Live Action Mafia https://mafia.mit.edu/ 

Preordained solution? https://mafia.mit.edu/viewtopic.php?f=13&t=198 
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Author:  mats_a [ Wed Jan 12, 2011 7:02 am ] 
Post subject:  Preordained solution? 
Is it possible to craft a winning strategy for the townspeople at the beginning of this Mafia game, assuming that townspeople are willing to sacrifice themselves? For example, say we had a trusted randomization thing sort players into a list, that only spat out one name each night right before lynching, that everyone obediently lynched, that excluded the randomlyselected mayor. (players lynch anyone who votes out of line, and guard the mayor with their lives) Sure it wouldn't win all the time, but I wonder from a gametheoretic standpoint if this would 'work'  granted it defeats the entire purpose of playing Mafia. =p Just some food for thought. 
Author:  Mark Velednitsky [ Wed Jan 12, 2011 1:44 pm ] 
Post subject:  i don't think it's work 
There is probably some interesting math that could be considered rigorously, but here is a very, very rough estimate. Think of it as a list of the 27 players in some random order and you lynch those players in that order. Obviously, if a Mafia player is last or second to last, they win (the chance of that is about 1/4). But keep in mind that the Mafia gets to kill people. By symmetry, I would estimate that each kill gives them about a 1/5 chance of killing someone that falls after all of them on the list, thereby potentially pushing them closer to the end. That suggests that at least one kill will be a kill of someone near the end of the list, which means the mafia just need to be among the last three in the list, which ups their chances to roughly 1/3. Furthermore, if the mafia is playing intelligently, then they would kill every night until their is one mafia left. When there is one mafia left, they would conscript a civilian at random. Playing with this strategy probably increases their chances to about 1/2. Again, that's practically a guess and does not account for all of the special roles and abilities (just sum), but if I had to bet I'd say the probability is somewhere in the range from 0.3 to 0.7, so you might as well just flip a coin and decide the game on that. 
Author:  Alex [ Wed Jan 12, 2011 1:51 pm ] 
Post subject:  
If you want asymptotics, I can show that in the high limit of a vanilla Mafia game (meaning lots of players), if people lynch randomly and Mafia kill daily, the probability of winning is determined by the ratio of Mafia squared to total players. So, a balanced game will have the number of Mafia grow as the square root of the number of players times a fixed constant that my technique is too weak to determine. 
Author:  Mark Velednitsky [ Wed Jan 12, 2011 2:25 pm ] 
Post subject:  see wikipedia 
Wikipedia contains some analysis of the vanilla game under "Game technique and strategy: Mathematical study." It basically reaches the same conclusions as Alex. 
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