will be in jeopardy.Swofford’s future will be in jeopardy.

• Quite possibly, the president would select Quite possibly, the president would select dd22 or or dd33 to avoid the possibility of incurring a to avoid the possibility of incurring a $50,000 loss.$50,000 loss.

• A reasonable conclusion is that, if a loss of A reasonable conclusion is that, if a loss of even $30,000 could drive Swofford out of even $30,000 could drive Swofford out of business, the president would select business, the president would select dd33, , believing that both investments A and B are believing that both investments A and B are too risky for Swofford’s current financial too risky for Swofford’s current financial position.position.

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The Meaning of UtilityThe Meaning of Utility

UtilitiesUtilities are used when the decision criteria are used when the decision criteria must be based on more than just expected must be based on more than just expected monetary values.monetary values.

UtilityUtility is a measure of the total worth of a is a measure of the total worth of a particular outcome, reflecting the decision particular outcome, reflecting the decision maker’s attitude towards a collection of factors. maker’s attitude towards a collection of factors.

Some of these factors may be profit, loss, and Some of these factors may be profit, loss, and risk.risk.

This analysis is particularly appropriate in cases This analysis is particularly appropriate in cases where payoffs can assume extremely high or where payoffs can assume extremely high or extremely low values.extremely low values.

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Steps for Determining the Utility of MoneySteps for Determining the Utility of Money

Step 1:Step 1:

Develop a payoff table using monetary values.Develop a payoff table using monetary values.

Step 2:Step 2:

Identify the best and worst payoff values and Identify the best and worst payoff values and assignassign

each a utility value, witheach a utility value, with

UU(best payoff) > (best payoff) > UU(worst payoff).(worst payoff).

Step 3:Step 3: Define the lottery. The best payoff is Define the lottery. The best payoff is obtained obtained with probability with probability pp; the ; the worst is obtained with worst is obtained with probability (1 – probability (1 – pp).).

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Step 1:Step 1: Develop payoff table. Develop payoff table.

Monetary payoff table on earlier slide.Monetary payoff table on earlier slide.

Step 2:Step 2: Assign utility values to best and worst Assign utility values to best and worst payoffs.payoffs.

Utility of Utility of $50,000 = $50,000 = UU((50,000) = 50,000) = 0 0Utility of $50,000 = Utility of $50,000 = UU(50,000) = 10(50,000) = 10

Step 3:Step 3: Define the lottery. Define the lottery.

Swofford obtains a payoff of $50,000 withSwofford obtains a payoff of $50,000 withprobability probability pp and a payoff of and a payoff of $50,000 $50,000

withwithprobability (1 probability (1 pp).).

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Step 4:Step 4:

For every other monetary value For every other monetary value MM in the payoff in the payoff table:table:

4a:4a: Determine the value of Determine the value of pp such that the such that the decision decision maker is indifferent maker is indifferent between a guaranteed between a guaranteed payoff of payoff of MM and and the lottery defined in step 3.the lottery defined in step 3.

4b:4b: Calculate the utility of Calculate the utility of MM::

UU((MM) = ) = pUpU(best payoff) + (1 – (best payoff) + (1 – pp))UU(worst (worst payoff)payoff)


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Establishing the utility for the payoff of $30,000:Establishing the utility for the payoff of $30,000:

Step 4a: Step 4a: Determine the value of Determine the value of pp..

Let us assume that when Let us assume that when pp = 0.95, = 0.95, Swofford’sSwofford’s

president is indifferent between the president is indifferent between the guaranteedguaranteed

payoff of $30,000 and the lottery.payoff of $30,000 and the lottery.

Step 4b:Step 4b: Calculate the utility of Calculate the utility of MM..

UU(30,000)(30,000) == pU pU(50,000) + (1 (50,000) + (1 pp))UU((50,000)50,000)

= 0.95(10) + (0.05)(0)= 0.95(10) + (0.05)(0)

= 9.5= 9.5


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Step 5:Step 5:

Convert the payoff table from monetary values to Convert the payoff table from monetary values to utility values.utility values.

