GAME THEORY Terminology Example : Game with Saddle point Dominance Rules: (Theory-Example) Arithmetic method – Example Algebraic method - Example Matrix method - Example Graphical method - Example

1. Operations research
Chapter: GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example

2. What is Game Theory ?
At its most basic level, game theory is the study of how people or
companies (referred as players) determine strategies in different
situations despite of competing strategies acted out by other
players.
What is two person zero sum game?
The simplest type of competitive situation. These games involve
only two players. This game is called zero-sum game because one
player wins whatever the other player loses.

3. What is Saddle Point?
If Maximin = Minimax
What is strategy? :
A move or moves chosen by a player.

4. What is optimal strategy?
An optimal strategy is one that provides the best payoff for a
player in a game.
The strategy that most benefits a player.
Optimal Strategy = A strategy that maximizes a player's expected
payoff.
What is Value (expected value) of game?:
The amount representing the result when the best possible
strategy is played by each player.

5. What is Payoff?
It is an outcome of game.
An amount showing as an element in the payoff matrix, which
indicates the amount gained or lost by the row player.
What is payoff matrix?
In game theory, a payoff matrix is a table in which strategies of
one player are listed in rows and strategies of the other player are
listed in columns and the cells show payoffs to each player
A matrix whose elements represent all the amounts won or lost
by the row player.

6. Pure strategy: A player always chooses the same strategy-same
row or column.
Mixed strategy: A player chooses the strategy with some fix
probabilities.
A player changes the choice of row or column with different plays
or turns.

7. Method 1 : Saddle Point Steps (Rule)
Step-
1:
1. Select the minimum element from each row and write them in
Row Minimum – last column.
2. Select the maximum element from Row Minimum column and
enclose it in [ ]. It is called Row MaxiMin.
Step-
2:
1. Select the maximum element from each column and write
them in Column Maximum- last row.
2. Select the minimum element from Column Maximum row and
enclose it in ( ). It is called Column MiniMax.
Step-
3:
1. Find out the elements that is same in rectangle [ ] and circle ( ).
2. If Column MiniMax = Row MaxiMin then the game has saddle
point and it is the value of the game.

8. Player APlayer B B1 B2 B3 B4
A1 20 15 12 35
A2 25 14 8 10
A3 40 2 10 5
A4 -5 4 11 0
Example-1
Find Solution of game theory problem.
Following payoff matrix is given in data.

9. Player B
B1 B2 B3 B4
Player A
A1 20 15 12 35
A2 25 14 8 10
A3 40 2 10 5
A4 -5 4 11 0
Solution:
Apply the maximin (minimax) principle to analyze the
game. Saddle point testing

10. Player B
B1 B2 B3 B4
Row
Minimum
Player A
A1 20 15 [(12)] 35 [12]
A2 25 14 8 10 8
A3 40 2 10 5 2
A4 -5 4 11 0 -5
Column
Maximum
40 15 (12) 35
Maximin
Minimax

11. select minimum from the maximum of columns
Column MiniMax = (12)
Select maximum from the minimum of rows
Row MaxiMin = [12]

12. ANSWER
Here, Column MiniMax = Row MaxiMin = 12
This is game with saddle point
 value of the game V= 12
The optimal strategies for both players are
The player A will always adopt strategy A1
The player B will always adopt strategy B3

13. Example-2
Player APlayer B B1 B2 B3
A1 -2 14 -2
A2 -5 -6 -4
A3 -6 20 -8
Find Solution of game theory problem using saddle point

14. Player B
B1 B2 B3
Player A
A1 -2 14 -2
A2 -5 -6 -4
A3 -6 20 -8
Solution:
Saddle point testing

15. Player B
B1 B2 B3
Row
Minimum
Player A
A1 [(-2)] 14 -2 [-2]
A2 -5 -6 -4 -6
A3 -6 20 -8 -8
Column
Maximum
(-2) 20 -2
We apply the maximin (minimax) principle to analyze the game.
Minimax
Maximin

