Robin and Cathy play a game of matching fingers. On a predetermined signal, both players simultaneously extend 1, 2, or 3 fingers from a closed fist. If the sum of the number of fingers extended is even, then Robin receives an amount in dollars equal t that sum from Cathy. If the sum of the numbers of fingers extended is odd, then Cathy receives an amount in dollars equal to that sum from Robin. (a) Construct the payoff matrix for the game. (Assume Robin is the row player and Cathy is the column player.) 1 2 3 1 3 (b) Find the maximin and the minimax strategies for Robin and Cathy, respectively. The maximin strategy for Robin is to play row The minimax strategy for Cathy is to play column (c) Is the game strictly determined? Yes O No (d) If the answer to part (c) is yes, what is the value of the game? (If it is not strictly determined, enter DNE.)
Robin and Cathy play a game of matching fingers. On a predetermined signal, both players simultaneously extend 1, 2, or 3 fingers from a closed fist. If the sum of the number of fingers extended is even, then Robin receives an amount in dollars equal t that sum from Cathy. If the sum of the numbers of fingers extended is odd, then Cathy receives an amount in dollars equal to that sum from Robin. (a) Construct the payoff matrix for the game. (Assume Robin is the row player and Cathy is the column player.) 1 2 3 1 3 (b) Find the maximin and the minimax strategies for Robin and Cathy, respectively. The maximin strategy for Robin is to play row The minimax strategy for Cathy is to play column (c) Is the game strictly determined? Yes O No (d) If the answer to part (c) is yes, what is the value of the game? (If it is not strictly determined, enter DNE.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Game Theory Exercise: Matching Fingers Game**
Robin and Cathy play a game of matching fingers. On a predetermined signal, both players simultaneously extend 1, 2, or 3 fingers from a closed fist. If the sum of the number of fingers extended is even, then Robin receives an amount in dollars equal to that sum from Cathy. If the sum of the numbers of fingers extended is odd, then Cathy receives an amount in dollars equal to that sum from Robin.
**Tasks:**
(a) **Construct the payoff matrix for the game.** *(Assume Robin is the row player and Cathy is the column player.)*
Payoff Matrix:
\[
\begin{array}{c|ccc}
& 1 & 2 & 3 \\
\hline
1 & & & \\
2 & & & \\
3 & & & \\
\end{array}
\]
(b) **Find the maximin and the minimax strategies for Robin and Cathy, respectively.**
- The maximin strategy for Robin is to play row \_\_\_\_\_.
- The minimax strategy for Cathy is to play column \_\_\_\_\_.
(c) **Is the game strictly determined?**
- \( \text{O Yes} \)
- \( \text{O No} \)
(d) **If the answer to part (c) is yes, what is the value of the game?** *(If it is not strictly determined, enter DNE.)*
- Value of the game: \_\_\_\_\_
**Explanation of the Payoff Matrix:**
In the payoff matrix, the rows represent the number of fingers Robin can choose to show (1, 2, or 3), and the columns represent the number of fingers Cathy can choose to show (1, 2, or 3). The values in the cells represent the amount of dollars won or lost by Robin, based on the sum of the fingers shown. If the sum is even, Robin gains that amount from Cathy. If the sum is odd, Cathy gains that amount from Robin. The goals are to determine the optimal strategy for each player and assess whether the game is strictly determined and to calculate the game's value accordingly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77247d15-844d-4e77-b205-212be8c6419e%2F0edd1d9e-f64c-479b-a2e2-e86c08e2a60e%2F2fkn6kn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Game Theory Exercise: Matching Fingers Game**
Robin and Cathy play a game of matching fingers. On a predetermined signal, both players simultaneously extend 1, 2, or 3 fingers from a closed fist. If the sum of the number of fingers extended is even, then Robin receives an amount in dollars equal to that sum from Cathy. If the sum of the numbers of fingers extended is odd, then Cathy receives an amount in dollars equal to that sum from Robin.
**Tasks:**
(a) **Construct the payoff matrix for the game.** *(Assume Robin is the row player and Cathy is the column player.)*
Payoff Matrix:
\[
\begin{array}{c|ccc}
& 1 & 2 & 3 \\
\hline
1 & & & \\
2 & & & \\
3 & & & \\
\end{array}
\]
(b) **Find the maximin and the minimax strategies for Robin and Cathy, respectively.**
- The maximin strategy for Robin is to play row \_\_\_\_\_.
- The minimax strategy for Cathy is to play column \_\_\_\_\_.
(c) **Is the game strictly determined?**
- \( \text{O Yes} \)
- \( \text{O No} \)
(d) **If the answer to part (c) is yes, what is the value of the game?** *(If it is not strictly determined, enter DNE.)*
- Value of the game: \_\_\_\_\_
**Explanation of the Payoff Matrix:**
In the payoff matrix, the rows represent the number of fingers Robin can choose to show (1, 2, or 3), and the columns represent the number of fingers Cathy can choose to show (1, 2, or 3). The values in the cells represent the amount of dollars won or lost by Robin, based on the sum of the fingers shown. If the sum is even, Robin gains that amount from Cathy. If the sum is odd, Cathy gains that amount from Robin. The goals are to determine the optimal strategy for each player and assess whether the game is strictly determined and to calculate the game's value accordingly.
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