Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Game Theory in Political Campaign Strategy**
Bella Robinson and Steve Carson are running for a seat in the U.S. Senate. The outcome of their campaigns depends on where they decide to focus their efforts: the major cities or the rural areas. The expected voting percentages based on their campaign strategies are as follows:
1. If both candidates campaign only in the major cities of the state, Robinson is expected to get 70% of the votes.
2. If both candidates campaign only in the rural areas, then Robinson is expected to get 55% of the votes.
3. If Robinson campaigns exclusively in the city and Carson campaigns exclusively in the rural areas, then Robinson is expected to get 20% of the votes.
4. If Robinson campaigns exclusively in the rural areas and Carson campaigns exclusively in the city, then Robinson is expected to get 35% of the votes.
**Tasks:**
1. (a) Construct the payoff matrix for the game. (Enter each percentage as a decimal.)
| | Carson | |
|------------|-------------------|----------------|
| | City | Rural |
| Robinson | | |
| City | **0.70** | **0.20** |
| Rural | **0.35** | **0.55** |
Is the game strictly determined?
- Yes
- No
2. (b) Find the optimal strategy for both Robinson (row) and Carson (column).
\( P = \, [_] \)
\( Q = \, [_] \)
**Explanation of Diagram:**
The table shown is a payoff matrix for the game. The matrix includes different scenarios where each candidate can focus their campaign efforts either in the city or in rural areas.
- The rows represent Robinson's strategies: campaigning in the **City** or in **Rural** areas.
- The columns represent Carson's strategies: campaigning in the **City** or in **Rural** areas.
- Each cell in the matrix shows the expected percentage of votes that Robinson will receive, given the combination of both candidates' strategies.
For example:
- If both Robinson and Carson campaign in the city, Robinson is expected to get 70% of the votes (0.70).
- If Robinson campaigns in the city while Carson campaigns in rural areas, Robinson is expected to get 20% of the votes (0.20).](https://content.bartleby.com/qna-images/question/77247d15-844d-4e77-b205-212be8c6419e/9e094790-7529-4b3b-803e-19adc96acea7/t5s6d8_processed.jpeg)
Transcribed Image Text:**Game Theory in Political Campaign Strategy**
Bella Robinson and Steve Carson are running for a seat in the U.S. Senate. The outcome of their campaigns depends on where they decide to focus their efforts: the major cities or the rural areas. The expected voting percentages based on their campaign strategies are as follows:
1. If both candidates campaign only in the major cities of the state, Robinson is expected to get 70% of the votes.
2. If both candidates campaign only in the rural areas, then Robinson is expected to get 55% of the votes.
3. If Robinson campaigns exclusively in the city and Carson campaigns exclusively in the rural areas, then Robinson is expected to get 20% of the votes.
4. If Robinson campaigns exclusively in the rural areas and Carson campaigns exclusively in the city, then Robinson is expected to get 35% of the votes.
**Tasks:**
1. (a) Construct the payoff matrix for the game. (Enter each percentage as a decimal.)
| | Carson | |
|------------|-------------------|----------------|
| | City | Rural |
| Robinson | | |
| City | **0.70** | **0.20** |
| Rural | **0.35** | **0.55** |
Is the game strictly determined?
- Yes
- No
2. (b) Find the optimal strategy for both Robinson (row) and Carson (column).
\( P = \, [_] \)
\( Q = \, [_] \)
**Explanation of Diagram:**
The table shown is a payoff matrix for the game. The matrix includes different scenarios where each candidate can focus their campaign efforts either in the city or in rural areas.
- The rows represent Robinson's strategies: campaigning in the **City** or in **Rural** areas.
- The columns represent Carson's strategies: campaigning in the **City** or in **Rural** areas.
- Each cell in the matrix shows the expected percentage of votes that Robinson will receive, given the combination of both candidates' strategies.
For example:
- If both Robinson and Carson campaign in the city, Robinson is expected to get 70% of the votes (0.70).
- If Robinson campaigns in the city while Carson campaigns in rural areas, Robinson is expected to get 20% of the votes (0.20).
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