Mixed Strategies

Games which are not strictly determined can be further analysed by considering what happens when R and C play a game many times. Instead of choosing the same row (or column) on each occasion, R and C can adopt ``mixed'' strategies and thereby improve their expected payoff.

Definition 6..10   A strategy for the player R is a vector u in $\mathbb {R}$^m with $\sum_{{i=1}}^{m}$u_i = 1 and u$\ge$ 0. A strategy for the player C is a vector v in $\mathbb {R}$ⁿ with $\sum_{{j=1}}^{n}$v_j = 1 and v$\ge$ 0.

What we have in mind is that if R has strategy u then he chooses row a_i with probability u_i. That is, over a series of games, he chooses a_i with the frequency that this specifies. Similarly, if C has strategy v then he chooses column a^j with the frequency given by the probability v_j.

Definition 6..11   For strategies u for R and v for C, the expectation E(u,v) is given by

E(u,v) = $$\sum_{{i=1}}^{m}$$$$\sum_{{j=1}}^{n}$$u_ia_ijv_j.

The expectation gives the 'expected value' of the payoff that R receives from C. Using matrix notation, one has E(u,v) = u^TAv. Note that Rowman is sure to win $\min_{{\v}}^{}$E(u,v) if he uses mixed strategy u. If he now maximises his certain winnings by choosing u sensibly, he can be sure of winning

$$\alpha^{*}_{}$$ = $$\max_{{\u}}^{}$$$$\min_{{\v}}^{}$$E(u,v).

In the same way, Columnman, using strategy v, will lose at most $\max_{{\u}}^{}$E(u,v), and he can reduce this loss as much as possible by choosing his strategy v sensibly; he then looses

$$\beta^{*}_{}$$ = $$\min_{{\v}}^{}$$$$\max_{{\u}}^{}$$E(u,v).

Definition 6..12   Let

$$\alpha^{*}_{}$$ = $$\max_{{\u}}^{}$$$$\min_{{\v}}^{}$$E(u,v) and $$\beta^{*}_{}$$ = $$\min_{{\v}}^{}$$$$\max_{{\u}}^{}$$E(u,v).

Then $\alpha^{*}_{}$ and $\beta^{*}_{}$ are the optimum lower and optimum upper values of the game A. Any u such that

$$\min_{{\v}}^{}$$E(u,v) = $$\alpha^{*}_{}$$

is an optimum strategy for R and any v such that

$$\max_{{\u}}^{}$$E(u,v) = $$\beta^{*}_{}$$

is an optimum strategy for C.