GAME THEORYGAME THEORY“Trust None. For oaths are straws, men’s faiths are wafer-cakes.” - William Shakespeare (Henry V)

Prisoners’ DilemmaSanya and Cinque: Two Players (Two robbers of Hibernia Savings Bank)Payoff Matrix

“Trust None. For oaths are straws, men’s faiths are wafer-cakes.” - William Shakespeare (Henry V)

Prisoners’ DilemmaSanya and Cinque: Two Players (Two robbers of Hibernia Savings Bank)Payoff Matrix

Bill

Confess Not Confess

AlConfess 3, 3 1, 10

Not Confess 10, 1 2, 2

NASH EQULIBRIUMNASH EQULIBRIUM

Nash Equilibrium (Game theory) - A stable state of a system that involves several interacting participants in which no participant can gain by a change of strategy as long as all other participants remain unchanged.

Nash Equilibrium (Game theory) - A stable state of a system that involves several interacting participants in which no participant can gain by a change of strategy as long as all other participants remain unchanged.

Game Theory is basically a strategic interaction between mutually aware players. It is based on the fact that “You are self-interested and selfish” So is everyone else....

Terminologya) No. of players: Two-person game and n-person gameb) Sum of gains and losses: zero-sum game and non-zero sum

gamec) Strategy: It is set of rules which a person should adopt at

each playPure-strategy - If a player knows what course of action he is

going to adopt based on the knowledge about opponent’s course of action, then he always selects a particular course of action (known with certainty)

Mixed-strategy - When both the players are guessing as to which course of action is to be selected on a particular occasion with certain probability then it is a mixed-strategy game

Game Theory is basically a strategic interaction between mutually aware players. It is based on the fact that “You are self-interested and selfish” So is everyone else....

Terminologya) No. of players: Two-person game and n-person gameb) Sum of gains and losses: zero-sum game and non-zero sum

gamec) Strategy: It is set of rules which a person should adopt at

each playPure-strategy - If a player knows what course of action he is

going to adopt based on the knowledge about opponent’s course of action, then he always selects a particular course of action (known with certainty)

Mixed-strategy - When both the players are guessing as to which course of action is to be selected on a particular occasion with certain probability then it is a mixed-strategy game

d) Two-person, Zero-sum games: A game with two players where the loss of one player is equal to gain of the other with net gain being zero. e) Payoff Matrix:Row designations are Player A’s strategies and Column designations are Player B’s strategies.

d) Two-person, Zero-sum games: A game with two players where the loss of one player is equal to gain of the other with net gain being zero. e) Payoff Matrix:Row designations are Player A’s strategies and Column designations are Player B’s strategies.

Player B’s Strategies

B1 B2...........

.Bn

Player A’s

Strategies

A1 a11 a12 ............ a1n

A2 a21 a22 ............ a2n

.

. ..

.

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

..

..

Am am1 am2 ............ amn

Pure strategiesPure strategiesMaximin and Minimax Strategies

Payoff matrix for A Firm B’s Strategy B 1 2 -2 Firm A’s A 2 -3 -4 Strategy

0 -1 2 Player B

Solve the following:

Player A

Maximin and Minimax Strategies

Payoff matrix for A Firm B’s Strategy B 1 2 -2 Firm A’s A 2 -3 -4 Strategy

0 -1 2 Player B

Solve the following:

Player A

1 2

1 20 -6

2 8 2

3 -4 1

B1 B2 B3 B4

A1 1 7 3 4

A2 5 6 4 5

A3 7 2 0 3

saddle pointsaddle pointIf Maximin = Minimax = Value of the game, then we have a Saddle Point. Saddle Point of a payoff matrix is the position of an element which is minimum in its row and maximum in its column. Player B

Player A

Player B

Player A

If Maximin = Minimax = Value of the game, then we have a Saddle Point. Saddle Point of a payoff matrix is the position of an element which is minimum in its row and maximum in its column. Player B

Player A

Player B

Player A

8 7 15 12

9 14 8 10

10 12 14 13

-7 -4

7 -3

8 -2

2) Player B

Player A

3) Firm B

Firm A

2) Player B

Player A

3) Firm B

Firm A

B1 B2 B3 B4 B5

A1 -1 0 0 5 3

A2 3 2 3 2 2

A3 -4 -3 0 2 6

A4 5 3 -4 2 6

B1 B2 B3 B4 B5

A1 3 -1 4 6 7

A2 -1 8 2 4 12

A3 16 8 6 14 12

A4 1 11 -4 2 1

4) Assume that two firms are competing for the market share for a particular product. Each firm is considering what promotional strategy to employ for the coming period. Assume that the following payoff matrix describes the increase in market share of Firm A and the decrease in market share for Firm B. Determine the optimal strategies for each firm.

a) Which firm would be winner, in terms of market share?b) What might the two firms do to maximize their profits or minimize their losses?

4) Assume that two firms are competing for the market share for a particular product. Each firm is considering what promotional strategy to employ for the coming period. Assume that the following payoff matrix describes the increase in market share of Firm A and the decrease in market share for Firm B. Determine the optimal strategies for each firm.

a) Which firm would be winner, in terms of market share?b) What might the two firms do to maximize their profits or minimize their losses?

Firm B

No Promotion

ModeratePromotion

More Promotion

Firm A

No Promotion 5 0 -10

ModeratePromotion 10 6 2

More Promotion 20 15 10

Mixed strategiesMixed strategiesConsider this problemConsider this problem

20 8 -6

12 10 2

3 5 6

1/31/3 2/32/3

1/21/2

1/21/2

Algebraic Method (2 x 2) games5) Solve the following 2 x 2 games without saddle points B B

A A

6) Two players A & B without showing each other, put a coin on a table, with head or tail up. A wins Rs.8 when both coins show head and Re.1 when both are tails. B wins Rs.3 when the coins do not match. Given the choice of being a matching player (A) and non-matching player (B), which one would you choose and what would be your strategy?