Step 6:Step 6:

Apply the expected utility approach to the utility Apply the expected utility approach to the utility table developed in step 5, and select the decision table developed in step 5, and select the decision alternative with the highest expected utility.alternative with the highest expected utility.

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Step 5:Step 5: Convert payoff table to utility values. Convert payoff table to utility values.

States of NatureStates of Nature Real Estate Prices:Real Estate Prices:

Go Up Remain Same Go Go Up Remain Same Go DownDown

Decision AlternativeDecision Alternative ss11 ss22 ss33

Make Investment A, Make Investment A, dd11 9.5 9.0 9.5 9.0 0 0

Make Investment B, Make Investment B, dd22 10.0 5.5 10.0 5.5 4.0 4.0

Do Not Invest, Do Not Invest, dd33 7.5 7.5 7.5 7.5 7.5 7.5

Probability .3 .5 Probability .3 .5 .2 .2

UTILITY TABLEUTILITY TABLE


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Expected Utility ApproachExpected Utility Approach

Once a utility function has been determined, Once a utility function has been determined, the optimal decision can be chosen using the the optimal decision can be chosen using the expected utility approachexpected utility approach. .

Here, for each decision alternative, the utility Here, for each decision alternative, the utility corresponding to each state of nature is corresponding to each state of nature is multiplied by the probability for that state of multiplied by the probability for that state of nature. nature.

The sum of these products for each decision The sum of these products for each decision alternative represents the expected utility for alternative represents the expected utility for that alternative. that alternative.

The decision alternative with the highest The decision alternative with the highest expected utility is chosen.expected utility is chosen.

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Step 6: Step 6: Apply the expected utility approach. Apply the expected utility approach.

The expected utility for each of the The expected utility for each of the decision alternatives in the Swofford problem decision alternatives in the Swofford problem is:is:

EV(EV(dd11) = .3( 9.5) + .5(9.0) + .2( 0 ) = ) = .3( 9.5) + .5(9.0) + .2( 0 ) = 7.357.35

EV(EV(dd22) = .3(10.0) + .5(5.5) + .2(4.0) = ) = .3(10.0) + .5(5.5) + .2(4.0) = 6.556.55

EV(EV(dd33) = .3( 7.5) + .5(7.5) + .2(7.5) = ) = .3( 7.5) + .5(7.5) + .2(7.5) = 7.507.50

Considering the utility associated with each Considering the utility associated with each possible payoff, the optimal decision is possible payoff, the optimal decision is dd33 with with an expected utility of 7.50.an expected utility of 7.50.


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DecisionDecision ExpectedExpected ExpectedExpected

AlternativeAlternative UtilityUtility ValueValue

Do Not Do Not InvestInvest

7.507.50 00

Investment Investment AA

7.357.35 9,0009,000

Investment Investment BB

6.556.55 -1,000-1,000

Comparison of EU and EV ResultsComparison of EU and EV Results

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Risk Avoiders Versus Risk TakersRisk Avoiders Versus Risk Takers

A A risk avoiderrisk avoider will have a concave utility will have a concave utility function when utility is measured on the function when utility is measured on the vertical axis and monetary value is measured vertical axis and monetary value is measured on the horizontal axis. Individuals purchasing on the horizontal axis. Individuals purchasing insurance exhibit risk avoidance behavior.insurance exhibit risk avoidance behavior.

A A risk takerrisk taker, such as a gambler, pays a , such as a gambler, pays a premium to obtain risk. His/her utility function premium to obtain risk. His/her utility function is convex. This reflects the decision maker’s is convex. This reflects the decision maker’s increasing marginal value of money.increasing marginal value of money.

A A risk neutral decision makerrisk neutral decision maker has a linear utility has a linear utility function. In this case, the expected value function. In this case, the expected value approach can be used.approach can be used.

