2385: Final Exam
Title text: For those of you also taking Game Theory, your grade in that class will be based on how close your grade on this exam is to 80% of the average.
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In this comic, Ponytail appears to be administering a group sitting for cybersecurity exam. However, at the beginning of the exam, she informs her students that they have all failed, despite not having taken a test yet. She then informs them that their grades are stored on the department server and will be submitted the next day. The implication here is that the true test, rather than being a traditional exam, is actually whether the students can hack into the server and change their grades. This may be a jab at education security which is known to be vulnerable to assault (not the first time XKCD has made such a joke). In real life, students have attempted to change their grades in this manner, with occasional success.
The title text adds a twist to this. In order for a student to get a good grade in the game theory class, they need to get a below-average grade on this final exam. This incentivizes the students to also change the grades of other students when they change their grade. However, this is more complicated than it seems, and depends on various factors, such as the fraction of students who take game theory in addition to cybersecurity. If, for example, half of the students also take game theory, then for all of them to get 80% of the average score, even assuming that all their non-game-theory classmates get maximum possible score, they would have to target for 2/3 (or about 67%) of the maximum possible score, to get 80% of the final average. While that would make their game theory grade perfect, it might noticeably worsen their cybersecurity grade. This gets progressively worse with the increasing fraction of students who take game theory along with cybersecurity.
In the extreme case of all cybersecurity students also taking game theory class, this degenerates into another common game theory problem: Guess 2/3 of the average of everybody's guesses. The only winning strategy is, of course, for everyone to guess 0, which means that 2/3 of the average will be 0. This assumes perfect rationality of all players with respect to the game theory problem. The catch is that here we have the same number as a grade for the cybersecurity exam and for the game theory guess. We'd like one to be as high as possible, and the other to be zero or close to zero, which are obviously conflicting goals.
To improve their overall results, students could resort to various compromises and strategies, such as increasing other students' scores against their will, or making alliances with students who might not mind taking a hit to their game theory grade (perhaps in exchange for other incentives) - these are all topics that the game theory class would have been dealing with. Specifically, this test seems to refer to the prisoner's dilemma and tragedy of the commons; if one student changes their grade to 80% of the average, they will receive high marks, but if more and more students attempt this, the gain for each one drops and tends towards zero.
The combination with cybersecurity adds another layer of complexity, in that students could, for example, also attempt to lock each other out of the server to achieve maximum control over the results to their benefit.
In the strip, there is no actual test to take. But if there was one, there would still be strategies to optimize performance without hacking the grades. One option would be to take the test normally, and then change every fifth answer to the bubble below it; using this strategy your overall grade will drop to 80% if you were at 100%, and may even raise your score if a student performed particularly poorly. The trick, though, is that other students (assuming rationality) would try this strategy as well; thus, a student may need to overcorrect more, weigh the possibilities of whether any of their classmates had followed this as well, and perform this recursively until it is most likely that the score is 80 percent of the average.
Note: All of the above is based on the assumption that the game theory mark will be directly (and not inversely) proportional to how close the cybersecurity grade is to 80% of the average. This is left ambiguous in the formulation.
Note: The above also assumes the system accepts a maximum of 100%. If (as is likely) the system allows for extra credit you could reach a Nash equilibrium by setting the non-game theory students to an arbitrary, but very high, number (say 2000%) C and then the game theory students to (C*g)/(.25+g) where g is the percentage of students not in game theory.
Note: The solution becomes trivial if the game-theory grade is stored on the same server but submitted after the cybersecurity grade. Students would simply give themselves full marks on cybersecurity, then edit the game-theory grade after cybersecurity has been submitted.
- [Ponytail is standing in front of a whiteboard addressing someone off-panel to the left.]
- Ponytail: Welcome to your final exam.
- Ponytail: The exam is now over.
- Ponytail: I'm afraid all of you failed.
- Ponytail: Your grades have been stored on our department server and will be submitted tomorrow.
- Ponytail: Class dismissed.
- [Caption below the panel:]
- Cybersecurity final exams
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