Doing the calculations, it's relatively easy to see that for Rock/Paper/Scissors, all strategies are equally good.
Huh? Clearly throwing rock all the time is worse than throwing each with equal probability, because it is countered perfectly by throwing paper all the time. Unless there is something I am not understanding?
There are different types of equilibria. I don't know too much about it, but both players playing randomly is the 'strongest' equilibrium. However, if you know that your opponent will play randomly, it doesn't matter how you play, so all those are equilibria too, even though your opponent could improve his play by deviating.
I've been trying to use game theory in mafia ever since D7 of DS9. I don't have any staggering results, and applying it to large initial setups is computationally not-fun. I did do some stuff a while back to determine who I should kill in a multiball game (fire and ice), since there were a few funky rules there but only one town role, who was known.
Here's a nice puzzle:
3 players are alive during the day: A vanilla townsperson, a Mafia goon, and a Serial Killer. All three know the alignments of the others. Voting is as normal; players can place votes at any time, and two votes is sufficient for a lynch. Let's say all the players are online, all the time. Better yet, IRL. If an hour goes by without a lynch, the day ends and night starts. In the night, the mafia goon and the serial killer have the opportunity to kill somebody.
If the Mafia and the Serial Killer die, the townsperson wins. If one of the mafia/serial killer die, the other wins. If just the townsperson dies, the mafia and serial killer tie, which is halfway between winning and losing. If the mafia or SK must lose, they would very slightly rather the town win.
What happens? Note that this is not a zero-sum game, but that doesn't really matter. Assume a win is worth 1, a loss 0, and a tie 1/2. Say that the Mafia/SK get .01 from the town winning. Here's a variant that I haven't solved yet. Say that instead the Mafia/SK get x from the other bad guy winning, where 0 < x < 1/2. Now what happens, in terms of x? This came from D7 of DS9, where I was the townsperson in this scenario. However, I thought just one of them was a bad guy, so that didn't work out so well.
Here's a different puzzle, more commonly applicable to mafia games.
3 players are alive during the day; two townspeople, and a mafia. Neither townsperson has any clue who the mafia is. Town wins if they lynch the mafia, mafia wins if they lynch a townsperson or no lynch happens. Say that the game is IRL again, and that if nobody votes for an entire hour, no lynch happens. It takes two votes for a lynch. What are the optimal strategies and expected payoffs for each player? Note that this is a zero-sum game, if we assume that the town is one player.
And here's a puzzle I posted in the maths thread, already solved there but you can try it too! Not related to mafia, but rock-paper-scissors!
We're playing a game of Rock, Paper, Scissors, with a twist; I am never allowed to throw paper. If we both throw the same thing, we throw again. If we throw different things, the winner gets +1, the loser -1, and the game ends. If we both throw the same thing n times, for some whole number n which is fixed ahead of time, the game ends and I get +1, you -1. Only the last throw of the game matters.
What is the optimal strategy for each player and what is your expected payoff for this game, in terms of n?