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Risk Avoiders Versus Risk TakersRisk Avoiders Versus Risk Takers

Most individuals are risk avoiders for some amounts Most individuals are risk avoiders for some amounts of money, risk neutral for other amounts of money, of money, risk neutral for other amounts of money, and risk takers for still other amounts of money. and risk takers for still other amounts of money.

This explains why the same individual will purchase This explains why the same individual will purchase both insurance and also a lottery ticket.both insurance and also a lottery ticket.

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Utility Example 1Utility Example 1

Consider the following three-state, three-decision Consider the following three-state, three-decision problem with the following payoff table in dollars:problem with the following payoff table in dollars:

ss11 ss22 ss33

dd11 +100,000+100,000 +40,000+40,000 -60,000-60,000

dd22 +50,000 +50,000 +20,000+20,000 -30,000-30,000

dd33 +20,000 +20,000 +20,000+20,000 -10,000-10,000

The probabilities for the three states of nature The probabilities for the three states of nature are: are: P(P(ss11) = .1, P() = .1, P(ss22) = .3, and P() = .3, and P(ss33) = .6.) = .6.

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Risk-Neutral Decision MakerRisk-Neutral Decision Maker

If the decision maker is risk neutral the If the decision maker is risk neutral the expected value approach is applicable.expected value approach is applicable.

EV(EV(dd11) = .1(100,000) + .3(40,000) + .6(-60,000) = -) = .1(100,000) + .3(40,000) + .6(-60,000) = -$14,000$14,000

EV(EV(dd22) = .1( 50,000) + .3(20,000) + .6(-30,000) = -) = .1( 50,000) + .3(20,000) + .6(-30,000) = -$ 7,000$ 7,000

EV(EV(dd33) = .1( 20,000) + .3(20,000) + .6(-10,000) = ) = .1( 20,000) + .3(20,000) + .6(-10,000) = +$ 2,000+$ 2,000

The optimal decision is The optimal decision is dd33..


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Decision Makers with Different UtilitiesDecision Makers with Different Utilities

Suppose two decision makers have the Suppose two decision makers have the following utility values:following utility values:

UtilityUtility UtilityUtility

AmountAmount Decision Maker IDecision Maker IDecision Maker IIDecision Maker II $100,000$100,000 100 100 100100 $ 50,000$ 50,000 94 94 58 58 $ 40,000$ 40,000 90 90 50 50 $ 20,000$ 20,000 80 80 35 35 -$ 10,000-$ 10,000 60 60 18 18 -$ 30,000-$ 30,000 40 40 10 10 -$ 60,000-$ 60,000 0 0 0 0

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Graph of the Two Decision Makers’ Utility Curves

Decision Maker IDecision Maker I

Decision Maker IIDecision Maker II

Monetary Value (in $1000’s)Monetary Value (in $1000’s)

UtilityUtility

-60 -40 -20 0 20 40 60 80 100-60 -40 -20 0 20 40 60 80 100

100100

6060

4040

2020

8080

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Decision Maker I• Decision Maker I has a concave utility function.Decision Maker I has a concave utility function.• He/she is a He/she is a risk avoiderrisk avoider. .

Decision Maker II• Decision Maker II has convex utility function.Decision Maker II has convex utility function.• He/she is a He/she is a risk takerrisk taker..

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OptimaOptimall

decisiodecisionn

is is dd33

LargestLargestexpectexpect

ededutilityutility

Expected Utility: Decision Maker IExpected Utility: Decision Maker I

ExpectedExpected

ss11 ss22 ss33 Utility Utility

dd11 100100 90 90 0 0 37.037.0

dd22 94 94 80 80 40 40 57.457.4

dd33 80 80 80 80 60 60 68.068.0

Probability .1Probability .1 .3 .3 .6 .6

Note: Note: dd44 is dominated by is dominated by dd22 and hence is not and hence is not consideredconsidered

Decision Maker I should make decision Decision Maker I should make decision dd33..

