Let’s ignore all the wand stuff (I heard this question with guns a long time ago) and just doing the odds of hitting math. The miss on purpose strategy being right might depend on the order of the shots and who has the best aim. I might be missing some easy shortcut, but the math seems really complicated. Obviously if you all had the same chance of hitting, you’d definitely all want to pass. But when the odds of hitting are different, I’m not sure that’s the case.
I don’t know about the values in this video, I think I originally heard it as playing 1 has a 30% to kill, player 2 has a 70%, player 3 has a 90% chance of hitting. So I made a simulation because the markov chains were getting too complex for napkins. I compared two strategies - fire at the person with the highest chance to hit vs. pass unless there is only one other person.
If all three go “highest” then the odds of winning at 29%, 58%, 13% for players 1, 2, 3.
If player one switches their strategy to “pass unless 2” then the odds of winning change to 36%, 45%, 19% - a nice improvement for player 1.
But if player two follows suit then the odds of winning change to 32%, 0%, 68%.
That’s not rounding, it’s literally zero. Because player three will always kill player two, no matter how many rounds it takes. Basically you hand it off to the third player and say, “Do you prefer a 100% chance of drawing or a 68% chance of winning”. In a game to the death that decision will be made by whether a draw means you die or live (but if a draw means you live then obviously no one should shoot anyone else, you don’t even need math to figure that out). But usually in any game I wouldn’t think someone would take a draw over a 68% play to win.
I tried with some different numbers and orders and it came out as I expected: It’s right for the person with the lowest chance of success to pass until there are two players, but for the middle and high player, they should do the “obvious” thing and shoot at the other player with the highest chance of winning.
