On the Linear Ordering Problem and the Rankability of Data
This work addresses the challenge of assessing rankability in data, such as sports or college rankings, but is incremental as it builds on existing concepts.
The paper tackles the problem of quantifying a dataset's inherent ability to be meaningfully ranked by using the linear ordering problem to analyze rankability, showing that optimal rankings maximize hindsight accuracy and developing a method to measure diversity among optimal rankings without full enumeration.
In 2019, Anderson et al. proposed the concept of rankability, which refers to a dataset's inherent ability to be meaningfully ranked. In this article, we give an expository review of the linear ordering problem (LOP) and then use it to analyze the rankability of data. Specifically, the degree of linearity is used to quantify what percentage of the data aligns with an optimal ranking. In a sports context, this is analogous to the number of games that a ranking can correctly predict in hindsight. In fact, under the appropriate objective function, we show that the optimal rankings computed via the LOP maximize the hindsight accuracy of a ranking. Moreover, we develop a binary program to compute the maximal Kendall tau ranking distance between two optimal rankings, which can be used to measure the diversity among optimal rankings without having to enumerate all optima. Finally, we provide several examples from the world of sports and college rankings to illustrate these concepts and demonstrate our results.