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Expected Utility: Decision Maker IIExpected Utility: Decision Maker II

ExpectedExpected

ss11 ss22 ss33 Utility Utility

dd11 100100 50 50 0 0 25.025.0

dd22 58 58 35 35 10 10 22.322.3

dd33 35 35 35 35 18 18 24.824.8

ProbabilityProbability .1 .1 .3 .3 .6 .6

Note: Note: dd44 is dominated by is dominated by dd22 and hence is not and hence is not considered.considered.

Decision Maker II should make decision Decision Maker II should make decision dd11..

OptimaOptimall

decisiodecisionn

is is dd11

LargestLargestexpectexpect

ededutilityutility

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Suppose the probabilities for the three Suppose the probabilities for the three states of nature in Example 1 were changed to: states of nature in Example 1 were changed to:

P(P(ss11) = .5, P() = .5, P(ss22) = .3, and P() = .3, and P(ss33) = .2.) = .2.

• What is the optimal decision for a risk-neutral What is the optimal decision for a risk-neutral decision maker?decision maker?

• What is the optimal decision for Decision Maker What is the optimal decision for Decision Maker I?I? . . . for Decision Maker II?. . . for Decision Maker II?

• What is the value of this decision problem to What is the value of this decision problem to Decision Maker I? . . . to Decision Maker II? Decision Maker I? . . . to Decision Maker II?

• What conclusion can you draw?What conclusion can you draw?

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Risk-Neutral Decision MakerRisk-Neutral Decision Maker

EV(EV(dd11) = .5(100,000) + .3(40,000) + .2(-60,000) = ) = .5(100,000) + .3(40,000) + .2(-60,000) = 50,00050,000

EV(EV(dd22) = .5( 50,000) + .3(20,000) + .2(-30,000) = ) = .5( 50,000) + .3(20,000) + .2(-30,000) = 25,00025,000

EV(EV(dd33) = .5( 20,000) + .3(20,000) + .2(-10,000) = ) = .5( 20,000) + .3(20,000) + .2(-10,000) = 14,00014,000

The risk-neutral optimal decision is The risk-neutral optimal decision is dd11..

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Expected Utility: Decision Maker IExpected Utility: Decision Maker I

EU(EU(dd11) = .5(100) + .3(90) + .2( 0) = 77.0) = .5(100) + .3(90) + .2( 0) = 77.0

EU(EU(dd22) = .5( 94) + .3(80) + .2(40) = ) = .5( 94) + .3(80) + .2(40) = 79.079.0

EU(EU(dd33) = .5( 80) + .3(80) + .2(60) = 76.0) = .5( 80) + .3(80) + .2(60) = 76.0

Decision Maker I’s optimal decision is Decision Maker I’s optimal decision is dd22..

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Expected Utility: Decision Maker IIExpected Utility: Decision Maker II

EU(EU(dd11) = .5(100) + .3(50) + .2( 0) = ) = .5(100) + .3(50) + .2( 0) = 65.065.0

EU(EU(dd22) = .5( 58) + .3(35) + .2(10) = 41.5) = .5( 58) + .3(35) + .2(10) = 41.5

EU(EU(dd33) = .5( 35) + .3(35) + .2(18) = 31.6) = .5( 35) + .3(35) + .2(18) = 31.6

Decision Maker II’s optimal decision is Decision Maker II’s optimal decision is dd11..

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Value of the Decision Problem: Decision Maker IValue of the Decision Problem: Decision Maker I

• Decision Maker I’s optimal expected utility is Decision Maker I’s optimal expected utility is 79. 79.

• He assigned a utility of 80 to +$20,000, and a He assigned a utility of 80 to +$20,000, and a utility of 60 to -$10,000. utility of 60 to -$10,000.

• Linearly interpolating in this range 1 point is Linearly interpolating in this range 1 point is worth $30,000/20 = $1,500. worth $30,000/20 = $1,500.