Algebraic Method (2 x 2) games5) Solve the following 2 x 2 games without saddle points B B

A A

6) Two players A & B without showing each other, put a coin on a table, with head or tail up. A wins Rs.8 when both coins show head and Re.1 when both are tails. B wins Rs.3 when the coins do not match. Given the choice of being a matching player (A) and non-matching player (B), which one would you choose and what would be your strategy?

5 1

3 4

6 -3

-3 0

The Rules of Dominance: If a row or column is dominated by another row or column in terms of pay-offs, then the dominated row or column can be deleted to reduce a ‘n x m’ matrix to a ‘2 x 2’ matrix. Player BFor ex.:

Player A

Player B

Player A

The Rules of Dominance: If a row or column is dominated by another row or column in terms of pay-offs, then the dominated row or column can be deleted to reduce a ‘n x m’ matrix to a ‘2 x 2’ matrix. Player BFor ex.:

Player A

Player B

Player A

I II III

I -4 6 3

II -3 -3 4

III 2 -3 4

B1 B2 B3 B4

A1 3 2 4 0

A2 3 4 2 4

A3 4 2 4 0

A4 0 4 0 8

7) In an election campaign, the strategies adopted by the ruling and opposition party along with payoffs (ruling party’s % share in votes polled) are given below. Assume a zero-sum game. Find optimum strategies for both parties and expected payoff to ruling party.

7) In an election campaign, the strategies adopted by the ruling and opposition party along with payoffs (ruling party’s % share in votes polled) are given below. Assume a zero-sum game. Find optimum strategies for both parties and expected payoff to ruling party.

Opposition Party’s Strategies

Campaigning one day in each

city

Campaigning two days in each

city

Campaigning two days in large

rural area

Ruling Party’s Strategi

es

Campaigning one day in each

city55 40 35

Campaigning two days in each city

70 70 55

Campaigning two days in

large rural area75 55 65

8) A steel company is negotiating with its union for revision of wages to its employees. The management, with the help of a mediator, has prepared a payoff matrix shown below. Plus sign indicates a wage increase, while a negative sign indicates a wage decrease. Union has also constructed a table which is comparable to that developed by management. What strategies are best for the management and union and what is the value of the game?

8) A steel company is negotiating with its union for revision of wages to its employees. The management, with the help of a mediator, has prepared a payoff matrix shown below. Plus sign indicates a wage increase, while a negative sign indicates a wage decrease. Union has also constructed a table which is comparable to that developed by management. What strategies are best for the management and union and what is the value of the game? Union Strategies

U1 U2 U3 U4

Steel Co. Strategie

s

C1 2.5 2.7 3.5 -0.2

C2 2.00 1.60 0.80 0.80

C3 1.40 1.20 1.50 1.30

C4 3.00 1.40 1.90 0

Arithmetic Method9) Solve the following game:

10) Two breakfast food mfgers, ABC and XYZ are competing for an increased market share. The pay-off matrix, shown in the following table, shows the increase in market share for ABC and decrease in market share for XYZ. Simplify the problem by rule of dominance and find optimum strategies and value of game.

Arithmetic Method9) Solve the following game:

10) Two breakfast food mfgers, ABC and XYZ are competing for an increased market share. The pay-off matrix, shown in the following table, shows the increase in market share for ABC and decrease in market share for XYZ. Simplify the problem by rule of dominance and find optimum strategies and value of game.

Player B

B1 B2 B3

Player A

A1 1 7 2

A2 6 2 7

A3 5 2 6

Give coupons

Decrease Price

Maintain Present Strategy

Increase advertising

Give coupons 2 -2 4 1

Decrease Price 6 1 12 3

Maintain Present Strategy

-3 2 0 6

Increase advertising 2 -3 7 1

graphical methodgraphical methodGraphical method is used to convert a m x 2 and 2 x n matrices to a 2 x 2 matrix.For a 2 x n matrix problem, we have to find a maximin point and for a m x2 matrix problem we have to find a minimax point.

2 x n matrix problem B’s Strategies11) Solve the 2 x 3 game graphically A’s strategies

12) Solve the 4 x 2 game graphicallym x 2 matrix problem A

Graphical method is used to convert a m x 2 and 2 x n matrices to a 2 x 2 matrix.For a 2 x n matrix problem, we have to find a maximin point and for a m x2 matrix problem we have to find a minimax point.

2 x n matrix problem B’s Strategies11) Solve the 2 x 3 game graphically A’s strategies

12) Solve the 4 x 2 game graphicallym x 2 matrix problem A

I II III

I 1 3 11

II 8 5 2

I II

I 2 4

II 2 3

III 3 2

IV -2 6

13) A soft drink company calculated the market share of two products against its major competitor having three products and found out the impact of additional advertisement in any one of its products against the other.

What is the best strategy for the company as well as the competitor? what is the payoff obtained by the company and the competitor in the long run? Use graphical method to obtain the solution.

13) A soft drink company calculated the market share of two products against its major competitor having three products and found out the impact of additional advertisement in any one of its products against the other.

What is the best strategy for the company as well as the competitor? what is the payoff obtained by the company and the competitor in the long run? Use graphical method to obtain the solution.

Competitor B

B1 B2 B3

Company A

A1 6 7 15

A2 20 12 10