Michael Cameron’s blog Sex, Drugs and Economics had an interesting post regarding game theory and cheating in online assessment. He mentions a paper by Eren Bilena Alexander Matros entitled ‘Online cheating amid COVID-19’ in the Journal of Economic Behavior and Organization. They use a simple game theory model below to show the payoffs of the student and the professor with the student cheating or being honest.
In the sequential-move game, the student chooses to either cheat or be honest. The professor observes the student choice and decides either to report the student for cheating or not. There are four outcomes in this game, but the professor and the student rank these outcomes differently – see table below.
The table (right) gives an example of players’ payoffs. This game has a unique mixed-strategy equilibrium, which means that the student and the professor should randomise between their two actions in equilibrium. Thus cheating as well as reporting is a part of the equilibrium.
To find the Nash equilibriums you use the ‘best response method – for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the textbook definition of Nash equilibrium). In this game, the best responses are:
- If the student chooses to cheat, the professor’s best response is to report the student (since 3 is a better payoff than 2);
- If the student chooses not to cheat, the professor’s best response is not to report the student (since 4 is a better payoff than 1);
- If the professor chooses to report the student, the student’s best response is to not cheat (since 2 is a better payoff than 1); and
- If the professor chooses not to report the student, the student’s best response is to cheat (since 4 is a better payoff than 3).
A Nash equilibrium occurs where both players’ best responses coincide – note that there isn’t actually any case where both players are playing a best response.
In cases such as this, we say that there is no Nash equilibrium in pure strategy. However, there will be a mixed strategy equilibrium, where the players randomise their choices of strategy. The student should cheat with some probability, and the professor should report the student with some probability.