• Thus a utility of 79 is worth about $20,000 - Thus a utility of 79 is worth about $20,000 - 1,500 = $18,500.1,500 = $18,500.

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Value of the Decision Problem: Decision Maker IIValue of the Decision Problem: Decision Maker II

• Decision Maker II’s optimal expected utility is 65. Decision Maker II’s optimal expected utility is 65. • He assigned a utility of 100 to $100,000, and a He assigned a utility of 100 to $100,000, and a

utility of 58 to $50,000. utility of 58 to $50,000. • In this range, 1 point is worth $50,000/42 = In this range, 1 point is worth $50,000/42 =

$1190. $1190. • Thus a utility of 65 is worth about $50,000 + Thus a utility of 65 is worth about $50,000 +

7(1190) = $58,330.7(1190) = $58,330.

The decision problem is worth more to DecisionThe decision problem is worth more to Decision Maker II (since $58,330 > $18,500).Maker II (since $58,330 > $18,500).

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Expected Monetary ValueExpected Monetary ValueVersus Expected UtilityVersus Expected Utility

Expected monetary value and expected utility Expected monetary value and expected utility will will alwaysalways lead to identical recommendations if lead to identical recommendations if the decision maker is risk neutral.the decision maker is risk neutral.

This result is This result is generallygenerally true if the decision true if the decision maker is maker is almostalmost risk neutral over the range of risk neutral over the range of payoffs in the problem.payoffs in the problem.

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Expected Monetary ValueExpected Monetary ValueVersus Expected UtilityVersus Expected Utility

Generally, when the payoffs fall into a Generally, when the payoffs fall into a “reasonable” range, decision makers express “reasonable” range, decision makers express preferences that agree with the expected preferences that agree with the expected monetary value approach.monetary value approach.

Payoffs fall into a “reasonable” range when the Payoffs fall into a “reasonable” range when the best is not too good and the worst is not too best is not too good and the worst is not too bad.bad.

If the decision maker does not feel the payoffs If the decision maker does not feel the payoffs are reasonable, a utility analysis should be are reasonable, a utility analysis should be considered.considered.

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Introduction to Game TheoryIntroduction to Game Theory

In In decision analysisdecision analysis, a single decision maker , a single decision maker seeks to select an optimal alternative.seeks to select an optimal alternative.

In In game theorygame theory, there are two or more decision , there are two or more decision makers, called players, who compete as makers, called players, who compete as adversaries against each other.adversaries against each other.

It is assumed that each player has the same It is assumed that each player has the same information and will select the strategy that information and will select the strategy that provides the best possible outcome from his provides the best possible outcome from his point of view.point of view.

Each player selects a strategy independently Each player selects a strategy independently without knowing in advance the strategy of the without knowing in advance the strategy of the other player(s).other player(s).

continuecontinue

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Introduction to Game TheoryIntroduction to Game Theory

The combination of the competing strategies The combination of the competing strategies provides the provides the value of the gamevalue of the game to the players. to the players.

Examples of competing players are teams, Examples of competing players are teams, armies, companies, political candidates, and armies, companies, political candidates, and contract bidders.contract bidders.

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Two-personTwo-person means there are two competing means there are two competing players in the game.players in the game.

Zero-sumZero-sum means the gain (or loss) for one means the gain (or loss) for one player is equal to the corresponding loss (or player is equal to the corresponding loss (or gain) for the other player.gain) for the other player.

The gain and loss balance out so that there is a The gain and loss balance out so that there is a zero-sum for the game.zero-sum for the game.

What one player wins, the other player loses.What one player wins, the other player loses.

Two-Person Zero-Sum GameTwo-Person Zero-Sum Game

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Competing for Vehicle SalesCompeting for Vehicle Sales

Suppose that there are only two vehicle Suppose that there are only two vehicle dealer-ships in a small city. Each dealership dealer-ships in a small city. Each dealership is consideringis considering

three strategies that are designed to take three strategies that are designed to take sales ofsales of

new vehicles from the other dealership over anew vehicles from the other dealership over a

four-month period. The strategies, assumed four-month period. The strategies, assumed to beto be

the same for both dealerships, are on the the same for both dealerships, are on the next slide.next slide.

