Unformatted text preview: Minimax Theorem Notice ﬁrst that both (Pg) and (PM) are feasible problems. We can choose any
p E P and sufﬁciently small 0: to get a feasible solution to (P1), and any q E Q and
sufﬁciently large {3 to get a feasible solution to (PM). Moreover, they are dual problems. Their coefﬁcient matrices are transposes of one
another and their righthand sides and objective vectors switch. The unconstrained
(.1 corresponds to the equality constraint Z (,3 = 1, etc. Since both are feasible, strong
duality tells us that both have optimal solutions, and their optimal values are equal.
Hence player I has a maximin mixed strategy 13‘, II has a minimax mixed strategy
9‘. and r=a*='p“Aq* =l3*=ﬁ This is John von Neumann’s Minimax Theorem (1928):
Every 11'}. x n twoperson zerosum game with payoff matrix A has a value in mixed
strategies, 1‘; := min max wk; 2 u := max min pAq. qEQ pEP pep QEQ Further results We can use other results we know from LP to help our understanding of zerosum
game theory. For instance, the complementary slackness theorem tells us: Suppose p* is a max
irnin strategy for I, and p‘AJ > n“ for somej (n‘ = 1.1 = i7 is the value of the game).
Then there is strict inequality in the jth constraint of (Fir) at optimality. Hence
the jth dual variable is zero in every optimal solution of (PH), so or; = [1 for every
minimax strategy q‘ for II; 11 doesn’t use her jth pure strategy. Note in the example,
where p‘ = (U,U.4,U.6) and U" = 0.2, that $3013 = 0.4 > n“ = 0.2. Hence we could
have known immediately that in the minjmax strategy (1* we would have q; = 0.
Similarly, #144 = 0.5 > 1." = 0.2; hence q; = 0. 3 ...
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 Spring '08
 SHMOYS

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