Game Theory in Among Us
Introduction:
When we started talking about game theory, my first though was how do i weaponize this knowledge to absolutely demolish my friends at social deception games. The classic simple social deception game is mafia/werewolf. This form of game recently got popularized by Among Us, a game by Innersloth that takes the core concepts of mafia/werewolf and expands on it.
For the few people that have no idea how Among Us is played, here is a simple rundown on the game play. In Among Us, you are either a crewmate (C) or an imposter (I). Crewmates want to complete their tasks, vote out the imposters, and generally survive. Imposters are aiming to kill all the cremates while remaining undetected. Imposters have special abilities that can sabotage crewmates tasks. The game is essentially split into the voting aspect and the exploration aspect. While voting, all the players discuss who they think the imposter(s) are and get to vote 1 player off the space ship. When not voting, the players explore a map where the crewmates can do their tasks and the imposters can kill.
So, as the game play is split into 2 parts, we will analyze each part individually. First we will explore the exploration aspect. We will reduce this aspect into a mathematical game then perform Nash equilibrium analysis on it. Afterwards, we will do the same to the voting aspect.
Exploration phase:
The crewmates have 2 options, do their tasks or search for clues. Each option pertains to one of their win conditions. The first win condition, finish all the tasks, is usually the better option as finding who the imposters are is harder. So we can assign arbitrary point values to each action. How ever, the imposters also have 2 actions. They can either sabotage, or they can kill. Killing would be less effective when crew mates are searching for clues as that would give them more info. Sabotaging is generally a good idea but better when the crewmates are looking for clues as it would disrupt them more.
We can use all the above relations to model a game. I have assigned the points as such:
Finding the Nash equilibrium in this game, I found that the Crew mates should do their task 2/3’s of time and look for clues 1/3 of the time. The imposters should evenly kill and sabotage (ie 50/50). This somewhat lines up with the gameplay i have witnessed. Crewmates tend to do their tasks more often then just looking for info. Imposters seem to kill and sabotage about evenly, depending on ones style.
Voting phase:
This phase is harder to turn into a game as it is not very adversarial. The 2 options for each player is Vote or skip. The Imposters, because they know who their fellow imposters are, should always vote for a random crewmate. That is a very easy thing to figure out. For the imposters, skipping achieves nothing, and voting for a fellow imposter is a huge negative.
For crewmates, it gets a heck of a lot more complicated. For simplicity’s sake, lets assume that everyone has decided to only vote/skip on one person. There are 4 possible outcomes now: the crewmate votes and the person was an imposter, the crewmate skips and the person was an imposter, the crewmate votes and the person was not an imposter, and the crewmate skips and the person was not an imposter. Lets call them A,B.C, and D respectively.
A:
This is the best possible outcome. The crewmates get rid of one possible imposter and get many steps closer to their victory. lets assign this 3 points.
B:
This isn’t the worst thing to happen, however, its not a good thing either. The crewmate has potentially squandered their ability to remove and imposter. lets give this -2 points
C:
This is the worst possible outcome. The crewmates have killed one of their own. This advances the imposters win condition. Lets give this -6 points
D:
This is a neutral outcome. Nothing has happened. Lets just give it 0 points.
To make things even simpler and assume the cremates choose an imposter 50% of the time. This means the avg payoff for choosing to vote (options A and C) is .5(3) + .5(-6)=-1.5. The avg payoff for choosing to vote (options A and C) is .5(0) + .5(-2)=-1. So its better for the crewmates to always skip.
Conclusion
Well, this was a hyper simplified form of Among Us. This failed to take into account the social aspect of the game, or the fact that Crewmates have access to other information. In reality, games are way more complicated to model. I still found it interesting that the mixed strategy found for the exploration phase was more accurate then i thought. However the strategy found in the voting analysis was super inaccurate. Partly due to the fact that we simplified the game, but also we forgot to factor in the point of the game. Games (not in a mathematical sense) serve to entertain. Single mindedly employing a boring but effective strategy doesn’t really happen. People tend to go for more fun and chaotic strategies even if its sub-optimal
Sources:
Kaustubh. “Among Us and Game Theory.” Medium, Medium, 25 Oct. 2020, https://kaustubh-q.medium.com/among-us-and-game-theory-f74c8ac9f05.