CMA Final · Strategic Cost Management · Game Theory
In a zero-sum game, A (maximiser) chooses between A1 and A2, and B chooses between B1 and B2. A's payoffs are: A1 gets 5 against B1 and 1 against B2; A2 gets 3 against B1 and 4 against B2. What is the value of the game?
The value is 3.4. No saddle point exists because maximin is 3 and minimax is 4. Equalising A's expected payoff against both of B's strategies gives A1 probability 0.2, and the resulting expected payoff is 3.4 whichever column B chooses.
- A3.4Correct
- B3.5
- C3.0
- D4.0
Explanation
Row minima are 1 and 3 (maximin 3); column maxima are 5 and 4 (minimax 4), so there is no saddle point. Let A play A1 with probability p: 5p+3(1-p) = p+4(1-p), so 3+2p = 4-3p and p = 0.2. Value = 3+0.4 = 3.4. Check with B's q = 0.6: 5(0.6)+1(0.4) = 3.4. Taking the average of 3 and 4 (3.5) is wrong because the mixed value is not a simple midpoint.
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