yunruse · February 24, 2026 17:46
diff --git a/analysis.ipynb b/analysis.ipynb
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    "# Fair team scoring in a 8:2 ratio game of _Among Us_\n",
    "\n",
    "## Abstract\n",
    "\n",
    "A dataset containing N=435 games of the social deduction game _Among Us_ were analysed. Due to the high amount of modded games, we filter to the modal \"standard case\" of 8 \"crewmates\" to 2 \"impostors\". In this subset (N=89), we find that impostors win about 55% of the time, and as such recommend that, if scored by team, a winning \"impostor\" team should receive 4 points and \"crewmate\" team 5 points.\n",
    "\n",
    "As players are randomly distributed in these teams to preserve hidden information, division of these team points amongst individual players is nontrivial and was not discussed.\n",
    "\n",
    "## Introduction\n",
    "\n",
    "_Among Us_ is a social deduction game in the vein of _Mafia_ (_Werewolf_). It is comprised of two roles:\n",
    "\n",
    "- the crewmates, who wish to survive, complete tasks, and vote out impostors,\n",
    "- the impostors, who are able to kill crewmates, and wish to confuse the crewmates\n",
    "\n",
    "Typically the ratio is such that there are only two of the latter.\n",
    "\n",
    "## Data\n",
    "\n",
    "A source [has laboriously documented 435 games of Among Us](https://moresidemen.vercel.app) as played by a YouTuber group.\n",
    "\n",
    "Taking the Google sheet document and stripping out irrelevant columns, we have:"
   ]
  },
  {
   "cell_type": "code",
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    {
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       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Winner</th>\n",
       "      <th>Players, Roles and Tasks</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Crewmate</td>\n",
       "      <td>Simon - Crewmate,Josh - Crewmate,Vik - Imposte...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Imposter</td>\n",
       "      <td>Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Imposter</td>\n",
       "      <td>Simon - Crewmate,Josh - Crewmate,Vik - Imposte...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Crewmate</td>\n",
       "      <td>Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Imposter</td>\n",
       "      <td>Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>430</th>\n",
       "      <td>Crewmate</td>\n",
       "      <td>Simon - Operative Sheriff - (7/8),Josh - Grena...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>431</th>\n",
       "      <td>Imposter</td>\n",
       "      <td>Simon - Doctor - (2/8),Josh - Engineer - (3/8)...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>432</th>\n",
       "      <td>Imposter</td>\n",
       "      <td>Simon - Swapper - (3/8),Josh - Dumper,Vik - Go...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>433</th>\n",
       "      <td>Imposter</td>\n",
       "      <td>Simon - Operative Crewmate - (6/8),Josh - Doct...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>434</th>\n",
       "      <td>Imposter</td>\n",
       "      <td>Simon - Grenadier,Josh - Engineer - (0/8),Vik ...</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>435 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "       Winner                           Players, Roles and Tasks\n",
       "0    Crewmate  Simon - Crewmate,Josh - Crewmate,Vik - Imposte...\n",
       "1    Imposter  Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...\n",
       "2    Imposter  Simon - Crewmate,Josh - Crewmate,Vik - Imposte...\n",
       "3    Crewmate  Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...\n",
       "4    Imposter  Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...\n",
       "..        ...                                                ...\n",
       "430  Crewmate  Simon - Operative Sheriff - (7/8),Josh - Grena...\n",
       "431  Imposter  Simon - Doctor - (2/8),Josh - Engineer - (3/8)...\n",
       "432  Imposter  Simon - Swapper - (3/8),Josh - Dumper,Vik - Go...\n",
       "433  Imposter  Simon - Operative Crewmate - (6/8),Josh - Doct...\n",
