Equilibria may also change when we relax the assumption that only the cheater student is punished and when we allow the "cheating technology" to be imperfect, such that the grade of student who copied is lower than that of the one who has the exam copied. In the former the possibility of an equilibrium in which both students cheat arises, while the latter increases the chance of a virtuous one. Yet, when we consider the case in which the utility of the student who has the exam copied is affected by the cheating behavior, and he has the option of avoiding that his classmate copies his exam, Nash equilibrium in pure strategies may no exist.
We also consider the case when there exists a further punishment for the dishonest student, that is, if the student is caught cheating, he is punished by losing a constant level of utility-due to failing grade in the course, suspension or expulsion, for example-in addition to zero grade in the exam. Under complete information, we find that a harder punishment decreases the minimum probability required to students choose not to cheat.
Types of cheats
This results resembles those of the classical analysis of Economics of Crime Becker, , in which the probability of being caught and the magnitude of the punishment drive the incentives of potential offenders. A more interesting extension is one that relaxes the assumption of complete information and allows the lenient professor to send a signal to students through a commitment that he will apply such a harder punishment in case of catching cheaters.
A final extension allows students to be heterogeneous. They can be heterogeneous in terms of disutility of effort, for instance. This implies they choose different levels of effort and thus receive different grades. Different grades in turn implies different potential benefits for the cheater: the student with the lower grade has more to gain by cheating than the one with higher grade. This can be seen through the fact that the minimum probability that induces the student to play fair is decreasing in his own grade and increasing in the grade of the another student. Once again our model shows a feature of Economics of Crime, namely the higher the potential benefit of the offense, the more prone to commit it the individual is.
In fact, the difference in terms of grades may be so large that playing not to cheat may be a dominant strategy for the student with higher grade regardless the probability of being caught cheating. Although our framework presents similarities with the seminal model proposed by Gary Becker, there is an important distinction between them. For instance, the set of potential outcomes includes one in which both students choose to cheat, case in which they both copy the exam of the other student and have his own copied by the other.
On the contrary, in the classical version of Becker's model Becker, there is no active role for victims. Given that his framework is not a game, the only agent to play is the potential criminal. Observe that if we considered that each student has his own crib note, made previously at home, and he used it to cheat, his choice about cheating or playing fair would depend exclusively on the probability of being caught. Thus the other student's choice no longer would affect his decision and no student would be a "victim".
Unlike our baseline model, this version would be very similar to the one provided in Becker's seminal paper. Furthermore, even police or law enforcer has no role in Becker's model. In our game, however, their role is played by the professor, which makes the probability of the cheater being caught endogenous.
In fact, once we allow for the existence of two different types of professor, the value of the probability of catching cheaters depends on whether the "law enforcer in classroom" is lenient or severe. Even when the professor is severe, there is a possibility that his level of effort does not achieve the minimum required to create incentives for students to play fair. The actual professor type therefore affects directly the game's equilibrium. This feature is even more salient when we consider the extended model with incomplete information and possibility of signaling.
In this case, the active role of the professor can also affect the level of uncertainty that students face-when the lenient chooses to signal in order to mimic the severe's behavior, for example. The existence of different types of "law enforcers", and the consequent uncertainty about which the true one is, is a novel contribution of our paper when compared to the standard literature on economics of crime, and to Becker's paper in particular. Our contribution to the literature is to provide a theoretical framework that is able to capture all the strategic features of students' choice of cheating, and professor's choice of how much effort to exert in order to catch cheaters.
The authors provide a relevant discussion about the use of mathematical utility modeling-and thus game theory-with respect to ethics. Based on the reasoning developed in works such as Gibson , they argued that incorporation of games such as the Prisoner's Dilemma into the ethics issues-and thus in cheating as well-may be useful to better understand costs and benefits involved. However, their model focuses in collaboration in take-home tasks rather than in-class activities, and thus is substantially different from the framework we develop in this paper.
