While grading a final exam, an economics professor discovers that two students h
ID: 1177604 • Letter: W
Question
While grading a final exam, an economics professor discovers that two students have virtually identical answers. She is convinced the two cheated but cannot prove it. The professor speaks with each student separately and offers the following deal:
Sign a statement admitting to cheating. If both students sign the statement, each will receive an %u201CF%u201D for the course. If only one signs, he is allowed to withdraw from the course while the other student is expelled. If neither signs, both receive a %u201CC%u201D because the professor does not have sufficient evidence to prove cheating.
a. Draw the payoff matrix.
b. Which outcome do you expect? Why?
Explanation / Answer
The expected outcome depends on what value you assign to each outcome. Suppose a C is worth 2 points and an F is worth 0 points. Being expelled is probably worse than getting an F, so let it be worth -1 point for now. If withdrawing from the course is better than getting a C, you have the classic "prisoner's dilemma". Let a W be worth 3 points. The choices of students A and B are:
A/B:
[ clam/clam] [ clam/ rat ]
[ rat /clam ] [ rat / rat ]
with payoffs of
A/B:
[ 2 / 2 ] [ -1 / 3 ]
[ 3 / -1 ] [ 0 / 0 ]
Consider A's options: If B confesses ("rat"), A is better off confessing ("rat"; 0 points) than keeping quiet ("clam"; -1 point). If B keeps quiet, A is still better off confessing (3 points) than keeping quiet (2 points). So, A confesses. The payoff matrix is symmetric, so B will reach the same conclusion, and both will confess. Notice that you get the same result as long as being expelled is even a teensy bit worse than getting an F (say, -0.0001), and dropping the course is only a teensy bit better than getting a C (say, 2.0001).
If getting a C is better than dropping the course, though, the students will choose differently. Let withdrawal be worth only 1 point now, and expulsion worth -1 point as before. Now the payoffs are:
A/B:
[ 2 / 2 ] [ -1 / 1 ]
[ 1 / -1 ] [ 0 / 0 ]
If student B confesses, then student A is again better off confessing (0 points) than not (-1 point). However, if student B doesn't confess, then student A is better off keeping quiet as well (2 points). A's choice (and of course B's as well) will depend on the expected value of each choice. If A figures that B will probably (60%) keep quiet, then the expected payoff if s/he confesses is
E(rat) = 1 x 60% + 0 x (1 - 60%) = 0.6
and if s/he doesn't is
E(clam) = 2 x 60% + -1 x (1 - 60%) = 0.8
so s/he plays the odds and clams up. (Of course, both A and B were cheating on an economics exam, so it's not certain that both of them will be able to compute an expected value, or if they can, that they will both get the same answer if they do.)
In the stable Nash equilibrium the exact size of the payoff didn't matter. With an unstable equilibrium and one chance to play, it matters a great deal. If being expelled is horrible, say -2 points horrible, then the expected value of clamming up drops, and A will elect to confess.
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