gaming theory - study notes

7/29/2019 Gaming Theory - Study Notes

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CHAPTER 6

Game Theory

6.1 INTRODUCTION

The theory of games (or game theory or competitive strategies) is a mathematical theory that

deals with the general features of competitive situations. This theory is helpful when two or

more individuals or organisations with conflicting objectives try to make decisions. In such

situations, a decision made by one decision-maker affects the decision made by one or more of

the remaining decision-makers and the final outcome depends upon the decision of all the

parties. Such situations often arise in the fields of business, industry, economics, sociology and

military training. This theory is applicable to a wide variety of situations such as two players

struggling to win at chess, candidates fighting an election, two enemies planning war tactics, fu-

ms struggling to maintain their market shares, launching advertisement campaigns by companies

marketing competing product, negotiations between organisations and unions, etc. These

situations differ from the ones we have discussed so far wherein nature was viewed as a

harmless opponent

The theory of games is based on the min-max principle put forward by J. von Neumann which

implies that each competitor will act so as to minimize has maximum loss (or maximize his

minimum gain) or achieve best of the worst. So far only simple competitive problems have been

analysed by this mathematical theory. The theory does not describe how a game should be

played; it describes only the procedure, and principles by which plays should be selected.

Though the theory of games was developed by von Neumann (called father of game theory) in

1928, it was only after 1944 when he and Morgenstern published their work named Theory of

Games and Economic Behaviour', that the theory received its proper attention. Since, so far the

theory has been capable of analysing very simple situations only, there has remained a wide gap

between what the theory can handle and the most actual situations in business and industry. So,

the primary contribution of game theory has been its concepts rather than its formal application,

to the solution of real problems.


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6.2 CHARACTERISTICS OF GAMES

A competitive game has the following characteristics

(i) There are finite number of participants or competitors. If the number of participants is 2, the

game is called two-person game for number greater than two, it is called n-person game.

(ii) Each participant has available to him a list of finite number of possible courses of action. The

list may not be same for each participant.

(iii) Each participant knows all the possible choices available to others but does not know which

of them is going to be chosen by them.

(iv) A play is said to occur when each of the participants chooses one of the courses of action

available to him. The choices are assumed to be made simultaneously so that no participant

knows the choices made by others until he has decided his own.

(v) Every combination of courses of action determines an outcome which results in gains to the

participants. The gain may be positive, negative or zero. Negative gain is called a loss.

(vi) The gain of a participant depends not only on his own actions but also those of others.

(vii) The gains (payoffs) for each and every play are fixed and specified in advance and are

known to each player. Thus each player knows fully the information contained in the payoff

matrix.

(viii) The players make individual decisions without direct communication.

GAME MODELS

There are various types of game, models. They are based on the factors like the number of players

participating, the sum of gains-or losses and the number of strategies available, etc.

1. Number of persons : If a game involves only two players, it is called two-person game; if there-

are more than two players, it is named n-person game. An n-person game does not imply that

exactly n players are involved in it. Rather it means that the participants can be classified into n

mutually exclusive groups, with all members in a group having identical interests.

2. Sum of payoff's : If the sum of payoffs (gains and losses) to the players is zero, the game iscalled zero-sum or constant-sum game-, otherwise non zero-sum game.

3. Number of strategies : If the number of strategies (moves or choices) is finite, the game is

called a finite game; if not, it is called infinite game.


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6.2.1 DEFINITIONS

1. Game : It is an activity, between two or more persons, involving actions by each one of them

according to a set of rules, which results in some gain (+ve, -ve or zero) for each.

If in a game the actions are determined by skills, it is called a game of strategy, if they are

determined by chance, it is termed as a game of chance. Further a game may be finite or infinite.

A finite game has a finite number of moves and choices, while an infinite game contains an

infinite number of them.

2. Player : Each participant or competitor playing a game is called a player. Each player is

equally intelligent and rational in approach.

3. Play : A play of the game is said to occur when each player chooses one of his courses of

action.

4. Strategy : It is the predetermined rule by which a player decides his course of action from his

list of courses of actions during the game. To decide a particular strategy, the player need not

know the-other's strategy.

5. Pure strategy : It is the decision rule to always select a particular course of action. It is

usually represented by a number with which the course of action is associated.

6. Mixed strategy : It is decision, in advance of all plays, to choose a course of action for each

play in accordance with some probability distribution. Thus, a mixed strategy is a selection

among pure strategies with some fixed probabilities (proportions). The advantage of a mixed

strategy, after the pattern of the game has become evident, is that the opponents are kept

guessing as to which course of action will be adopted by a player.

Mathematically, a mixed strategy of a player with m possible courses of actions is a set X ofm

non-negative numbers whose sum is unity, where each number represents the probability with

which each course of action (pure strategy) is chosen. Thus if pi is the probability of choosing

course i, then

)p,.......p,p,p(X m321= ,

where 1p.......ppp m321 =++++

and m.....,3,2,1i0p i =

Evidently a pure strategy is a special case of mixed strategy in which all but one xi are zero. A

player may be able to choose only m pure strategies but he has an infinite number of mixed

strategies to choose from.


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7. Optimal strategy : The strategy that puts the player in the most preferred position irrespective

of the strategy of his opponents is called an optimal strategy. Any deviation from this strategy

would reduce his payoff.

8. Zero-sum game : It is a game in which the sum of payments to all the players, after the play

of the game, is zero. In such a game, the gain of players that win is exactly equal to the loss of

players that lose e.g., two candidates fighting elections, wherein the gain of votes by one is the

loss of votes to the other.

9. Two-person zero-sum game : It is a game involving only two players, in which the gain of

one player equals the loss to the other. It is also called a rectangular game or matrix game

because the payoff matrix is rectangular in form. If there are n players and the sum of the game

is zero, it is called n-person zero-sum game. The characteristics of a two-person zero-sum game

are

(a) only two players are involved,

(b) each player has a finite number of strategies to use,

(c) each specific strategy results in a payoff,

(d) total payoff to the two players at the end of each play is zero.

10. Nonzero-sum game : Here a third party (e.g. the 'house' or a 'kitty') receives or makes some

payment. A payoff matrix for such a game is shown below. The left-hand entry in each cell is the

payoff to A, and the right-hand entry is the payoff to B

TABLE 6.1 Payoff Matrix of Player A and Player B

11. Payoff: It is the outcome of the game. Payoff (gain or game) matrix is the table showing the

amounts received by the player named at the left-hand-side after all possible plays of the game.

The payment is made by player named at the top of the table.

Let player A have m courses of action and player B have n courses of action. Then the game

can be described by a pair of matrices which can be constructed as described below.


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(a) Row designations for each matrix are the courses of action available to player A.

(b) Column designations for each matrix are the courses of action available to player B.

(c) The cell entries are the payments to A for one matrix and to B for the other matrix. The cell

entry aij is the payment to A in A's payoff matrix when A chooses the course of action i and B

chooses the course of action j.

.

TABLE 6.2 Player As Payoff Matrix

TABLE 6.3 Player Bs Payoff Matrix

(d) In a two-person zero-sum game, the cell entries in B's payoff matrix will be the negative of


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the corresponding cell entries in A's payoff matrix. A is called maximizing player as he would

try to maximize the gains, while B is called minimizing player as he would try to minimize his

losses.

Thus the sum of payoff matrices for A and B is a null matrix. Here, the objective is to determine

the optimum strategies of both the players that result in optimum payoff to each, irrespective of

the strategy used by the other.

EXAMPLE 6.1 Consider the game in which two players simultaneously reveal 1, 2, or 3 fingers

each. If the sum of the revealed fingers is even, player B pays to player A the sum in dollars, if

the sum is odd, player A pays to player B the sum in dollars.

For this very simple two-person, zero-sum game, the pure strategies can be identified with the

individual activities. Furthermore, both players have the same set of pure strategies, {1, 2, 3}.

