Game Theory 2person Zero-Sum Game

8/10/2019 Game Theory 2person Zero-Sum Game

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Chua Soon Jie 1081104073

Keh Yan Shun 1081103316Lim Kok Siong 1101109878

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Content

Introductiono What is Game Theory?

o Assumptions

o Classifications- number of players, types of games

o Pure-strategy game

Example 1

Example 2 Real Case Problem

Reference2


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Game Theory andDecision Analysis

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What is Game Theory?

Game theory

Tool for decision under uncertainty

Player optimizehis own decisionat the expense of the other

player.

Game theory is the study of how optimal strategies areformulated in the conflict.

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ASSUMPTIONS

There are finite number of players.

There is conflicts of interests between them.

Players know all possible available choices but doesnot know which one is going to be chosen

Players simultaneously select their respectivecourses of action.

Their payoff is fixed and determine in advance.

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CLASSIFICATION Two-Person Game- A game with 2players.

N-Person Game- A game with Nnumber of player,where n>2.

Zero-Sum Game- A game in which gains are equalto sum of losses.

Non-Zero Sum Game- A game in which sum ofgains and losses is not equal.

Pure-Strategy Game- Each player plays onestrategy throughout the game.

Mixed-Strategy Game- Each player employdifferentstrategies at different times in the game.

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Pay Off

It is the sum of gains and losses from the game thatare available to the players.

If in a game sum of the gains to one player is

exactly equal to the sum of losses to another player,so that the sum of the gains and losses equals zerothen the game is said to be a zero-sum game.

There are also games in which the sum of the

players gains and losses does not equal to zero,and these games are denoted as non-zero sumgames.

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Two-person, Zero-sumGame- Pure Strategy

Characteristics:

1. Two-players with finite set of strategies.

2. Losses of one player is the exact gain of the others

3. One strategy is repeatedly recommended to eachplayer

4. Bargaining is not allowed. There could be nomutually advantageous agreement.

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Solution of Pure Strategy

Games MaximinMaximum in Row Minimum

MinimaxMinimum in Column Maximum

A two-person sum game has a saddle point if and only ifMaximin = Minimax

Minimax is the decision rule to minimize the possible loss for

a worst case (maximum loss).

Maximin is the decision rule to minimize the opponentmaximum payoff and maximizing ones own minimum gain.

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Method of solving 2-personzero-sum games

Saddle point exists----- pure strategies

No saddle point ---- mixed strategies

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Example 1:

Two Person Zero Sum Game With Single Saddle Point

Find optimal strategies for person A & person B and findvalue of game

B1 B2 B3 B4

A1 -5 4 1 20

A2 5 5 4 6

A3 3 -2 0 -5

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Step 1: Determine the row minimum

B1 B2 B3 B4Row

Minimum

A1 -5 4 1 20

A2 5 5 4 6

A3 3 -2 0 -5

-5

4

-5

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Step 2: Determine the column maximum

B1 B2 B3 B4Row

Minimum

A1 -5 4 1 20 -5

A2 5 5 4 6 4

A3 3 -2 0 -5 -5

Column

Maximum

5 5 4 20

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Step 3: Determine the Maximinand Minimax

B1 B2 B3 B4 RowMinimum

A1 -5 4 1 20 -5

A2 5 5 4 6 4

A3 3 -2 0 -5 -5

ColumnMaximum

5 5 4 20

Saddle point is at A2,B3where the value is 4,Where saddle point is the best strategy for both A and B

maximin

minimax

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B1 B2 B3

A1 7 -1 2

A2 4 4 6

A3 6 3 0

A4 7 4 5

Example 2:Two Person Zero Sum Game With Two Saddle Point

Find optimal strategies for person A & person B andfind value of game

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B1 B2 B3Row

Minimum

A1 7 -1 2 -1A2 4 4 6 4

A3 6 3 0 0

A4 7 4 5 4

ColumnMaximum

7 4 6

Repeat step1 to step 3

Since there are twoMaximin and oneMinimax, the game has 2 saddle points,(A2,B2)&(A4,B2)

Thus, optimal strategy A is A2&A4and for B is B2

maximin

minimax

Example 2:Two Person Zero Sum Game With Two Saddle Point

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Real Case Problem

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Given Data:

UnionStrategies

Company Strategies

C1 C2 C3 C4

U1 2.0 1.5 1.2 3.5

U2 2.5 1.4 0.8 1.0

U3 4.0 0.2 1.0 0.5

U4 -0.5 0.4 1.1 0.0

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Step 1 : Draw extra column and row

UnionStrategies

Company Strategies

C1 C2 C3 C4 RowMinimum

U1 2.0 1.5 1.2 3.5

U2 2.5 1.4 0.8 1.0

U3 4.0 0.2 1.0 0.5

U4 -0.5 0.4 1.1 0.0

Columnmaximum

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Step 2 : Identify minimumvalue of each row

