Paper #2: a Game Theoretical Framework for Understanding Academic Cheating

Ryan Zhang

11/8/12

Math 89s GTD

Paper #2: A Game Theoretical Framework for Understanding Academic Cheating

This past fall, Harvard College was rocked by a cheating scandal that was, in the words of dean of undergraduate education Jay Harris, “unprecedented in its scope and magnitude.” A total of 125 students enrolled in the class Introduction to Congress were accused of working together on a take-home final exam. In the fallout from the scandal, dozens of students were forced to withdraw from the university for the upcoming academic year1.

Over the summer, the prestigious Stuyvesant High School suffered a cheating scandal that attracted widespread attention both regionally and nationally. The scandal centered on the actions of a ring of over 70 students who communicated answers to one another during the New York State Regents exams. As a result, several students were suspended and the administration experienced an upheaval after the sudden retirement of Principal Stanley Teitel2.

In the spring of 2007, 34 students at Duke University’s Fuqua School of Business were alleged to have cheated on a take-home final exam, marking one of the most large-scale violations of the honor code in university history. Nine of the students were expelled, while the remaining students were suspended and/or given failing grades3.

In each of these cases, the accused students cheated knowing that they would face severe consequences if they were caught, and indeed they did: Harvard students who previously boasted spotless academic records now have a transgression they must explain to any future employer or graduate school; Stuyvesant seniors whose sights had been set on top colleges around the country now face diminished prospects and potentially awkward interviews; Duke MBA students were, in some cases, forced to completely change their path in life after being expelled. In fact, it can be argued that the fallout from each of these scandals, though not crippling, certainly changed the lives of the students involved by a significant amount, probably negatively.

And yet, according to studies conducted within the last decade, between 60% and 86% of students admit to having cheated in college. This raises the question: why do so many people cheat despite the tangible (and intangible) risks involved?

It is logical to assume that, for students who do cheat, the potential reward (in other words, the “expected value of cheating”) outweighs the potential risk. Even in our daily lives, we are constantly weighing the two against one another, perhaps by making pros and cons lists, or maybe by consciously simulating future outcomes in our minds. It stands to reason, then, that determining the reasoning behind student cheating first requires us to identify both the rewards and risks attached.

In a 2009 study by Simkin and McLeod4, three motivational factors and three deterrent factors were identified through an analysis of previous literature. The motivational factors selected for study were “availability,” “getting ahead,” and “time demands” while the deterrent factors were “cultural,” “moral,” and “risk.” In that study, “getting ahead” was determined to be one of the most relevant factors influencing cheating, an unsurprising result considering that the tangible benefits of cheating often concern “getting ahead” in some way, whether by achieving a top grade or perhaps causing others to achieve lower grades. Also unsurprisingly, “moral” and “risk” factors were determined to be the most significant deterrents from cheating.

To identify which of the two deterrent factors is more important in deterring cheating, I created a preferential ballot survey and presented it to 18 people using the We Vote Here website. Rather than directly ask whether or not academic risk or moral values was more important to people in the issue of cheating, I phrased the question in such a way that also considered strength of opinion.

The following represents the text of the survey:

In which of the following scenarios would you be most likely to cheat? Rank the following in order of which you would be most likely to cheat in.

A)I would never cheat, regardless of circumstance

B)If the professor never finds out, but other students find out

C)If the professor finds out, but other students never find out

D)If there was a 50% chance of everyone finding out and a 50% chance of no one finding out

The ranked pairs ordering of the survey was A => B => D => C. Of the 18 people surveyed, 12 claimed that they would never cheat – that cheating, in fact, was strictly a last option. This surprised me given that the vote was conducted anonymously. The 33% of respondents acknowledging that they would ever even consider cheating was just half of what might be expected based on prior research.

The ordering of the remaining three choices was more interesting. Choice B finished second, suggesting that, of those who would cheat given the opportunity, people would rather have other students find out than have the professor find out; the obvious reason for this is that they place a greater emphasis on the tangible academic consequences than on any sort of individual moral code. Furthermore, the option that offered a 50% chance of no one finding out – despite also carrying a 50% risk that everyone, including the professor, finds out – was favored over the option in which the professor finds out while no one else finds out, an option that would minimize academic risk but hurt one’s moral reputation among peers. This seems to indicate that the professor’s opinion matters far more than anyone else’s opinion, and thus that academic risk may serve as a greater deterrent to cheating than infringement of moral values.

If we then accept the primary motivational factor to be “getting ahead” and the primary deterrent factors to be theconcrete “risk” factor, a scenario in which a student chooses to cheat must be one in which theprobability of being caught multiplied by some constant of detriment is less than some constant of reward.

In each of the real-world scandals listed at the beginning of the paper, the cheating was exposed when answers seemed too similar upon comparison. This makes sense. Professors have a limited amount of information, mostly based on comparisons of student tests with online sources. However, by comparing student test with student test, they can gain additional information that, in the cases above, served to confirm the occurrence of cheating.

This lends itself to the idea of a two-student model of cheating, where the actions of each and every student determines not only the final grades of each student, but also the probability that the professor catches on to any cheating. Two students would be used to simplify matters, as two would be the minimum number of students in a class for a professor to compare tests.