The payoff matrix is given in Table 6.4.

TABLE 6.4 payoff matrix (Example 6.1)

The termstrategy refers to the decision by the Player A (or Player B) to choose a particular row

(or column). If each player uses the same strategy every time the game is played, then the game

is said to involve pure strategies. In the next section, we will look at games involving mixed

strategies, where the players try to outguess each other and vary their strategies each time the

game is played.

Pure StrategiesConsider a two-person, zero-sum game involving a payoff matrix P, in which the

Player A chooses a row and the Player B chooses a column. The entry in the chosen

row and column is the amount that the Player B must pay the Player A. The pure

strategy for each player is as follows.

1. Player A: Determine the smallest entry in each row. Choose the row containing the

largest of these smallest entries.

2. Player B. Determine the largest entry in each column. Choose the column

containing the smallest of these largest entries.


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REMARKBe sure you see that these guidelines are only for strategies that are pure (to be used

each time the game is played). We will see that for many games, using the same strategy each

time is not the best (or optimal) strategy.

Usually, a player's strategy is to try to make a choice that will win the largest amount (or lose the

smallest amount). Such a strategy is called an optimal strategy. We will see that pure strategies

may or may not be optimal.

Examples 6.2 and 6.3 show how to find the pure strategies for various types of payoff matrices.

EXAMPLE 6.2Finding Pure Strategies

Determine the pure strategies for the payoff matrix A.

=

12

75A

discussed at the beginning of this section. If each player uses a pure strategy, what will the

payoff be? For this game, is the pure strategy optimal?

SOLUTION

1. For the Player A, the smallest entry in the first row is a 12 = -7 and the smallest entry in the

second row is a22 = 1.

Since the largest of these two is 1, the Player A should choose the second row.

2. For the Player B, the largest entry in the first column is a11 = 5 and the largest entry in the

second column is a22 = 1.


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Since the smallest of these two is 1, the Player B should choose the second column.

Assuming the Player A chooses the second row and the Player B chooses the second column, the

payoff for the game is p22 = 1, which means that the Player Bmust pay the Player A $1 every

time the game is played.

Note that if either player uses a strategy other than the pure strategy, then that player will not do

as well. Thus, thepure strategy is the optimal strategy for both players.

REMARKRemember that entries in the payoff matrix are always listed in terms of amounts

that the Player B must pay the Player A. Thus, positive numbers are good for the Player A, and

negative numbers are good for the Player B.

EXAMPLE 6.3 Finding Pure Strategies

Determine the pure strategies for the following payoff matrix as shown in Table 6.5.


If each player uses a pure strategy, what will the payoff be?

SOLUTION

1. For the Row Player, the smallest entry in the first row is a12 = - 6, the smallest entry in the

second row is a22 = -1, and the smallest entry in the third row is a32 = -7.

Since the largest of these three is -1, the Row Player should choose the second row.

2. For the Player B, the largest entry in the first column is a21 = 5, the largest entry in the second

column is a22=- 1, and the largest entry in the third column is a23 =- 6.


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Since the smallest of these three is -1, the Player B should choose the second column.

Assuming the Player A chooses the second row and the Player B chooses the second column,the payoff for the game is a22 = -1, which means that the Player A must pay the Player B $1

every time the game is played.

In Examples 6.2 and 6.3, note that the payoff is both the smallest entry in its row and the largest

entry in its column. We call such an entry a saddle point.

This result tells us that if a payoff matrix has a saddle point, then each player should use a pure

strategy, choosing the row or column that contains the saddle point.

It is possible for a payoff matrix to have more than one saddle point. For instance, the payoff

matrix

=

91102

31153

P

saddle point

Consider a two-person, zero-sum game involving a payoff matrix P. If the matrix P bas an

entry that is both smallest entryin its row and the largest entry in its column, then that entry iscalled a saddle point of the payoff matrix. Games that have a saddle point are called strictly

determinedand in such games, the optimal strategy for each player is, the, pure strategy.


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has two saddle points: a11 = 3 and a14 = 3. Both of these entries are the least entries in their row

and the greatest entries in their column. Thus, the pure strategy for the Player A is to choose the

first row, and the pure strategy for the Player B is to choose either the first column or the fourth

column.

Many payoff matrices do not have saddle points, and are therefore not strictly determined. We

illustrate this type of matrix in Example 6.4.

EXAMPLE 6.4A Payoff Matrix with No Saddle Point

Show that the payoff matrix given in Table 6.6


=

4331

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P

has no saddle point. If each player uses a pure strategy, what will the payoff be?

SOLUTION

To begin, we determine the pure strategy for each player.

1. For the Rowr A, the smallest row entries are a11=-1 (first row), a22 = 0 (second row), a32 = -3

(third row), and a41=-3 (fourth row). Since the largest of these is 0, the pure strategy for the

Player A is to choose the second row.

2. For the Player B, the largest column entries are a21 = 6 (first column) and a12 = 5 (second

column). Since the smallest of these is 5, the pure strategy for the Player B is to choose the

second column.

If each player uses a pure strategy, the payoff will be a22 = 0, which means that no money is

exchanged between the two players. Note that a22 = 0 is not a saddle point because it is not the

largest entry in the second column.

If you study the payoff matrix in Example 6.4, you can see why this game is not strictly

determined (that is, why a pure strategy might not be the best for each player). For instance,

suppose you are the Player A. Knowing that the Player B is likely to choose the second column,

you might try altering your strategy and choose the first row to get a payoff of $5. On the other

hand, the Player B knows that you might alter your strategy and choose the first row, so the

Player B might also decide to use an alternative strategy and choose the first column, which


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would yield $1 to the Player B.

We will say more about this type of "mixed strategy" in the next section. For now the point is

this. If a payoff matrix has a saddle point, then the best strategy for each player is to use a pure

strategy every time the game is played. If a payoff matrix does not have a saddle point, then the

game becomes more interesting because one player may be able to outguess the other player and

increase his or her payoff.

We summarize these ideas as follows.

6.3 CREATING MODELS FOR GAMES

So far in this section, we have described games by listing a payoff matrix. In practice, most

applications of game theory involve an initial stage in which you must create the payoff matrix.

In order fora game to qualify as a two-person, zero-sum game, the game must have the following

features.

1. The game must have two players, usually called the Row Player and the Column Player.

2. Each time the game is played, the Row Player can make any of m different moves, and the

Column Player can make any of n different moves. Moreover, these moves are made

simultaneously so that neither player knows the move that the other player is going to make.

3. After each player has made a move, the rules of the game determine the payoff. This can be an

amount that the Column Player pays the Row Player (listed as a positive amount), an amount

that the Row Player pays the Column Player (listed as a negative amount), or a tie (listed as

zero).

Guidelines for Choosing a Strategy

Consider a two-person, zero-sum game involving a payoff matrix P. To determine the best

strategy for the players, use the following steps.

1. Determine the pure strategy for the Row Player.

2. Determine the pure strategy for the Column Player.

3. Determine the payoff that would result if both players used a pure strategy. If this payoff

is a saddle point of the matrix P, then the best strategy for both players is to use a pure

strategy every time the game is played. U the payoff is not a saddle point of the matrix P,

then the players should use a mixed strategy (which will be discussed in Section 6.##).


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6.3.1 The Maxmin-Minmax Principle

This principle is used for the selection of optimal strategies by two players. Consider two player

A and B. A is a player who wishes to maximize his gain while player B wishes to minimize his

losses. Since A would like to maximize his minimum gain, we obtain for player A, the value

calledMaxmin Value and the corresponding strategy is called the maxmin strategy.

On the other hand, since player B wishes to minimize his losses, a value called the Minmax

Value which is the minimum of the maximum losses is, found. The corresponding strategy is

called the minmax strategy. When these two are equal (maxmin value = minmax value), the

corresponding strategies are called optimal strategy and the game is said to have a saddle point.

