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Thread: #21: Game Design - Core Balance (by Apoc)

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    #21: Game Design - Core Balance (by Apoc)

    Article #21: Game Design - Core Balance
    - written by Apoc

    Hello! This is the first of a series of articles containing my thoughts on mafia/werewolf game design.

    The goal of this article will be to explain the fundamentals of balancing a setup.

    This guide does not discuss whether certain roles or designs are fun, but focuses strictly on what is balanced/fair. Fun/Feel is a very different area of design, worthy of its own article.

    ---

    Introduction
    A lot of game design and balancing is typically done by feel or experience, a somewhat indecipherable process of guesswork and intuition. By distilling mafia games down to a few "core" building blocks, I hope to make the process a little more accessible for everyone.

    When designing a game, I think it's good to have a solid starting point. I like to go all the way back to the original design, the thing all other setups were built upon, Mountainous. Let's examine the characteristics of a Vanilla setup and then layer other considerations on top.


    1. Team Sizes
    Typical team sizes are 7v2, 10v3, 13v4.

    Which can be described in a number of ways:
    • Player Count = (Number of mafia x 4) + 1
    • Num of Town = (Number of mafia x 3) + 1
    • Percent Mafia = Around 23%


    I'm going to refer to the above as the "+1 Rule".
    Note that the "+1 Rule" doesn't scale, so with higher player counts the percentage keeps increasing slightly. e.g. (15 mafia x 4) + 1 = 61. 15/61 = 24.5%.

    Percentage is the easiest to use. Something around 25% is usually about right.

    You can pick a different percentage if you like, but too low and your game might become frustrating or boring...too high and it becomes difficult to balance.



    2. Wolf Win Condition
    Let's start this section by looking at the Village Win Condition:
    How many wolves do the village need to lynch to win? Easy, all of them.
    Everyone knows and immediately understands this. The number also never changes.

    Less often, designers think about the Wolf Win Condition:
    How many villagers do the wolves need to lynch to win? This one requires a bit more thinking.


    2.1 Number of Mislynches
    Let's examine a 9er, 7 villagers vs 2 wolves...

    The answer is not always obvious, so I typically do something like this:

    D1: 7v2
    Lynch Villager
    Night Kill

    D2: 5v2
    Lynch Villager
    Night Kill

    D3: 3v2
    Lynch Villager
    Parity!


    The answer is 3.


    Add in a vigilante, and the answer changes.
    Add a doctor, and the answer is variable.
    Add another player and sometimes the answer changes...but sometimes it doesn't!

    The village win condition is constant, but the wolf win condition is dependent on an endless number of factors. This is what makes the wolf win condition one of the most important things to consider when designing a game.


    For now, let's just try and quantify it for our Vanilla setup.

    Number of Mislynches = Number of Wolves + 1

    That +1 is the extra villager we added with the "+1 Rule".
    Let's get it is as a percentage of villagers that need to be lynched:

    9er: 42% (3/7)
    13er: 40% (4/10)
    17er: 38% (5/13)
    21er: 37% (6/16)

    This percentage is not really that important on its own, it's just an important preface for our next topic.


    2.2 Number of Non-Clears
    We now know how many villagers need to be lynched... but how many are actually lynchable? In the above example, the answer is "all of them".
    But as we move away from a Vanilla setup we have to consider that some villagers will have roles that mean they cannot be lynched. We call these players "Mechanical Clears". We can call the remaining villagers "Non-Clears".

    More important than counting mislynches, is counting your non-clears.


    Wolves think of their win condition as "How many villagers do we need to mislynch?".

    Designers need to think of the win condition as "How many non-clears do the wolves need to mislynch?"

    In my opinion, this number effectively represents the village's accuracy, which should be the deciding factor in who wins and loses.


    I typically aim for around 50% of non-clears in my games, but the nice thing about this number is that you can easily adjust it to match your site's skill level or meta. The biggest reason a setup fails during review is because this percentage is too high (often more than 100%!).

    The most important thing, is that you have a number, and that you check it.


    2.3 Example Analysis

    Cop 13er with a n0

    On Day 1 there are 2 clears, 8 non-clears, and 3 wolves. Wolves need to lynch 4 non-clears (50%).

    Let's assume only Vanilla Towns get lynched and every Night Kill is on the cop's peek.
    On Day 2 there are 2 clears and 6 non-clears. Wolves need to lynch 3 non-clears (50%).
    On Day 3 there are 2 clears and 4 non-clears. Wolves need to lynch 2 non-clears (50%).

    But now let's assume every Night Kill is on a non-clear.
    On Day 2 there are 3 clears and 5 non-clears. Wolves need to lynch 3 non-clears (60%).
    On Day 3 there are 4 clears and 2 non-clears. Wolves need to lynch 2 non-clears (100%).

    Only two missed NKs and the wolves are already in a very bad situation. This analysis doesn't even account for Seer Hunt Clears.

    Everyone knows this already, but (imo) this shows this setup is won or lost in the Night Phase.
    Kill the cop, or some of his peeks, and you have a fair game...but whiff your NKs and you'll be in trouble very fast.
    If you've ever been a wolf in this setup, you know the dread.


