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 fingers extended is odd, then Cathy receives an amount in dollars equal to that sum from Robin.
Find the minimax and maximin strategies for Robin and Cathy, respectively.

A) Maximin strategy for Robin is to extend 1,2 or 3 fingers whereas minimax strategy for Cathy is to extend 1 or 2 fingers
B) Maximin strategy for Robin is to extend 1,2 or 3 fingers whereas minimax strategy for Cathy is to extend 1 finger
C) Maximin strategy for Robin is to extend 1 finger whereas minimax strategy for Cathy is to extend 1 finger
D) Maximin strategy for Robin is to extend 1 finger whereas minimax strategy for Cathy is to extend 1 or 2 fingers
E) Maximin strategy for Robin is to extend 1 or 2 fingers whereas minimax strategy for Cathy is to extend 1 finger

Question

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 fingers extended is odd, then Cathy receives an amount in dollars equal to that sum from Robin. ​
Find the minimax and maximin strategies for Robin and Cathy, respectively.
​
A) Maximin strategy for Robin is to extend 1,2 or 3 fingers whereas minimax strategy for Cathy is to extend 1 or 2 fingers
B) Maximin strategy for Robin is to extend 1,2 or 3 fingers whereas minimax strategy for Cathy is to extend 1 finger
C) Maximin strategy for Robin is to extend 1 finger whereas minimax strategy for Cathy is to extend 1 finger
D) Maximin strategy for Robin is to extend 1 finger whereas minimax strategy for Cathy is to extend 1 or 2 fingers
E) Maximin strategy for Robin is to extend 1 or 2 fingers whereas minimax strategy for Cathy is to extend 1 finger

Quizplus · Accepted Answer

To find the minimax and maximin strategies, we need to analyze the game using a payoff matrix.

Robin's strategies are to extend 1, 2, or 3 fingers, and Cathy's strategies are to extend 1, 2, or 3 fingers. There are 9 possible outcomes:

- If Robin extends 1 finger and Cathy extends 1 finger, the sum is even and Robin gets $2 from Cathy.
- If Robin extends 1 finger and Cathy extends 2 fingers, the sum is odd and Cathy gets $1 from Robin.
- If Robin extends 1 finger and Cathy extends 3 fingers, the sum is even and Robin gets $4 from Cathy.
- If Robin extends 2 fingers and Cathy extends 1 finger, the sum is odd and Cathy gets $1 from Robin.
- If Robin extends 2 fingers and Cathy extends 2 fingers, the sum is even and Robin gets $4 from Cathy.
- If Robin extends 2 fingers and Cathy extends 3 fingers, the sum is odd and Cathy gets $3 from Robin.
- If Robin extends 3 fingers and Cathy extends 1 finger, the sum is even and Robin gets $4 from Cathy.
- If Robin extends 3 fingers and Cathy extends 2 fingers, the sum is odd and Cathy gets $3 from Robin.
- If Robin extends 3 fingers and Cathy extends 3 fingers, the sum is even and Robin gets $6 from Cathy.

Here is the payoff matrix:

1      2       3
      --------------------
  1  |  2,-1 | -1, 1 |  4,-3 |
      --------------------
  2  | -1, 2 |  4,-1 | -3, 3 |
      --------------------
  3  |  4,-3 | -3, 3 |  6,-5 |
      --------------------

To find Robin's minimax strategy, we need to find the minimum value in each row and choose the strategy that maximizes that minimum value. The minimum values are -1, -1, and -5, so Robin should choose to extend 1, 2, or 3 fingers with equal probability, since all three strategies have the same maximum minimum value of -1.

To find Cathy's maximin strategy, we need to find the maximum value in each column and choose the strategy that minimizes that maximum value. The maximum values are 4, 3, and 3, so Cathy should choose to extend 1 finger to minimize her maximum loss. If she extends 2 or 3 fingers, her maximum loss could be even greater. Therefore, the answer is D.

Robin and Cathy Play a Game of Matching Fingers