Among Us and Game Theory

A game-theoretic explanation for Among Us

Gamer or Not a gamer you’d know the incredibly popular mobile/PC game Among Us. The game has a sheer monopoly over the YouTube gamer community, at the time of writing this, 8 out of 10 trending videos in the gaming subsections of YouTube feature Among Us. The biggest creators on platforms like Twitch and YouTube have jumped on the Among Us bandwagon, playing with their fellow creators in turn skyrocketing the game’s popularity. Even mainstream creators like James Charles, KSI, and many more have sought to capitalize on this movement. One of the reasons for its popularity is that the ideology behind it is quite novel, a player has to use strategy more than skills, which doesn’t require long hours, so getting the grasp of the game and being quite decent at it wouldn’t take as much time as other competitive games such as CounterStrike or Fortnite. Most of the games that have survived the test of time are incredibly strategic and require the use of the mind more than tactile agility like Chess, Go, Uno, Poker, etc. Coincidently most of these games use Game Theory or can be explained by game-theoretic tools, the focus of this piece is the same, explaining the game Among Us using Game Theory. The article itself is divided into sections, the first, a brief overview of the game itself, second, a refresher on game-theory, and finally Among Us with Game Theory.

How Does Among Us Work?

This one is for the misanthropes of 2020, the game starts in a lobby of ideally 10 players, the players get divided into 2 categories randomly: crewmates and imposters. The population of crewmates is larger than the population of the imposters which range from 1 to 3 in a game. The host of the game gets the liberty of choosing the overall structure of the game (this is important, we’ll circle back to this) which includes tasks, player movement, kill distance, and much more. Anyways, just before the game starts you’ll get to know which group you belong to crewmate or imposter. Initially, all players start in the Cafeteria, which contains a red button for emergency meetings which can be triggered only a limited number of times in the entirety of the game. Now, for the Crewmates to win they have two options, first, to repair the ship or to figure out the imposters, as for the imposters they can win only by killing the entire crewmate team. This is the initial premise of the game, an assist for the imposter team is the “sabotage” feature, where they can sabotage different parts of the ship just to create distractions and adding more repairs for the crewmates team. Hence the team that achieves their goal first wins. The game itself has some tidbits that assist either of the teams, the players can view the locations of fellow players using the admin map but they wouldn’t know exactly who it is, an imposter can hide or travel to certain locations using vents.

As you can probably see that advantage to a group is balanced with either with a disadvantage to the same group or a similar advantage to the other group so as to make the competition fair. Also, the visuals of the game work on the principle of ray-casting, which restricts the vision of a player making it more realistic.

Now when an imposter kills a crewmate, a fellow crewmate can report the body to call for a vote where all the players collectively decide who the impostor is and eliminating the same, an imposter after killing also has a choice of reporting himself, the motive behind the same is deception as other players first tend to listen to the person who called in the vote and if the imposter is able to make a compelling argument here they are virtually safe from elimination. The voting part of the game itself can be considered as a game (subgame, this will come in the Game Theory part), this is the chance for the crewmate team to eliminate one of the alleged imposters, the decision of elimination is ideally discussed through discord or built-in message exchange, the discussion entails pointing fingers on suspects and those suspects trying to deflect or deny the accusations made. Once the vote is cast, the person who gets the most amount of votes as a suspect gets eliminated, and the next round starts and goes on until the whole imposter team has been eliminated or the crewmates have finished their task or the imposter team has killed off the entire crewmate team. This was the gist of the game, we will get into the nitty gritties in due time.

Basics of Game Theory

If you are a movie fanatic you might remember Game Theory from the Russel Crowe movie “A Beautiful Mind”, if you haven’t watched it I highly recommend it, great cinema. It’s a biography on mathematician John Nash one of the greatest minds of the 20th century, but he wasn’t the one at the birth of this field. It started with Plato when he described the Battle Of Delium in his piece Laches, the first mathematical formulation was done by John von Neuman, where non-mathematically speaking we can always find an equilibrium in a finite two-player game(I will define all these buzzwords, don’t worry) and John Nash who proved that equilibrium exists even in an n-player game. In this article, our focus will be on John Nash’s contributions as Among Us is a multiplayer game, we will also look into groups and game theory. I wouldn’t dive very deep into the mathematics of things as I want to keep this interesting for readers who have a high school mathematics background. And there is no way anyone can fit even the basics of Game Theory in an article, but I’ll try my best not to deviate from the topic and keep it short. So let’s start with some very basic definitions:

A Game

A game in a very “non-gamer” sense is a situation whose results depend on the actions of the parties involved.