    C9++, Random Setup

    I randed this once, here's what I got:

    99, 57, 61, 79, 11, 72, 87

    B = Roleblocker
    CC = Cop
    V = 1-Shot Vig
    T = Goon + Roleblocker + Godfather, Serial Killer (Investigation Immune OR 1-Shot Bulletproof)
    D = Doctor
    M = Innocent Child

    The Setup:
    Cop
    Doc
    Roleblocker
    Vig 1-Shot
    Inno
    4 Vanilla Town
    1 SK (Investigation Immune OR 1-Shot Bulletproof)

    3 Wolves (Goon + Roleblocker + Godfather)


    There are 5 mechanical clears in this setup. 3 of them (cop, inno, vig) are guaranteed. Maybe the Doctor or Roleblocker get lynched, but usually not.

    There are 5 non-clears on Day 1. Wolves need to lynch 4 of them (80%).
    The setup should be nigh unwinnable, and I didn't even consider what all the town PRs are capable of, just that they were unlynchable.


    In the name of science, I went for 3 more rands:


    Let's go again.

    79, 4, 87, 62, 67, 66, 25

    Cop 1-Shot
    Doc
    Doc 1-Shot
    Vig 1-Shot
    Inno
    5 Vanilla Town

    3 Wolves

    % of non-clear lynches to win: 80%;



    And again:

    33, 8, 23, 64, 57, 41, 87

    Cop
    Inno
    8 Vanilla

    3 Wolves (Roleblocker + 2 Goons)

    % of non-clear lynches to win: 50%. (This one is a cop 13er)



    Okay, last one:

    24, 79, 42, 14, 63, 95, 72

    Doctor
    Roleblocker
    1-Shot Roleblocker
    1-Shot Vigilante
    5 Vanilla

    Serial Killer

    3 Wolves (2 Goons + Roleblocker)

    % of non-clear lynches to win: 67%

    C9++ sucks C9++ is way too town-sided. For a lot of people, this is probably not new information, just an easy example.



    3. Kill Power (KP)
    Nothing changes your game's wincon like additional KP, sometimes in ways you might not even realize until you think it through properly.

    When calculating how many non-clears the wolves need to kill, count all village controlled KP as "mislynches" or "miskills". Let's learn by example...

    Here's the rundown of a Vanilla 17er:
    Wolves need to miskill 5/13 non-clears to win (38%).

    13 town v 4 wolves.

    13v4
    Day 1 Mislynch
    Night 1 Kill

    11v4
    Day 2 Mislynch
    Night 2 Kill

    9v4
    Day 3 Mislynch
    Night 3 Kill

    7v4
    Day 4 Mislynch
    Night 4 Kill

    5v4
    Day 5 Mislynch
    Parity at day end.


    Now let's consider a Vig 17er:
    Wolves need to miskill 6/12 non-clears to win (50%).

    Vig 17er
    12 Vanilla Town
    1 Town Vigilante
    4 Mafia Goon

    13 town v 4 wolves. 12 non-clears.

    13v4
    Day 1 Mislynch
    Night 1 Vigilante Shot on Town
    Night 1 Factional Kill

    10v4
    Day 2 Mislynch
    Night 2 Vigilante Shot on Town
    Night 2 Factional Kill

    7v4
    Day 3 Mislynch
    Night 3 Vigilante Shot on Town
    Night 3 Factional Kill

    Parity at day start.


    The Town Vigilante changed quite a bit:
    • It added 1 mechanical clear to the setup.
    • It added 1 extra village-controlled kill before parity.
    • Combined, these increased the wolf wincon from miskilling 38% of non-clears, to 50%!

    Town vigilantes remove non-clears from the game, meaning wolves have to lynch a higher percentage of them to win. They are somewhat equivalent to a free mislynch, or a night without a mafia kill. They also tend to remove the least townie players, making mislynches harder (a slightly intangible benefit).

    Wolf Vigilantes are basically the exact opposite, they reduce the number of villagers the wolves need to miskill/mislynch. A good time to add a wolf vig is in a game with too many clears.

    Example: Balancing clears with wolf KP
    Consider the following setup:

    10 town v 3 scum
    4 Innocent Children
    6 Vanilla Town
    3 Mafia Goon

    Start of Day 1, 6 non-clears. 4 need to be mislynched (66%). Probably a difficult game for the wolves.

    But, adding a 2-Shot Mafia Vigilante let's scum remove two clears during the game, and lowers the number of mislynches needed to reach parity.
    Now only 3 of the 6 non-clears need to be mislynched (50%).

    The above is all relatively straightforward, but less obvious is an important concept I'll call "KP Ratio".


    3.1 KP Ratio
    Back to Vanillas: The only Kill Power in the game are lynches and night kills. If we say the villagers "control" the lynch, then both teams share 50% of the game's KP. I don't think many people think about KP this way, but 50% each...that's my baseline number.
    Now think about In-Thread Attacks for a second, town control 75% of them!

    One effect of this ratio is how much control each team has on the game, and whether the townie players are dying faster than the scummy ones.

    A much less intuitive effect...is that with an uneven KP Ratio, the number of mislynches before parity changes over time. The longer a game lasts, the more it changes. This can be a real pain to balance around; My best advice is to keep the ratio relatively even.

    Vig 17er Example (Continued):
    In this example, a couple of wolves dying prolongs the game and the unbalanced KP Ratio increased the number of miskills to 7 (58% of non-clears).
    Let's assume the wolves never kill the vigilante.

