Game Theory

Game Theory and Strategy

ContentTwo-persons Zero-Sum GamesTwo-Persons Non-Zero-Sum GamesN-Persons Games

IntroductionAt least 2 playersStrategiesOutcomePayoffs

Two-persons Zero-Sum GamesPayoffs of each outcome add to zeroPure conflict between 2 players

Two-persons Zero-Sum Games

Colin

A

B

C

D

Rose

A

7, -7

-1, 1

1, -1

0, 0

B

5, -5

1, -1

6, -6

-9, 9

C

3, -3

2, -2

4, -4

3, -3

D

-8, 8

0, 0

0, 0

8, -8


Colin

A

B

C

D

Rose

A

7

-1

1

0

B

5

1

6

-9

C

3

2

4

3

D

-8

0

0

8

Dominance and Dominance Principle Definition: A strategy S dominates a strategy T if every outcome in S is at least as good as the corresponding outcome in T, and at least one outcome in S is strictly better than the corresponding outcome in T.Dominance Principle: A rational player would never play a dominated strategy.

Saddle Points and Saddle Points PrincipleDefinition: An outcome in a matrix game is called a Saddle Point if the entry at that outcome is both less than or equal to any in its row, and greater than or equal to any entry in its column.Saddle Point Principle: If a matrix game has a saddle point, both players should play a strategy which contains it.

Value Definition: For a matrix game, if there is a number such that player A has a strategy which guarantees that he will win at least v and player B has a strategy which guarantees player A will win no more than v, then v is called the value of the game.


Colin

A

B

C

D

Rose

A

7

-1

1

0

B

5

1

6

-9

C

3

2

4

3

D

-8

0

0

8

Saddle Points

Minimax

Colin

A

B

C

D

Row minimum

Rose

A

4

3

2

5

2

Maximin

B

-10

2

0

1

-10

C

7

5

2

3

2

Maximin

D

0

8

-4

-5

-5

Column Maximum

7

8

2

5

Saddle Points0 saddle point1 saddle pointmore than 1 saddle points

Mixed Strategy

Colin

A

B

Rose

A

2

-3

B

0

3

Mixed StrategyColin plays with probability x for A, (1-x) for BRose A: x(2) + (1-x)(-3) = -3 + 5xRose B: x(0) + (1-x)(3) = 3 - 3xif -3 + 5x = 3 - 3x => x = 0.75Rose A: 0.75(2) + 0.25(-3) = 0.75Rose B: 0.75(0) + 0.25(3) = 0.75

Mixed StrategyRose plays with probability x for A, (1-x) for BColin A: x(2) + (1-x)(0) = 2xColin B: x(-3) + (1-x)(3) = 3 - 6xif 2x = 3 - 6x => x = 0.375Colin A: 0.375(2) + 0.625(0) = 0.75Colin B: 0.375(-3) + 0.625(3) = 0.75

Mixed Strategy0.75 as the value of the game0.75A, 0.25B as Colins optimal strategy0.375A. 0.625B as Roses optimal strategy

Mixed Strategy

Colin

Row difference

Rose oddments

Rose probabilities

A

B

Rose

A

2

-3

2 - (-3) = 5

3

3/8

B

0

3

0 3 = -3

5

5/8

Column difference

2 0 = 2

-3 3 = -6

Colin oddments

6

2

Colin probabilities

6/8

2/8

Minimax TheoremEvery m x n matrix game has a solution. There is a unique number v, called the value of game, and optimal strategy for the players such thati) player As expected payoff is no less that v, no matter what player B does, andii) player Bs expected payoff is no more that v, no matter what player A doesThe solution can always be found in k x k subgame of the original game

Minimax Theorem (example)

Colin

A

B

C

D

E

Rose

A

1

12

13

9

10

B

11

2

8

14

5

C

6

7

3

4

15

Minimax Theorem (example)There is no dominance in the above exampleFrom arrows in the graph, Colin will only choose A, B or C, but not D or E.So the game is reduced into a 3 x 3 subgame