Let’s break it down. The parties involved are called players, these are the entities that take actions that affect the state of the game. Now, why are players even interested in taking an action, well they’ll get a reward if they take an action that benefits himself/herself. This reward is defined by a utility function or payoff function, which is just a mapping of the actions of a player to a real number. So a finite n-person game is a tuple with (N, A,u) where N is the number of players, A is the action set(the set of all actions that will be taken by any player, you can also call it action space) and u which is the utility function. Mathematically this can be expressed as follows:

I’m borrowing this from one of my previous articles: here

Now that we have a naive understanding let’s look at the things that actually matter. We know that a player gets a reward for playing a good strategy, so the goal is to maximize the utility function by using the actions available in the action space, now if I know or have some intuition on what the other players will do, I can tailor my actions in order to get the highest utility, this is known as Best Response, which basically is the means that the reward I will get by playing the best response action is will be greater than any other possible set of actions. So, to understand this in a more mathematical sense,

Please don’t get intimidated by the symbols

If every player plays his/her best response, we achieve a Nash Equilibria. Another common phrase in Game Theory is a zero-sum game, which basically means that the benefit of one group corresponds to the loss of the other group. Now that we have some sense of the words of Game Theory let’s make sentences.

If we do a high-level glance on Among Us, there are groups in place of individuals: imposters and crewmates, but individuality exists between the crewmates, by this statement I mean that crewmates do not know any information about the other players hence for them the situation would count as an imperfect-information game. But for the imposter group, it is a perfect information game.

Let’s now look at multiplayer games, these games are just an extension of two-player games for example the prisoner’s dilemma is an example of a two-player game, in which the players either cooperate(snitch on the other player) or defect. Visually the matrix looks like this,

The numbers in the matrix represent the years each player has to spend in prison, if we look at Bob, if he decides to cooperate, he can either 3 years or zero years rather than 5 and 1 year if he decides to defect, hence he will choose to confess same goes for Alice as the matrix is symmetric, hence the Nash Equilibrium in this situation is for both the player to confess and the dilemma here is that both could’ve gotten a shorter sentence if they both defect, but that state would be more unstable that the confession state. This is a very small-scale example and there are only two players. Hence we need to generalize our definitions for n-players who are divided into two groups.

Before we do that there we need to investigate more into the concept of cooperators and defectors, especially in an n-player game. Also, we need to redefine Equilibria, as with the increase in the number of players it gets harder and harder to find a Nash Equilibria also games with infinitely many strategies might not even have an Equilibrium, and Among Us is one of those games where there are a lot of strategies both at an individualistic level and at the group level.

A Viable Nash Equilibrium

It’s quite trivial to state this but for most economic problems finding a viable equilibrium is more important than finding equilibria at all. But what does a viable equilibria mean?

The Equilibria should be focal amongst the players ie. with experience they know if they adopt the strategies that this equilibrium entails they might win.

Hence if a new equilibrium is introduced in the game, it should pass 2 tests, one the sustainability test, in which while adopting the new equilibrium strategy the players of both groups have equal chances to win, and second the formation test, equilibrium should have a realistic chance to be a focal strategy.

Here n is the total number of players, this means a player either deflects or follows an underlying strategy. This is very important for understanding why players might deflect from the herd mentality.

Game-theoretic explanation for Among Us

In order to avoid chaos in this section, I will divide this into 3parts, first will be about the two teams crewmates and impostors, the second will be about the voting subgame and finally, the third would be my very generalized closing arguments. Before we start, I’m just putting it out there, Among Us is a zero-sum game, and I encourage you to figure out why.