Two-Person Zero-Sum Game ExampleTwo-Person Zero-Sum Game Example

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Strategy ChoicesStrategy Choices

Strategy 1: Offer a Strategy 1: Offer a cash rebatecash rebate on a new on a new vehicle.vehicle. Strategy 2: Offer Strategy 2: Offer free optional equipmentfree optional equipment on aon a

new vehicle.new vehicle. Strategy 3: Offer a Strategy 3: Offer a 0% loan0% loan on a new on a new vehicle.vehicle.


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2 2 2 2 1 1

CashCashRebateRebate

bb11

0%0%LoanLoan

bb33

FreeFreeOptionsOptions

bb22

Dealership BDealership B

Payoff Table: Number of Vehicle SalesPayoff Table: Number of Vehicle Sales Gained Per Week by Gained Per Week by

Dealership ADealership A (or Lost Per Week by (or Lost Per Week by

Dealership B) Dealership B)

-3 3 -3 3 -1 -1 3 -2 3 -2 0 0

Cash Rebate Cash Rebate aa11

Free Options Free Options aa22

0% Loan 0% Loan aa33

Dealership ADealership A


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Step 1:Step 1: Identify the minimum payoff for each Identify the minimum payoff for each

row (for Player A).row (for Player A).

Step 2:Step 2: For Player A, select the strategy that For Player A, select the strategy that providesprovides

the maximum of the row minimums the maximum of the row minimums (called(called

the the maximinmaximin).).


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Identifying Maximin and Best Strategy Identifying Maximin and Best Strategy

RowRowMinimumMinimum

11

-3-3

-2-2

2 2 2 2 1 1


bb11

0%0%LoanLoan

bb33


bb22


-3 3 -3 3 -1 -1 3 -2 3 -2 0 0





Best Best StrategyStrategy

For Player AFor Player A

MaximinMaximinPayoffPayoff


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Step 3:Step 3: Identify the maximum payoff for each Identify the maximum payoff for each columncolumn

(for Player B).(for Player B). Step 4:Step 4: For Player B, select the strategy that For Player B, select the strategy that

providesprovides

the minimum of the column the minimum of the column maximumsmaximums

(called the (called the minimaxminimax).).


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Identifying Minimax and Best Identifying Minimax and Best Strategy Strategy

2 2 2 2 1 1


bb11

0%0%LoanLoan

bb33


bb22


-3 3 -3 3 -1 -1 3 -2 3 -2 0 0





Column MaximumColumn Maximum 3 3 3 3

1 1

Best Best StrategyStrategy

For Player BFor Player B

MinimaxMinimaxPayoffPayoff


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Pure StrategyPure Strategy

Whenever an optimal Whenever an optimal pure strategypure strategy exists: exists: the maximum of the row minimums equals the maximum of the row minimums equals

the minimum of the column maximums the minimum of the column maximums (Player A’s (Player A’s maximinmaximin equals Player B’s equals Player B’s minimaxminimax))

the game is said to have a the game is said to have a saddle pointsaddle point (the (the intersection of the optimal strategies)intersection of the optimal strategies)

the value of the saddle point is the the value of the saddle point is the value of value of the gamethe game

neither player can improve his/her outcome neither player can improve his/her outcome by changing strategies even if he/she learns by changing strategies even if he/she learns in advance the opponent’s strategyin advance the opponent’s strategy

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11

-3-3

-2-2


bb11

0%0%LoanLoan

bb33


bb22


-3 3 -3 3 -1 -1 3 -2 3 -2 0 0





Column MaximumColumn Maximum 3 3 3 3

1 1

Pure Strategy ExamplePure Strategy Example

Saddle Point and Value of the Saddle Point and Value of the GameGame

2 2 2 2 1 1

SaddleSaddlePointPoint

Value of Value of thethe

game is 1game is 1

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Pure Strategy ExamplePure Strategy Example

