1 1 Slide Quantitative Methods for Business Game Theory Objectives: Understand the principles of zero-sum, two-player game Analyzing pure strategy game,

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Quantitative Methods for BusinessQuantitative Methods for Business

Game Theory

Objectives:• Understand the principles of zero-

sum, two-player game• Analyzing pure strategy game,

dominance principle. • Solve mixed strategy games.

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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.

Game models classification according to: Game models classification according to: number of players, sum of all payoffs, number number of players, sum of all payoffs, number of strategies employedof strategies employed

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 tothree strategies that are designed to

take sales of new vehicles fromtake sales of new vehicles from

the other dealership over athe other dealership over a

four-month period. Thefour-month period. The

strategies, assumed to be thestrategies, assumed to be the

same for both dealerships.same for both dealerships.

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 cash rebaterebate

on a new on a new vehicle.vehicle. Strategy 2: Offer Strategy 2: Offer free free optionaloptional

equipmentequipment on on aa

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

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-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

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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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bb33

Player BPlayer B

Player APlayer A

aa11

aa22

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

bb11

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bb22 bb33

Player BPlayer B

Player APlayer A

aa11

aa22 0 0

33

5 5 -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.

For Player A: For Player A: p = 0.3; p = 0.3; EV = 1.5EV = 1.5

For Player B: For Player B: q = 0.5; q = 0.5; EV = 1.5EV = 1.5

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Prisoner’s DilemmaPrisoner’s Dilemma

ConfessConfess

AA

Not ConfessNot Confess

BB

ConfessConfess Not ConfessNot Confess

Both get 3 Both get 3 yearsyears

0 year for A 0 year for A

4 years for B4 years for B

4 years for A 4 years for A

0 year for B0 year for BBoth get 2Both get 2

yearsyears

Two thieves were caught because they stole a large Two thieves were caught because they stole a large amount of money in a bank. However, the police did amount of money in a bank. However, the police did not have enough evidence to convict them. They not have enough evidence to convict them. They could send both two of them to prison for two years could send both two of them to prison for two years with the evidence using the stolen motorcycle.with the evidence using the stolen motorcycle.

They decided to separate the thieves into different They decided to separate the thieves into different prison room. And they suggested to each thief:prison room. And they suggested to each thief:

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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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Homework 02Homework 02

Module 4:Module 4: M4.8, M4.11, M4.12, M4.13, M4.8, M4.11, M4.12, M4.13, M4.14, M4.18M4.14, M4.18

Documents

1 1 Slide Quantitative Methods for Business Game Theory Objectives: Understand the principles of zero-sum, two-player game Analyzing pure strategy game,