CMA Final · Strategic Cost Management · Game Theory
In a 2x2 zero-sum game, pay-offs to Player P are: P1 against Q1 = 8, P1 against Q2 = 2, P2 against Q1 = 3, P2 against Q2 = 6. What is the value of the game?
The value of the game is about 4.67, obtained as (8×6 − 2×3) divided by (8+6−2−3), which equals 42/9. No pure saddle point exists, so the mixed-strategy formula applies.
- A4.5Correct
- B5.0
- C4.0
- D5.5
Explanation
No saddle point exists since the maximin is 3 and the minimax is 6. The value is (ad − bc)/(a + d − b − c) = (8×6 − 2×3)/(8+6−2−3) = 42/9 = 4.67. Check against the probabilities: p for P1 satisfies 8p+3(1−p) = 2p+6(1−p), so 5p+3 = −4p+6, p = 1/3, and the value is 8/3+2 = 4.67. Option A (4.5) is not equal to this, so the answer key is mistaken.
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