Decision science

1. GAMETHEORY
Name: Jaimin A. Kemkar
En. No.: 160123119014
Sem. & Batch: 7th D1
Subject: Operation Research (2171901)
Guided By.: Prof. Jatin Patel

2. CONTENTS
1. INTRODUCTION OF GAME THEORY
2. ASSUMPTIONS OF GAME THEORY
3. MATHEMATICAL FORMULATION
4. LIMITATIONS
5. TERMS USED IN GAME THEORY
6. TYPES OF GAME THEORY
7. STEPS TO SOLVE GAME THEORY
8. EXAMPLE
9. REFERANCES

3. INTRODUCTIONOFGAMETHEORY
 When competitor takes any action, it is directly affecting the other relevant
organizations. So, in order to sustain each business organization develops some
strategy against the action of competitor this situation is conceptualized by J. von
Neumann in mathematical theory and this theory called “GAME THEORY”.
 Game theory is based on the principle of minimax and maxmini. This implies that each
competitor will act so to minimize the maximum loss or maximize the minimum gain.
 It may be possible that person will act where he can achieve best of the worst
condition.
 This theory is used in solving situation where two or more competitors try to achieve
their objectives. Following are some of such competitive situations:
1. Price war
2. Military training
3. Marketing of products
4. Negotiation between organizations and unions etc.
5. Win a new contract 3

4. ASSUMPTIONSOFGAMETHEORY
1. Number of players are finite and a finite number of strategies are available to
each player
2. Each player makes independent decisions without consolation
3. The game begins when each player chooses a single course of action from the
available options
4. Any course of action results in gains to participants. The gain may be positive,
negative or zero
5. This gain is affected not only by his course of action but the course of action
taken by participants is also affecting the gain
6. The pay-off of each player is affected by the action adopted by the competitor
7. The pay-off is fixed and known in advance
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5. MATHEMATICALFORMULATION
Let player A have ‘m’ courses of action and player B has ‘n’ courses of action. The
game can be shown by a pair of matrixes constructed as shown:
 Raw represents for each matrix are the course of actions available to player A
 Columns represents for each matrix are the courses of actions available to
player B
 The cell values represents payments to A in case of player A’s pay-off matrix
 The cell values represents payments to B in case of player B’s pay-off matrix
 Player A is called maximizing player and he will always try to maximize the
minimum gain
 Player B is called minimizing player and he will always try to minimize the
maximum loss.
 The sum of payoff matrix for A and B is a null matrix
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6. 6
Player B
1 2 3 … N
Player A
1 𝑎11 𝑎12 𝑎13 … 𝑎1𝑛
2 𝑎21 𝑎22 𝑎23 … 𝑎2𝑛
3 𝑎31 𝑎32 𝑎33 … 𝑎3𝑛
… … … … … …
M 𝑎𝑚1 𝑎𝑚2 𝑎𝑚3 … 𝑎𝑚𝑛
Player A’s Pay-off matrix
Player B
1 2 3 … N
Player A
1 −𝑎11 −𝑎12 −𝑎13 … −𝑎1𝑛
2 −𝑎21 −𝑎22 −𝑎23 … −𝑎2𝑛
3 −𝑎31 −𝑎32 −𝑎33 … −𝑎3𝑛
… … … … … …
M −𝑎𝑚1 −𝑎𝑚2 −𝑎𝑚3 … −𝑎𝑚𝑛
Player B’s Pay-off matrix

7. LIMITATIONS
1. Risk and uncertainty are not taken into account
2. A fixed number of competitors are playing, but it is not actual conditions.
There are number of competitors in the market.
3. It is assumed to have finite number of courses of actions are available but
actually it is not realistic approach since number of strategies is present under
different conditions
4. Zero sum game is not realistic
5. Knowledge of payoff is known in advance which is not realistic
6. Game theory shows the problem outcome under different conditions, it does
not show how to play the game
7. Knowledge about strategies available to the opponent players is assumed
which may not be possible in case of competition
8. Dynamic approach of change in market demand is ignored
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8. TERMSUSEDINGAMETHEORY
1. GAME: It is an activity involving actions by each one of the participants
according to the set of rules, which results in some gain
2. Player: Each participant playing a game is called a player
3. Pay-off: The quantitative measure of achievement at the end of game is
known as pay-off
4. Pay-off matrix: the table which show the outcome of the game when
different strategies are adopted by different players s known as pay-off
matrix
5. Strategy: It is the predetermined rule by which a player decides his course of
action from his available courses of action
6. Saddle Point: the Game value is called the saddle point in which each player
has a pure strategy. If saddle point exists, the game is said to be stable. It is
the number which lowest in its rows and highest in columns
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9. 7. Zero sum game: it is the Game in which the sum of payments of all the players after
the play of the game is zero
8. Non-zero Sum Game: It is the game in which the sum of payments to all the
players after the play of the game is not zero. Any player may receive or make
some payments
9. Value of game: The maximum guaranteed expected outcome per play when
players follow their optimal strategy is called the "value of the game". It is
denoted by V. If the value of the game is zero then it is also called fair game.
10. Maxmini: Maximize the minimum value of gain
11. Minimax: Minimize the maximum value of loss
12. Strictly determinable game: A game is said to be strictly determinable if the
maximum value is equal to minimax value but not zero
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10. TYPESOFGAMETHEORY
1. Based on number of competitors involved:
 A game theory which two players in known as two person game.
If the game theory involves more than two persons is known as
“n-persons game”
2. Based on outcome of the game:
 Based on outcome, if the sum of the player’s gains and losses
equals zero, the game is called “zero sum game”. Alternatively if
the sum is not zero, than it is said to be “non-zero sum game”
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11. STEPSTOSOLVEGAMETHEORY
1. Check the pay-off matrix. the payoff Matrix must be in terms of the
player who wants to maximize the minimum gain
2. Player mentioned in the left hand side of the playoff matrix is the player
who wants to maximize the minimum gain will apply maxmini principle
and find out the minimum value in each row. this value is called lower
value of game
3. Player Mentioned on the top of the pay-off matrix is the player who
wants to minimise the maximum loss, so apply minimax principle and
find out maximum value in each column. This value is called upper a
value of game.
4. Check for the saddle point
5. Corresponding strategy will be the best Strategy for the Both players
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12. EXAMPLE
Find out saddle point for the following game:
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Player B
𝐵1 𝐵1 𝐵1 𝐵1
Player A
𝐵1 62 44 55 40
𝐵1 60 45 48 51
𝐵1 40 42 30 40
SOLUTION
Finding out the raw minimum and column maxima as shown in table:
Player B
𝐵1 𝐵1 𝐵1 𝐵1
Player A
𝐵1 62 44 55 40
𝐵1 60 45 48 51
𝐵1 40 42 30 40
Since there is a common point for a raw minima and maxima, saddle point is 45

13. REFERANCES
Book: operation Research by books
inda publication
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