Introduction
Among Us is one of the most popular games, especially during the entire pandemic. The game takes place on a spaceship where each player is told at the beginning if they are the imposter or not. There are more cremates than imposters. The imposter’s job is to kill all the cremates. The crewmates job is to complete given tasks and vote out any imposters, they achieve this by holding a meeting where they can discuss what they saw and who they think is the imposter. The imposters can choose to perform sabotages which include, damaging repaired tasks, and closing and opening doors. The objective of the game for the crewmates is to complete all the tasks before importers kill everyone, or vote out all imposters. The imposter’s objective is to kill all crewmates.
Analysis
Crewmates have the majority in a game of Among Us. For example in a game of 10 players, there are 3 imposters and 7 crewmates. So if one is a crewmate then the probability that any random player is a crewmate is 66%. We can formulate this into a payoff matrix where we have 2 players, the crewmate, and the imposter. Crewmates can either do tasks (Tsk) or figure out who the imposters are (Imp). The imposters can either kill crewmates, or create distractions by using sabotages (Sbt). The payoff of the crewmate team for finding the imposter is 2 when the imposters are out to sabotage, compared to 5 when the imposters are out to kill. This makes sense because it is much easier to get caught in the act of killing in the game, compared to being caught sabotaging. The payoff for a crewmate on doing a task when the imposters are out to kill is 2, and when the imposter team is out to sabotage is 6. Completing a task is harder when imposters are directly planning on killing since when doing a task you cannot see what’s going on around you. The payoff for the imposter team for killing while the crewmate team is trying to complete tasks is 5 and 3 when the crewmates are looking out for the imposters. The payoff for sabotaging when crewmates are doing their tasks is 4, and when they are on the lookout payoff is 5.
The payoff matrix looks as follows:
Now we can try to find a Nash Equilibrium. If crewmates are choosing the strategy of doing tasks the imposters will choose the strategy of kill. If crewmates are choosing a strategy of figuring out who the imposters are, then the imposters will choose the strategy of sabotaging. If imposters are choosing the strategy of sabotaging, then the crewmates will choose the strategy of doing tasks. If the imposters are choosing the strategy of killing, then the crewmates will choose the strategy of figuring out who the imposters are. We can see that no pair of strategies are the best responses to each other, so no pure-strategy Nash Equilibrium exists for this game. There is however a mixed strategy Nash Equilibrium.
Crewmates choose probability p for strategy tasks, making probability for strategy of figuring out who the imposters are, 1-p. The expected payoff for imposters using strategy sabotage is:
The expected payoff for imposters using strategy kill is:
Imposters are indifferent between it’s strategies when:
Imposters choose probability q for strategy sabotage, making probability for strategy kill 1-q
The expected payoff for crewmates using strategy tasks is:
The expected payoff for crewmates using the strategy of figuring out who the imposters are is:
Crewmates are indifferent between it’s strategies when:
Mixed Nash:
So, optimally the crewmates would choose strategy tasks with probability of 67%, and choose strategy of figuring out who the imposters are with probability 33%. The imposters will choose strategy sabotage with probability of 57%, and choose strategy of kill with probability of 43%.
Crewmates are more likely to choose doing tasks, and imposters are more likely to choose killing. Giving imposters generally higher payoffs, if both crewmates and imposters play rationally.
Conclusion
Using game theory we can see that Among Us can be displayed using a payoff matrix, and using tools we learned in class can analyze this game to see that there are no clear dominant strategies for either crew members or imposters. However, when playing as crewmate always slightly prioritize doing tasks and when imposter always prioritize killing is the optimal way to play this game. This has given me insight for the next time I decide to play Among Us, and I hope it also gave everyone reading insight into this as well!
Resources
Kaustubh. (2020, October 25). Among us and game theory. Medium. Retrieved November 18, 2022, from https://kaustubh-q.medium.com/among-us-and-game-theory-f74c8ac9f05