The value of the game is given by the saddle point.

The selection of maxmin and minmax strategies by A and B is based upon the so-called

maxmin-minmax principle which guarantees the best of the worst results.

Saddle Point A saddle point is a position in the payoff matrix where the maximum of row

minima coincides with the minimum of column maxima. The payoff at the saddle point is called

the value of the game.

We shall denote the maxmin value by the minmax value of the game by and the value of the

game by .

Note (i) A game is said to be fair if

maxmin value=minmax value =0, i.e., if .=

(ii) A game is said to be strictly determinable if maxmin value= minmax value 0. .=

EXAMPLE 6.5 Solve the game whose pay off matrix is given by Table 6.7



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SOLUTION

B1 B2 B3 Row minima

1-4-1

Column maxima

A1A2A3

151

340

131

1 5 1

Maxi (minimum) = Max (1,-4,-1)=1 Mini (maximum) = Min (1, 5, 1) = l .

ie., Maxmin value 1= =Minmax value

Saddle point exists. The value of the game is the saddle point which is 1. The optimal strategy is

the position of the saddle point and is given by (A1, B1,).

EXAMPLE 6.6 For what value of , the game with the following matrix is strictly

determinable?

B1 B2 B3

A1A2A3

42

71

26

SOLUTION

B1 B2 B3 Row minima2-7-2

Column maxima

A1A2A3

42

71

26

-1 6 2

The game is strictly determinable, if

== Hence 2= , 1=

21

EXAMPLE 6.7 Determine which of the following two person zero sum games arc strictly


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determinable and fair. Given the optimum strategy for each player in the case of strictly

determinable games.

(a)

B1 B2

A1A2 47

25

(b)

B1 B2

A1A2 34

11

SOLUTION

Maxi (minimum)= = Max (-5,-7)= -5 Mini (maximum)= =Min (-5, 2)=-5

(a)

B1 B2

A1A2 47

25

Since 5= , 5= , the game is strictly determinable. There exists a saddle point = -5. Hence

the value of the game is -5. The optimal strategy is the position of the saddle point given by

(A1, B1).

(b)

B1 B2

A1A2 34

11

Maxi (minimum) = = Max (1, - 3) = 1.

Mini (maximum) = = Min (4, 1) = 1.

Since 1=== , the game is strictly determinable. The value of game is 1. The optimal

strategy is (A1, B2).


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EXAMPLE 6.8Solve the game whose payoff matrix is given in Table 6.8.


62435

62034

22123

35002

SOLUTION

TABLE 6.9 Payoff matrix of player A with Maxmin and Minmax values

B1 B2 B3 Row minima-21-4-6

Column maxima

A1A2A3

62435

62034

22123

35002

5 3 1 5 6

Maxi (minimum) = = Max (-2, 1, -4, -6)=1

Mini (maximum) = = Min (5, 3, 1, 5, 6) = 1.

Since 1=== , there exists a saddle point. the value of the game is 1. The position of the

saddle point is the optimal strategy and is given [A2, B3]

Applications to Business

Game theory has many applications other than "games." In the next two examples, we show how

game theory can be used to develop optimal strategies in business situations.

EXAMPLE 6.9 Leasing a Computer

A data processing firm rents computer time to several other companies. The firm has decided to

offer a new service which will require the purchase or lease of an additional computer system.

The management has already determined to lease (rather than buy) and has narrowed the choiceto three types: a large computer system, a medium-sized computer system, and a small computer

system. For each system, the firm's clients might have a high acceptance (meaning that 75% of

the clients would use the service), or a low acceptance (meaning that only 25% of the clients

would use the service). For each of these possibilities, the accounting department has predicted

the following profits for the firm given in Table 6.10.


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TABLE 6.10 Payoff matrix of Example 6.9

Computer

Size

High

Acceptance

Low

Acceptance

Large $250,000 $30,000

Medium $200,000 $40,000

Small $150,000 $75,000

If you were required to decide which system to lease, what would your decision be?

SOLUTION

In practice, of course, you would probably try to predict the likelihood of either a high or low

acceptance. If there was a strong likelihood of a high acceptance for the large system, then

ordering the large system would be very tempting because it offers the greatest possible profit.

For this problem, however, we assume that no other information is available to aid you in

making your decision. Assuming that you are the Row Player, the payoff matrix is given in

Table 6.11.


=

000,75000,150

000,40000,200

000,30000,250

A

Since you are the Row Player, your pure strategy is to choose the third row (the small system).

REMARKBe sure you see that the decision that was made in Example 6.9 corresponds to a

pure strategy. If you take a course in decision theory, you will see that other strategies could

have been employed, and these might lead to different choices.

EXAMPLE 6 .10 :Hard-Sell or Soft-Sell?

Your company, Company A, is planning an advertising campaign fora new product. Your major

competitor, Company B, is also producing new advertisements for its product. Each company

can gear its ads toward a "hard-sell" which would emphasize the benefits of its product and the

shortcomings of the competitor's product, or gear its ads toward a "soft-sell" that would only

emphasize the benefits of its product. The market research department has predicted the


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following market shares (for your company) for hard-sell, soft-sell combinations as in Table

6.12.


Which type of advertising would you recommend for your company? Is this "game" strictly

determined?

SOLUTION

For this "game," the payoff matrix is as follows.

=

45.030.0

55.040.0

P

The pure optimal strategy for Company A, the Player A, is to choose the first row, and the pure

optimal strategy for Company B, the Player B, is to choose the first column. If both companies

use a pure optimal strategy, then the payoff will be p11 = 0.40. Moreover, since this payoff is a

saddle point, the game is strictly determined.

EXERCISES

1. Determine which of the following two-person zero sum games are strictly determinable and

fair. Give the optimum strategies for each player in the case of strictly determinable games

(a)

20

05(b)

41

20

[Ans: Not strictly determinable] [Ans: Fair]

2. For the game with payoff matrix

646

221

determine the best strategies for players A and B and also the value of the game for them. Is this


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game (1) fair, (ii) strictly determinable. [Ans:Value of game is -2. Game is not fair but strictly

determinable]

3. Determine the optimal minmax strategies for each player in the following game as in Table

6.13.

TABLE 6.13 Payoff matrix

3204

8465

7025

[Ans: 4= , (A2, B3) is the optimum strategy].

In Exercises 4-7, consider a two-person game in which each person simultaneously chooses an

integer: 1, 2, or 3. Construct a payoff matrix for the given game.

4. If the sum of the two numbers is odd, then the Player A pays the Player B that sum (in

dollars). If the sum of the two numbers is even, then the Player B pays the Player A $4.

5. If both players choose the same number, then no money is exchanged. Otherwise, the player

with the higher number wins $1.

6. If both players choose the same number, then no money is exchanged. If the two numbers

differ by 1, then the player with the higher number wins $2. If the two numbers differ by 2, then

the player with the higher number wins $1.

7. If the players choose the same number, no money is exchanged. If the sum of the two numbersis even, then the player with the higher number wins $2. If the sum of the two numbers is odd,

then the player with the lower number wins $2.

8. Coin-Tossing Game Two players each have a coin. Simultaneously, each player chooses to

lay his or her coin on the table (heads up or tails up). If both players choose heads, then the

Player Bwins $2. If both players choose tails, then. the Player A wins $3. If the players choose

different sides, then the player who chooses heads wins $1. What is the pure strategy for each

player? Is this game strictly determined?

9. Two players each have a coin. Simultaneously, each player chooses to lay his or her coin on

the table (heads up or tails up). If both players choose heads, then the Player A wins $2. If both

players choose tails, then the Player A wins $3. If the coins differ, then the Player B wins $1 or

$2, depending on whether the Player A's coin is heads or tails, respectively. What is the pure

strategy for each player? Is this game strictly determined?


