Motivation:
1-1-1 is not an interesting setup unless there have already been at least one night phase, otherwise it's just a named townie game. (EV 1/2). The game becomes interesting when the cop has a chance to get a result. Let's assume this 1-1-1 is the conclusion of a larger setup. It could have crazy rules in it, like 4 lynches per day or 3 mafia nightkills or the cop investigates 5 people each night, but assume that through careful analysis of the different possible outcomes and assuming optimal play etc, we are able to compute the probability that the cop is a supercop given that there is exactly 1 cop 1 VT and 1 mafia at the start of this day, and we call this value x. What would the EV of this situation be? Note:
I had a hard time understanding exactly what people were calculating in what post in the last thread. There were a lot of things that I didn't find interesting (mafia with cheaty reflexes =/= perfect and symmetric players) So I did everything manually and I will post it here. Please correct if I made an error X(Strategy:
it's 1-1-1. We're going to be playing the pointing game detailed in the last thread. The players are arbitrarily named A B and C. Townies point randomly. Cop points at the mafia, or randomly if he has no result. Mafia points randomly because he can't cheat. What configurations might we see?Code: Select all
1: A -> B | A will always point to B since 1 person will always point to another and the people are arbitrary. But now that this link is defined, we build around it.
===== 1/2: A -> B -> C
========== 1/4 A -> B -> C -> A | Triangle case
========== 1/4 A -> B <-> C | 2v1 case
===== 1/2: A <-> B
========== 1/4 A <-> B <- C | 2v1 case
========== 1/4 C -> A <-> B | 2v1 caseSo we will see A -> B -> C -> A (Triangle case) with probability 1/4 and A <-> B <- C (2v1 case) with probability 3/4. In order to be indistinguishable from the real cop, mafia will claim to be a supercop with probability x and ignorant cop with probability 1 - x, unless his supercop claim would be a guilty result on a confirmed townie, where he will claim ignorant cop. If he doesnt claim cop with these probabilities idk what happens
but I hope town just catches on and adjusts how they handle 50/50 scenarios accordingly until it goes back to the 50/50 equilibrium. like maybe if he claims ignorant cop too much they notice and start lynching ignorant cops a bit more often so let's assume they are so damn good they will sniff out any improbably cop claims or something like wtf do we even do without this assumption X(Triangle case (1/4)
Strategy: A claims, B claims, C claims. If someone claims a supercop result on a non-contradictory claim, they evil, kill em. M = mafia. C = cop. T = townie.
Code: Select all
There is a super cop:
a. C -> M -> T | Mafia can adjust claim | ev = 1/2
b. M -> T -> C | Mafia dies if he claims SC, coinflip otherwise | ev = x + (1 - x)/2
c. T -> C -> M | ev = 1/2
= 1/3 + x/3 + (1-x)/6
There is an ignorant cop:
d. C -> M -> T | Mafia can adjust claim | ev = 1/2
e. M -> T -> C | Mafia dies if he claims SC, coinflip otherwise | ev = x + (1 - x)/2
f. T -> C -> M | Mafia can adjust claim | ev = 1/2
g. C -> T -> M | Mafia can adjust claim | ev = 1/2
h. M -> C -> T | ev = 1/2
i. T -> M -> C | ev = 1/2
= 5/12 + x/6 + (1-x)/12
Total:
= [(x)(1/3 + x/3 + (1-x/6) + (1-x)( 5/12 + x/6 + (1-x)/12) ]
2v1 case (3/4)
Strategy: A <-> B <- C. B claims first. If he claims supercop on A, and C claims cop, B dies.
Code: Select all
There is a super cop:
j. C <-> M <- T | ev 1/2
k. M <-> C <- T | ev 1/2
l. T <-> M <- C | Mafia dies if he claims SC, coinflip otherwise | ev = x + (1 - x)/2
= 1/3 + x/3 + (1-x)/6
There is an ignorant cop:
m. C <-> M <- T | ev 1/2
n. M <-> C <- T | ev 1/2
o. T <-> M <- C || Mafia dies if he claims SC, coinflip otherwise | ev = x + (1 - x)/2
p. C <-> T <- M | Mafia can adjust claim | ev = 1/2
q. T <-> C <- M | ev 1/2
r. M <-> T <- C | Mafia can adjust claim | ev = 1/2
= 5/12 + x/6 + (1-x)/12
Total:
= [(x)(1/3 + x/3 + (1-x/6) + (1-x)( 5/12 + x/6 + (1-x)/12) ]
Overall total:
Code: Select all
= 1/4[(x)(1/3 + x/3 + (1-x/6) + (1-x)( 5/12 + x/6 + (1-x)/12) ] + 3/4[(x)(1/3 + x/3 + (1-x/6) + (1-x)( 5/12 + x/6 + (1-x)/12) ]
= (1/4 + 3/4)[(x)(1/3 + x/3 + (1-x/6) + (1-x)( 5/12 + x/6 + (1-x)/12) ]
=1[(x)(1/3 + x/3 + (1-x/6) + (1-x)( 5/12 + x/6 + (1-x)/12) ]
=(x)(1/3 + x/3 + (1-x/6) + (1-x)( 5/12 + x/6 + (1-x)/12)
= (x^2 + x + 6)/12
What does this result mean?
At x = 0 (cop never has a result) ev = 0 + 0 + 6/12 = 1/2 which makes you wonder why they are pointing at all
At x = 1 (confirmed supercop) ev = 8/12 = 2/3 This is actually the only inaccurate result that this equation produces. Why? Because it assumes there is always a chance for mafia to change his claim to ignorant cop if he will cause a contradiction otherwise. However this is not OK when its not an option to claim ignorant cop.
So we should redefine this function:
_______{ (x^2 + x + 6)/12 ; 0 <= x < 1
ev(x) = { 3/4 ; x = 1
_______{ 0 ; otherwise
Which leads to another strange conclusion: If the probability that there is a supercop is arbitrarily close to 1, the EV approaches 2/3 instead of 3/4. The reasoning behind this is that it is perfectly acceptable for the mafia to adjust his claim to ignorant cop in 4 cases of triangle case, and 2 cases of 2v1 case (a, b, f, g, p, r) . However, cases (b, f, g, p, r) assume ignorant cop, and since x is arbitrarily close to 1, let's write these off as highly improbable. This means that in case (a), which occurs with probability arbitrarily close to but less then 1/12, mafia claims ignorant cop, which is much much greater then the infinitesimal value that is 1 - x and town sees absolutely no problem with this. Perhaps the assumption that mafia can always adjust his fakeclaim is unreasonable? Perhaps there should be some additional argument that takes into account believability of fakeclaim that mathematically smoothes out the curve from 1/2 to 3/4, and in terms of gameplay, means the town takes into consideration these special cases and lynches ignorant cops more often, forcing the mafia to adapt.
I also suspect that there would be a point where the probability that mafia will claim ignorant cop would line up with how often they would want to claim ignorant cop VIA claim adjustment. I think it would be difficult to find that point.