Crewmates and Impostors

Crewmates have the monopoly when it comes to population, in a vanilla game(10 players, 3 imposters, 7 crewmates), if you are a crewmate the probability of a random player being a fellow crewmate is 0.66 which is high but not quite. For now, we’ll ignore the individuality of the players and look at crewmates as a team, let’s first figure out the payoff matrix as we saw earlier in Prisoner’s dilemma.

Before conjuring a matrix, I would like you to give some pointers on how the matrix is logically sound:

1. The matrix has two teams crewmates(left) and impostors(top), the crewmates can either do a task(Tsk) or try to figure out who the impostor is (Imp) and the imposter team can either kill crewmates(Kill) or sabotage to create distractions or win(Sbt).
2. The payoff of the crewmate team on finding the impostor when the impostor team is out to sabotage is 2 and 5 when the impostor team is out to kill, the reason for this distribution is that it is easier to find someone red-handed when they aren’t creating distractions opposed to catching someone amidst the chaos.
3. The payoff for the crewmate on doing a task while the impostor team is just focussed on sabotaging is 6 and 2 when the impostor team is explicitly out to kill, again the reason is that completing a task while murderers are at loose is harder.
4. Similarly for the impostors, killing while the crewmate team is trying to complete tasks gives a payoff of 5 and a payoff of 3 when the crewmate team is out on the lookout for killers.
5. And finally, the payoff for sabotaging when crewmates are doing their tasks is 4 and 5 when they are out on the lookout, as when the crewmate team is already doing tasks, adding one more task won’t put them underpressure compared to sabotaging when they are out on the lookout.

Hence the matrix would finally look like,

The numbers are just arbitrary, the easier the job higher the payoff.

Now, let’s try to find the dominant strategy for the crewmate team, if the crewmate team does the tasks their payoff can either be 6 or 2, but if they investigate to find the impostors their payoff will be significantly higher, ie. 2 or 5 hence the dominant strategy for the crewmate team will be in completing their tasks. Similarly, let’s look at the impostor team if they just sabotage their payoff will either 4 or 5 but if they kill their payoff would be somewhat smaller ie. 5 and 3, hence the dominant strategy for impostors is to win by sabotaging. Making the top left column our Nash Equilibria. But if you’ve played the game how boring both of them are, yet it even holds good when some of the crewmate members have died, and we haven’t even looked at individualistic decisions yet. This is what causes the deviations in this game, individualistic benefit rather than group benefit. The individualistic goal being not to die for the crewmates and finding the impostor and killing for the impostors, both deviating from the Nash Equilibrium. The individualistic goal isn’t for winning but for keeping things “interesting”.

We can look at this problem from a different angle too if we assume that all players were to play a strategy π which yields the win for the crewmate team. The people who are destined to lose if they deviate from π would be the crewmate team, hence D(π) = 7 and from equation 1, we get F(π) = 3. Now in our game, we can assume the rate of decay for D(π) is greater than the rate of decay for F(π). For the sake of a simplistic discussion we’ll look into 2 cases:

1. The rate of decay for F(π) is 0, ie the number of impostors don’t change over time, this means that either impostors are good at killing or both impostors and crewmates are living in harmony(this is boring and extremely unstable) or impostors are good in putting the blame on someone. So as impostors start killing off the value of D(π) will depreciate which means the ability to sustain the π-equilibrium will fall, and the game will switch to let’s say π’, which is the winning strategy for the impostor team. Makes sense right if the impostors kill, the game will switch to their favor.
2. Now let’s say the rate of decay for D(π) is 0, ie. the number of crewmates don’t change, here the things get interesting, this might be because they are either very good at pointing out who the impostors are(reducing F(π)) or they move in groups making it hard for impostors to kill or even majority of members of the crewmate teams are skipping their votes leading to ties(we’ll come back to this). If crewmates are good at spotting the killers they are essentially depreciating the value of F(π), which would make the strategy π even stronger.
3. But realistically things aren’t as vanilla, we’d usually have a tradeoff between the two, and the one which is higher, in the end, might win. Also, this brings out the concept of a mixed strategy, where a group regularly changes their strategy in order to fool the opponent, making a probabilistic distribution, this means that in one round the impostors might be out to kill and sometimes they might focus on sabotaging. We will not indulge in this, it will only complicate things more.

