Two companies complete for a share of the soft drink market. Each has worked with an advertising agency to develop alternative advertising strategies for the coming year. A variety of television advertisements, newspaper advertisement, product promotions and in-store displays have provided four different strategies for each company. The payoff table summarizes the gain in market share for company A projected for the various combinations of Company A and Company B strategies. What is the optimal strategy for each company? What is the value of the game?
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To determine the optimal strategy for each company and the value of the game, we can analyze the payoff table. The payoff table shows the gain in market share for Company A based on the combinations of strategies chosen by both companies, A and B.
First, let's visualize the payoff table:
| Company B Strategies
Company A Strategies | Strategy 1 | Strategy 2 | Strategy 3 | Strategy 4
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Strategy 1 | 3 | 5 | 4 | 1
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Strategy 2 | -2 | 6 | 7 | 3
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Strategy 3 | 0 | 7 | 6 | 4
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Strategy 4 | 1 | 2 | 9 | 5
In this table, the numbers represent the gain in market share for Company A.
To find the optimal strategy for each company, we need to apply a decision-making approach called the minimax (maximin) strategy. This approach maximizes the minimum possible gain for each company.
1. Find the minimum value for each row (Company A's gain):
Row 1: Minimum value = 1
Row 2: Minimum value = -2
Row 3: Minimum value = 0
Row 4: Minimum value = 1
2. Among the minimum values obtained in step 1, select the maximum value. In this case, the maximum value is 1.
3. Determine the company's strategy that corresponds to the maximum value obtained in step 2.
For Company A, the maximum value of 1 is achieved with Strategy 1 (Row 4). Therefore, the optimal strategy for Company A is Strategy 1.
4. Repeat steps 1-3 for Company B, but in this case, find the maximum value for each column (Company B's gain) and then select the minimum value among them.
Column 1: Maximum value = 3
Column 2: Maximum value = 7
Column 3: Maximum value = 9
Column 4: Maximum value = 5
5. Among the maximum values obtained in step 4, select the minimum value. In this case, the minimum value is 3.
6. Determine the company's strategy that corresponds to the minimum value obtained in step 5.
For Company B, the minimum value of 3 is achieved with Strategy 1 (Column 2). Therefore, the optimal strategy for Company B is Strategy 1.
The value of the game is the gain in market share for Company A when both companies play their optimal strategies. In this case, since both companies choose Strategy 1, the gain in market share for Company A is 3.
So, the optimal strategy for Company A is Strategy 1, the optimal strategy for Company B is Strategy 1, and the value of the game is 3.