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Preference orderings are orderings of a set of items according to thepreferences (of judges). Such orderings arise in a variety of domains,including group decision making, consumer marketing, voting and machinelearning. Measuring the mutual information and extracting the common patternsin a set of preference orderings are key to these areas. In this paper we dealwith the representation of sets of preference orderings, the quantification ofthe degree to which judges agree on their ordering of the items (i.e. theconcordance), and the efficient, meaningful description of such sets.  We propose to represent the orderings in a subsequence-based feature spaceand present a new algorithm to calculate the size of the set of all commonsubsequences - the basis of a quantification of concordance, not only for pairsof orderings but also for sets of orderings. The new algorithm is fast andstorage efficient with a time complexity of only $O(Nn^2)$ for the orderings of$n$ items by $N$ judges and a space complexity of only $O(\min\{Nn,n^2\})$.  Also, we propose to represent the set of all $N$ orderings through a smallestset of covering preferences and present an algorithm to construct this smallestcovering set.  The source code for the algorithms is available athttps://github.com/zhiweiuu/secs

Concordance and the Smallest Covering Set of Preference Orderings