    13v4
    Day 1 Mislynch
    Night 1 Vigilante Shot on Town
    Night 1 Factional Kill

    10v4
    Day 2 Mislynch
    Night 2 Vigilante Shot on Town
    Night 2 Factional Kill

    7v4
    Day 3 Mislynch
    Night 3 Vigilante Shot on Scum
    Night 3 Factional Kill

    5v3
    Day 4 Mislynch
    Night 4 Vigilante Shot on Scum
    Night 4 Factional Kill


    3v2
    At this point, 6 villagers have died to village KP...and they still haven't lost.

    A Day 5 Mislynch gives Parity

    A long Vanilla 17er for comparison:
    It doesn't matter how long this setup lasts...the number of miskills never changes.
    13v4
    Day 1 Mislynch
    Night 1 Kill

    11v4
    Day 2 Mislynch
    Night 2 Kill

    9v4
    Day 3 Mislynch
    Night 3 Kill

    7v4
    Day 4 Mislynch
    Night 4 Kill

    5v4
    Day 5 Lynch Scum
    Night 5 Kill

    4v3
    Day 6 Lynch Scum
    Night 6 Kill

    3v2
    Day 7 Mislynch
    Parity at day end.


    Number of Mislynches: 5.



    Summary
    We've identified three core aspects of a game's design: Team Size, Wolf Win Condition, and Kill Power.

    Your chosen team sizes are effectively just a starting point for your wolf win condition, and the most important aspect of Kill Power is how it changes your number of lynches. Ultimately, almost everything comes back to the Wolf Win Condition.

    We defined the Wolf Win Condition as: "percentage of non-clears wolves need to mislynch". If you can relate a role, mechanic, or idea back to this percentage, then you can quantify its impact on your game and balance around it accordingly. Using this idea, we can make a wide range of alterations to a setup but analyze them all in a similar way.

    What if we run a game as 15v2 instead of 13v4?
    Check your non-clears/mislynches and see what effect it had.
    With 13 villagers, wolves need 5/13 lynches (38%)
    With 15 villagers, wolves need 7/15 lynches (46%)
    If for some reason you decided that 38% was the right number, you now know how much tweaking to do to get back to that number...all while running a game with 2 less wolves. For example, adding a 1-Shot wolf vig would give you 6/15 lynches (40%), pretty close to 38%.

    What if we add masons to a setup?
    Remove two non-clears and recheck your Wolf Win Condition. Then tweak the percentages as needed.

    What if we add a Vigilante?
    Check your Wolf Win condition is ok, then check that your KP ratio doesn't ruin it.

    You get the idea.

    That's it for Part 1. Hopefully you found it useful. Feel free to request a topic for future articles.

    ---

    Core Design TL;DR
    • Around 25% of players are Werewolves
    • The number of mislynches wolves need to win should be 50% of all non-clear villagers
    • Both teams should control 50% of the game's kill power
    • All deviations should be compensated for in some way
    Last edited by Newcomb; August 14th, 2019 at 08:27 PM.

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    GOAT Tier LordQuas's Avatar
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    too much math
    :wiwe

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    GOAT Tier Jaleb's Avatar
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    Was thinking about typing up something similar to this on how vigilantes shooting town is still a pro-town action. The whole statement of "you're helping mafia get to their win condition," while also removing a person that mafia wouldn't kill except by using town's lynch is pretty bad.
    Quote Originally Posted by Empoof (#4961)
    it was jaleb

    @Jaleb is a god
    ??Frog ??Last Thursday at 8:11 AM
    ok dudes
    you can discuss the event here if you want

    Jaleb (RB/1% ITA)Last Thursday at 8:05 AM
    Done, will take my prize now
    I do have a 1900 rating in solving tactics :smiley:

    AndrewGreve(night vig/39%ITA)Last Thursday at 8:10 AM
    hey
    i dont know chess tactics
    should I try?

    Jaleb (RB/1% ITA)Last Thursday at 8:10 AM
    1. Bf6 2. Qh7 3. Rf7 4. Nh6 5. Re5 6. ... Qh3 7. Qd8 8. h7 9. ... Ne5+; fxe5 h5 10. ... Bg1; Kg3 Qf2; Kh3 Qh2 i you want the answer, not that it matters
    if*

    ??Frog ??Last Thursday at 8:11 AM
    Jaleb, you're lowkey Magnus

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    Wants It Most Apoc's Avatar Game Manager
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    Morning Nerds!

    I have the best part of 2 and a half articles written, it's just a matter of figuring out the structure & order of content for the next few. Would appreciate some feedback about what people are interested in. Right Now Part 2 is Role Balance, which will surely be a more contentious topic

    This first article is pretty simple, but it made sense as a stopping point for part 1. It's aimed at total design newbs so i tried to add a lot of detail/examples that might be overkill for some

    To the people who learned something new from this (hopefully someone!?), let me know if the examples were easy to follow, we can update/improve the article based on feedback.

    Thanks

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    Wants It More LanMisa's Avatar
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    Interesting articles, easy to follow, now if only I was actively hosting games...

    What it does though is give me a general idea of what should be balanced and what isn't so as to avoid certain games that are unbalanced from the start.

