by user3475234
Last Updated January 18, 2018 05:19 AM

I'm assuming a game of Russian Roulette where you have a gun with 1 bullet and six chambers, and two people playing.

The rules are:

You must shoot at least once on your turn.

After shooting once, you can continue shooting as many times as you want until you decide to pass it to the other player.

Whoever gets shot loses (obviously?)

The chamber is spun once at the start of the game and the shots are taken in order from there without spinning. If the bullet is chamber 4 whoever shoots 4th will lose.

What are the optimal strategies for player 1 and player 2? Is it possible to figure this out or are there too many variables?

How about just for the first move? How many times should Player 1 shoot before passing it for an optimal chance at winning?

What are the optimal strategies for player 1 and player 2? How many times should Player 1 shoot before passing it for an optimal chance at winning?

Once the barrel is spun, the position of the bullet is fixed, which means that there is a $\frac{1}{6}$ probability to be shot at turn number $i$, $i \in [|1,6|]$. As a result, the optimal strategy for player 1 and player 2 is to shoot only once on their turn. Following this strategy, the probability that player 1 wins is the same as the probability that player 2 wins, i.e. $0.5$, regardless of who starts.

Here is an equivalent game:

There is a stack of 6 cards. The top card has the number 1, the second has the number 2, and so on through the bottom card, which has the number 6.

The rules are:

- You must take at least one card on your turn.
- After taking one card, you can continue taking as many cards as you want until you decide to pass your turn to the other player.
- After all cards are taken, a 6-sided die is rolled. Whoever holds the corresponding card loses.

There is only one difference between this game and the Russian roulette game. In this game, the losing card is not revealed until after all cards have been chosen; in the Russian roulette game, the losing chamber is revealed immediately when that chamber is chosen. But this doesn't matter for choosing a strategy, because none of the choices matter any more once the losing card/chamber has been taken.

In the card game, the probability of losing is clearly proportional to the number of cards taken, so the winning strategy is simply to take as few cards as possible.

This means that the winning strategy for both games is:

- If there are an even number of cards/chambers remaining, take only one card/shot.
- If there are an odd number of cards/chambers remaining, take either one or two cards/shots; it makes no difference.

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