# Nash Equilibrium of cheating a test($N$-player game)

Consider a classroom with $N$ students. All the students are taking a test. Each student has 2 strategies. They can either "cheat" or be "honest"(meaning they don't cheat). The payoffs are as follows

• If more than $k$ students ($k<N$) decide to cheat, then the invigilator will suspect something fishy going on, and will start checking each student, and in this process, all students who cheated will be caught and get a payoff of $-1$, while those who did not cheat will get a payoff of $1$.

• If $k$ or less students decide to cheat, then the invigilator will not suspect any malpractice and the students who cheated will get away with higher scores. This means that the students who cheated will get a payoff of $2$ while those who took the test honestly will get get $1$.

What is the Nash Equilibrium for this $N$-player game? And please mention any other interesting characteristics of this game, if any. Are there other examples which are similar to this problem?

There are ${N\choose k}$ pure Nash equilibria, corresponding to the different possible subsets of $k$ students that cheat. When $k$ (the maximum) students cheat, nobody wants to change their strategy; the cheaters and non-cheaters alike would do worse if they switched.