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Part 3 Linear Programming. 3.6 Game Theory. Example. 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.
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Part 3 Linear Programming 3.6 Game Theory
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 frequencyx2=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.
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.
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).
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.
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.