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In particular, Briggs et al. The probability of succeeding in cheating is endogenous, given that whenever at least one student tattles, the cheater is caught and punished. Observe that the model presented by those authors focuses on collaborative cheating rather than individual cheating, which makes it quite different from ours. Moreover, there is no active role for the professor, since the punishment is exogenously defined and the probability of cheaters being caught is determined by students's behavior.
The game-theoretic approach we employ allows us to provide both positive and normative conclusions. Second, it also provides insights that can help reduce cheating on campuses. Some of them had already been found effective by the literature, such as harsh penalties imposed by both the institution and the professor. Others, however, have not received much attention, including hiring high effort professors, who value fairness in classroom.
As we show below, a necessary condition for the existence of a virtuous equilibrium without cheating is that the professor do not be lenient. The rest of the paper is organized as follows. In the following section we present our baseline model, composed by two identical students and a professor. We discuss the incentives each one faces and describe their processes of choice.
This section also establishes necessary and sufficient conditions for the existence of a Nash equilibrium without cheating. Section 3 extends the model in several directions. Section 4 concludes and suggests some other extensions. The proofs of propositions omitted in the text are shown in the Appendix.
Our baseline model is composed by two identical students, A and B , and one professor or teacher. There will be an exam in the course that the professor is in charge. Each student must choose his level of effort in studying and the professor must choose his effort to detect and punish in-class cheating. Whenever a student chooses to cheat, he does not study, such that his level of effort is equal to zero. All these actions are chosen before the exam happens, there is no communication among the players and the information is complete, such that we can model this strategic situation as a three-players static game.
We consider only one type of academic dishonesty, namely the action of one student of copying from the another student's exam without his consent or knowledge. Therefore, we rule out common cheating practices such as helping someone on the exam and using a crib note. We also do not consider academic cheating in take-home tasks, such as representing someone else's work as your own e. Given this assumption, whenever the professor detects a dishonesty action, he can punish only the student that copied the exam.
We assume the punishment sets the dishonest student's grade equal to zero. If the student succeeds in cheating, his grade is equal to that of the other student. Moreover, given that players are identical, they have the same utility as well as the same grade function. There are four possible cases to consider:.
Student's grade is increasing in his effort in studying. However, the return of the effort is decreasing. We also assume some other conditions on the behavior of this function, which may be seen equal to the Inada conditions. The assumption below summarizes and formalizes the features of the grade function. Assumption 1. The first order condition of his problem is then given by. We discuss the existence of such an optimal choice below. Proposition 2. Suppose that the student's utility function has the following further characteristics:.
Thus, our model rules out "very lazy" students. Although the assumptions made about the second derivatives are quite standard when one want to guarantee concavity, it is possible to think about at least two cases in which they could be relaxed. The second case happens when the student is of the type which always wants to achieve the maximum grade. This may fits students who want to graduate with honors, for example. The same comment about the previous case, concerning the potential changes in the optimal choice, applies to here. However, it is possible to guarantee the concavity of U if the magnitude of the second derivative is relatively small see equation A-2 in the proof of proposition 2.
In fact, this alternative may also ensure a well-behaved solution in the case with a minimum grade. The assumption that the mixed partial derivative is non-positive means that the marginal utility of the grade is higher when the level of effort is low than when it is high, ceteris paribus. The ideia behind this assumption is that the student gets a higher marginal pleasure when the grade can be achieved with less effort.
The alternative assumption, namely positive mixed partial second derivative, suggests a behavior in which the student feels that his effort is rewarding, such that the marginal utility increases with the level of effort. Once again, it is possible to obtain the concavity of U with such alternative assumption as long as we impose bounds in the magnitude of the derivative. Proposition 3. The student's optimal level of effort is a function that:. We must impose some further regularities in the professor behavior.
Assumption 4. The characteristics of the probability function are quite standard and satisfy the Inada conditions, as we highlight below. Assumption 5.