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    ༼ つ ;-; ༽つ give smith another day mhsmith0's Avatar Game Manager
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    Quote Originally Posted by Jaleb (#3)
    Was thinking about typing up something similar to this on how vigilantes shooting town is still a pro-town action. The whole statement of "you're helping mafia get to their win condition," while also removing a person that mafia wouldn't kill except by using town's lynch is pretty bad.
    town vig roles empower really $#@!ty villagers who just lulz shoot obvious villagers, and that's obviously antitown when it happens
    but otherwise I don't think it's particularly prowolf. In sites with really $#@!ty town meta, it's more wolf-friendly than on sites with better town metas.

    PS I think that "what % of non-clears do wolves need to mislynch" is an excellent way to look at game balance, though I'm not sure that 50% is necessarily the correct number (I think figuring out a "normal" % is probably an EXCELLENT way of trying to understand balancing standards for a given community though). I'd probably also say that vigs shooting villagers is less town-helpful than the actual lynch situation would be (partially because they could be PRs, partially because even mislynches can yield information in terms of wagon composition, who's willing to defend/attack the lynchbaits, etc), so I might be cautious in applying the same "value" to each, but idk what a better # would be so *shrugs*
    Life is simply unfair... don't you think?
    http://wiki.mafiascum.net/index.php?title=mhsmith0

    BATMAN: Only jugg was the only one we didn’t scum read at least not me

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    GOAT Tier Jaleb's Avatar
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    Quote Originally Posted by mhsmith0 (#6)
    town vig roles empower really $#@!ty villagers who just lulz shoot obvious villagers, and that's obviously antitown when it happens
    but otherwise I don't think it's particularly prowolf. In sites with really $#@!ty town meta, it's more wolf-friendly than on sites with better town metas.
    Explains why town can never win in the yearly daily mafia game.
    Quote Originally Posted by Empoof (#4961)
    it was jaleb

    @Jaleb is a god
    ??Frog ??Last Thursday at 8:11 AM
    ok dudes
    you can discuss the event here if you want

    Jaleb (RB/1% ITA)Last Thursday at 8:05 AM
    Done, will take my prize now
    I do have a 1900 rating in solving tactics :smiley:

    AndrewGreve(night vig/39%ITA)Last Thursday at 8:10 AM
    hey
    i dont know chess tactics
    should I try?

    Jaleb (RB/1% ITA)Last Thursday at 8:10 AM
    1. Bf6 2. Qh7 3. Rf7 4. Nh6 5. Re5 6. ... Qh3 7. Qd8 8. h7 9. ... Ne5+; fxe5 h5 10. ... Bg1; Kg3 Qf2; Kh3 Qh2 i you want the answer, not that it matters
    if*

    ??Frog ??Last Thursday at 8:11 AM
    Jaleb, you're lowkey Magnus

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    Wants It Most Apoc's Avatar Game Manager
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    Quote Originally Posted by mhsmith0 (#6)
    PS I think that "what % of non-clears do wolves need to mislynch" is an excellent way to look at game balance, though I'm not sure that 50% is necessarily the correct number (I think figuring out a "normal" % is probably an EXCELLENT way of trying to understand balancing standards for a given community though)
    Ya more than anything else, the most important take away is finding a number that works for you and using that as your benchmark

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    Thread Analyst Helz's Avatar
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    I love the topic and the approach. This was something that was critically needed and presented in a pragmatic way. Much respect Apoc

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    The Self Narrator Owner Of A Lonely Heart's Avatar
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    We design very differently, but it was informative so thank you.

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    Soul Reader kaempfer13's Avatar
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    Assuming you were to step away from the formula for the number of wolves, how would that affect the game balance (naturally, more wolves means more wolfsided, but whats the formula)? Especially wrt the nonclear villas.

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    Wants It Most Apoc's Avatar Game Manager
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    Quote Originally Posted by kaempfer13 (#11)
    Assuming you were to step away from the formula for the number of wolves, how would that affect the game balance (naturally, more wolves means more wolfsided, but whats the formula)? Especially wrt the nonclear villas.
    It doesn't really matter how many wolves there are, but adding more will generally mean less mislynches.

    Your formula is:
    Number of Non-Clears / Number of Mislynches = %

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    Wants It Most Apoc's Avatar Game Manager
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    There an example at the end comparing 15v2 & 13v4

    It's kind of the reverse of what you're asking

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    Soul Reader kaempfer13's Avatar
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    But that would imply that 1vs10 is just as balanced as 3vs10. That doesn't strike me as realistic. There is no way to gain towncred through bussing and you can just die randomly. I think the number of times town has to lynch correctly is completely ignored here.

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    Wants It Most Apoc's Avatar Game Manager
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    Quote Originally Posted by kaempfer13 (#14)
    But that would imply that 1vs10 is just as balanced as 3vs10. That doesn't strike me as realistic. There is no way to gain towncred through bussing and you can just die randomly.
    There's a big difference between 15v2 and 13v4 compared to 10v1 and 10v3

    Besides, 15v2 required some tweaking to make it roughly similar to 13v4. Did you read the example?

    I wouldn't agree that anything i said indicates that 10v1 & 10v3 are the same...


    10v3 is normal mountainous. Wolves need to lynch 40% of non-clears. Slightly scum sided game.

    10v1 is barely even playing mafia anymore. But Scum need to lynch 50% of non-clears. Balanced game, but probably not very fun!