       "434  Imposter  Simon - Grenadier,Josh - Engineer - (0/8),Vik ...\n",
       "\n",
       "[435 rows x 2 columns]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "games = pd.read_csv('games.csv')\n",
    "games"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2cab0aa8",
   "metadata": {},
   "source": [
    "As can be seen, roles exist that are not present in classical (unmodded) versions of the game. In addition, a careful analysis (by use of Python's `Counter`) shows that a significant amount of games have a standard layout:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "dedc2e00",
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    {
     "data": {
      "text/plain": [
       "89"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from typing import Counter\n",
    "\n",
    "standard_game = Counter({'Crewmate': 8, 'Imposter': 2})\n",
    "\n",
    "def is_standard_game(game: pd.Series):\n",
    "    roles = [player.split(' - ')[1] for player in game['Players, Roles and Tasks'].split(',')]\n",
    "    roles = Counter(roles)\n",
    "    return roles == standard_game\n",
    "\n",
    "\n",
    "standard_games = games[games.apply(is_standard_game, axis=1)]\n",
    "len(standard_games)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bae461d1",
   "metadata": {},
   "source": [
    "Of the 89 unmodded \"standard games\", we can trivially see that:"
   ]
  },
  {
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   "execution_count": 38,
   "id": "eb4c797e",
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    {
     "data": {
      "text/plain": [
       "Counter({'Imposter': 49, 'Crewmate': 40})"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "winners = Counter(standard_games.Winner)\n",
    "winners"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a67c45f8",
   "metadata": {},
   "source": [
    "Not accounting for skill of the player, we desire to score roles such that the expected value of the score for the role (as a team) is equal among roles.\n",
    "\n",
    "For a two-team game, the numerics cancel out, and a _fair score ratio_ is inversely proportional to the _win ratio_.\n",
    "\n",
    "As such, if impostors have a 49/40 ratio of wins, they should have 40/49 of the score crewmates have.\n",
    "\n",
    "Expressed as a simpler fraction, we might say that impostors gain 4 points for a win, whereas crewmate team should gain 5 points for a win.\n",
    "\n",
    "## Discussion\n",
    "\n",
    "435 games of _Among Us_ were analysed, a significant proportion modded, and the modal game, a standard unmodded game, was found to have impostors win 49/89 (55%) of the time.\n",
    "\n",
    "For fairness among teams, we thus propose that a crewmate win should give 5 points, whereas an imposter win should give 4 points.\n",
    "\n",
    "We note the limitation that the above applies to points given to _teams_, not players. As the standard game had 8 crewmates and 2 impostors, we propose that, if scored per player, an impostor should receive 2 points for every 0.625 player points. Renormalised, this is 1 point for all winning crewmates and 3.2 points for all winning impostors.\n",
    "\n",
    "## Limitations\n",
    "\n",
    "The above study very briefly explores a standard game, which has no additional roles. As such, it does not explore the game probability in the case of a game that has more or less of a given role, and should be taken only for this case.\n",
    "\n",
    "The recommendations of scoring apply only to a team. While this score is trivially divisible amongst players, the structure of the game is such that each player might be removed (by 'murder' or by social eviction). As this is a more complex dynamic, a scoring system in respect of this model - such that longer-surviving players gain a higher score - was not explored.\n",
    "\n",
    "As players are randomly distributed in these teams to preserve hidden information, division of these team points amongst individual players is nonetheless somewhat important for competition purposes. Therefore the naïve equal player distribution lacks nuance that may sway a competition comprising a limited amount of games."