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10. Two players play "Stone, Scissors, and Paper," a game in which each player simultaneously

chooses either a stone, a pair of scissors, or paper. If there is a tie, there is no payoff. A stone

"breaks" scissors and is paid $2. Paper "covers" stone and is paid $3. Scissors "cut" paper and is

paid $4. What is the pure strategy for each player? Is this game strictly determined?

11. Repeat Exercise 10 using the following payoffs. A stone "breaks" scissors and is paid $3.

Paper "covers" stone and is paid $2. Scissors "cut" paper and is paid $4.

12. Two players each have three cards numbered 2, 3, and 4. If both players choose the same

number, no money is exchanged. Otherwise, the player with the higher number wins the sum of

the two numbers. What is the pure strategy for each player? Is this game strictly determined?

13.Two players simultaneously choose an integer from 1 to 3. If they choose the same number,

the Row Player wins $1. If the two numbers differ by 1, the player with the higher number wins

$3. If the two numbers differ by 2, the player with the higher number wins $2. What is the pure

strategy for each player? Is this game strictly determined?

14. A person buys a large tract of land, which can be used for condominiums, a hotel, or a

restaurant. There is a possibility that a highway will be built near this land. The weekly profit (in

dollars) for each type of construction is asin Table 6.14.


Highway No Highway

Condominiums -1000 6000Hotel 8000 -3000

Restaurant 6000 5000

What is the pure strategy for this "player?" Is this game strictly determined?

15. In a presidential campaign, there are two candidates and three major issues. Each candidate

must decide to campaign as "pro" or "anti" with respect to each of the three issues. The

"payoffs" (listed in terms of Candidate A) are as follows.

16. In a presidential campaign, there are two candidates and three major issues. Each candidate

must decide to campaign as "pro" or "anti" with respect to each of the three issues. The

"payoffs" (listed in terms of Candidate A) are in Table 6.15.


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Candidate B

Issue 1 Issue 2 Issue 3

Candidate A

Issue 1 3 -2 -1

Issue 2 -1 2 4

Issue 3 2 3 1

What is the pure strategy for Candidate A? Is this game strictly determined?

17.There are three construction sites in town. Two companies supply snacks from trucks. Site 1

has 50% of the customers, Site 2 has 30%, and Site 3 has 20%. If both trucks go to the same site,

they split the customers equally. If they go to different sites, each gets the entire business at that

site, and they split the business at the third site. Construct the payoff matrix and determine where

the trucks should go.

18. Repeat Exercise 17 assuming that Site 1 has 35% of the customers, Site 2 has 40% of the

customers, and Site 3 has 25% of the customers.

19. Several firms are building new offices and are taking bids from the two most prominent

construction companies in the area: Company A and Company B. When bidding, the companies

choose between three strategies to win the bid: cost, time, and quality. From past history, the

market share for Company A is given in Table 6.16.

TABLE 6.16 Payoff matrixCompany B

Cost Time Quality

Company A

Cost 50% 40% 30%

Time 55% 55% 45%

Quality 50% 60% 60%

What is the pure strategy for Company A? Is this game strictly determined?

20.Repeat Exercise 19 using the following assumptions. Company A's quality has fallen. As a

result, if Company A chooses quality as its strategy, its market share will be 10% less than

before.


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6.4 GAMES WITHOUT SADDLE POINTS(MIXED STRATEGIES)

6.4.1 Expected Value of a Game

In Section 6.3, we looked at several examples of games whose payoff matrices have saddle

points. As we have already mentioned, such games are strictly determined and both players

should use a pure strategy every time the game is played.

For payoff matrices that do not have saddle points, the choice of a best strategy is more difficult.

To see why, let's consider the following simple payoff matrix.

=

13

24A

The pure strategy for the Row Player would be to choose the first row, and the pure strategy for

the Column Player would be to choose the second column. These two choices would result in a

payoff of a12 = -2, which is not a saddle point because it is not the largest entry in the second

column.

Note that for this payoff matrix, if the Column Player chooses the second column each time the

game is played, then the Row Player would soon catch on, and switch to the second row. On the

other hand, if the Row Player started choosing the second row, then a smart Column Player

would start choosing the first column. The idea is simply this: if a payoff matrix does not have a

saddle point, then each player should use a mixed strategy.

For instance, suppose that the probability that the Row Player will choose the first row is 0.40,

and the probability that the Column Player will choose the first column is 0.25. Over the long

run, what would the payoff for this game be? To determine this, we find the expected value of

the game as follows.

Payoff is a11 = 4: Probability = p1q1 = (0.4)(0.25) = 0.10

Payoff is a12 =2: Probability = pIq2 = (0.4)(0.75) = 0.30

Payoff is a21 = -3: Probability = p2q1 = (0.6)(0.25) = 0.15

Payoff is a22 = 1: Probability = p2q2 = (0.6)(0.75) = 0.45

Thus, the expected value for this game is

E= p1q1a11 + pIq2aI2 +p2qI a21 +p2q2a22

= (0.4)(0.25)(4) + (0.4)(0.75)(-2) + (0.6)(0.25)(-3) + (0.6)(0.75)(1)

= -0.2.

In other words, if this game is played many times, then the average payoff per game would be


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22

-0.2. In matrix notation, we can write the formula for the expected value as

[ ]

=

2

1

2221

121121

q

q

aa

aappE

In this definition, the matrix [ ]m321 p...pppP =

is called the strategy for the Player A because it gives the probabilities that the Player A will

make each of the m possible "row-moves." Similarly, the matrix

=

n

3

2

1

q....

...

q

q

q

Q

is called the strategy for the Player B because it gives the probabilities that the Player B will

make each of the n possible "column-moves."

Note that the entries in a "strategy matrix" must be nonnegative because they represent

probabilities. Moreover,

1q....qqqand1p....ppp n321m321 =++++=++++

Expected Value of a Game

Consider a two-person, zero-sum game with a payoff matrix P. If the probability that the Row

Player will choose the ith row is ri, and the probability that the Column Player will choose the

jth column is cp then the expected value of the game is

[ ]

n

3

2

1

mnmj3m2m1m

inij3i2i1i

n3j3333231

n2j2232221

n1j1131211

m321

q....

...

q

q

q

a......a......aaa

a......a.....aaa

.....................

a.......a......aaa

a......a....aaa

a......a...aaa

p...ppp


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A strategy is called a mixed strategy if at least two entries in the strategy matrix are not zero. If

one of the entries is 1 (and the other entries are zero), then the strategy is called a pure strategy.

For instance,

0.00

P = [0.25 0.10 0.65] and /]40.030.030.000.0[Q = are mixed strategies, whereas

P = [0 1 0] and /]1000[Q = are pure strategies.

EXAMPLE 6.11Finding the Expected Value of a Game

A two-person, zero-sum game has a payoff matrix of

=

12

35A

Find the expected value of the game if the players use the following mixed strategies.

(a)

=

50.0

50.0Q]50.050.0[P

(b)

=

75.0

25.0Q]60.040.0[P

(c)

=

60.0

40.0Q]75.025.0[P

SOLUTION

(a) For these strategies, the expected value is

25.050.0

50.0]01.150.1[

50.0

50.0

12

35]50.050.0[PAQE

=

=

==

(b) For these strategies, the expected value is

25.075.0

25.0]60.080.0[

75.0

25.0

12

35]60.040.0[PAQE

=

=

==

(c) For these strategies, the expected value is


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10.060.0

40.0]00.025.0[

60.0

40.0

12

35]75.025.0[PAQE

=

=

==

Note in Example 6.11 that the expected value of the game can change, depending on the

strategies that are used by the Player A and the Player B.

EXAMPLE 6.12 Comparing Expected Values for Mixed Strategies A two-person, zero-sum

game has a payoff matrix of

=

124

142

213

A

The Player A knows that the strategy used by the Player B will be

=

50.0

20.0

30.0

Q

Which of the following strategies would be better for the Row Player?