We consider two types of professor, depending on the relative magnitude of his disutility of effort. This means that any effort to catch dishonest behaviors does not leave the lenient professor better off, regardless students' grades. In other words, for this type of professor the benefit from the increase in the probability of catching is not higher than the desutility of effort.
Now an initial effort is worth, because the marginal benefit of increasing the probability p is higher than the disutility caused by such an effort. However, this case presents a further complexity, such that we may have to consider the effect that the higher probability has on the students' grades. We must later analyze whether such an effect is large enough to overcome the other two, which would make the severe professor mimic the lenient's choice.
The first order condition of his problem when student A cheats and student B plays fair is given by. When student A plays fair and student B cheats, the professor's FOC is similar to 2 , except by the exchange of subscripts. For the other two cases both students cheat and both students play fair , the FOC is also similar to 2 , but now without the effect on the grades.
Proposition 6. Suppose that the severe professor's utility function satisfies Assumption 4. The assumptions assumed in order to assure the existence of the global maximizer above are quite standard, as we have already discussed. We can now establish comparative statics results for the professor's optimal choice. Proposition 7.
The severe professor's optimal level of effort is a function that:. The above result states that when marginal benefits or marginal costs change, the optimal choice changes as well. While items i and ii are related to marginal costs, items iii and iv are associated to marginal benefits.
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This explains why in the former the relationship is inverse and in the latter it is direct. Item ii deserves some attention. Observe that the students' grades affect the professor's level of effort only if one of them chooses to cheat. In this case, there is a negative effect: higher effort implies higher probability, which in turn decreases the cheater's grade.
We can sum up the game in the two payoff matrices below. Let us consider the decision of student A. As we are assuming symmetry, the same results are valid for student B. In this case, student A chooses to play fair, because. If student B chooses to cheat, student A chooses to play fair, because his payoffs are the same as those given by 3. However, if student B chooses to play fair, the choice of student A depend on the probability of being caught cheating. In fact, student A plays fair if and only if. Proposition 8. We can sum up the students' best choices with the support of the above two payoff matrices.
In order to do so, let us disregard the professor's decision for a while. We must now study the professor's best choice. Let us start with the lenient one. This covers all possibilities. Despite the larger complexity of the behavior of the severe professor, there are two straightforward cases. First, suppose that both students choose to cheat.
Suppose now that student A cheats and student B plays fair. A practice of getting unfair advantage in competitive video games has been legally addressed in Korea and is now punishable by law. Basically, this is the practice where a rookie player pays a pro gamer to play and win games with their account, artificially inflating the rankings and putting them in the big leagues, without actually doing the work to get there. This practice is strictly forbidden and frowned upon by both the gaming community and the game developers themselves, viewed as extremely detrimental to the game experience.
Why is this so bad? Well, on top of being a form of cheating, buying a status without having the skill and dedication to earn it, ruins the games for other players as well. But it gets worse because the one who bought their way on the ladder is now in a team of four or five other players who rely on that player and depend on their communication, tactics, and skill. Aggression is at the heart of the game, which has a rich history of violence. The violence follows naturally enough from the pioneer nature of the game.
View all New York Times newsletters. Cheating matters, and in some times and places it has been central. There really were riverboat gamblers and they really did play some nasty and ingenious tricks. Green claimed that every manufactured deck was marked, asking volunteers from the audience to present a deck purchased at any local shop. He broke the seal and, inspecting the backs, he correctly declared the suit and number of every card before turning them over. It emerged that he was using a tiny mirror at the edge of his stage table to read the undersides.
Other cultural and historical strands McManus covers include game theory, online poker and the friendly game among local regulars, played for harmless stakes. The televised tournaments created stars and sold many copies of many books, how-to and anecdotal. Computer programs designed to improve skills have sold well, too. I sigh with nostalgia. McManus writes with verve and knowledge, though his book is a little sprawling. Few readers will wish it were longer. The thoughts on poker terms and principles in global politics, and on the application of game theory to fields like cancer research, are interesting, although McManus does sometimes exaggerate or stretch a point.
A review on Nov.