16. ANSWER:
Here,
Column MiniMax = Row MaxiMin = -2
∴ This game has a saddle point and value of the game is -2
The optimal strategies for both players are
The player A will always adopt strategy 1
The player B will always adopt strategy 1
Select minimum from the maximum of columns
Column MiniMax = (-2)
Select maximum from the minimum of rows
Row MaxiMin = [-2]

17. Operation research
GAME THEORY
Dominance Rule : (Theory-Example)

18. Dominance Rules: (method Steps)
Step-1: If all the elements of Column-i are greater than or equal to the
corresponding elements of any other Column-j, then the Column-i is
dominated by the Column-j and it is removed from the matrix.
eg. If all values of Column-2 ≥ Column-4, then remove Column-2
Step-2: If all the elements of a Row-i are less than or equal to the
corresponding elements of any other Row-j, then the Row-i is
dominated by the Row-j and it is removed from the matrix.
eg. If all values of Row-3 ≤ Row-4, then remove Row-3
Step-3: If strategy k is dominated by average of any two strategy i and j than
delete column or row strategy k
Why Dominance Rules used?
To reduce size of payoff matrix in game theory

19. Player APlayer B B1 B2 B3 B4
A1 3 5 4 2
A2 5 6 2 4
A3 2 1 4 0
A4 3 3 5 2
Example-3
Reduce matrix size of game theory problem using dominance
method and Find Solution.

20. Player APlayer B B1 B2 B3 B4
A1 3 5 4 2
A2 5 6 2 4
A3 2 1 4 0
A4 3 3 5 2
Solution:
Dominance rule to reduce the size of the payoff matrix
Row-3 ≤ Row-4,
so remove Row-3

21. Player B
B1 B2 B3 B4
Player A
A1 3 5 4 2
A2 5 6 2 4
A4 3 3 5 2
Column-2 ≥ Column-4,
so remove Column-2

22. Player B
B1 B3 B4
Player A
A1 3 4 2
A2 5 2 4
A4 3 5 2
Column-B1 ≥ Column-B4,
so remove Column-B1

23. Player B
B3 B4
Player A
A1 4 2
A2 2 4
A4 5 2
Row-1 ≤ Row-3,
so remove Row-1

24. Player B
B3 B4
Player A
A2 2 4
A4 5 2
Now,
Find out solution  Optimum strategy and value of game

25. Player APlayer B B1 B2 B3
A1 1 7 2
A2 6 2 7
A3 5 1 6
Example 4:
Reduce matrix size of game theory problem using
dominance method and Find Solution.

26. Player B
B1 B2 B3
Player
A
A1 1 7 2
A2 6 2 7
A3 5 1 6
Solution:
Apply Dominance rule to reduce the size of the payoff matrix
All Values of row-3 ≤ row-2, so remove row-3

27. Player B
B1 B2 B3
Player
A
A1 1 7 2
A2 6 2 7
All Values of column-3 ≥ column-1, so remove column-3

28. Player B
B1 B2
Player
A
A1 1 7
A2 6 2

29. Operations research
GAME THEORY
Arithmetic method – Example
Algebraic method - Example

30. Mixed strategy: A player chooses more than one strategy with
some fix probabilities.
Game without Saddle Point
If Maximin ≠ Minimax
It is also known as mixed strategy problem

31. Solution Methods of Game without Saddle Point problem
For 2*2 matrix size problems four methods are used to
find solution:
1. Arithmetic method
2. Algebraic method
3. Matrix method
4. Alternate-calculus method

32. Arithmetic method Steps (Rule)
Step-1: Find the difference between the two values of Row-1 and put
this value against the Row-2, ignore the sign.
Step-2: Find the difference between the two values of Row-2 and put
this value against the Row-1, ignore the sign.
Step-3: Find the difference between the two values of Column-1 and put
this value against the Column-2, ignore the sign.
Step-4: Find the difference between the two values of Column-2 and put
this value against the Column-1, ignore the sign.
Step-5: Find probabilities of each by dividing their sum
Step-6: Find value of the game by algebraic method.