Example

9-Police

9-0

8-1

7-2

6-3

5-4

7-Guerrillas

7-0

1/2

1/2

1/2

1

1

6-1

1

1/2

1/2

1/2

1

5-2

1

1

1/2

1/2

4-3

1

1

1

1/2

0

Example

9-Police

7-2

6-3

5-4

7-Guerrillas

7-0

1/2

1

1

6-1

1/2

1/2

1

5-2

1/2

1/2

4-3

1

1/2

0

Example

9-Police

7-2

6-3

5-4

7-Guerrillas

7-0

1/2

1

1

4-3

1

1/2

0

Example

9-Police

7-2

5-4

7-Guerrillas

7-0

1/2

1

4-3

1

0

Mixed Strategy

9-Police

Row difference

Guerrillas oddments

Guerrillas probabilities

7-2

5-4

7-Guerrillas

7-0

1/2

1

-1/2

1

2/3

4-3

1

0

1

1/3

Column difference

1/2

1

Police oddments

1

1/2

Police probabilities

2/3

1/3

Utility Theory

Colin

A

B

Rose

A

U

V

B

W

X

C

Y

Z

Utility TheoryRoses order is u, w, x, z, y, vColins order is v, y, z, x, w, u

Colin

A

B

Rose

A

6

1

B

5

4

C

2

3

Utility Theory

v

w

x

u

i)

0

20

40

60

80

100

ii)

-1

0

1

2

3

4

iii)

17

19

21

23

25

27

Utility TheoryTransformation can be done using a positive linear function, f(x) = ax + bin this example, f(x) = 0.5(x - 17)

-------->

Colin

A

B

Rose

A

27, -5

17, 0

B

19, -1

23, -3

Colin

A

B

Rose

A

5, -5

0, 0

B

1, -1

3, -3

Two-Persons Non-Zero-Sum GamesEquilibrium outcomes in non-zero-sum games ~ saddle points in zero-sum games

Prisoners Dilemma

Colin

Confess

Dont

Rose

Confess

10, 10

0, 20

Dont

20, 0

1, 1

Nash EquilibriumIf there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium

Dominant Strategy EquilibriumIf every player in the game has a dominant strategy, and each player plays the dominant strategy, then that combination of strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.

Pareto-optimalIf an outcome cannot be improved upon, ie. no one can be made better off without making somebody else worse off, then the outcome is Pareto-optimal

Pareto PrincipleTo be acceptable as a solution to a game, an outcome should be Pareto-optimal.

Prudential Strategy, Security Level and Counter-Prudential StrategyIn a non-zero-sum game, player As optimal strategy in As game is called As prudential strategy.The value of As game is called As security levelAs counter-prudential strategy is As optimal response to his opponents prudential strategy.

Example

Colin

A

B

Rose

A

2, 4

1, 0

B

3, 1

0, 4

Exampleconsider only Roses strategysaddle point at AB

Colin

A

B

Rose

A

2

1

B

3

0

Exampleconsider only Colins strategy

Colin

A

B

Rose

A

4

0

B

1

4

Column difference

4-1=3

0-4=-4

Colin oddments

4

3

Colin probabilities

4/7

3/7

Example

Rose strategy

Colin strategy

Rose payoff

Colin payoff

prudential

prudential

11/7

16/7

prudential

Counter-prudential

2

4

Counter-prudential

Prudential

12/7

16/7

Counter-prudential

Counter-prudential

3

1

Example

Rose prudential

A

Colin Prudential

4/7A, 3/7B

Rose Counter-prudential

B

Colin Counter-prudential

A

Example

BB AA

Equilibrium

BA AB

Co-operative Solution

Negotiation Set

Co-operative SolutionConcerns are Trust and Suspicion

N-Person GamesMore important and common in real lifen is assumed to be at least three