    Arguably there is no TMI in a 10v1 game, though some people still feel guilt even as a lone wolf and get caught anyway (even without awkward teammate interactions to analyze). In a game without TMI...lynching is probably just random guessing.
    If you randomly lynch 5 people from 11, town wins 45% of the time, which is good enough balance for most setups! (Again, that doesn't mean it's a fun setup)

    This feels like a weird math proof for my preferred non-clear percentage of 50%

    QED?



    That was fun. I don't think we learned anything tbh...but I'm sure @LordQuas enjoyed all the numbers and percentages!

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    Wants It Most Apoc's Avatar Game Manager
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    Anyway! Let's talk "Number of correct lynches".

    Quote Originally Posted by kaempfer13 (#14)
    I think the number of times town has to lynch correctly is completely ignored here.
    It's not ignored at all.

    Number of Correct Lynches is just the "Number of Mafia" (see: Team Sizes). Too few mafia and it's not really a game of mafia anymore. Too many and it becomes really difficult for town to win. I felt like this idea was obvious enough that it wasn't really worth more than the couple of lines I gave it in the article.

    You can also look at Correct Lynches as one definition of a "Town Win Condition". My article focuses almost exclusively on the Wolf Win Condition. I think i explained why well enough, but let's go over it a bit more

    I focus on Wolf Win Condition because:
    - Designers never overlook the village win condition (case in point: you! )
    - Designers regularly overlook the wolf win condition
    - imo, the wolf win condition captures the whole game and is a useful indicator of balance
    - imo, the town win condition does not!


    Number of correct lynches inherently seems like a useful metric for game balance...but it's actually quite useless.

    Consider our Mountainous 13er
    Correct Lynches to win (for town): 3
    Non-clears to lynch (for wolves): 40%


    Now let's add 8 innocent children!
    The setup is 3 scum and 8 innocent children and 2 vanilla town.

    Correct Lynches to win (for town): 3
    Non-clears to lynch (for wolves): 200%


    The number of correct lynches didn't change, but the PoE is 5 people and town have 6 lynches!
    The game is unwinnable for scum..and yet your metric (correct lynches) didn't capture that at all.

    Percent of non-clears to lynch is 200% though, it's immediately obvious this setup is broken.


    Number of Correct Lynches doesn't say anything about number of PRs, Clears, Non-Clears, or overall game difficulty for town.

    It's more like a game rule than anything else.
    Score 3 goals to win, but how hard is it to score? We have no idea, because the number doesn't tell us anything about it.

    The Wolf Win Condition (as defined in the article) tells of everything, so that's the one to focus on.




    We could try to define a Town Win Condition based around percentage of correct reads in relation to non-clears or something...
    But i think we'd inevitably just come up with either the inverse of the formulas already in the article or some variation of what we already have?

    I'll probably give it a go later tbh, but now i gotta get back to work

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    Wants It Most Apoc's Avatar Game Manager
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    For a 13er

    Towns win condition is lynch 3 scum in 6 lynches.
    Or dont lynch more than 3 towns.
    Or townclear 7 towns.

    I dunno what makes the most sense.


    If town win condition is clearing 7/10 non-clears (70%), then the town loss condition (i.e. wolf win condition) is if they only clear 6/10 towns (& misclear 1 scum), which is 60%.

    This is just the inverse of "Number of Non-clears to mislynch" for wolf win condition: 40%.


    So you can look at balance from the point of view of correct lynches but it's just going to be the exact inverse of the Wolf Win Con (mislynches).
    As expected

    The important common denominator is that you are counting non-clears.
    Last edited by Apoc; August 22nd, 2019 at 10:41 AM.

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    Wants It Most Apoc's Avatar Game Manager
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    I'm going to start over.

    Quote Originally Posted by kaempfer13 (#14)
    I think the number of times town has to lynch correctly is completely ignored here.
    Number of times to lynch correctly is just the exact opposite of number of times you can lynch incorrectly, i.e. mislynches.

    Number of Mislynches is covered pretty thoroughly in the article

    TY for your question
    Last edited by Apoc; August 22nd, 2019 at 10:44 AM.

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    Soul Reader kaempfer13's Avatar
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    Yh, my mistake was to not account for the additional nightkill that occurs after lynching scum. as such 8v1 would theoretically have the same balance as 10v3.

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    Wants It Most Apoc's Avatar Game Manager
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    Quote Originally Posted by kaempfer13 (#19)
    Yh, my mistake was to not account for the additional nightkill that occurs after lynching scum. as such 8v1 would theoretically have the same balance as 10v3.
    Gotcha

    That's ok i enjoy talking to myself about balance
    Last edited by Apoc; August 22nd, 2019 at 11:44 AM.

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    Wants It More LanMisa's Avatar
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    Quote Originally Posted by Apoc (#20)
    Quote Originally Posted by kaempfer13 (#19)
    Yh, my mistake was to not account for the additional nightkill that occurs after lynching scum. as such 8v1 would theoretically have the same balance as 10v3.
    Gotcha

    That's ok i enjoy talking to myself about balance
    Well it is informative AND entertaining. You should start a YouTube channel.

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    GOAT Tier LordQuas's Avatar
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    Quote Originally Posted by Apoc (#15)
    Quote Originally Posted by kaempfer13 (#14)
    But that would imply that 1vs10 is just as balanced as 3vs10. That doesn't strike me as realistic. There is no way to gain towncred through bussing and you can just die randomly.
    There's a big difference between 15v2 and 13v4 compared to 10v1 and 10v3

    Besides, 15v2 required some tweaking to make it roughly similar to 13v4. Did you read the example?