(a) P = [0.30 0.40 0.30] (b) P = [0.20 0.60 0.20]

SOLUTION

(a) For the strategy P= [0.30 0.40 0.30], the expected value of the game is

[ ] 02.2

50.0

20.0

30.0

124

142

213

30.040.030.0PAQ =

=

(b) For the strategy R [0.20 0.60 0.20], the expected value of the game is

[ ] 98.1

50.0

20.0

30.0

124

142

213

20.060.020.0PAQ =

=

Since larger payoffs are better for the Row Player, it would be better for the Row Player to use

the first strategy.

In Example 6.12, can you think of a strategy for the Player A that is better than either of the two

given? Can you think of a strategy for the Player B that is better than the one given? The


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25

following theorem, called the Fundamental Theorem of Game Theory tells us that there is

always a best (or optimal) strategy for the Player A and a best (or optimal) strategy for the

Player B.

VAQPAQPandVAQPPAQ ****** ==

To see why P* and Q* are called optimal, remember that large payoffs are good for the Player

A. Thus, the statement

*** AQPPAQ

means that if the Player A chooses any strategy P other than the optimal strategy P*, then the

payoff will be less than or equal to the payoff for the optimal strategy. Similarly, since small

payoffs are good for the column player, the statement

*** AQPAQP

means that if the Column Player chooses any strategy Q other than the optimal strategy Q*, then

the payoff will be greater than or equal to the payoff for the optimal strategy.

6.4.2 Finding Optimal Mixed Strategies

Note that the Fundamental Theorem of Game Theory tells us of the existence of the strategies P*

and Q*, but does not tell us how to find these optimal strategies. Of course, if the game is strictly

determined, then we know that the optimal strategy is to use a pure strategy.

Finding optimal mixed strategies fora game can be a tedious process. In Section 6.7##, we will

show how linear programming can be used to find optimal mixed strategies for an m x n payoff

matrix. For a 2 x 2 payoff matrix, however, the problem is straightforward, as seen in the

following result.

Fundamental Theorem of Game TheoryConsider a two-person, zero-sum game with a payoff matrix A. There exist optimal

strategies P* and Q* and a unique payoff P*AQ* = V such that for any other strategies P

and Q,

VAQPAQPandVAQPPAQ ****** ==

The number V is called the value of the game. H the value of the game is zero, then the

game is called fair.


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EXAMPLE 6.13 Finding Optimal Mixed Strategies

In Example 6.11, we found the expected value for three different pairs of strategies for the

payoff matrix

=

12

35A

Find the optimal strategies for this payoff matrix. What is the value of the game? Is the game

fair?

SOLUTION

First, you should check to see that this payoff matrix has no saddle points. Next, applying the

formula for the optimal row strategy produces

11

3

)2()3(15

)2(1

)aa()aa(

aap

21122211

21221 =+

=

++

=

which implies that

.11

8p1p 12 ==

Similarly, applying the formula for the optimal column strategy produces

11

4

)2()3(15

)3(1

)aa()aa(

aaq

21122211

12221 =+

=

++

=

Optimal Mixed Strategy for 2 x 2 Payoff Matrix

Consider a 2 x 2 payoff matrix

21122211

2221

1211aaaa,

aa

aaA +

=

that has no saddle points. The optimal strategy for the Player A is to play the first tow with a

probability of

)aa()aa(

aap

21122211

21221 ++

= and the second row with a probability of .p1p 12 = The

optimal strategy for the Player B is to play the first column with a probability of

)aa()aa(

aaq

21122211

12221 ++

= , and the second column with a probability of 12 q1q =


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27

which implies that

.11

7q1q 12 ==

Thus, the optimal strategies for the payoff matrix are

=11

8

11

3P* and

=

11

711

4

Q*

Using these strategies, we find the value of the payoff matrix to be

.09.011

1

11

711

4

12

35

11

8

11

3AQPV ** ==

== 3

Because the value of the game is not zero, we conclude that the game is not fair. (It favors the

Player B.)

EXAMPLE 6.14Finding Optimal Mixed Strategies

Two people play a card-matching game as follows. Both players have two cards, a 2 and a 3.

Simultaneously, each player chooses one of the cards and lays it face up on the table. The

payoffs for the game are as in Table 6.17.

TABLE 6.17 payoff Matrix (Example 6.14)

Player A Player B Payoff

2 2 Player 2 pays Player 1 $4




What are the optimal strategies for each player? Is this game fair?

SOLUTION

The payoff matrix for this game is

=

93

124A

Moreover, this payoff matrix has no saddle point, so the game is not strictly determined (which


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28

means that pure strategies are not appropriate). The optimal strategy for the Row Player is given

by

7

3

28

12

)3()12(94

)3(9

)aa()aa(

aap

21122211

21221 ==+

=

++

= and

.7

4p1p 12 ==

The optimal strategy for the Column Player is given by

4

3

28

21

)3()12(94

)12(9

)aa()aa(

aaq

21122211

12221 ==+

=

++

= and

.4

1q1q 12 ==

Thus, the optimal strategies for the two players are

=7

4

7

3P* and

=

4

14

3

Q*

Using these strategies, we find the value of the payoff matrix to be

0

4

14

3

93

124

7

4

7

3AQP ** =

=

Because the value of the game is zero, we conclude that the game is fair.

EXAMPLE 6.15 Solve the following game and determine the value of the game

44

44

SolutionIt is clear that the pay off matrix does not possess any saddle point. The players will

use mixed strategies.

The optimum mixed strategy for the players are

Where2

1

16

8

)44()44(

)4(4

)aa()aa(

aap

21122211

21221 ==+

=

++

= , and

2

1p1p 12 ==


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29

2

1

16

8

)44()44(

)4(4

)aa()aa(

aaq

21122211

12221 ==+

=

++

= and

2

1q1q 12 ==

The value of the game is =V 0)44()44(

)4()4(44=

+

The optimum mixed strategies

=2

1

2

1P* and

=

2

12

1

Q*

The value of the game is 0V = .

EXAMPLE 6.16 In a game of matching coins with two players suppose A wins one unit of

value when there are two heads, wins nothing when there are two tails and losses 1/2 unit of

value when there are one head and one tail Determine the payoff matrix, the best strategies for

each player and the value of game to A.

SOLUTIONThe payoff matrix for the player A is given by

H T

H

T

02

12

11

Let this be

2221

1211

aa

aaand The optimum mixed strategies



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30

=

21

21

App

AAS and

=

21

21

Bqq

BBS

Where4

1

)2/12/1()01(

)2/1(0

)aa()aa(

aap

21122211

21221 =+

=

++

= , 1pp 21 =+

4

3p1p 12 ==

4

1

)2/12/1()01(

)2/1(0

)aa()aa(

aaq

21122211

12221 =+

=

++

=

, 1qq 21 =+ ,4

3q1q 12 ==

The value of the game is =V

8

1

)2/12/1()01(

)2/1()2/1(01=

+


=4

3

4

1P* and

=

4

34

1

Q*

The value of the game is81V =

EXERCISES

1. For a game with the following payoff matrix, determine the optimal strategy and the value ofthe game.

(i)

03

36

[Ans.

=2

1

2

1P* and

=

4

34

1

Q* The value of the game is4

3V = ]

(ii)

14

52


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31

[Ans.

=4

3

4

1P* and

=

3

13

2

Q* The value of the game is 3V = ]

2. Two players A and Bmatch coins. If the coins match, then A wins two units of value. If coinsdo not match, thenB wins two units of value. Determine the optimum strategies for the players

and the value of the game.

=2

1

2

1P* and

=

2

12

1

Q* The value of the game is 0V = .

In Exercises 19-24, find the value of the given payoff matrix, and determine whether the game is

fair. If it is not fair, which player does it favor?

19.

=

65

43A 20.

=

10

12A

21.

=

43

12A 22.

=

33

14A

23.

=

98

1016A 24.