Pure Strategy SummaryPure Strategy Summary Player A should choose Strategy Player A should choose Strategy aa11 (offer a (offer a

cash rebate).cash rebate). Player A can expect a Player A can expect a gaingain of of at leastat least 1 1

vehicle sale per week.vehicle sale per week. Player B should choose Strategy Player B should choose Strategy bb33 (offer a (offer a

0% loan).0% loan). Player B can expect a Player B can expect a lossloss of of no more thanno more than

1 vehicle sale per week.1 vehicle sale per week.

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Mixed StrategyMixed Strategy

If the maximin value for Player A does not equal If the maximin value for Player A does not equal the minimax value for Player B, then a pure the minimax value for Player B, then a pure strategy is not optimal for the game.strategy is not optimal for the game.

In this case, a In this case, a mixed strategymixed strategy is best. is best. With a mixed strategy, each player employs With a mixed strategy, each player employs

more than one strategy.more than one strategy. Each player should use one strategy some of Each player should use one strategy some of

the time and other strategies the rest of the the time and other strategies the rest of the time.time.

The optimal solution is the relative frequencies The optimal solution is the relative frequencies with which each player should use his possible with which each player should use his possible strategies.strategies.

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Mixed Strategy ExampleMixed Strategy Example

bb11 bb22

Player BPlayer B

11 11 55

aa11

aa22

Player APlayer A

4 4 88

Consider the following two-person zero-sum Consider the following two-person zero-sum game. The maximin does not equal the game. The maximin does not equal the minimax. There is not an optimal pure minimax. There is not an optimal pure strategy. strategy.

ColumnColumnMaximumMaximum 11 11

88


44

55

MaximinMaximin

MinimaxMinimax

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pp = the probability Player A selects strategy = the probability Player A selects strategy aa11

(1 (1 pp) = the probability Player A selects strategy ) = the probability Player A selects strategy aa22

If Player B selects If Player B selects bb11::

EV = 4EV = 4pp + 11(1 – + 11(1 – pp))

If Player B selects If Player B selects bb22::

EV = 8EV = 8pp + 5(1 – + 5(1 – pp))

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44pp + 11(1 – + 11(1 – pp) = 8) = 8pp + 5(1 – + 5(1 – pp))

To solve for the optimal probabilities for Player ATo solve for the optimal probabilities for Player Awe set the two expected values equal and solve forwe set the two expected values equal and solve forthe value of the value of pp..

44pp + 11 – 11 + 11 – 11pp = 8 = 8pp + 5 – 5 + 5 – 5pp

11 – 711 – 7pp = 5 + 3 = 5 + 3pp

-10-10pp = -6 = -6pp = .6 = .6

Player A should select:Player A should select: Strategy Strategy aa11 with a .6 with a .6 probability andprobability and Strategy Strategy aa22 with a .4 with a .4 probability.probability.

Hence,Hence,(1 (1 p p) ) = .4= .4

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qq = the probability Player B selects strategy = the probability Player B selects strategy bb11

(1 (1 qq) = the probability Player B selects strategy ) = the probability Player B selects strategy bb22

If Player A selects If Player A selects aa11::

EV = 4EV = 4qq + 8(1 – + 8(1 – qq))

If Player A selects If Player A selects aa22::

EV = 11EV = 11qq + 5(1 – + 5(1 – qq))

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44qq + 8(1 – + 8(1 – qq) = 11) = 11qq + 5(1 – + 5(1 – qq))

To solve for the optimal probabilities for Player BTo solve for the optimal probabilities for Player Bwe set the two expected values equal and solve forwe set the two expected values equal and solve forthe value of the value of qq..