    I wouldn't agree that anything i said indicates that 10v1 & 10v3 are the same...


    10v3 is normal mountainous. Wolves need to lynch 40% of non-clears. Slightly scum sided game.

    10v1 is barely even playing mafia anymore. But Scum need to lynch 50% of non-clears. Balanced game, but probably not very fun!


    Arguably there is no TMI in a 10v1 game, though some people still feel guilt even as a lone wolf and get caught anyway (even without awkward teammate interactions to analyze). In a game without TMI...lynching is probably just random guessing.
    If you randomly lynch 5 people from 11, town wins 45% of the time, which is good enough balance for most setups! (Again, that doesn't mean it's a fun setup)

    This feels like a weird math proof for my preferred non-clear percentage of 50%

    QED?



    That was fun. I don't think we learned anything tbh...but I'm sure @LordQuas enjoyed all the numbers and percentages!
    grrrrrrrr
    :wiwe

  23. ISO #23
    ༼ つ ;-; ༽つ give smith another day mhsmith0's Avatar Game Manager
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    % of non-clears to lynch is probably a reasonable enough metric given standard-ish team sizes, it just breaks down when you go well off the beaten path (which, incidentally, is true of most to all metrics).

    Another sometimes useful metric is win %'s assuming random lynching (obviously you never see truly random lynches/night kills, but you can get a sense of how accurate lynches are in your site meta and tweak the calculation accordingly - few sites have lynch accuracy wildly worse than or wildly better than rand, if you're lynching 20% above rand then people need to learn how to wolf, and if 20% below rand then severe lolwillage, etc ).

    10v1 mountainous with random lynching / mandatory night kills gives you a wolf win only 37% of the time. If you presume town is just barely competent enough to exceed random lynching, then the % goes down further, and is not a well-balanced game and debatably not a playable game, which makes sense, because 10/1 is aids as $#@!.

    11v2 mountainous with random lynching / mandatory night kills gives you a wolf win 37% of the time AFTER A PRESUMED DAY ONE WOLF LYNCH. That's a good deal more reasonable. It's 41% if you presume d1 mislynch, d2 correct lynch. 46% if you mislynch d1-2, correct lynch d3.

    Obviously that's not a perfect calculation, but you can kind of squint and get something that tells you that 11/2 isn't completely insane, but probably hard on wolves given a decent town.

    To put it a different way...
    Quote Originally Posted by kaempfer13 (#19)
    Yh, my mistake was to not account for the additional nightkill that occurs after lynching scum. as such 8v1 would theoretically have the same balance as 10v3.
    they're not the same balance though
    8/1 is (numbers-wise at least) the same as a 10/3 where you correctly lynched on the first 2 days
    by lynching wolves you swing the numbers in your favor and make the game much easier
    a 10/3 mountainous is wolfsided, a 8/1 is townsided
    Last edited by mhsmith0; August 22nd, 2019 at 04:47 PM.
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    Wants It Most Apoc's Avatar Game Manager
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    Quote Originally Posted by mhsmith0 (#23)
    10v1 mountainous with random lynching / mandatory night kills gives you a wolf win only 37% of the time.
    explain the math on this one

    this just doesn't seem intuitive, but sometimes math isn't?


    initially i was lazy and did 45% of all players are lynched. 55% chance the scum isn't one of them.


    Just now I tried:

    D1 = 10/11 (Odds of Town lynch)
    AND
    D2 = 8/9 (Odds of Town lynch)
    AND
    D3 = 6/7 (Odds of Town lynch)
    AND
    D4 = 4/5 (Odds of Town lynch)
    AND
    D5 = 2/3 (Odds of Town lynch)
    ---
    10/11 x 8/9 x 6/7 x 4/5 x 2/3 = 66.666% = Odds of scum never getting lynched?
    I forgot to click enter on the calculator. LOL.

    10/11 x 8/9 x 6/7 x 4/5 x 2/3 = 37% = Odds of scum never getting lynched?


    But then i did just ran through final 3 and I had:

    Odds of townie #1 voting right (50%) AND Odds of townie #2 voting right (50%)
    = Odds of both townies voting right (25%).

    75% in scums favour is a bit better than the above calculation of 66%...maybe the math needs to be a bit more complex


    @mhsmith0
    Last edited by Apoc; August 22nd, 2019 at 05:13 PM.

  25. ISO #25
    ༼ つ ;-; ༽つ give smith another day mhsmith0's Avatar Game Manager
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    You can complicate the math for sure, but the main thing is that "random player gets lynched" is a reasonable quick and dirty proxy for a game state in a bunch of site metas (MU lynches above rand, but not like insanely above it I don't think). Blindly voting gives you 3/4 mislynch in f3, but "random lynching" effectively assumes town is a LITTLE better than random voting, but the difference cancels the wolf info advantage to get at 2/3 (f3), 4/5 (f5), etc.

    I think random lynching is a mediocre balance metric overall but it can sometimes help with certain unusual structures.

    Or take it to a fully absurd level, 1000/1 is balanced by the standard of "what % of willagers do you need to mislynch" but not really balanced in pratice (you could make it 1000/3 to make it "mafia" with a "team" and the balance doen'st much change ).
    Life is simply unfair... don't you think?
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    Wants It Most Apoc's Avatar Game Manager
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    I dunno if i did this right but the odds of townies agreeing is far lower than the odds of choosing a player at random.


