Game Theory: Two friends who are students of an Econ class got hold of the
answer key to the graded homework problems. Suppose that over the quarter, the Professor poses two graded homework assignments. Since these homework assignments are very difficult, the friends are tempted to cheat and submit the answers from the answer key to the first homework assignment. Each of them believes that with probability µ professor checks. If he checks and a student cheats, then he will figure out that the student cheats. Moreover, the Professor checks the first homework if and only if he checks the second one. To each friend, the payoff to being caught cheating is -c with c > 0. The payoff for doing the first homework themselves and submitting their own answer is 0. The payoff from cheating successfully without being caught cheating is 1. If a student is caught cheating on the first homework, then both friends know that the Professor checks. Hence both know that if they cheat on the second homework, they would surely be caught cheating and thus do not cheat on the second homework.
If at least one of them cheats on the first homework and is not caught, then both learn that the professor does not check. If neither cheats on the first day, each continues to believe that with probability µ Professor will check.
Hence, in this case, a student cheats on the second homework if -µc + 1 - µ > 0 and does not cheat if -µc + 1 - µ < 0, thus receiving an expected payoff
of max {-µc + 1 - µ, 0}.
Consider this situation as a strategic game in which both students decide whether or
not to cheat on the first homework assignment. For example, if a student cheats on the first homework assignment, then the student is caught with probability µ, in which case the student does not cheat on the second homework assignment. If a student cheats on the first homework assignment, then this student is not caught cheating with probability 1 - µ, in which case the student cheats on the second homework assignment as well.
Thus, the expected payoff to the student if he cheats on the first homework assignment is µ(-c + 0) + (1 - µ)(1 + 1) = -µc + 2(1 - µ), independent of the other students's action. Find the mixed Nash equilibrium of the game (depending on µ and c). Does the existence of the friend makes it more likely that the students decides to cheat on the first homework assignment.