How effective can simple ordinal peer grading be?

Ordinal peer grading has been proposed as a simple and scalable solution for computing reliable information about student performance in massive open online courses. The idea is to outsource the grading task to the students themselves as follows. After the end of an exam, each student is asked to rank -- in terms of quality -- a bundle of exam papers by fellow students. An aggregation rule then combines the individual rankings into a global one that contains all students. We define a broad class of simple aggregation rules, which we call type-ordering aggregation rules, and present a theoretical framework for assessing their effectiveness. When statistical information about the grading behaviour of students is available (in terms of a noise matrix that characterizes the grading behaviour of the average student from a student population), the framework can be used to compute the optimal rule from this class with respect to a series of performance objectives that compare the ranking returned by the aggregation rule to the underlying ground truth ranking. For example, a natural rule known as Borda is proved to be optimal when students grade correctly. In addition, we present extensive simulations that validate our theory and prove it to be extremely accurate in predicting the performance of aggregation rules even when only rough information about grading behaviour (i.e., an approximation of the noise matrix) is available. Both in the application of our theoretical framework and in our simulations, we exploit data about grading behaviour of students that have been extracted from two field experiments in the University of Patras.