   ]
  }
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diff --git a/games.csv b/games.csv
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	"cell_type": "markdown",
	"id": "ee1ee44b",
	"metadata": {},
	"source": [
	"# Fair team scoring in a 8:2 ratio game of _Among Us_\n",
	"\n",
	"## Abstract\n",
	"\n",
	"A dataset containing N=435 games of the social deduction game _Among Us_ were analysed. Due to the high amount of modded games, we filter to the modal \"standard case\" of 8 \"crewmates\" to 2 \"impostors\". In this subset (N=89), we find that impostors win about 55% of the time, and as such recommend that, if scored by team, a winning \"impostor\" team should receive 4 points and \"crewmate\" team 5 points.\n",
	"\n",
	"As players are randomly distributed in these teams to preserve hidden information, division of these team points amongst individual players is nontrivial and was not discussed.\n",
	"\n",
	"## Introduction\n",
	"\n",
	"_Among Us_ is a social deduction game in the vein of _Mafia_ (_Werewolf_). It is comprised of two roles:\n",
	"\n",
	"- the crewmates, who wish to survive, complete tasks, and vote out impostors,\n",
	"- the impostors, who are able to kill crewmates, and wish to confuse the crewmates\n",
	"\n",
	"Typically the ratio is such that there are only two of the latter.\n",
	"\n",
	"## Data\n",
	"\n",
	"A source [has laboriously documented 435 games of Among Us](https://moresidemen.vercel.app) as played by a YouTuber group.\n",
	"\n",
	"Taking the Google sheet document and stripping out irrelevant columns, we have:"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 22,
	"id": "8776803b",
	"metadata": {},
	"outputs": [
	{
	"data": {
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	"<div>\n",
	"<style scoped>\n",
	" .dataframe tbody tr th:only-of-type {\n",
	" vertical-align: middle;\n",
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	" .dataframe tbody tr th {\n",
	" vertical-align: top;\n",
	" }\n",
	"\n",
	" .dataframe thead th {\n",
	" text-align: right;\n",
	" }\n",
	"</style>\n",
	"<table border=\"1\" class=\"dataframe\">\n",
	" <thead>\n",
	" <tr style=\"text-align: right;\">\n",
	" <th></th>\n",
	" <th>Winner</th>\n",
	" <th>Players, Roles and Tasks</th>\n",
	" </tr>\n",
	" </thead>\n",
	" <tbody>\n",
	" <tr>\n",
	" <th>0</th>\n",
	" <td>Crewmate</td>\n",
	" <td>Simon - Crewmate,Josh - Crewmate,Vik - Imposte...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>1</th>\n",
	" <td>Imposter</td>\n",
	" <td>Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>2</th>\n",
	" <td>Imposter</td>\n",
	" <td>Simon - Crewmate,Josh - Crewmate,Vik - Imposte...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>3</th>\n",
	" <td>Crewmate</td>\n",
	" <td>Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>4</th>\n",
	" <td>Imposter</td>\n",
	" <td>Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>...</th>\n",
	" <td>...</td>\n",
	" <td>...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>430</th>\n",
	" <td>Crewmate</td>\n",
	" <td>Simon - Operative Sheriff - (7/8),Josh - Grena...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>431</th>\n",
	" <td>Imposter</td>\n",
	" <td>Simon - Doctor - (2/8),Josh - Engineer - (3/8)...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>432</th>\n",
	" <td>Imposter</td>\n",
	" <td>Simon - Swapper - (3/8),Josh - Dumper,Vik - Go...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>433</th>\n",
	" <td>Imposter</td>\n",
	" <td>Simon - Operative Crewmate - (6/8),Josh - Doct...</td>\n",
	" </tr>\n",
	" <tr>\n",
	" <th>434</th>\n",
	" <td>Imposter</td>\n",
	" <td>Simon - Grenadier,Josh - Engineer - (0/8),Vik ...</td>\n",
	" </tr>\n",
	" </tbody>\n",
	"</table>\n",
	"<p>435 rows × 2 columns</p>\n",
	"</div>"
	],
	"text/plain": [
	" Winner Players, Roles and Tasks\n",
	"0 Crewmate Simon - Crewmate,Josh - Crewmate,Vik - Imposte...\n",
	"1 Imposter Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...\n",
	"2 Imposter Simon - Crewmate,Josh - Crewmate,Vik - Imposte...\n",
	"3 Crewmate Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...\n",
	"4 Imposter Simon - Crewmate,Josh - Crewmate,Vik - Crewmat...\n",
	".. ... ...\n",
	"430 Crewmate Simon - Operative Sheriff - (7/8),Josh - Grena...\n",
	"431 Imposter Simon - Doctor - (2/8),Josh - Engineer - (3/8)...\n",
	"432 Imposter Simon - Swapper - (3/8),Josh - Dumper,Vik - Go...\n",
