Bella Robinson and Steve Carson are running for a seat in the U.S. Senate. If both candidates campaign only in the major cities of the state, then Robins candidates campaign in only rural areas, then Robinson is expected get 55% of the votes; if Robinson campaigns exclusively in the city and Carson camp Robinson is expected to get 20% of the votes; finally, if Robinson campaigns exclusively in the rural areas and Carson campaigns exclusively in the city, votes. (a) Construct the payoff matrix for the game. (Enter each percentage as a decimal.) Carson Robinson City Rural City Is the game strictly determined? Yes No P= Rural (b) Find the optimal strategy for both Robinson (row) and Carson (column).

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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).
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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