=

712

88A

25. Card Game Two players have two cards each: a 3 and a 4. They each choose one of the

cards. The payoffs are as in Table 6.18.

TABLE 6.18 payoff Matrix

Player 1 Player 2 Payoff





What is the optimal strategy for each player? What is the value of the game? Is the game fair?

26. Card Game Two players have two cards each: a face card and a 10. They each choose one of

the cards. The payoffs are as in Table 6.19.


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Player 1 Player 2 Payoff

face card face card Player 2 pays Player 1 $5

face card ten Player 1 pays Player 2 $8

ten face card Player 1 pays Player 2 $5

ten ten Player 2 pays Player 1 $10

What is the optimal strategy for each player? What is the value of the game? Is the game fair?

27. Card Game Two players have one card each. Simultaneously, they each lay their card on the

table: either face-up or face-down. If both choose the same side, then Player 1 wins $10. If

Player I's card is face-up and Player 2's card is face-down, then Player 2 wins $5. If Player I's

card is face-down and Player 2's card is face-up, then Player 2 wins $10. What is the optimal

strategy for each player? What is the value of the game? Is the game fair?

28. Guessing Game Two people play the following guessing game. One player thinks of a

number: either 1 or 2. The second player tries to guess the number. If the guess is correct, the

second player wins $5. If the guess is incorrect, the first player wins $8. What is the optimal

strategy for each player? What is the value of the game? Is the game fair?

29. Career Choice You are offered a promotion to manage a new branch office. You assign

values to the possible outcomes of accepting or not accepting the promotion as in Table 6.20.

TABLE 6.20 payoff MatrixBranch

Succeeds

Branch

Fails

Accept Promotion 10 4

Decline Promotion 2 9

Based on these payoffs, should you accept the promotion?

30. Political Campaign A politician is debating whether to visit a certain city during the

campaign. The politician assigns the following numerical values to the possible outcomes.

Visit and win city's support: 7

Visit and lose city's support: -5

Do not visit and win city's support: 9

Do not visit and lose city's support: -6

If you were this person, would you visit the city?


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31. Concert Attendance Assume that a concert is being held and you are debating whether or not

to go. Assign numerical values to the following possible outcomes. Based on your assigned

values, should you attend the concert?

You attend and the concert is good.

You attend and the concert is bad.

You do not attend and the concert is good.

You do not attend and the concert is bad.

32. Committee Chair Choice Assume that you are asked to be chairperson of a committee. If you

do not accept the position, your employer will ask one of your coworkers to chair the committee.

Assign numerical values to the following possible outcomes. Based on your assigned values,

should you accept the assignment?

You chair the committee and the outcome is good. You chair the committee and the outcome is

bad. Your coworker chairs the committee and the outcome is good.

Your coworker chairs the committee and the outcome is bad.

33. Computer Game Use the computer program given in Appendix A to play the following game

10 times. Two people have two cards, a 2 and a 3. Simultaneously, each player chooses one of

the cards and lays it face up on the table. The payoffs for the game are as in Table 6.21.


Player 1 Player 2 Payoff 2 2 Player 2 pays Player 1 $4




You are Player 1, the Row Player. How much money did you win? Compare your results with

the expected value for playing this game ten times.

6.5 DOMINANCE PROPERTY

In some games, it is possible to reduce the size of the payoff matrix by eliminating redundant

rows (or columns). If a game has such redundant rows (or columns), those rows or columns are

dominated by some other rows (or columns), respectively. Such property is known as dominance

property.


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EXAMPLE 6.17

Players A and B play a game in which each player has three coins (20p, 25p and 50p). Each of

them selects a coin without the knowledge of the other person. If the sum of the values of the

coins is an even number, A wins B's coin. If that sum is an odd number, B wins A's coin.

(a) Develop a payoff matrix with respect to Player A.

(b) Find the optimal strategies for the players.

SOLUTION

The payoff matrix with respect to Player A is shown in Table 6.22.

Dominance property for columns

(a) In the payoff matrix of Player A, if all the entries in a column (X) are lesser than to the

corresponding entries of another column (Y), then column Y is said to be strictly dominatedby

column X. Under such situation, the column Y of the payoff matrix can be deleted.

(b) In the payoff matrix of Player A, if all the entries in a column (X) are lesser than or equal to

the corresponding entries of another column (Y), then column Y is is said to be weekly

dominated by column X. Under such situation, the column Y of the payoff matrix can be

deleted.

Dominance property for rows

(a) In the payoff matrix of Player A, if all the entries in a row (X) are greater than to the

corresponding entries of another row (Y), then row Y is said to be strictly dominatedby rowX. Under such situation, row Y of the payoff matrix can be deleted.

(b) In the payoff matrix of Player A, if all the entries in a row (X) are greater than or equal to

the corresponding entries of another row (Y), then row Y is said to be weekly dominatedby

row X. Under such situation, row Y of the payoff matrix can be deleted.


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35

The maxmin value (-20) is not equal to the minmax value (20). Hence, the game has no saddle

point. As a result, the game has mixed strategies.

Check for dominance property. Row III is weekly dominatedby row I and hence row III is to be

deleted. The resultant matrix after deleting row III is shown in Table 6.23.

TABLE 6.23 Payoff Matrix After Deleting Row III

IN Table 6.23 the column III is weekly dominatedby the column I and hence, column III is to be

deleted. The resultant matrix after deleting column III is shown in Table 6.24. Now payoff

matrix is reduced to 2x2 matrix and appply usual method to find optimal mixed strategies with

respect to paalayer A and player B.TABLE 6.24 Payoff Matrix After Deleting Column III


Where 9

4

90

50

)2520()2520(

)25(25

)aa()aa(

aap

21122211

21221 ==+

=++

= , and

9

5p1p 12 ==

2

1

90

45

)2520()2520(

)20(25

)aa()aa(

aaq

21122211

12221 ==+

=

++

= and


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36

2

1q1q 12 ==

The value of the game is =V 0)2520()2520(

)20()25(2520=

+


=9

5

9

4P* and

=

2

12

1

Q*

The value of the game is 0V = .

EXAMPLE 6.18 Consider the 4 x 4 game as shown in Table 6.25 which represents the payoff

matrix of the Player A. Solve it optimally.

TABLE 6.25 Payoff Matrix for Example 6.18

4343

1424

4243

0423

SOLUTION The check for saddle point is done as shown in Table 6.26. Since, the maxmin

value (3) is not equal to the minmax value (4), the game has no saddle point. This means that the

game has mixed strategies for both the players.

TABLE 6.26 Payoff Matrix with Check for Saddle Point

Check for dominance property. In Table 6.26, row 3 dominates row 1 and the corresponding

reduced payoff matrix is shown in Table 6.27. In Table 6.27, column 3 dominates column I and


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the corresponding reduced payoff matrix is shown in Table 6.28. In Table 6.28, row 4 dominates

row 2 and the corresponding reduced payoff matrix is shown in Table 6.29. In Table 6.29,

column 4 dominates column 2 and the corresponding reduced payoff matrix (2 x 2 matrix) is

shown in Table 6.30.

TABLE 6.27 Payoff Matrix after Deleting Row 1

Player B

1 2 3 4

2 3 4 2 4

Player A 3 4 2 4 1

4 3 4 3 4

TABLE 6.28 Payoff Matrix after Deleting Column I

Palyer B

2 3 4

2 4 2 4

Player A 3 2 4 1

4 4 3 4


Palyer B

2 3 4

Player A 3 2 4 1

4 4 3 4

TABLE 6.30 Payoff Matrix after Deleting Column 2

Player B

3 4

Player A 3 4 1

4 3 4



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Where4

1

)13()44(

)3(4

)aa()aa(

aap

21122211

21223 =++

=

++

= , and

4

3p1p 34 ==

4

3

)13()44(

14

)aa()aa(

aaq

21122211

12223 =++

=

++

= and

4

1q1q 34 ==

The value of the game is =V4

13

)13()44(

1344=

++


=4

3

4

100P* and

=

4/1

4/3

0

0

Q*

The value of the game is4

13V = .