    Final 5
    5 players. Wolf always votes to save themself.
    3 need to vote the wolf to kill him...or 2-1-1 split works too (coinflip)...but a 2-2 split doesn't (wolf self pres). Let's just look at maj for now...

    25% percent chance town #1 votes correctly
    25% percent chance town #2 votes correctly
    25% percent chance town #3 votes correctly
    25% percent chance town #4 votes correctly


    all correct = 0.00390625
    1,2,3 correct = 0.01171875
    1,2,4 correct = 0.01171875
    1,3,4 correct = 0.01171875
    2,3,4 correct = 0.01171875
    =
    0.05859375


    ~6% chance enough randomly townies agree and vote scum

    ???

    odds of a random death hitting scum: 20%
    Last edited by Apoc; August 22nd, 2019 at 05:33 PM.

  27. ISO #27
    Wants It Most Apoc's Avatar Game Manager
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    Quote Originally Posted by mhsmith0 (#25)
    You can complicate the math for sure, but the main thing is that "random player gets lynched" is a reasonable quick and dirty proxy for a game state in a bunch of site metas (MU lynches above rand, but not like insanely above it I don't think). Blindly voting gives you 3/4 mislynch in f3, but "random lynching" effectively assumes town is a LITTLE better than random voting, but the difference cancels the wolf info advantage to get at 2/3 (f3), 4/5 (f5), etc.

    I think random lynching is a mediocre balance metric overall but it can sometimes help with certain unusual structures.

    Or take it to a fully absurd level, 1000/1 is balanced by the standard of "what % of willagers do you need to mislynch" but not really balanced in pratice (you could make it 1000/3 to make it "mafia" with a "team" and the balance doen'st much change ).
    random voting seems more valid than random lynching

    in the same way that random lynching seems more valid that my original "quick and dirty" 5/11 players get lynched, therefore 55% chance scum survives


    My original idea doesn't account for narrowing lynch pool via NKs.
    Random lynching doesn't account for the teamwork/coordination required to actually lynch one person.
    Last edited by Apoc; August 22nd, 2019 at 05:38 PM.

  28. ISO #28
    ༼ つ ;-; ༽つ give smith another day mhsmith0's Avatar Game Manager
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    Quote Originally Posted by Apoc (#26)
    I dunno if i did this right but the odds of townies agreeing is far lower than the odds of choosing a player at random.


    Final 5
    5 players. Wolf always votes to save themself.
    3 need to vote the wolf to kill him...or 2-1-1 split works too (coinflip)...but a 2-2 split doesn't (wolf self pres). Let's just look at maj for now...

    25% percent chance town #1 votes correctly
    25% percent chance town #2 votes correctly
    25% percent chance town #3 votes correctly
    25% percent chance town #4 votes correctly


    all correct = 0.00390625
    1,2,3 correct = 0.01171875
    1,2,4 correct = 0.01171875
    1,3,4 correct = 0.01171875
    2,3,4 correct = 0.01171875
    =
    0.05859375


    ~6% chance enough randomly townies agree and vote scum

    ???

    odds of a random death hitting scum: 20%
    Sure, but in real games there's also the existence of a bunch of information from earlier in the game (which goes up against the fact that the late game willagers tend to be less effective than the earlier game willagers).

    You could also pretty reasonably set up a mechanism whereby collective scumspect A chooses person B to thunderdome. If you rand the accuracy there, then it's 60% likely neither of those two is the wolf. If one IS the wolf, then you have 3 townies left, and you need a majority to pick correctly.

    Odds of wolf survival in that structure = 60% (chances he isn't part of the thunderdome) + 40% (odds he is part) * 20% = 0.6 + 0.2 = 0.8, which brings back to random lynching.

    Maybe my math is off, but at the least I think even sort of random town odds are better than the pure rand you envision. It's not really reasonable for a village to see votes go like 1-1-1-1-1 in f5 (yes, yes, I see you gamer dude ), so random calculations that rely on a construct where that's considered realistic (or at least equally realistic as various other possible vote distributions) are imo flawed.

    (which is i guess a different way of saying that, yes, it DOES make sense to try and consolidate votes compared to just spreading them out for no particularly good reason )
    Last edited by mhsmith0; August 22nd, 2019 at 05:43 PM.
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    Soul Reader kaempfer13's Avatar
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    Yh, I knew something was off when looking at only non-clear%. as mhs said a 10v3 is more scumsided than 8v1 unless villagers manage a wolflynch on both the first days (otherwise the gamestate on day 3 will be worse than day 1 of 8v1, although this doesnt account for time that village spent on talking in those 2 days).

    About random lynching: If you want to take the randomvotes path include some basic wagonomics. there will only be 2-3 wagons at the end of the day (with the exception of a few stubborn people) and the probability of wolves voting villagers only really jumps to 100% at lylo (arguably not even then when villagers can still vote and its not the last wolf), as otherwise wagonomics will catch them later on.

  30. ISO #30
    Wants It Most Apoc's Avatar Game Manager
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    Quote Originally Posted by kaempfer13 (#29)
    Yh, I knew something was off when looking at only non-clear%

    Quote Originally Posted by Newcomb
    Core Design TL;DR
    • Around 25% of players are Werewolves
    • The number of mislynches wolves need to win should be 50% of all non-clear villagers
    • Both teams should control 50% of the game's kill power
    • All deviations should be compensated for in some way
    Bullet point 1

    25% of players should be mafia

    Kaemper's examples:
    - games with 8 to 12% mafia.


