Is the imposter selection in Among Us truly random?

I played "Among Us" with a group of seven friends last night. At one point, I was the imposter FIVE straight times. This seemed crazy, and started a lot of conspiracy-theory discussion about how the game takes ping into account when choosing imposter, takes winning into account, on and on. After browsing reddit about imposter selection in this game, it sounds like there are LOTS of anecdotal complaints about this kind of thing:

• "I played all night and didn't get to be imposter once!"
• "This one guy was imposter half the time!"

I watched a GREAT slide deck on "Stats for Hackers," and decided that this would be an excellent opportunity to run a simple simulation and see just how probable it is, in a night of gaming, for someone to be the imposter so many times in a row, for someone to never be imposter, etc.

My plan is to run a simulation of imposter selection a represetative amount of times (probably 10k at least) and collect some statistics each time. We played about 20 games last night, so this means I will choose 20 imposters, 100k times, and keep track of the following:

1. The longest unbroken string of being the imposter (was someone imposter 3 times in a row? 4?). How likely is it that someone would get to be imposter so many times in a row?
2. The person who was the imposter the most times, not necessarily in a row. Is it statistically crazy that someone might be imposter 10 times in a 20 game session?
3. The person who was the imposter the least times. How likely is it that someone doesn't get to be imposter at all?
4. The median imposter number. This is boring, but doing stats on this will give us some idea of the distribution.

Let's get cracking!

from random import randrange
from typing import Tuple
from statistics import median
import pandas as pd # i plan to use altair eventually to visualize the results, and you need to load your data in a dataframe to do so

def run_single_simulation(games: int, players: int) -> Tuple[int, int, int, int]:
    imposter_ids: list = []
    longest_string: int = 1
    current_string: int = 1
    previous_imposter: int = -1
    for g in range(games):
        imposter = randrange(0, players) # randint has inclusive bounds, so start at 1
        if imposter == previous_imposter:
            current_string += 1
            longest_string = max(current_string, longest_string) # update longest string if the current imposter has a high count
        else:
            current_string = 1
            previous_imposter = imposter
        imposter_ids.append(imposter)
    imposter_counts: list = []
    for p in range(players):
        imposter_counts.append(imposter_ids.count(p)) # keep a list of how many games each participant was imposter
    max_count = max(imposter_counts) # what is the highest number? we don't care who, just the value
    min_count = min(imposter_counts)
    median_count = int(median(imposter_counts)) # using int to round down
    return longest_string, max_count, min_count, median_count

def run_many_simulations(simulations: int, games: int, players: int) -> pd.DataFrame:
    longest_unbroken_strings: list = [] # for each simulation, what was the longest string of someone being the imposter?
    max_imposters: list = [] # for each sim, what was the most times anyone was imposter?
    min_imposters: list = [] # for each sim, what was the least times anyone was imposter?
    median_imposters: list = [] # for each sim, what was the median imposter count for all players?

    for i in range(simulations):
        longest, max_, min_, median_ = run_single_simulation(games=games, players=players)
        longest_unbroken_strings.append(longest)
        max_imposters.append(max_)
        min_imposters.append(min_)
        median_imposters.append(median_)

    # calculate the statistics now:
    longest_prob = []
    max_prob = []
    min_prob = []
    median_prob = []
    for g in range(games):
        longest_prob.append(1.0*longest_unbroken_strings.count(g)/simulations)
        max_prob.append(1.0*max_imposters.count(g)/simulations)
        min_prob.append(1.0*min_imposters.count(g)/simulations)
        median_prob.append(1.0*median_imposters.count(g)/simulations)

    # load data into dataframe. again, this will make plotting easier
    df = pd.DataFrame()
    df["LongestString"] = longest_prob
    df["LongestStringCumSum"] = df["LongestString"].cumsum()
    df["LongestStringPVal"] = 1 - df["LongestStringCumSum"] + df["LongestString"] #one sided p value - what is the probability of being equal to or greater than this value?
    df["Max"] = max_prob
    df["MaxCumSum"] = df["Max"].cumsum()
    df["MaxPVal"] = 1 - df["MaxCumSum"] + df["Max"]
    df["Min"] = min_prob
    df["MinCumSum"] = df["Min"].cumsum()
    df["MinPVal"] = 1 - df["MinCumSum"] + df["Min"]
    df["Median"] = median_prob
    df["MedianCumSum"] = df["Median"].cumsum()
    df["MedianPVal"] = 1 - df["MedianCumSum"] + df["Median"]
    df["Counts"] = list(df.index)

    return df

# let's test this out! we need pandas so we can visiualize in altair
import pandas as pd

simulations: int = 100000 # how many simulations to run
games: int = 20 # how many games are in each simulation
players: int = 7 # how many people are playing?

df_imposter_probabilities = run_many_simulations(simulations, games, players)
df_imposter_probabilities

