Game theory ppt

quantitative techniques

1. GAME THEORY PRESENTED BY: AKANKSHA SHARMA AKANSHA BHARGAWA ANKITA DHEER ANUSHKA KAPOOR PRAJAL RITURAJ SINGH

2. Game theory Developed by Prof. John Von Neumann and Oscar Morgenstern in 1928 game theory is a body of knowledge that deals with making decisions. The approach of game theory is to seek, to determine a rival’s most profitable counter-strategy to one’s own best moves.

3. Classification Two-Person Game – A game with 2 number of players. Zero-Sum Game – A game in which sum of amounts won by all winners is equal to sum of amounts lost by all losers. Non-Zero Sum Game – A game in which the sum of gains and losses are not equal. Pure-Strategy Game – A game in which the best strategy for each player is to play one strategy throughout the game. Mixed-Strategy Game – A game in which each player employs different strategies at different times in the game.

4. (1) Saddle point method: At the right of each row, write the row minimum and underline the largest of them. At the bottom of each column, write the column maximum and underline the smallest of them. If these two elements are equal, the corresponding cell is the saddle point and the value is value of the game.

5. Example : The pay off matrix of a two person zero sum game is:- Solution:

6. (2) Dominance method It states that if the strategy of a player dominates over the other strategy in all condition, the later strategy can be ignored. Rule 1 : If all the elements in a row of a pay-off matrix are “<” or “=” to the corresponding elements of other row then comparative row will be deleted Rule 2 : If all elements in a column in a pay-off matrix are “>” or “=” to the corresponding elements of other column then comparative column will be deleted.

7. Example : consider a game with a pay-off matrix: b1 b2 b3 b4 A1 42 72 32 12 A2 40 30 25 10 A3 30 8 -10 0 A4 45 10 0 15 Solution : b1 b2 b3 b4 A1 42 72 32 12 A2 40 30 25 10 A3 30 8 -10 0 A4 45 10 0 15 => B3 B4 A1 32 12 A2 0 15

8. B3 B4 1 11 111 A1 32 12 20 15 15/35 A2 0 15 15 20 20/35 32 3 11 3 32 111 3/35 32/35 Applying odoment method: Value of game= 32x15 + 0x20 = 96/7 35

9. Graphical method It is helpful in finding out which of the two strategies can be used. point area Case (a): mx2 mini max Case(b): 2xn maxi min

10. Example : consider a game with a pay-off matrix B1 B2 b3 B4 B5 A1 2 -4 6 -3 5 A2 -3 4 -4 1 Solution: By applying dominance rule we can cut of the following columns: B1 B2 b3 B4 B5 A1 2 -4 6 -3 5 A2 -3 4 -4 1

11. So we have: B1 B4 A1 2 -3 A2 -3 1 8 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 8 7 6 5 4 3 2 1 -1 -2 -3 -4 -5

12. Applying oddoment method: B1 B2 A1 2 -3 A2 -3 1 1 5 4 11 4 5 111 4/9 5/9 1 11 111 5 4 4/9 4 5 5/9 Value of game = 2x(4/9) – 3x(5/9) = 7/9

13. Algebraic method: This method is used for 2*2 games which do not have any Saddle Point. As it does not have any saddle point so mixed strategy has to be used. Players selects each of the available strategies for certain proportion of time i.e., each player selects a strategy with some probability.

14. Example : consider a game with a pay-off matrix B1 B2 A1 1 3 A2 7 -5 Let, p= probability that A uses strategy A1, q= probability that B uses strategy B1 So, 1-p= probability that A uses strategy A2, 1-q = probability that B uses strategy B2 V=px1+ (1-p)x7------------------------(1) V=px3+ (1-p)x(-5)---------------------(2) V=qx1+(1-q)x3-------------------------(3) V=qx7+(1-q)x(-5)----------------------(4) Solution: From equation (1) and (2) we get : p= 6/7 & (1-p)= 1/7 Strategy of A is 6/7 1/7 From equation (3) and (4) we get : q= 4/7 & ( 1-q)= 3/7 Strategy of B is 4/7 3/7 Value of game : V= 6/7x1 + 1/7x7= 13/7

15. Limitations of game theory: The assumptions that each player has the knowledge about his own pay-offs and pay-off’s of the opponent is not practical The method of solution becomes complex with the increase in no. of players In the game theory it is assumed that both the players are equally wise and they behave in a rational way ,this assumption is also not possible.