[Theoretical EV] Recursive Multiball

[Awoo]

This game would almost certainly never be feasible to actually play due to the numbers going way out of whack, so I present it as a thought exercise for looking at the maths/EV/asymptotic behavior of it's balance. It doesn't seem too hard to write a program that would calculate EV for this setup, and setups of this family.

Recursive Multiball:

X Townies

Y Mafia Loyalists

Z Mafia Rebels

X > Y > Z

Note that there are

X Townies

Y + Z Mafia members

Welcome, Awoo, you are a townie who wants to kill all the mafia.

If at any point in the game the mafia has numbers equal to or greater then the town, all townies die (endgamed), and the phase skips to day.
You win when all the

mafia loyalists

and

mafia rebels

are eliminated and at least one

townie

is alive.

Welcome, Awoo, you are a mafia loyalist, you wish to see the mafia take over the town. You are a mafia member.

The members of the mafia are []. Z of them are Mafia Rebels.
While there are townies alive, the mafia share a factional nightkill which can be used on any townie.

When the number of

mafia members

is greater than or equal to the number of

townies

, all the

townies

will die, and the phase will jump to day.
You win when all the

townies

and

mafia rebels

are dead.

Welcome, Awoo, you are a mafia rebel, you wish to see the mafia take over the town, but you also want to rule over the mafia yourselves. You are a mafia member.

The members of the mafia are []. The mafia rebels are {}.
While there are townies alive, the mafia share a factional nightkill which can be used on any townie.
While there are no more townies alive, the mafia rebels share a factional nightkill which can be used on any mafia loyalist.

When the number of

mafia members

is greater than or equal to the number of

townies

, all the

townies

will die, and the phase will jump to day.
You win when all the

townies

and

mafia loyalists

are dead.

So basically it's a setup where the mafia are playing a mafia game inside the mafia game, and if the mafia kill all the townies, they become the new town and they need to hunt for the mafia within the mafia in order to win. I imagine bussing would not go over well

I am curious as to what numbers would make this setup balanced 33/33/33 with... say... two mafia rebels. I wonder what the proportions would look like? I'll probably run some numbers later this week if no one beats me to it.

[Sukima]

I assume the last PM should be "You are a Mafia Rebel".

[Something_Smart]

How is the general mafia nightkill determined?

[Awoo]

General mafia nightkill? It kills a random (aka the mafia members factionally decide) townie. The rebels can't nightkill the loyalists until the townies are all dead. If you mean mechanically, uh... I guess they vote on it at night if there's a disagreement? idk, this isn't a setup that humans should play.



Anyways, got some results.

These results are not at all what I expected, I thought there would be a LOT more loyalists in 33/33/33 setups. I chose specific town numbers that would set their winrate to as close as 33% as possible, fixed rebels to a constant, then iterated over loyalists.
I'll come back again later with a faster script, I just wanted to see how far brute force could take me. I left the computer on for a few hours, but the script can't seem to get past 6 mafia without outlasting my patience.


Sanity check / test cases:

(Town, Loyalists, Rebels) : (Town EV, Loyalist EV, Rebel EV)
(3, 1, 1) : (0.13333333333333333, 0.13333333333333336, 0.7333333333333334)
(5, 1, 0) : (0.4666666666666667, 0.5333333333333333, 0)
(6, 5, 1) : (0, 0.4666666666666667, 0.5333333333333333)
(7, 2, 1) : (0.11428571428571425, 0.3587301587301587, 0.526984126984127)

Interesting results:

(Town, Loyalists, Rebels) : (Town EV, Loyalist EV, Rebel EV)
(9, 1, 1) : (0.35209235209235207, 0.18759018759018764, 0.4603174603174604)

(18, 2, 1) : (0.3320765921385116, 0.3294511474078037, 0.3384722604536846)

(31, 3, 1) : (0.3291981889117276, 0.40465039840729394, 0.26615141268097825)
(48, 4, 1) : (0.33010024598442966, 0.4521368492499652, 0.21776290476560495)
(69, 5, 1) : (0.3318359869496648, 0.4843605817823275, 0.1838034312680077)

(18, 1, 2) : (0.3320765921385116, 0.10681001145397433, 0.5611133964075141)
(31, 2, 2) : (0.32919818891172764, 0.20581646068242432, 0.46498535040584793)
(48, 3, 2) : (0.33010024598442955, 0.2779622016838137, 0.3919375523317566)

(69, 4, 2) : (0.3318359869496648, 0.33102263116136216, 0.33714138188897286)

(31, 1, 3) : (0.3291981889117276, 0.07429999791366297, 0.5965018131746092)
(48, 2, 3) : (0.3301002459844296, 0.1473758113171153, 0.522523942698455)
(69, 3, 3) : (0.3318359869496648, 0.20815016118743934, 0.4600138518628956)

... maybe the next one will be 33/33/33?

