Game theory Introduction Lectures 4.pdf

Game theory lecture

1. Lecture 4
GAME THEORY

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

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

4. 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.

5. Player A\
Player 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

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

7. We apply the maximin (minimax) principle to analyze the
game.
Player A\
Player 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

8. 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.

9. Player A
\
Player BB1B2B3
A1 105-2
A2 131215
A3 161410
Apply Dominance rule to reduce the size of the payoff matrix
row-1≤row-3, so remove row-1

10. Player A
\
Player BB1B2B3
A2 131215
A3 161410
row-1≤row-3, so remove row-1
column-1≥column-2,
so remove column-1

11. Player A
\
Player BB2B3
A2 1215
A3 1410
column-1≥column-2,
so remove column-1

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

13. Probabilities
P(A2)=4/7=0.57
P(A3)=3/7=0.43
Hence, FirmAshould adopt
strategyA2andA3with 57% of time and
43% of time respectively.

14. Probabilities
P(B2)=5/7=0.71
P(B3)=2/7=0.29
FirmBshould adopt
StrategyB2andB3with 71% of time and
29% of time respectively.

15. Value of Game:
Expected gain of Firm A

16. Expected loss of Firm B

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

18. PlayerB
B1 B2 B3
Row
Minimum
PlayerA
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.

19. 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.

20. Apply Dominance rule to reduce the size of the payoff matrix
row-3≤row-2, so remove row-3
PlayerB
B1B2B3
Player
A
A1 1 7 2
A2 6 2 7

21. column-3≥column-1, so remove column-3
PlayerB
B1B2
Player
A
A1 1 7
A2 6 2

23. Hence, firmAshould adopt strategyA1andA2with
40% of time and 60% of time respectively.

24. Similarly, firmBshould adopt strategyB1andB2with
50% of time and 50% of time respectively.

25. Value of Game:

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

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

29. Matrix Method

30. Example

32. wherep1andp2represent the probabilities of player A's, using his
strategiesA1andA2respectively.

34. whereq1andq2represent the probabilities of player B's, using
his strategiesB1andB2respectively.

35. Hence, Value of the gameV=
(Player A's optimal strategies)×(Payoff matrixPij)×(Player B's
optimal strategies)

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

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