Game theory

Game theory

1. KULDEEP MATHUR M.B.A . JIWAJI UNIVERSITY GWALIOR GAME THEORY:

2. Game is defined as an activity between two or more persons involving activities by each person according to a set of rules , at the end of which each person receives some benefit or satisfaction or suffers loss. The set of rules define the game . Going through the set of rules once by the participants defines a play.

3. Game theory attempts to study “ decision making in situations where two or more intelligent and rational opponents are involved under conditions of conflict and competition. ”

4. Terminology: Strategy : A strategy for a player is defined as a set of rules or alternative courses of action available to him in advance , by which player decides the course of action that he should adopt. TYPES: Pure strategy Mixed strategy Optimal strategy

5. Pure strategy: pure strategy is a decision rule always to select a particular course of action. Mixed strategy: Mixed strategy is a selection among pure strategies with fixed probabilities. Optimal strategy: A course of action or play which puts the player in the most preferred position , irrespective of the strategy of his competitors is called an optimal strategy.

6. Number of players: If a game involves n players , then it is called a n-person game . Payoff: Outcomes of a game when different alternatives are adopted by the competing players are called the payoffs.

7. Payoff matrix: The payoffs in terms of gains or losses when player select their particular strategies can be represented in the form of a matrix called the payoff matrix.

8. Zero-sum games: If the players make payments only to each other i ..e “ loss of one is the gain of others ” , then the competitive game is called zero-sum game .

9. Two person zero-sum game: A game of two person in which the gains of one player are the losses of the other player is called a two person zero-sum game.

10. Example: Player A: gainer Player B: loser All the payoffs are assumed in terms of player A. a ij : the payoff which player A gains from the player B chooses strategy A i and player B chooses strategy B j Negative entry in the table means that the payments are to be made by A to B.

11. Player A: Maximizing player Player B: Minimizing player Maximin Principle: The player A decides to play that strategy which corresponds to the maximum of the minimum gains for his different courses of action. Minimax Principle: The player B would like to play that strategy which corresponds to the minimum of the maximum losses for his different courses of action.

12. Saddle point: The maximin value = The minimax value We get a saddle point. The saddle point is the solution of the game . Therefore , The strategies of A and B corresponding to saddle point are the optimal strategies of A and B .

13. Value of the game: It is the expected payoff of play when all the players of the game follow their optimum strategies. The game is called fair if the value of the game is zero and strictly determinable if it is non zero.

14. Problem: Player A can choose his strategies from (A1,A2,A3) only .while B can choose from the set (B1,B2) only. The rules of the game state that the payments should be made in accordance with the selection of strategies : Strategies pair selected Payment to be made (A1,B1) Player A pays Re 1 to player B (A1,B2) Player B pays Rs 6 to player A (A2,B1) Player B pays Rs 2 to player A (A2,B2) Player B pays Rs 4 to player A (A3,B1) Player A pays Rs 2 to player B (A3,B2) Player A pays Rs 6 to player B What strategies should A and B play in order to get the optimum benefit of the play

15. Rectangular games without saddle point: The game has no saddle point The concept of optimum strategies can b extended to all matrix games by introducing a probability with choice and mathematical expectation with payoff.

16. Let player A chooses a particular activity i such that 1 < i < m. then the set x=(x i , 1 < i < m) of probability constitute the strategy of A. Similarly, The set y=( yi , 1 < i < n) of probability constitute the strategy of B. Thus the vector x=(x1,x2,… x m ) of the non negative numbers satisfying x1+x2+…+ x m =1 is called mixed strategy of the player A. Similarly, the vector y=(y1,y2,… y n ) of the non negative numbers satisfying y1+y2+…+ y n =1 is called mixed strategy of the player B.

17. The mathematical expectation of the payoff function E(x , y) in a game whose payoff matrix is V ij is defined by E(x , y)=∑∑ x i v ij y i where x and y are the mixed strategies of the player A and player B. Thus, The player A chooses x so as to maximize his minimum expectation and the player B should chooses y so as to minimize the player A ‘s greatest expectation. The player A tries max min E( x,y ) The player B tries min max E( x,y )

18. Strategic saddle point: If min max E( x,y )=E( x o ,y o )=max minE ( x,y ) Then ( x o ,y o ) is called the strategic saddle point of the game where x o and y o define the optimum strategies and v=E( x o ,y o ) is the value of the game.

19. Principle of dominance: The concept of dominance is especially useful for the evaluation of two person zero-sum games where a saddle point does not exist . RULE : When all element in a row of a payoff matrix are less than or equal to the corresponding elements in another row, then the former row is dominated by the latter and can be deleted from the matrix.

20. When all element in a column of a payoff matrix are greater than or equal to the corresponding elements in another column, then the former row is dominated by the latter and can be deleted from the matrix. A pure strategy may be dominated if it is inferior to average of two or more other pure strategies.

21. Graphical method for 2×n and m×2games: The graphic method consists of two graphs: The payoff available to player A versus his strategies, and The payoff available by player B versus his strategies

22. Graphical method for 2×n: Graph for the player A: The highest point on the lower boundary of these lines will give maximum expected payoff among the minimum expected payoffs on the lower boundary and the optimal value of the probability p1 and p2. Now, the two strategies of player B corresponding to these lines passes through the maximin point can be determined. It helps in reducing the size of the game (2×2).

23. Graphical method for m×2: Graph for the player B: The lowest point on the upper boundary of these lines will give minimum expected payoff among the maximum expected payoffs on the upper boundary and the optimal value of the probability q1 and q2. Now, the two strategies of player A corresponding to these lines passes through the minimax point can be determined. It helps in reducing the size of the game (2×2).