Based on the two-student model, we can construct a number of simplistic payoff matrices based on expected outcomes. In the following, it is assumed that the two students each have two possible behaviors: either cheat or don’t cheat. The outcomes represent hypothetical grades relative to each other (where each grade is based off of a class curve).

The first matrix represents a scenario in which the professor is unaware of cheating behavior.

Matrix A / Student2 Behavior
Cheat / Don’t cheat
Student 1
Behavior / Cheat / 4.0 / 4.0 / 4.0 / 2.0
Don’t cheat / 2.0 / 4.0 / 3.0 / 3.0

In Matrix A, each student has a clear dominant strategy: cheat. By cheating, both students ensure themselves 4.0s, while not cheating results in a 3.0 at best.

The second matrix represents a scenario in which the professor is aware of any and all cheating behavior.

Matrix B / Student 2 Behavior
Cheat / Don’t cheat
Student 1
Behavior / Cheat / 0.0 / 0.0 / 0.0 / 3.0
Don’t cheat / 3.0 / 0.0 / 3.0 / 3.0

In Matrix B, there is once again a clear dominant strategy: don’t cheat.

The third matrix does not display grades as outcomes, but rather the likelihood that cheating will go unnoticed by the professor. This is based on a polling result that suggested that 95% of cheaters do not get caught5. The probabilities that the students will not be caught decrease to some positive k if both students are cheating.

Matrix C / Student 2 Behavior
Cheat / Don’t cheat
Student 1
Behavior / Cheat / k%/ k% / 95% / 100%
Don’t cheat / 100% /95% / 100% / 100%

If we multiply the percentages in matrix C by the grades in matrix A we get the following “expected values” (or adjusted grades):

Matrix C / Student 2 Behavior
Cheat / Don’t cheat
Student 1
Behavior / Cheat / 4.0 * k% / 4.0 * k% / 3.8 / 2.0
Don’t cheat / 2.0 / 3.8 / 3.0 / 3.0

For not cheating to be the dominant strategy, the average of 4.0 * k% and 3.8 must be some number lower than the average of 2.0 and 3.0. When solved, k comes out to equal 30. This means that, for cheating to be a dominated strategy in a simple model as shown above, there must be a 70% chance that working together will cause the professor to catch the cheating – a fairly unlikely assumption. Other numbers may be substituted in at various points, but the gist of it is that cheating remains a viable option for most students despite recent publicized scandals, so long as the student can overlook the moral quandary that cheating produces.

The purpose of this paper thus far has been to create a framework for considering academic cheating from a game theoretical perspective. However, this framework applies very specifically to a single situation, in which multiple students are taking the same test. Cheating takes many different forms, and is often impossible to model simply. The complex forms that cheating can take might be attributable to the results of a Duke study that found that cheaters are more likely to be people who rate highly on measures of creativity.

Cheating is an issue that hits close to home with me (quite literally, as you’ll see in a moment), and I will certainly acknowledge that the risk/reward balance is almost invariably skewed toward the reward, no matter what parameters the cheating takes place within. I’ve been looking at situations where multiple students take one test – but how about a situation where one student took multiple tests?

A student from Long Island’s Great Neck North High School (the rival school of my own Great Neck South) made national headlines last year when it was discovered that he had taken the SAT for several classmates6. That student, Samuel Eshaghoff, was able to repeatedly cheat for other students simply by replacing the picture on the student ID. Because he had such little protection from being caught, the risk was significantly higher than the 5% of our earlier model. It should come as no surprise, then, that his expected reward had to be proportionally higher in order for him to expect a profit. Eshaghoff’s tests each came at an exorbitant price (usually over $2,500 per test), a number that was apparently enough to offset the dangerous risks associated with the cheating.

If one only considers the tangible risks involved, it may seem that cheating is always an appropriate solution. However, when one considers the intangible risks – the feelings of regret or guilt, the questioning of one’s own values, and the implications of cheating behavior on one’s interactions with wider society – the morality of cheating once again must be considered. But that is beyond my mathematical scope.

It is sufficient instead to conclude with an observation about Samuel Eshaghoff from an interview he gave shortly after the scandal broke. Eshaghoff, now a student at Emory University, claims that he would never have cheated if he could start over. However, the immorality of his cheating must have bothered him while he was doing it, because one statement he made seems to suggest a possible moral self-justification.Eshaghoff claimed at one point that his test-taking was “saving lives.” When prompted, he clarified:

“I mean a kid who has a horrible grade point average, who no matter how much he studies is gonna totally bomb this test, by giving him an amazing score, I totally give him like, a new lease on life. He's gonna go to a totally new college, he's gonna be bound for a totally new career and a totally new path in life.”

The interviewer promptly pointed out the students who worked harder for that path in life, and without cheating, to which Eshaghoff had no response. In many cases, cheaters can get away with their actions, earning higher GPAs on average, or perhaps getting a slightly higher starting salary than they would have otherwise. But when it comes to moral currency, cheaters never prosper.

Works Cited

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