Article Contents
Article Contents

# On the linear ordering problem and the rankability of data

• 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.

Mathematics Subject Classification: 52B12, 90C09, 90C27, 90C35.

 Citation:

• Figure 1.  Linear Ordering Polytope: $P^{3}_{ \rm{LO}}$

Figure 2.  Simple Digraphs on $3$ vertices

Figure 3.  Optimal Solution(s) on Linear Ordering Polytope

Figure 4.  College Features from U.S. World News 2013 Rankings

Figure 5.  The degree of linearity and the hindsight accuracy of NFL rankings

Figure 6.  Correlation between the degree of linearity and the hindsight accuracy of NFL rankings

Figure 7.  The absolute difference between the foresight accuracy of two optimal rankings

Figure 8.  Diversity among NFL playoff teams in two optimal rankings for years 1992 and 1999

Figure 9.  NFL playoff progression in 1992; the opaque team lost the indicated matchup

Figure 10.  NFL playoff progression in 1999; the opaque team lost the indicated matchup

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