I am supposed to be working and not procrastinating so here is my mathematical theory on policy in guns and hookers. The most common scenario this occurs in is with cop dead.
Here are the pure probabilities of town win vs. mafia win with the following very large caveats:
Every shot and lynch is random.
Gunsmith gunned alive.
Mafia always hides gun.
Gunsmith always guns in 4way.
Hooker always hooks gunsmith if both are alive.
Without policy:
Town wins 24.27% of the time [1165/4800]
Mafia wins 75.73% of the time [3635/4800]
With policy:
Town wins 30.99% of the time [119/384]
Mafia wins 69.01% of the time [265/384]
Further breakdowns:
Without policy & villager hooked = 27.5% town win
Without policy & gunsmith hooked = 14.58% town win
With policy & villager hooked = 36.45% town win
With policy & gunsmith hooked = 14.58% town win
As you can see, the main reason policy is effective is that it's more likely that villager is hooked than gunsmith. If gunsmith is hooked then you are at the same odds either way. So never scumhunt and just use policy and you will increase your random win odds by almost 6 percent!
I mean, if you don't want to remake it, I can just do it in SAS and post it, however I don't think people want to download a 10mb gretl file breaking up the probabilities of each choice.
In my "model" (lol) mafia just takes the lynch and doesn't use gun. If you reveal the gun and shoot gs I doubt it would improve mafia's odds. It certainly wouldn't if nilla is being lynched. If hooker is being lynched it might but I scrapped the original document I had made so I would have to rework it to figure out the difference. I probably will since a few people are asking for the methodology.
Oh sorry, I missed the detail with cop dying N1, my questions are stupid, except for one: does GS say who's going to be gunned on N2 or not? If GS says, it might be more optimal for mafia to kill the gunned town unless there's still policy on D2.
IF* B. Is true, then A. isn't true, leading towards skewed results for town in a policy situation.
GS is alive, and they policy, and mafia is gunned, then the random assumption is incorrect, because mafia will reveal the gun and shoot GS, leading to statistically better outcome in the case of policy.
I'd actually be interested in seeing a pastebin of the branch logic if you did this in python or something.
Edit: Saw the comment on "Cop dies N1" Modified my post accordingly.
i'm not gonna take this as a good intention debate if you're gonna suggest hook-killing n1, but my caveats were listed. Another caveat would be mafia always kills GS n2 since they have gunned alive in this scenario and they have claimed.
Do you take only possibilities when hooker hooks GS even on N1? Is there policy lynch on D2? What about hook-killing N1? Does mafia hide the gun even on D2 when hooker is dead and GS alive?
I made a lil flow chart / tree then multiplied the odds at each node from the root to each leaf node then added it up. If that doesn’t make sense you can picture one branch like (1/5 chance cop died) x (1/4 chance gs hooked) x (3/5 chance villager lynched) = 3/100 chance town loss. And adding up all of those little probabilities you arrive at the numbers I gave. Which was just the cop dead branch.
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