	LongestString	LongestStringCumSum	LongestStringPVal	Max	MaxCumSum	MaxPVal	Min	MinCumSum	MinPVal	Median	MedianCumSum	MedianPVal	Counts
0	0.00000	0.00000	1.00000	0.00000	0.00000	1.000000e+00	0.29698	0.29698	1.00000	0.00000	0.00000	1.000000e+00	0
1	0.05412	0.05412	1.00000	0.00000	0.00000	1.000000e+00	0.57654	0.87352	0.70302	0.00417	0.00417	1.000000e+00	1
2	0.66672	0.72084	0.94588	0.00000	0.00000	1.000000e+00	0.12648	1.00000	0.12648	0.28474	0.28891	9.958300e-01	2
3	0.23723	0.95807	0.27916	0.00230	0.00230	1.000000e+00	0.00000	1.00000	0.00000	0.68307	0.97198	7.110900e-01	3
4	0.03628	0.99435	0.04193	0.20525	0.20755	9.977000e-01	0.00000	1.00000	0.00000	0.02801	0.99999	2.802000e-02	4
5	0.00490	0.99925	0.00565	0.42477	0.63232	7.924500e-01	0.00000	1.00000	0.00000	0.00001	1.00000	1.000000e-05	5
6	0.00062	0.99987	0.00075	0.24852	0.88084	3.676800e-01	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	6
7	0.00009	0.99996	0.00013	0.08885	0.96969	1.191600e-01	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	7
8	0.00004	1.00000	0.00004	0.02390	0.99359	3.031000e-02	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	8
9	0.00000	1.00000	0.00000	0.00513	0.99872	6.410000e-03	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	9
10	0.00000	1.00000	0.00000	0.00101	0.99973	1.280000e-03	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	10
11	0.00000	1.00000	0.00000	0.00025	0.99998	2.700000e-04	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	11
12	0.00000	1.00000	0.00000	0.00001	0.99999	2.000000e-05	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	12
13	0.00000	1.00000	0.00000	0.00001	1.00000	1.000000e-05	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	13
14	0.00000	1.00000	0.00000	0.00000	1.00000	2.220446e-16	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	14
15	0.00000	1.00000	0.00000	0.00000	1.00000	2.220446e-16	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	15
16	0.00000	1.00000	0.00000	0.00000	1.00000	2.220446e-16	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	16
17	0.00000	1.00000	0.00000	0.00000	1.00000	2.220446e-16	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	17
18	0.00000	1.00000	0.00000	0.00000	1.00000	2.220446e-16	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	18
19	0.00000	1.00000	0.00000	0.00000	1.00000	2.220446e-16	0.00000	1.00000	0.00000	0.00000	1.00000	1.110223e-16	19

The above dataframe looks right - the cum sums all end at 1 and the one-sided p values all go to 0. It will be easier to assess these results by plotting them.

# visualize!
import altair as alt

c_main = alt.Chart(df_imposter_probabilities)

c1 = c_main.mark_bar(size=15).encode(
    x=alt.X('Counts',title = "Consecutive games of any one person being the imposter"),
    y=alt.Y('LongestString',title = "Probability"),
    tooltip=['LongestString','LongestStringPVal']
).properties(
    title="Probability Distribution of Being the Imposter X Times in a Row, 7 players, 20 games",
)

c1.display()

The above plot is why I started this notebook. If you hover your mouse over the bar for 5 straight games, you will see the p-value of someone being the imposter 5 or more times is a smidge more than 0.5% (0.005). So, about 1 in 200. To me, this sounds like while it is unlikely that smeone gets to be imposter so many times in a row, it's not THAT crazy. Seeing someone be the imposter 4 or more times blasts that probability to 5%, which is really fairly common, and 3 times in a row is a colossal 29%. These events are really rather common.

An interesting side question might be "how many times do I need to play Among Us to see someone be the imposter 5 times in a row, given there is a 1/200 chance of it happening on any given gaming session (assuming you play 20 games per session, about 90 minutes worth). We can run a quick little side simulation to calculate this:

from random import random

def time_until_rare_event_occurs(chance: float) -> int:
    times: int = 0
    event_happened: bool = False
    while not event_happened:
        event_happened = random() < chance # only triggers if a random # from 0 to 1 is less than chance, a float from 0 to 1
        times += 1
    return times

def simulate_many_rare_events(simulations: int, chance: float) -> pd.DataFrame:
    times_when_event_triggered: list = []
    for s in range(simulations):
        time_until_event: int = time_until_rare_event_occurs(chance)
        times_when_event_triggered.append(time_until_event)
    max_time: int = max(times_when_event_triggered) # just get a reasonable max value
    counts_events: list = []
    for i in range(max_time):
        event_count = times_when_event_triggered.count(i)
        event_chance = 1.0*event_count / simulations
        counts_events.append(event_chance)
    df = pd.DataFrame(counts_events, columns = ["EventObservedThisTime"])
    df["EventObservedByThisTime"] = df.EventObservedThisTime.cumsum()
    df["Time"] = list(df.index)
    df["Time"] = df["Time"] + 1 #fix off by 1 error
    return df

sims: int = 100000
chance: float = 0.005

df_time_until_event = simulate_many_rare_events(sims, chance)

alt.Chart(df_time_until_event.iloc[:300]).mark_point(size=1).encode(
    x=alt.X('Time',title = "Time of rare event", scale = alt.Scale(domain=[0,300])),
    y=alt.Y('EventObservedByThisTime',title = "Cumulative Probability of Event"),
    tooltip=['EventObservedThisTime','EventObservedByThisTime','Time']
)

