# Two Player Zero-Sum Games

## Presentation on theme: "Two Player Zero-Sum Games"— Presentation transcript:

Two Player Zero-Sum Games

b S4 c d Given any two-player zero-sum game, can we construct an algebraic formula to determine the saddle point?

b S4 c d Given any two-player zero-sum game, can we construct an algebraic formula to determine the saddle point? Let’s assume that the saddle point does not occur in pure strategies. That is, assume the saddle point lies in mixed strategies…

p p Given any two-player zero-sum game, can we construct a formula to determine the saddle point in mixed strategies? The answer is yes. This is based on the assumption that the saddlepoint in mixed strategies will occur at the intersection of payoff functions defined as follows. S1 S2 S3 a b S4 c d q 1-q We will assign probability p to strategy S1 and thus 1-p to strategy S2 and likewise, we’ll assign probability q to strategy S3 and thus 1-q to strategy S4. Now the payoff functions for each player are as follows… ES1 = aq + c(1-q) = aq + c – cq = q(a-c) + c ES2 = bq + d(1-q) = bq + d – dq = q(b-d) + d ES3 = ap + b(1-p) = ap + b – bp = p(a-b) + b ES4 = cp + d(1-p) = cp + d – dp = p(c-d) + d

p p S1 S2 S3 a b S4 c d The optimal strategy for the column player will occur at the intersection of these two payoff functions: ES3 = p(a-b) + b ES4 = p(c-d) + d q 1-q …and the optimal strategy for the row player will occur at the intersection of these payoff functions. ES1 = q(a-c) + c ES2 = q(b-d) + d

p p ES3 = p(a-b) + b ES4 = p(c-d) + d Solving ES3 = ES4 for p we get… S1 S2 S3 a b S4 c d Note that this is undefined if the denominator is equal to zero.