# Russian Roulette (Redux!)

You are challenged to a game of Russian roulette with a six-chamber revolver, but you and your opponent are allowed to change the outcome of the game in different ways. Here's how the game will proceed:

• You may fill any of the chambers with a bullet. You must place at least 1 bullet into the revolver, but you have enough bullets to fill the entire revolver
• After you have chosen the configuration, your opponent may do 1 of 2 things:
• Choose whether to go first or second
• Move a single bullet, and flip a coin to decide whether they go first or second
• After the opponent action is complete, the cylinder is spun exactly once
• You both continue to fire the revolver until one player can no longer continue

Going "first" means that person is the first target of the revolver. Find a) the strategy/strategies that maximize(s) the odds of survival for yourself, and b) the corresponding strategy/strategies that maximize(s) the odds of survival for your opponent. Additionally, show that your opponent can bound your maximum probability of survival to 50%, and that you can bound your opponent's maximum probability of survival to 50%.

• is it allowed to give her anxiolytics just before starting the "choose whether to go" part? Commented Nov 8, 2019 at 21:33
• It's not allowed to interact with the other person. All you may do is choose whether each chamber has/does not have a bullet. The opponent is similarly forbidden from interacting with you. All they may do is choose first/second, or move a bullet and flip for first/second.
– Avi
Commented Nov 8, 2019 at 21:35
• sorry, this is just my english: moving a bullet means removing one bullet while seeing the whole cyliinder load, moving a bullet over to different hole, or turning a single bullet turn on the cylinder? Commented Nov 8, 2019 at 21:40
• Does the opponent get to see the configuration I chose before deciding which action to take?
– Bass
Commented Nov 8, 2019 at 21:40
• @Bass Yes - they can look in the gun as they choose which bullet to move. They will not shoot themselves accidentally while looking into the gun.
– Avi
Commented Nov 8, 2019 at 21:41

You always have a simple 50% strategy:

put 1 bullet in the gun.

Whatever the opponent chooses, there are exactly 3 good spin results and 3 bad ones for you.

Your opponent always has a simple 50+% strategy:

Go second.

Doing otherwise is always worse, or equal at best; there are no possible bullet configurations where there are more than 3 "good" spin results for the player going first.

Since your opponent always has a 50+% strategy, it's impossible for you to have better than 50% strategy, so these are best strategies for both.

• I may have made this question too easy facepalm...
– Avi
Commented Nov 8, 2019 at 21:57
• I was thinking the other way around; always put 6 bullets for a simple 50% strategy
– Alex
Commented Nov 8, 2019 at 22:00
• @Avi not at all, there are a lot of interesting variations that give rise to plenty of interesting strategies, and the opponent's strategy didn't feel at all trivial to find.
– Bass
Commented Nov 8, 2019 at 22:00
• @Alex isn't that an automatic loss though? The opponent would choose to go second.
– Bass
Commented Nov 8, 2019 at 22:02
• @Bass Ok, that's good to know :)
– Avi
Commented Nov 8, 2019 at 22:02