UnionStrategies

Company Strategies


U1 2.0 1.5 1.2 3.5

U2 2.5 1.4 0.8 1.0

U3 4.0 0.2 1.0 0.5

U4 -0.5 0.4 1.1 0.0

Columnmaximum

1.2

0.8

-0.5

0.2

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Step 3 : Identify maximumvalue of each column

UnionStrategies

Company Strategies


U1 2.0 1.5 1.2 3.5 1.2

U2 2.5 1.4 0.8 1.0 0.8

U3 4.0 0.2 1.0 0.5 0.2

U4 -0.5 0.4 1.1 0.0 -0.5

Columnmaximum

4.0 1.21.5 3.5

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Step 4 : Find the Maximin

UnionStrategies

Company Strategies


U1 2.0 1.5 1.2 3.5 1.2

U2 2.5 1.4 0.8 1.0 0.8

U3 4.0 0.2 1.0 0.5 0.2

U4 -0.5 0.4 1.1 0.0 -0.5

Columnmaximum

4.0 1.5 1.2 3.5

Maximin

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Step 5 : Find the Minimax

UnionStrategies

Company Strategies


U1 2.0 1.5 1.2 3.5 1.2

U2 2.5 1.4 0.8 1.0 0.8

U3 4.0 0.2 1.0 0.5 0.2

U4 -0.5 0.4 1.1 0.0 -0.5

Columnmaximum

4.0 1.5 1.2 3.5

Maximin

Minimax23


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UnionStrategies

Company Strategies


U1 2.0 1.5 1.2 3.5 1.2

U2 2.5 1.4 0.8 1.0 0.8

U3 4.0 0.2 1.0 0.5 0.2

U4 -0.5 0.4 1.1 0.0 -0.5

Columnmaximum

4.0 1.5 1.2 3.5

Maximin

Minimax 24

Step 6 : Identify the saddle point

Saddle point


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UnionStrategies

Company Strategies


U1 2.0 1.5 1.2 3.5 1.2

U2 2.5 1.4 0.8 1.0 0.8

U3 4.0 0.2 1.0 0.5 0.2

U4 -0.5 0.4 1.1 0.0 -0.5

Columnmaximum

4.0 1.5 1.2 3.5

Conclusion:

1. Saddle point has the value of 1.2.

2. Optimum solution for both the workers andcompany is U1 & C3 .

3. Union will gain 1.2 while company will loss 1.2. (Zero-sum )

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Limitation of Game Theory Analysis will be more complexas the number of players

increase.

The assumption that players know their own pay-offs and pay-

offs of others is not practical.

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References https://www.youtube.com/watch?v=R7Sbc8s3FxA

http://www.egwald.ca/operationsresearch/gametheory.p

hp3 https://www.youtube.com/watch?v=PqK6dXPlr-

8&list=UU_zylrAtEs3Kd7ENktYQxMQ

http://www.princeton.edu/~rvdb/542/lectures/lec8.pdf

http://www.encyclopediaofmath.org/index.php/Two-

person_zero-sum_game

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https://www.youtube.com/watch?v=R7Sbc8s3FxAhttp://www.egwald.ca/operationsresearch/gametheory.php3http://www.egwald.ca/operationsresearch/gametheory.php3https://www.youtube.com/watch?v=PqK6dXPlr-8&list=UU_zylrAtEs3Kd7ENktYQxMQhttps://www.youtube.com/watch?v=PqK6dXPlr-8&list=UU_zylrAtEs3Kd7ENktYQxMQhttp://www.princeton.edu/~rvdb/542/lectures/lec8.pdfhttp://www.encyclopediaofmath.org/index.php/Two-person_zero-sum_gamehttp://www.encyclopediaofmath.org/index.php/Two-person_zero-sum_gamehttp://www.encyclopediaofmath.org/index.php/Two-person_zero-sum_gamehttp://www.encyclopediaofmath.org/index.php/Two-person_zero-sum_gamehttp://www.encyclopediaofmath.org/index.php/Two-person_zero-sum_gamehttp://www.encyclopediaofmath.org/index.php/Two-person_zero-sum_gamehttp://www.encyclopediaofmath.org/index.php/Two-person_zero-sum_gamehttp://www.princeton.edu/~rvdb/542/lectures/lec8.pdfhttps://www.youtube.com/watch?v=PqK6dXPlr-8&list=UU_zylrAtEs3Kd7ENktYQxMQhttps://www.youtube.com/watch?v=PqK6dXPlr-8&list=UU_zylrAtEs3Kd7ENktYQxMQhttps://www.youtube.com/watch?v=PqK6dXPlr-8&list=UU_zylrAtEs3Kd7ENktYQxMQhttp://www.egwald.ca/operationsresearch/gametheory.php3http://www.egwald.ca/operationsresearch/gametheory.php3https://www.youtube.com/watch?v=R7Sbc8s3FxA


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Documents

Game Theory 2person Zero-Sum Game