Question

Alexa1 and Zoltan2 play the following game:

AZ-game:

Step 1: Alexa chooses a uniformly random element from the set {1,2,3}. Let a denote the element that Alexa chooses.
Step 2: Zoltan chooses a uniformly random element from the set {1, 2, 3}. Let z denote the element that Zoltan chooses.

Step 3: Using one of the three strategies mentioned below, Alexa chooses an element from the set {1, 2, 3} \ {a}. Let a′ denote the element that Alexa chooses. Step 4: Using one of the three strategies mentioned below, Zoltan chooses an element from the set {1, 2, 3} \ {z}. Let z′ denote the element that Alexa chooses.

The AZ-game is a success if a′ ̸= z′.

• MinMin Strategy: In Step 3, Alexa chooses the smallest element in the set {1, 2, 3}\{a}, and Zoltan chooses the smallest element in the set {1, 2, 3} \ {z}.

– Describe the sample space for this strategy.
– For this strategy, determine the probability that the AZ-game is a success.

• MinMax Strategy: In Step 3, Alexa chooses the smallest element in the set {1, 2, 3}\{a}, and Zoltan chooses the largest element in the set {1, 2, 3} \ {z}.

– Describe the sample space for this strategy.
– For this strategy, determine the probability that the AZ-game is a success.

• Random Strategy: In Step 3, Alexa chooses a uniformly random element in the set {1, 2, 3} \ {a}, and Zoltan chooses a uniformly random element in the set {1, 2, 3} \ {z}.

– Describe the sample space for this strategy.
– For this strategy, determine the probability that the AZ-game is a success.

Answer 1