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CMA Final · Strategic Cost Management · Game Theory

Two telecom firms, X and Y, compare campaign choices. The table shows the change in X's market-share index (Y's change is the negative). X1 gives +2 against Y1 and -3 against Y2. X2 gives -1 against Y1 and +4 against Y2. What are Y's optimal probability of playing Y1 and the value of the game?

Y should play Y1 with probability 0.7, and the game value is 0.5. There is no saddle point, so Y equalises X's expected payoff across X1 and X2, giving q = 0.7. Substituting back yields an expected gain of 0.5 for X.

  1. AProbability 0.7 for Y1; value 0.5Correct
  2. BProbability 0.3 for Y1; value 0.5
  3. CProbability 0.7 for Y1; value -1
  4. DProbability 0.5 for Y1; value 0.5

Explanation

Row minima are -3 and -1 (maximin -1) and column maxima are 2 and 4 (minimax 2), so no saddle point exists. Y plays Y1 with probability q: 2q-3(1-q) = -q+4(1-q), so 5q-3 = 4-5q and q = 0.7. Value = 2(0.7)-3(0.3) = 0.5. Check with X's p = 0.5: 2(0.5)-1(0.5) = 0.5. The 0.5 probability is X's, not Y's.

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