The Voting Subgame

Voting is possibly the most important part of the game, the reason I say this after proving that for both groups are better off doing their in-game tasks(tasks for crewmates and sabotaging for impostors), is that this is the only way the crewmate team can have a speedy victory. For the crewmate team, this is the second hurdle, first being not to get killed by the impostor team, but for impostors, this is the only hurdle, this is the only way they can be eliminated so the stakes are high for both the teams. Voting itself has a very simple premise, a player can either vote to kick a player or skip their vote. Voting can be called only if someone has reported the body or pressed the emergency button. In order to make this discussion uniform, we’ll first consider voting when a body gets reported.

Let’s see a payoff matrix, this one is quite different from the ones we saw above but this is actually more clearer as it involves only 1 group at a time.

1. This is the payoff matrix for an individual of the crewmate team. So, in the first call to vote after someone’s death, the probability of a fellow player being an impostor is 0.375 (3/8), so if the crewmate, we’ll call him Bob, votes and the person was the impostor he and the whole team gets a payoff of 2, for successfully eliminating an impostor, but Bob votes and it turns out they eliminated crewmate, it’s going to cost them as they reduced their own population and an impostor managed to bypass a voting round. The same goes for skipping the vote. If we just look at probabilities Bob’s guess at finding the impostor would be right almost 40% of the time. But it’s not just a game of probabilities, players converse and everyone has a chance to explain who they think is the impostor.

2. The payoff matrix for the impostors is very simple, they already know who the crewmates and the impostors are, hence their only job here is to divert the players from attacking their team. Hence if they convince players to vote against someone who is not an impostor they get a payoff of 2, as they have successfully eliminated a member of the other team, but if the skip then it wouldn’t affect them at all.

3. I hope the above 2 matrices made sense, if we combine the two, the dominant strategy for the Impostor, would be to vote, though for the crewmate team the dominant strategy is to skip as they incur less of a loss. But no Nash Equilibria exists. So crewmates should skip voting but only till the point impostors can’t caste the majority vote.

Though I can get more into the voting part, I feel like this is more based on human interactions than game theory, we already know the dominant strategy for both the teams but, they would just yield an infinite game, here is where the individualism comes into play. The crewmates are not playing a team, so they are unaware of what other players are doing, this uncertainty brings out the selfish side, all the crewmates want is to survive and play the next round, which might make them gullible and prone to fictitious play. So my advice to the crewmate team would be to play your dominant strategy till you have substantial certainty.

Closing Statements

Some things that I skipped over but play an important role in the game,

1. The complexity of the crewmate task.
2. The Admin Map for the Crewmaes aid.
3. Vents for the impostors.
4. The significance of the emergency button.
5. The hierarchy of the sabotage moves.

One of the most interesting things, in the game settings, is the “Confirm Eject” feature, if this is checked at the beginning of the game, after every round of the game if someone is ejected, the players get to know whether that player was an impostor or a crewmate, now if this feature is enabled, an individual crewmate has some knowledge about the probabilities, but if this feature is disabled, the crewmates virtually have no knowledge and the probabilities mentioned would just be valid for the first round, and then will become stochastic themselves, this is way out of scope for this article.

Finally, if you like it or not these are the dominant strategies for the players. These were derived assuming that players behave as a group rather than individuals.

During Gameplay

1. Crewmate: Finish all tasks rather than trying to find the impostor.
2. Impostor: Winning by sabotage.

While Voting

1. Crewmate: Skipping Votes(till certainty).
2. Impostor: Voting for a random Crewmate(no skipping and definitely not voting for a fellow impostor)

References

1. https://econ.biu.ac.il/files/economics/seminars/viable_nash_oct_10_19.pdf
2. https://link.springer.com/chapter/10.1007/3-540-44683-4_3
3. Collaborative Games
4. https://journals.sagepub.com/doi/abs/10.1177/002224378502200303
5. https://journals.sagepub.com/doi/abs/10.1177/002224379603300301?journalCode=mrja
6. https://www.nobelprize.org/uploads/2018/06/nash-lecture.pdf
7. An online tool to find out Nash Equilibria in a 2x2 payoff matrix, here.