The above visual is showing us that, if you play A LOT of Among Us, you will certainly eventually see someone get to be the imposter 5 straight times. At 20 sessions worth of gaming (playing every other night for 2 months-ish), there is a 10% chance that you will see this event occur. So, again, while someone being the imposter 5 straight times is definitely a rare event, it's not a "get struck by lightning while ALSO being attacked by a shark" sort of thing.

alt.Chart(df_imposter_probabilities).mark_bar(size=15).encode(
    x=alt.X('Counts',title = "Maximum times someone was the imposter"),
    y=alt.Y('Max',title = "Probability"),
    tooltip=['Max','MaxPVal']
).properties(
    title="Probability Distribution of Max Times someone was the imposter, 7 players, 20 games",
)

Max times that someone in a party of 7 will get to be the imposter, over 20 games. This is another interesting plot, since almost always someone is claiming that so-and-so is ALWAYS the imposter. The p-calue for 7 or more times is about 0.12, so it is nearly a dice-roll's chance that someone will be the imposter 7 or more times. Again, this is a pretty likely event.

alt.Chart(df_imposter_probabilities).mark_bar(size=15).encode(
    x=alt.X('Counts',title = "Minimum times someone was the imposter"),
    y=alt.Y('Min',title = "Probability"),
    tooltip=['Min','MinPVal']
).properties(
    title="Probability distribution of least times someone was the imposter, 7 players, 20 games",
)

I think this is the most interesting finding here. With 7 players, there is a whopping 30% chance that someone NEVER gets to be the imposter!! And a 8% chance that someone only gets to be the imposter once or never! This seems to be a very common complaint from people, and I believe this is why many people accuse the game of not actually having random selection. Alas, probability is harsh and cruel.

Interestingly, the above visual is VERY sensitive to player count. If you lower the amount of players by 1, you have a much smaller chance of "screwing" someone out of being the imposter:

simulations: int = 100000 
games: int = 20 
players: int = 6 # reduced from 7

df_imposter_probabilities_6 = run_many_simulations(simulations, games, players)

alt.Chart(df_imposter_probabilities_6).mark_bar(size=15).encode(
    x=alt.X('Counts',title = "Minimum times someone was the imposter"),
    y=alt.Y('Min',title = "Probability"),
    tooltip=['Min','MinPVal']
).properties(
    title="Probability distribution of least times someone was the imposter, 6 players, 20 games",
)

Yes, just removing one player almost doubles the chance that everyone gets to be imposter at least once.

Moving on to the median statistics...

alt.Chart(df_imposter_probabilities).mark_bar(size=15).encode(
    x=alt.X('Counts',title = "Median times someone was the imposter"),
    y=alt.Y('Median',title = "Probability"),
    tooltip=['Median','MedianPVal']
).properties(
    title="Probability distribution of median times someone was the imposter, 7 players, 20 games",
)

This is not so suprising - 20 / 7 is almost 3, so you would expect 3 times-ish per person. One possible takeaway is that with the p-value of 3 being about 70%, more often than not half the players playing don't really get to be imposter that much. So, on average, you shuoldn't expect to be imposter all that often.

What's the big takeaway?

The biggest thing is that random chance does not mean "uniform opportunity." While it may be tempting to call imposter selection in Among Us complete BS sometimes, it's really not uncommon for the same people to be imposter over and over, or for someone to never get a chance.

Additionally, it is important to distiniguish between SOMEONE having a rare event and YOU having a rare event. The cool "Among Us Odds Calculator" can show you the chances of certain things occurring (like YOU being the imposter 5 out of 20 games), but this doesnt report on the likelihood of it happening to SOMEONE in that game. Again, it is HIGHLY likely that SOMEONE is going to have something strange happen, they talk about it, and then everyone's perception is that the game has a weird random selection algorithm.

You may notice that I never used any fancy statistical equations here - no poisson distributions, no t-tests, nothing. This was all inspired by "Stats for Hackers,", and I encourage everyone to check it out!

If you want to play with this code, you can get the source jupyter notebook here: amongus_stats.ipynb. You will need to pip install pandas for the dataframes, and altair to make the visuals.

Thanks for reading!

-PBG