[EV] Protecting Mountainous

[BNL]

2 Mafia Goons
7 Vanilla Townies

Each Day, other than voting for a person to lynch, everyone also votes for a person to protect. The Mafia can't kill that player that Night.

Now, this mechanic doesn't seem to add much to the game. There is no way town can get additional information from the protection, and this can't prevent the Mafia from killing town every night, so town can't gain a mechanical advantage from this mechanic.

Or can they?

Now, suppose town uses this strategy: identify a Townie Day 1, and treat them as confirmed town for the rest of the game, protecting them every Day. If town do correctly identify a Townie, then this game becomes mountainous with a bulletproof IC, which has an EV of 29/64 if I didn't screw up my calculations. Since town has a 7/9 chance of correctly identifying a townie, the EV of this strategy is 203/576, which is 35.4%.

Compare this to ignoring the mechanic and playing mountainous, which according to mith's numbers is 94/315 or 29.8%
This result came as a shock to me. Town does gain EV from this mechanic, and while it isn't much, it certainly is not negligible.


Now, why does this increase the town EV at all? I have a few theories. First, town being able to block off a scum kill means that scum have less control over the night kill. While it may not seem like much, there is still a little bit of influence. Suppose town has more doctor shots, say for all but 3 of the alive players, then it is evident that town are basically directing the Mafia kills. This shows that town's influence on the night kill do affect the EV. Secondly, being able to link up the protects mean that town don't have to keep finding new townreads, and have to bank on only one town read being right throughout the game rather than risk the strongest townread of some day being wrong.

[popsofctown]

This is a really cool result. I got 203/576 too.

This kind of interacts with a dayplay policy I tout in a lot of my games about keeping some or all of my townreads secret to protect them from NKs. The mafia lose a lot of equity every time they shoot someone who was going to get mislynched anyway. Imagine a variant of mafia where Day 1 the town creates a list of all the lynches they want the whole game, in order, and those execute one at a time. The mafia would use the perfect knowledge to leave all town aligned players that are above a mafia player's name on the list alive, and shoot all the names on the list below the mafia players.
In standard mafia, the sequence of lynches is unknowable, so this happens "half" the time in the Calvinist fated sense.
In Protecting Mountainous, a name is moved to the bottom of the list, but simultaneously protected from that winning mafia strategy. So that benefit that they get about half the time drops to zero for that player. The "cost" of risking instaloss in identifying that player as conftown is apparently a wash, which is not incredibly intuitive but pretty easy to accept as true in a game where every lynch is a risk.

[callforjudgement]

The reason that this helps town is that it causes the scum kill to leak more information.

In a purely vanilla game, you don't get any information from the scum kill beyond what you get from the dead player's flip; under EV assumptions, the scum were killing a non-scum at random. The information you get from the scum kill makes every living player a little more likely to be scum (because you know in advance that scum were going to be shooting a townie, and there's a higher concentration of scum in the players left behind).

Now when one player is protected, scum is limited to shooting among the remaining townies. Anyone they could shoot and don't still becomes more likely to be scum. The player they couldn't shoot, though,

doesn't

; scum couldn't shoot them anyway. So they end up being more likely to be town, relative to the other players.

Here's another way to look at it: in a vanilla 7:2, each player is ~22% likely to be scum; after a mislynch and nightkill, we have a 5:2, so each player is now ~29% likely to be scum. In a protecting 7:2, each player is ~22% likely to be scum; after a mislynch and nightkill, the player who wasn't killed is

still

~22% likely to be scum, because as the scum couldn't interact with them, their kill doesn't reveal any information. So now additional information is being revealed about the other players to make the percentages add up, and their odds of being scum are higher than the ~29% in the purely vanilla setup.

This is effectively the same situation as the Monty Hall problem; someone's being forced to leak a specific amount of information, they have some subjects they can't leak information about, so it forces them to give more information on the subjects they

can

leak information about.

[popsofctown]

I'm not sure if my model is incorrect or a different way of presenting the same thing.

I love the Monty Hall problem though.

[FakeGod]

oh my goodness

this is beautiful