EXAMPLE 6.19A company is currently involved in negotiations with its union on the upcoming wage contract.

Positive signs in Table 6.31 represent wage increase while negative sign represents wage

reduction. What are the optimal strategies for the company as well as the union? What is the

game value ?

TABLE 6.31 Conditional Costs to the Company (Rs. in lakhs)

Union Strategies

U1 U2 U3 U4

C1

Company C2

strategiesC3

C4

0.25 0.27 0.35 -0.02

0.20 0.16 0.08 0.08

0.14 0.12 0.15 0.13

0.30 0.14 0.19 0.00

SOLUTION


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39

Since in a game matrix, player to its left is a maximizing player and me one at the top is a

minimizing player, Table 6.31 is transposed and rewritten as Table 6.32 since the companys

interest is to minimize the wage increase while union's interest is to get the maximum wage

increase.

TABLE 6.32 Payoff Matrix of Unions

Company strategies

C1 C2 C3 C4

U1

Union U2

Strategies U3

U4

0.25 0.20 0.14 0.30

0.27 0.16 0.12 0.14

0.35 0.08 0.15 0.19

-0.02 0.08 0.13 0.00

In Table 6.32, row U4 is dominated by row U1 as well as U3. It is, therefore, deleted to give

Table 6.33.


Company strategies

C1 C2 C3 C4

U1

Union U2

Strategies U3

0.25 0.20 0.14 0.30

0.27 0.16 0.12 0.14

0.35 0.08 0.15 0.19

In Table 6.33, column C1 is dominated by C2 as well as C3, while C4 is dominated by C3.

Deleting columns C1 and C4 we get

TABLE 6.34 Payoff Matrix after Deleting Columns 1 and 4.

Company strategies

U1

Union U2

Strategies U3

C2 C3

0.20 0.14

0.16 0.120.08 0.15

In Table 6.34, row U2 is dominated by U1 and is, therefore, deleting U2 we get,


Company strategies


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40

Union U1

Strategies U3

C2 C3

0.20 0.14

0.08 0.15


Where13

7

13.0

07.0

)14.008.0()15.020.0(

)08.0(15.0

)aa()aa(

aap

21122211

21221 ==++

=

++

= , and

13

6p1p 13 ==

13

1

13.0

01.0

)14.008.0()15.020.0(

14.015.0

)aa()aa(

aaq

21122211

12222 ==++

=

++

= and

13

12q1q 13 ==

The value of the game is =V13

188

)14.008.0()15.020.0(

14.008.015.020.0=

++

Arithmetic method yields the following conclusions

The optimum strategy for company: ]0,13/12,13/1,0[]q,q,q,q[ 4321 =

The optimum strategy for union: ]0,13/6,0,13/7[]p,p,p,p[ 4321 =

The value of the game is 13/188.RsV = .

EXERCISE

There are two competing departmental stores R and C in a city. Both stores have equal

reputation and the total number of customers is equally divided between the two. Both the stores

plan to run annual discount sales in the last week of December. For this, they want to attract

more number of customers by using advertisement through newspaper, radio and television. By

seeing the market trend, the store R constructed the following payoff matrix where the numbers

in the matrix indicate a gain or a loss of customers.

Store C

Store

R

40 50 -70

10 25 -10

100 30 60

(a)Check whether game is strictly determinable? If so find value of game.


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(b)Use Dominance property to reduce the given payoff matrix to 2x2.(c)Hence or otherwise, find optimal strategies for stores R and C and value of game.

6.7 GRAPHICAL METHOD FOR 2xn GAMES

Consider the Payoff matrix of following 2 x,n game in Table 6.31

TABLE 6.30 Payoff matrix of 2xn Game

B1 B2 B3 . .Bn

A1A2

n2232221

n1131211

a...aaa

a...aaa

Let the mixed strategy for player Abe given by

=

p1pAAS

21

A ,0p

Now for each of the pure strategies available to B, expected payoff for player Awould be as

follows.

B's pure move A's expected payoff function E(p)

BI )p1(apa)p(E 21111 +=

B2 )p1(apa)p(E 22122 +=

B3 )p1(apa)p(E 23133 +=

Bn )p1(apa)p(E n2n1n +=

The player B would like to choose that pure move Bj against SA for which Ej(p) is a minimum

for j= 1, 2 ... n. Let us denote this minimum expected payoff for A by.

V =Min (Ej(P)) j = 1, 2 ... n.

The objective of player A is to select p in such a way that V is as large as possible. This may be

done by plotting the straight lines.

j2j2j1j2j1j apaa)p1(apa)p(E +=+= j=1, 2, ....n.

as linear functions of p

The highest point on the lower boundary of these lines will give maximum value among the

minimum expected payoffs on the lower boundary (lower envelope) and the optimum value of

probability p.


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Now the two strategies of player B corresponding to those lines which pass through the

maxmin point can be determined. It helps in reducing the size of the game to (2 x 2).

Similarly, we can treat m x 2 games in the same way and get minmax point which will be the

lowest point on the upper boundary (upper envelope).

EXAMPLE 6.17

Solve the following 2 x 3game graphically.

258

1131

SOLUTION

Since the problem does not possess any saddle point let the player A play the mixed strategy

[ ]p1pP = ,0p

Against player B.

The A's expected payoff against B's pure move is given by

B's pure move A's expected payoff

BI 8p7)p1(8p)p(E1 +=+=

B2 5p2)p1(5p3)p(E2 +=+=

B3 2p9)p1(2p11)p(E3 +=+=

These expected payoff equations are then plotted as functions of p as shown in the Fig. 6.1,

which shows the payoffs of each column represented as points on two vertical axes 1 and 2of

unit distance apart.


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Fig 6.1 Player As Expected Payoff

Now since the player A wishes to maximize his minimum expected payoff, we consider the

highest point of intersection H on the lower envelope of A's expected payoff equation. The lines

B3 and B2 passing through H define the relevant moves B2 and B3 that alone need to play. The

solution to the origin 2 x 3 game reduces to

B2 B3

A1A2

25

113

The optimum strategy for A and B is given by

11

3

11

3

)511(23

52p =

=++

=

11

9

11

9

)511(23

112q1 =

=

++

= and 1qq 21 =+ ,11

2q 2 =

=11

8

11

3P* and

=

11

211

90

Q* The value of the game is11

49

11

556V =

= .


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EXAMPLE 6.18 Solve graphically

05

22

14

61

53

31

SOLUTION The given problem does not possess any saddle point. Therefore, let the player B

plays the mixed strategy

=q1

qQ

against player A.

The expected payoff equations are plotted in the following Fig. 6.2 with two axes I and II

vertically at unit distance apart.

Fig. 6.2 Player Bs Expected Payoff

Since the player B wishes to minimise his maximum expected payoff. We consider the lowest

point of the upper boundary of B's expected payoff equation. The point H (intersection of lines

A2 and A4) represents the minimax expected value of the game for player B.Hence, the solution


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to the original 6 x 2 game therefore reduces to the 2 x 2 payoff matrix.

The optimal mixed strategy for A and B is given by

2. The companies A and B are competing for the same product. Their different strategies are

given in the following payoff matrix. CompanyB

Determine the best strategies for the two companies.

Formulating games with mixed strategies

Consider the following example.

Example Two players A and B play a game in which they both simultaneously choose an

integer between one and three. Suppose A chooses x and B chooses y. If x = y, then A wins $x

from B. If yx , then B wins $x from A.

Each player has three possible strategies. For A, let Ai denote the strategy of choosing the

integer i, i = 1, 2,3; and for B, let Bj denote the strategy of choosing the integer j, j = 1, 2, 3.

Then the payoff matrix is given by the following table.

