Game theory question—iterated truel with private information
I was reading the discussion here of how truels with unlimited shots and symmetric complete information favor the weakest truelist. This set me to wondering about somewhat more complicated situations. Suppose you are in a world where there are daily truels for some substantial period of time, say 30 days. As with the linked problem, all hits are fatal and truelists are accurate 50%, 80%, or 100% of the time. Unlike the linked problem, however, rather than possessing complete information about the accuracy of other truelists, you only know your own true accuracy and the %hit of other truelists in earlier iterations, and there is no guarantee that there will be one truelist of each skill level in any given truel.
Now suppose you know that you are a perfect marksman. On which of the iterations would you intentionally miss your first shot? I definitely lack the math strength to offer a good strategy, but I’m sure many others here could do better.
Edit #2 -- I give up on the formatting.
How does your game resolve ties? That is, at the end of the day when nobody has fired a shot what happens to the players? Possible options include ‘they all live’, ‘if you miss X times in a row your gun explodes and you die’ and ’everybody is executed if more than 1 person is alive after X shots each”.
I assume you know your %hit from earlier iterations too? It makes a difference.
In this problem, you have perfect accuracy, and you know it. Although not knowing it would create a different and interesting problem.
Yes, but your record of how many times you’ve intentionally missed will influence your opponents’ strategy, so it’s something you should need to remember as well.
Very true. Thanks for the catch.
If you’re just doing one truel, you won’t get much chance to gather data. As more truels go by, you can gather data, but interpreting the data is very difficult, since you are unable to identify a difference between shot-to-miss and shot-to-kill-but-missed. It also depends on your own predictions of what strategies players are likely to employ, given what they know about their own accuracy.
As the original author stated, you do not particularly want to be the first person to kill an opponent, because then an unknown opponent will have their turn, and you will be their only target. Thus, in theory you should shoot to miss until one of your opponents has killed another. However, that strategy works for every player. People would just shoot to miss forever. Expecting this, a substantially weak player might actually try to shoot the other players, expecting only a small chance that they would actually get a hit. If another player in the game is playing with this strategy, that gives you a 50% chance of survival, since they will eventually hit one of the two others, and then if you survive, you will shoot them.
But, you know that as a talented player, you are not expected to shoot to kill. In which case, you can simply kill one opponent on the first round, and take your chances with the remaining player. The calculations here depend somewhat on your prior estimate of ability. If this was a simulated game theory tournament, and the accuracy of each player was determined by a pseudo-random number generator, then your expected value of survival is 50% again. On the other hand, if this is a physical duel with physical pistols, it is likely that your opponents will have an accuracy better than 50% (ie they’ve been practicing). In that case, you’re better off with the strategy of deliberately missing.
Win-win-win!
Markdown is for comments; html or the what-you-see-is-what-you-get editor for top-level posts.
Fixed it.