Alice and Betty play a safe version of Russian roulette, with a random device that does the role of a fake gun with three chambers. The bullet is placed randomly in a chamber. They take turns to pull the trigger and hand the gun over to the other player. They DO NOT spin the chambers in between. So, if the gun does not fire twice, the gun will fire at the third shot.

In this game, Alice and Betty decide simultaneously whether to go first or second. If both want to go first, they will flip a coin to know who will pull the trigger first. Likewise, if both want to go second, they flip a coin to know who will pull the trigger first.The payoff of loosing (if the gun fires) is 0, and of winning is 1.

1. (a)Check that being first yields a payoff of 1/3 on average.
2. (b)Show that, if Alice and Betty both want to start first, then their payoff will be 1/2 on average. Likewise, if both decide to go second, their payoff will be 1/2 on average.
3. (c)Using your previous results, check that the game can be represented as:

B

First Second

A First 1/2, 1/2 1/3, 2/3

Second 2/3, 1/3 1/2, 1/2

1. (d)Alice and Betty play twice with a discount factor (0,1). Find a Nash equilibrium and its respective payoffs.
2. (e)Alice and Betty play repeatedly that game forever with a discount factor of (0, 1). Find a the Nash equilibrium and its respective payoffs.