  31. ISO #31
    Wants It Most Apoc's Avatar Game Manager
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    Based on Smith's posts I have spent the morning comparing different metrics to see where they converge, and when each is most useful vs others

    It feels like a not small amount of work so it may take a while to actually figure it out (if that's even a thing that can be done!?)

  32. ISO #32
    Wants It Most Apoc's Avatar Game Manager
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    Preliminary Results (assuming i mathed right)

    Vanilla 9er
    Random lynch wolf win: 70.14%
    Non-clear percent: 42%

    Cop 9er (cop dies N1)
    Random Lynch wolf Win: 66.64%
    Non-clear percent: 60%

    Vanilla 13er
    Random Lynch Wolf Win: [@mhsmith0 probably knows this number by heart]%
    Non-clear percent: 40%

    Vanilla 17er
    Random Lynch Wolf Win: smith gimme this number too pls
    Non-clear percent: 38%


    Non-clear percent goes down with player count.
    I assume random lynching win % goes up with player count also.
    Seems like both agree with common consensus (more players is better for wolves)



    Random lynching doesn't seem to indicate a cop makes much difference in a 9er.
    Non-clears weights it as a decent sized difference.
    Maybe a second peek will change the results of this comparison (on the to-do list)



    Fun fact that when there are clears to be NKd (rather than non-clears) then you get this weird thing where randomly lynching exactly one wolf is as likely to happen on any given day.


    D1 wolf lynch: WTTT (2/7 x 5/6 x 4/5 x 2/3) = 0.126984126984127
    D2 wolf lynch: TWTT (5/7 x 2/6 x 4/5 x 2/3) = 0.126984126984127
    D3 wolf lynch: TTWT (5/7 x 4/6 x 2/5 x 2/3) = 0.126984126984127


    Normally it gets easier the more days you go without lynching a wolf:

    WTTT (2/9 x 6/7 x 4/5 x 2/3) = 0.1015873015873016
    TWTT (7/9 x 2/7 x 4/5 x 2/3) = 0.1185185185185185
    TTWT (7/9 x 5/7 x 2/5 x 2/3) = 0.1481481481481482


    #MathNerds
    Last edited by Apoc; August 23rd, 2019 at 09:13 AM.

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    JAMES MCCLOWN Frog's Avatar
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    Good intro article for mid/small games on site.

    Could add clears & semi-clears versus killing power per side; but that would be another topic entirely.

    (I see it)** edit
    Last edited by Frog; August 23rd, 2019 at 09:27 AM.

  34. ISO #34
    Soul Reader kaempfer13's Avatar
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    Quote Originally Posted by Apoc (#30)
    Quote Originally Posted by kaempfer13 (#29)
    Yh, I knew something was off when looking at only non-clear%

    Quote Originally Posted by Newcomb
    Core Design TL;DR
    • Around 25% of players are Werewolves
    • The number of mislynches wolves need to win should be 50% of all non-clear villagers
    • Both teams should control 50% of the game's kill power
    • All deviations should be compensated for in some way
    Bullet point 1

    25% of players should be mafia

    Kaemper's examples:
    - games with 8 to 12% mafia.

    Hence my initial question. Would be nice to be able to account for nontraditional games. I think you explained it fine for deviations in killpower, but its not immediately clear how wolfnumbers should affect this. (Tbf i would probably rather increase the number of wolves as otherwise almost everyone is salty when the wolf wins, but its easier to do calculations with 1 wolf)

  35. ISO #35
    ༼ つ ;-; ༽つ give smith another day mhsmith0's Avatar Game Manager
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    @Apoc I calc'd some stuff out a while ago on a similar concept so here goes...

    10/3 n0 cop game,
    wolves shoot cop n3 and miss cop's peeks n1/n2
    cop peeks villagers n1/n2
    town never lynches cop or his peeks

    random lynching then gives you town win odds of 52.3%, wolf win odds of 47.7%


    Town Sweep
    D1 mislynch odds = 73%
    8 uncleared villagers out of 11 uncleared slots

    D2 mislynch odds = 75%
    6 uncleared villagers out of 8 uncleared slots

    D3 mislynch odds = 80%
    4 uncleared villagers out of 5 uncleared slots

    total odds = 1.4% = (1 - 73%)* (1 - 75%) * (1 - 80%)

    Town wins with 1 mislynch
    D2: 27% chance a wolf lynch happened D1, 73% chance a mislynch happened D1
    if a wolf lynch, then mislynch odds = 75% (same as above), otherweise 63%

    D3: 48% exactly one correct lynch, one mislynch to get you in that state
    60% mislynch odds

    D4 (presume wolves shot cop prior night)
    25% odds of getting to the point of 2 correct lynches, 1 mislynch
    mislynch odds then = 75%

    odds of town winning with exactly one mislynch therefore = 25% * (1 - 75%) = 6.1%

    etc


    of course that mainly was an exercise to demonstrate how hard it gets on wolves when they let a cop live a long time (if random lynching calculation gets you near 50%, much above 50% for town win odds, it is a very very very hard game on wolves most of the time)
    Life is simply unfair... don't you think?
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