	"433 Imposter Simon - Operative Crewmate - (6/8),Josh - Doct...\n",
	"434 Imposter Simon - Grenadier,Josh - Engineer - (0/8),Vik ...\n",
	"\n",
	"[435 rows x 2 columns]"
	]
	},
	"execution_count": 22,
	"metadata": {},
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	}
	],
	"source": [
	"import pandas as pd\n",
	"\n",
	"games = pd.read_csv('games.csv')\n",
	"games"
	]
	},
	{
	"cell_type": "markdown",
	"id": "2cab0aa8",
	"metadata": {},
	"source": [
	"As can be seen, roles exist that are not present in classical (unmodded) versions of the game. In addition, a careful analysis (by use of Python's `Counter`) shows that a significant amount of games have a standard layout:"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 34,
	"id": "dedc2e00",
	"metadata": {},
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	{
	"data": {
	"text/plain": [
	"89"
	]
	},
	"execution_count": 34,
	"metadata": {},
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	}
	],
	"source": [
	"from typing import Counter\n",
	"\n",
	"standard_game = Counter({'Crewmate': 8, 'Imposter': 2})\n",
	"\n",
	"def is_standard_game(game: pd.Series):\n",
	" roles = [player.split(' - ')[1] for player in game['Players, Roles and Tasks'].split(',')]\n",
	" roles = Counter(roles)\n",
	" return roles == standard_game\n",
	"\n",
	"\n",
	"standard_games = games[games.apply(is_standard_game, axis=1)]\n",
	"len(standard_games)"
	]
	},
	{
	"cell_type": "markdown",
	"id": "bae461d1",
	"metadata": {},
	"source": [
	"Of the 89 unmodded \"standard games\", we can trivially see that:"
	]
	},
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	"cell_type": "code",
	"execution_count": 38,
	"id": "eb4c797e",
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	"Counter({'Imposter': 49, 'Crewmate': 40})"
	]
	},
	"execution_count": 38,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"winners = Counter(standard_games.Winner)\n",
	"winners"
	]
	},
	{
	"cell_type": "markdown",
	"id": "a67c45f8",
	"metadata": {},
	"source": [
	"Not accounting for skill of the player, we desire to score roles such that the expected value of the score for the role (as a team) is equal among roles.\n",
	"\n",
	"For a two-team game, the numerics cancel out, and a _fair score ratio_ is inversely proportional to the _win ratio_.\n",
	"\n",
	"As such, if impostors have a 49/40 ratio of wins, they should have 40/49 of the score crewmates have.\n",
	"\n",
	"Expressed as a simpler fraction, we might say that impostors gain 4 points for a win, whereas crewmate team should gain 5 points for a win.\n",
	"\n",
	"## Discussion\n",
	"\n",
	"435 games of _Among Us_ were analysed, a significant proportion modded, and the modal game, a standard unmodded game, was found to have impostors win 49/89 (55%) of the time.\n",
	"\n",
	"For fairness among teams, we thus propose that a crewmate win should give 5 points, whereas an imposter win should give 4 points.\n",
	"\n",
	"We note the limitation that the above applies to points given to _teams_, not players. As the standard game had 8 crewmates and 2 impostors, we propose that, if scored per player, an impostor should receive 2 points for every 0.625 player points. Renormalised, this is 1 point for all winning crewmates and 3.2 points for all winning impostors.\n",
	"\n",
	"## Limitations\n",
	"\n",
	"The above study very briefly explores a standard game, which has no additional roles. As such, it does not explore the game probability in the case of a game that has more or less of a given role, and should be taken only for this case.\n",
	"\n",
	"The recommendations of scoring apply only to a team. While this score is trivially divisible amongst players, the structure of the game is such that each player might be removed (by 'murder' or by social eviction). As this is a more complex dynamic, a scoring system in respect of this model - such that longer-surviving players gain a higher score - was not explored.\n",
	"\n",
	"As players are randomly distributed in these teams to preserve hidden information, division of these team points amongst individual players is nonetheless somewhat important for competition purposes. Therefore the naïve equal player distribution lacks nuance that may sway a competition comprising a limited amount of games."
	]
	}
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