44qq + 8 – 8 + 8 – 8qq = 11 = 11qq + 5 – 5 + 5 – 5qq

8 – 48 – 4qq = 5 + 6 = 5 + 6qq

-10-10qq = -3 = -3qq = .3 = .3

Hence,Hence,(1 (1 q q) ) = .7= .7

Player B should select:Player B should select: Strategy Strategy bb11 with a .3 with a .3 probability andprobability and Strategy Strategy bb22 with a .7 with a .7 probability.probability.

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Value of the GameValue of the Game

For Player A:For Player A:

EV = 4EV = 4pp + 11(1 – + 11(1 – pp) = 4(.6) + 11(.4) = 6.8) = 4(.6) + 11(.4) = 6.8

For Player B:For Player B:

EV = 4EV = 4qq + 8(1 – + 8(1 – qq) = 4(.3) + 8(.7) = 6.8) = 4(.3) + 8(.7) = 6.8

Expected Expected gaingain

per gameper gamefor Player Afor Player A

Expected Expected lossloss

per gameper gamefor Player Bfor Player B

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Dominated Strategies ExampleDominated Strategies Example


-2-2

00

-3-3

bb11 bb33bb22

Player BPlayer B

1 0 1 0 3 3 3 4 3 4 -3 -3

aa11

aa22

aa33

Player APlayer A

ColumnColumnMaximumMaximum 6 5 6 5

3 3

6 5 6 5 -2 -2

Suppose that the payoff table for a two-person Suppose that the payoff table for a two-person zero-zero-sum game is the following. Here there is no sum game is the following. Here there is no optimaloptimalpure strategy.pure strategy.

MaximinMaximin

MinimaxMinimax

54 54 Slide

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bb11 bb33bb22

Player BPlayer B

1 0 1 0 3 3

Player APlayer A

6 5 6 5 -2 -2

If a game larger than 2 x 2 has a mixed If a game larger than 2 x 2 has a mixed strategy, we first look for dominated strategy, we first look for dominated strategies in order to reduce the size of the strategies in order to reduce the size of the game.game.

3 4 3 4 -3 -3

aa11

aa22

aa33

Player A’s Strategy Player A’s Strategy aa33 is dominated by is dominated byStrategy Strategy aa11, so Strategy , so Strategy aa33 can be eliminated. can be eliminated.

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bb11 bb33

Player BPlayer B

Player APlayer A

aa11

aa22

Player B’s Strategy Player B’s Strategy bb22 is dominated by is dominated byStrategy Strategy bb11, so Strategy , so Strategy bb22 can be eliminated. can be eliminated.

bb22

1 0 1 0 3 3

6 5 6 5 -2 -2

We continue to look for dominated We continue to look for dominated strategies in order to reduce the size of the strategies in order to reduce the size of the game.game.

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bb11 bb33

Player BPlayer B

Player APlayer A

aa11

aa22 1 1

33

6 6 -2-2

The 3 x 3 game has been reduced to a 2 The 3 x 3 game has been reduced to a 2 x 2. It is now possible to solve algebraically x 2. It is now possible to solve algebraically for the optimal mixed-strategy probabilities.for the optimal mixed-strategy probabilities.

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Two-Person, Constant-Sum GamesTwo-Person, Constant-Sum Games

(The sum of the payoffs is a constant other (The sum of the payoffs is a constant other than zero.)than zero.)

Variable-Sum GamesVariable-Sum Games

(The sum of the payoffs is variable.)(The sum of the payoffs is variable.) nn-Person Games-Person Games

(A game involves more than two players.)(A game involves more than two players.) Cooperative GamesCooperative Games

(Players are allowed pre-play (Players are allowed pre-play communications.)communications.)

Infinite-Strategies GamesInfinite-Strategies Games

(An infinite number of strategies are available (An infinite number of strategies are available for the players.)for the players.)

Other Game Theory ModelsOther Game Theory Models

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End of Chapter 5End of Chapter 5

Documents

1 1 Slide © 2009 South-Western, a part of Cengage Learning Slides by John Loucks St. Edward’s University