[Irrelephant11]

(69, 4, 2)

[callforjudgement]

Strategy here could be interesting: scum aren't allowed to nightkill each other until all the townies are dead, so the rebels would likely be trying to eliminate loyalists and vice versa earlier in the game via bussing (which is the equivalent of lynching in a more normal game). Too much of that and it just ends up benefiting town. (As such, this game is somewhat self-balancing/"screwy" just like more normal multiball is.)

The reason the necessary number of loyalists is fairly small is that most scum deaths will happen while town is still alive. If we're using town vs. scum numbers that give town a 33% win rate, town are going to come close to wiping out the scum (all of them, both sides) in most games. That means there will likely be only a few scum left if it comes down to a Mafia civil war, and if there were a lot of loyalists, the remaining scum would probably be all loyalists at that point (it's a bit like playing Nightless in that respect). The rebels' big advantage from their recursive kill is that it makes it possible for them to win a 2:1-unfavourable endgame and gives them an automatic win in a 1:1 endgame, but it doesn't do much else for them.

I'm not sure if it'd be possible to make this into a playable setup; the problem is that you want the scum to largely remain alive if town loses, and yet give town a realistic chance of winning. (It almost seems like you'd want a victory condition

other than

eliminating opposing factions, which is an interesting idea for a game; I've had some thoughts along those lines in other setups but haven't found anything satisfactory yet.)

[Awoo]

Okay fixed my script. seems like rebels need to take up about 1/3 of the mafia in order for it to be balanced. so when you get numbers like 20 - 10 and 6 - 3 it's the most balanced. and when you get stuff like 5-3, 7-3 it's the least balanced.

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(Town, Loyalists, Rebels) : (Town EV, Loyalist EV, Rebel EV)

(9, 1, 1) : (0.35209235209235207, 0.18759018759018764, 0.4603174603174604)

(18, 2, 1) : (0.3320765921385116, 0.3294511474078037, 0.3384722604536846)

(48, 3, 2) : (0.33010024598442955, 0.2779622016838137, 0.3919375523317566)

(69, 4, 2) : (0.3318359869496648, 0.33102263116136216, 0.33714138188897286)

(121, 5, 3) : (0.33174639626390606, 0.29802805182551895, 0.3702255519105748)

(154, 6, 3) : (0.33396756785881304, 0.3309629328756226, 0.3350694992655642)

(189, 7, 3) : (0.33352047736820706, 0.35908353244816565, 0.30739599018362723)

(271, 8, 4) : (0.33388258156571027, 0.33117233258041506, 0.3349450858538745)

(316, 9, 4) : (0.33299709630826163, 0.3527829782362824, 0.31421992545545574)

(420, 10, 5) : (0.3334552455387918, 0.3313750428813027, 0.3351697115799053)

(477, 11, 5) : (0.33340363277512286, 0.3486669187193403, 0.3179294485055368)

(601, 12, 6) : (0.3330104427196188, 0.3315423764027638, 0.3354471808776174)

(670, 13, 6) : (0.3333704528132578, 0.3459556762391648, 0.3206738709475774)

(816, 14, 7) : (0.3331392787513769, 0.3315468783368802, 0.3353138429117427)

(895, 15, 7) : (0.33318614842027255, 0.34403481563590466, 0.3227790359438228)

(1065, 16, 8) : (0.3334815449548689, 0.33148719653294834, 0.3350312585121824)

(1154, 17, 8) : (0.3333266283361149, 0.3425028349908433, 0.3241705366730416)

(1344, 18, 9) : (0.3332571459475693, 0.3315617774162669, 0.335181076636163)

(1445, 19, 9) : (0.3333114298010203, 0.3413357407831198, 0.32535282941585886)

(1657, 20, 10) : (0.33328378445933304, 0.33156939744366754, 0.33514681809699814)

(1768, 21, 10) : (0.33321838902038364, 0.34041746896162167, 0.3263641420179937)