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%.

  • $\begingroup$ is it allowed to give her anxiolytics just before starting the "choose whether to go" part? $\endgroup$ – balazs.com Nov 8 '19 at 21:33
  • $\begingroup$ 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. $\endgroup$ – Avi Nov 8 '19 at 21:35
  • $\begingroup$ 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? $\endgroup$ – balazs.com Nov 8 '19 at 21:40
  • $\begingroup$ Does the opponent get to see the configuration I chose before deciding which action to take? $\endgroup$ – Bass Nov 8 '19 at 21:40
  • $\begingroup$ @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. $\endgroup$ – Avi Nov 8 '19 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.

  • $\begingroup$ I may have made this question too easy facepalm... $\endgroup$ – Avi Nov 8 '19 at 21:57
  • $\begingroup$ I was thinking the other way around; always put 6 bullets for a simple 50% strategy $\endgroup$ – Alex Nov 8 '19 at 22:00
  • $\begingroup$ @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. $\endgroup$ – Bass Nov 8 '19 at 22:00
  • 2
    $\begingroup$ @Alex isn't that an automatic loss though? The opponent would choose to go second. $\endgroup$ – Bass Nov 8 '19 at 22:02
  • $\begingroup$ @Bass Ok, that's good to know :) $\endgroup$ – Avi Nov 8 '19 at 22:02

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