1. Part 3 Linear Programming
3.6 Game Theory

2. Example

3. Mixed Strategy
It is obvious that X will not do the same thing every time, or Y would copy him and win everything. Similarly, Y cannot stick to a single strategy, or X will do the opposite.
Both players must use a mixed strategy, and furthermore the choice at every turn must be absolutely independent of the previous turns.
Assume that X decides that he will put up 1 hand with frequency x1 and 2 hands with frequency x2=1-x1. At every turn this decision is random. Similarly, Y can pick his probabilities y1 and y2=1-y1.
It is not appropriate to choose x1=x2=y1=y2=0.5 since Y would lose $20 too often. But the more Y moves to a pure 2-hand strategy, the more X will move toward 1 hand.

4. Equilibrium
Does there exist a mixed strategy y1 and y2 that, if used consistently by Y, offers no special advantage to X?
Can X choose the probabilities x1 and x2 that present Y with no reason to change his own strategy?
At such equilibrium, if it exists, the average payoff to X will have reached a saddle point. It is a maximum as far as X is concerned, and a minimum as far as Y is concerned.
To find such a saddle point is to “solve” the game.

5. Best Strategy for X

6. Best Strategy for Y

7. Significance of Solution
Such a saddle point is remarkable, because it means that X plays his 2-hand strategy only 2/5 of the time, even though it is this strategy that gives him a chance at $20.
At he same time, Y is forced to adopt a losing strategy – he would like to match X, but instead he uses the opposite probabilities 2/5 (1 hand) and 3/5 (2 hand).

8. Matrix Game
X has n possible moves to choose from, and Y has m. Thus, the dimension payoff matrix A is m by n. The entry aij in A represents the payment received by X when he chooses his jth strategy and Y chooses his ith. A negative entry means a win for Y.

9. Matrix Game

10. Example

11. Fair Game

12. Example

13. Equivalent Payoff Matrix
A matrix (E) that has every entry equal to 1. Adding a multiple of E to the payoff matrix, i.e. A→A+cE, simply means that X wins an additional amount c at every turn.
The value of the game is increased by c, but there is no reason to change the original strategies.

14. Minimax Theorem

15. Interpretation of Minimax Theorem from Player X’s Viewpoint

16. Interpretation of Minimax Theorem from Player Y’s Viewpoint

17. Connections Between Game Theory and Duality in LP – (1)

18. Connections Between Game Theory and Duality in LP – (2)

19. Connections Between Game Theory and Duality in LP – (3)