33. Player APlayer B B1 B2 B3
A1 10 5 -2
A2 13 12 15
A3 16 14 10
Example-1
Find Solution of game theory problem using arithmetic
method

34. Solution:
Saddle point testing
Player APlayer B B1 B2 B3
A1 10 5 -2
A2 13 12 15
A3 16 14 10

35. We apply the maximin (minimax) principle to analyze the
game.
Player APlayer B B1 B2 B3
Row
Minimum
A1 10 5 -2 -2
A2 13 [12] 15
[12]
Maximin
A3 16 (14) 10 10
Column
Maximum
16 (14)
Minimax
15

36. Select minimum from the maximum of columns
Column MiniMax = (14)
Select maximum from the minimum of rows
Row MaxiMin = [12]
Here, Column MiniMax ≠ Row MaxiMin
∴ This game has no saddle point.

37. Player A
Player B B1 B2 B3
A1 10 5 -2
A2 13 12 15
A3 16 14 10
Apply Dominance rule to reduce the size of the payoff matrix
row-1 ≤ row-3, so remove row-1

38. Player A
Player B B1 B2 B3
A2 13 12 15
A3 16 14 10
row-1 ≤ row-3, so remove row-1
column-1 ≥ column-2,
so remove column-1

39. Player A
Player B B2 B3
A2 12 15
A3 14 10
column-1 ≥ column-2,
so remove column-1

40. Using arithmetic method to get optimal mixed strategies for both
the firms.

41. Probabilities
P(A2)=4/7=0.57
P(A3)=3/7=0.43
Hence, Firm A should adopt
strategy A2 and A3 with 57% of time and
43% of time respectively.

42. Probabilities
P(B2)=5/7=0.71
P(B3)=2/7=0.29
Firm B should adopt
Strategy B2 and B3 with 71% of time and
29% of time respectively.

43. Value of Game:
Expected gain of Firm A

44. Expected loss of Firm B

45. Example-2
Find Solution of game theory problem using arithmetic
method
(Practice Problem: similar as problem 1)
Player APlayer B B1 B2 B3
A1 1 7 2
A2 6 2 7
A3 5 1 6

46. Player B
B1 B2 B3
Row
Minimum
Player A
A1 1 7 2 1
A2 (6) [2] 7 [2]
A3 5 1 6 1
Column
Maximum
(6) 7 7
We can apply the maximin (minimax) principle to analyze
the game.

47. Select minimum from the maximum of columns
Column MiniMax = (6)
Select maximum from the minimum of rows
Row MaxiMin = [2]
Here, Column MiniMax ≠ Row MaxiMin
∴ This game has no saddle point.

48. Apply Dominance rule to reduce the size of the payoff matrix
row-3 ≤ row-2, so remove row-3
Player B
B1 B2 B3
Player
A
A1 1 7 2
A2 6 2 7

49. column-3 ≥ column-1, so remove column-3
Player B
B1 B2
Player
A
A1 1 7
A2 6 2

51. Hence, firm A should adopt strategy A1 and A2 with
40% of time and 60% of time respectively.

52. Similarly, firm B should adopt strategy B1 and B2 with
50% of time and 50% of time respectively.

53. Value of Game:

55. 2.
Algebraic
method

61. Operations research
GAME THEORY
Lecture 4
Matrix method - Example
Graphical method - Example

62. Matrix Method

63. Example

65. where p1 and p2 represent the probabilities of player A's, using his
strategies A1 and A2 respectively.

67. where q1 and q2 represent the probabilities of player B's, using
his strategies B1 and B2 respectively.

68. Hence, Value of the game V =
(Player A's optimal strategies) × (Payoff matrix Pij) × (Player B's
optimal strategies)

69. Graphical method Steps (Rule)
Step-1: This method can only be used in games with no saddle
point, and having a pay-off matrix of type n×2 or 2×n.
Step-2: The example is used to explain the procedure.

70. Example

73. A1
A2
A1A2

76. Self Practice Problem

86. Summery
GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example