Monotonicity as a tool for differentiating between truth and optimality in the aggregation of rankings

- The aggregation of rankings, which is a common decision making problem in many fields of application, is addressed from a statistical point of view.
- The property of monotonicity of the profile, which serves as a tool for differentiating between truth an optimality in the aggregation of rankings, has been introduced.
- A noise model representing how people make mistakes, which results in both a measure of the most likely ranking and a statistical test for validating the real existence of such ranking, is proposed.
- The procedure has been illustrated with a real-life example concerning a decision making problem.

The choice of the ranking that best captures the preferences of several voters on a set of candidates has been a matter of study for centuries. An interesting point of view on this problem is centred on the notion of monotonicity. In this paper, we deal with an aspect of monotonicity that has not been addressed before: if there is a true ranking on the set of candidates and every voter expresses a ranking on the set of candidates, then the number of times that each ranking is expressed should decrease when we move away from this true ranking in terms of pairwise discordances. In addition, we propose a probabilistic model that allows to formulate the choice of the best ranking as a maximum likelihood estimation problem. A test for the validity of this monotonicity assumption is also proposed.