N-Person Games

Larry A

Colin

A

B

Rose

A

1, 1, -2

-4, 3, 1

B

2, -4, 2

-5, -5, 10

Larry B

Colin

A

B

Rose

A

3, -2, -1

-6, -6, 12

B

2, 2, -4

-2, 3, 1

N-Person Games

Colin and Larry

AA

BA

AB

BB

Rose

A

1

-4

3

-6

B

2

-5

2

-2

N-Person Games

Rose and Larry

AA

BA

AB

BB

Colin

A

1

-4

-2

2

B

3

-5

-6

3

N-Person Games

Rose and Colin

AA

BA

AB

BB

Larry

A

-2

2

1

10

B

-1

-4

12

-1

N-Person Games

Colin and Larry

BA

BB

Rose optimal

Rose

A

-4

-6

3/5

B

-5

-2

2/5

Colin and Larry optimal

4/5

1/5

Value = -4.4

N-Person Games

Rose and Larry

BA

AB

Colin optimal

Colin

A

-4

-2

1

B

-5

-6

0

Rose and Larry optimal

1

0

Value = -4

N-Person Games

Rose and Colin

AA

BA

Larry optimal

Larry

A

-2

2

3/7

B

-1

-4

4/7

Rose and Colin optimal

6/7

1/7

Value = -1.43

N-Person GamesThe result is

Rose v.s. Colin and Larry

-4.4, -0.64, 5.04

Colin v.s. Rose and Larry

2, -4, 2

Larry v.s. Rose and Colin

2.12, -0.69, -1.43

SuperaddictiveA characteristic function form game (N, v) is called superadditiveif v(S, T) >= v(S) + v(T) for any two coalitions S and T

N-Person Prisoners Dilemma

Number of others choosing C

0

1

2

3

4

Player chooses

C

-2

-1

0

1

2

D

-1

0

1

2

3

N-Person Prisoners DilemmaGeneral form of N-Person Prisoners Dilemmaeach of n players has two strategies, C and Dfor every player, D is a dominant strategyif all players choose D, add will be worse off than if all players had chosen C

Example

Blues

Red

A

B

B

C

C

D

D

A

ExampleSincere choice

Blues

Red

1st round

A

B

B

C

2nd round

C

D

D

A

ExampleOptimal choice

Blues

Red

2nd round

A

B

1st round

B

C

C

D

D

A

From the bottom up algorithmi) under optimal play, the Reds choice in last round will be the player who is last on the Blues preference list. Mark that player as the Reds last round choice and cross him off both teams listsii) the Blues choice in last round will be the player who is last on the Reds reduced list. Mark the player as Blues and cross him off both teams listsiii) continue like this, finding the choices in the next-to-last round, and on up to the first round


Blues

Red

A

B

B

C

2nd round

C

D

D

A


Blues

Red

A

B

1st round

B

C

2nd round

C

D

D

A

Example of N-Person Prisoners Dilemma

Blues

Red

Green

A

E

C

B

F

F

C

B

E

D

A

D

E

D

A

F

C

B

Example of N-Person Prisoners DilemmaSincere Choice

Blues

Red

Green

1st round

A

E

C

2nd round

B

F

F

C

B

E

D

A

D

E

D

A

F

C

B

Example of N-Person Prisoners DilemmaAfter Greens optimal Choice

Blues

Red

Green

1st round

A

E

C

2nd round

B

F

F

C

B

E

D

A

D

E

D

A

F

C

B

Example of N-Person Prisoners DilemmaAfter Reds optimal Choice

Blues

Red

Green

1st round

A

E

C

B

F

F

C

B

E

2nd round

D

A

D

E

D

A

F

C

B

Example of N-Person Prisoners DilemmaAfter Blues optimal Choice

Blues

Red

Green

2nd round

A

E

C

B

F

F

1st round

C

B

E

D

A

D

E

D

A

F

C

B

Game Theory

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