B1 B2 B3

A1

A2

A3333

222

111

Definition The minmax criterion. In a two-person zero-sum game, each player will act to

minimize his maximum expected loss.

Using the minmax criterion, A will choose his mixed strategy )p,p,p( 321 such that his

maximum expected loss is a minimum or, equivalently, such that his minimum expected gain is

as large as possible, where the minimum is taken over all possible strategies B1, B2, B3 for B.

Similarly, B will wish to choose his mixed strategy )q,q,q( 321 such that his maximumexpected loss is as small as possible, where the maximum is taken over all possible strategies A1,

A2, A3 for A.

Formulating the linear programming model

We shall show how the problem of finding the optimal mixed strategy for each player can be

modelled as a pair of dual linear programming problems. The formulation of this model requires


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that the guaranteed minimum gain to player A is positive. This will be the case when the matrix

D contains a positive row. When D does not have a positive row, we may add a positive real

number k to every entry of D to construct a new payoff matrix D* which does have a positive

row. Thus D* corresponds to playing a new game in which A always wins k more than in the

original game. Clearly the optimal mixed strategies for both players will be the same in both

games.

In the game described in Example , the guaranteed minimum gain to A is $(-1), and hence

we add 2 to each element of D, to form the matrix D*, in which the first row is positive. (We

could have added any constant k > 1 to create a positive row. We choose the smallest integer

value of k in order to make D* as nice as possible.) This gives the new payoff matrix

=

511

040

113

*D

Let v2 denote A's minimum expected gain when he plays the mixed strategy )p,p,p( 321 in the

revised game with payoff matrix D*. Then, we can calculate the expected payoff to A when B

chooses his strategy Bj, for j = 1, 2, 3. Since each of these expected payoffs must be at least A's

minimum expected gain v2, we obtain the following inequalities.

231

2321

231

vp5p

vpp4p

vpp3

+

(1)

Since )p,p,p( 321 are probabilities, the numbers pi, i = 1, 2, 3, satisfy the following additional

constraints.

1ppp 321 =++ (2)

0p,p,p 321 (3)

Now A's problem is to find Rp,p,p 321 to maximize v2, subject to the constraints (1), (2)

and (3). In order to express this problem as an LPP in standard form, we have to make a change

of variable. Since in the revised game, A has a strategy A1 that guarantees a gain of at least $1,

we can add the constraint V2 > 1 without changing the system. Thus we may assume that V2 is


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positive and so we can divide through each of the constraints (1), (2) and (3) by v2 while

preserving the direction of the inequalities. We then put

i

2

i yv

p= , i = 1, 2, 3, and 2

2

zv

1= .

Then constraint (2) becomes

22321 zv/1yyy ==++.

Now maximizing v2 is equivalent to minimizing z2. Hence we can formulate A's problem as the

following linear programming problem.

Find Ry,y,y 321 to minimize 321 yyyz ++=

subject to

231

2321

231

vy5y

vyy4y

vyy3

+

We can formulate the problem of finding an optimal mixed strategy )q,q,q( 321 for B in a

similar way. Let v1 denote B's maximum expected loss when he plays the mixed strategy)q,q,q( 321 in the revised game with payoff matrix D*. Then the expected loss to B when A

chooses his strategy Ai, for i = 1, 2, 3, must be at most B's maximum expected loss of v1. Thus

we obtain the following inequalities.

1321

12

1321

vq5qq

vq4

vqqq3

+

++

(5)

where

1qqq 321 =++ (6)

0q,q,q 321 (7)

Now B's problem is to find Rq,q,q 321 to minimize v1, subject to the constraints (5), (6)

and (7). By construction, a row of D* is positive (in this case, row 1). From the corresponding


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constraint in (5), vl is also positive. Thus, we can divide through each of the constraints (5), (6)

and (.7) by v1 without reversing the direction of the inequalities, and then put

i1i xv/q , i = 1, 2, 3, and 1/vi = z1. Then constraint (6) becomes

11321

zv/1xxx ==++

Now minimizing vl is equivalent to maximizing z1. Hence we can formulate B's problem as a

linear programming problem as follows.

Find Rx,x,x 321 to maximize z1 = x1 + x2 + x3,

subject to

1x5xx

1x4

1xxx3

321

2

321

+

++

0x,x,x 321

We complete the solution of the problem described in Example .

We add slack variables S1, S2, S3 to the first, second and third constraints respectively. Then,

after several iterations of the simplex algorithm, we obtain the following tableau.

3213

22

3211

321

S16

3S

32

1S

16

1

32

9x

S4

1

4

1x

S16

1S32

3S16

5

32

3x

S8

1S

16

3S

8

3

16

11z

=

=

++=

=

Thus, the optimal solution is xl = 5/32, x2 = 1/4, x3 = 9/32, and the maximum value of z1 is

11/16. Using the change of variables z1 = 1/vl and xi = qi/vi, i = 1, 2, 3, we deduce that the

optimal mixed strategy for B is (ql, q2, q3) = (5/22, 4/11, 9/22), and the minimum value of v1 is

16/11.Similarly, we may deduce from this tableau that the optimal solution to the dual problem to

is yi = 3/8, y2 = 3/16, y3 = 1/8, and the minimum value of z2 is 11/16. Using the change of

variables z2 = 1/v2 and iii v/pv = , i = 1, 2, 3, we deduce that the optimal mixed strategy for A

is (pl, p2, p3) = (6/11,3/11,2/11) and the maximum value of v2 is 16/11.

Hence, the value of the game with pay-off matrix D* is 16/11. Since D* was obtained from D by


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adding 2 to each entry, it follows that the value of the original game is given by

v = 16/11 - 2 = -6/11. 0

The negative value for v obtained for this game shows that it is not a fair game. We say that it is

biased in favour of B. The games described in Example 5.1 and Example 5.3 were both biased in

favour of A.

The optimal mixed strategy for A and B is given by

2. The companies A and B are competing for the same product. Their different strategies are

given in the following payoff matrix. CompanyB

Determine the best strategies for the two companies.

Since no row (or column) dominates another row (or column). The 3 x 2 game could now be

solved by graphical method. Since the playerB wishes to minimise his maximum loss, we find

the lowest point of the upper boundary. The expected payoff equations are then plotted as show

in Fig. 17.3.

Figure 17.3

The lowest point in the upper boundary is given by the intersection of lines A, and A,. The

solutionTn the original game is reduced to 2 x 2 matrix

B, B2

A,C

3 -2 A3

2 2

The optimum strategy for A andB is given by

Example 17.11 Solve the following game

PlayerB

1 7 2

Player A 6 2 7

5 1 6

Solution Since all the elements in the third row are less than or equal to the corresponding

elements of second row. Therefore, row III is dominated by row II. Delete this dominated row.

The reduced payoff matrix is given by

The optimum strategy for A andB is given by

Value of 3x2+(-2)x(2) _ 2

game y = 3+2-(-2+2) 5


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Example 17.11 Solve the following game

PlayerB

1 7 2

Player A 6 2 7

5 1 6

Solution Since all the elements in the third row are less than or equal to the corresponding

elements of second row. Therefore, row III is dominated by row II. Delete this dominated row.

The reduced payoff matrix is given by

Solution Since the game has no saddle point it is not a stable one. All the elements of the first

row and a second row are < to the corresponding elements of third row. Hence, these two rows

are dominated rows. Deleting these two rows from the payoff matrix, The reduced payoff matrix

is given by

PlayerB

Player A 13 4 1 1 4]

In this modified payoff matrix, we observe that all the elements of the second column are r to

the corresponding elements of the fourth column. Hence, this dominated column (2nd column) is

deleted from the payoff matrix. The reduced payoff matrix is given by

Player B

8 15 1

Player A

3 -1 4

Now we observe that no row or column dominates another row or column. However, we note

that a convex combination of 2"" and 3"' column is given by

15 x 2+ 1 x 2= 8


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PlayerB

Player _A 15 1 -1 4

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