On Discovering Bucket Orders from Preference Data

Abstract

The problem of ordering a set of entities which contain inherent ties among them arises in many applications. Notion of "bucket order" has emerged as a popular mechanism of ranking in such settings. A bucket order is an ordered partition of the set of entities into "buckets". There is a total order on the buckets, but the entities within a bucket are treated as tied.In this paper, we focus on discovering bucket order from data captured in the form of user preferences. We consider two settings: one in which the discrepancies in the input preferences are "local" (when collected from experts) and the other in which discrepancies could be arbitrary (when collected from a large population). We present a formal model to capture the setting of local discrepancies and consider the following question: "how many experts need to be queried to discover the underlying bucket order on n entities?". We prove an upperbound of O( √ log n). In the case of arbitrary discrepancies, we model it as the bucket order problem of discovering a bucket order that best fits the data (captured as pairwise preference statistics). We present a new approach which exploits a connection between the discovery of buckets and the correlation clustering problem. We present empirical evaluation of our algorithms on real and artificially generated datasets.

Median constrained bucket order rank aggregation

“…to one. The so-called optimal bucket order problem (Gionis et al, 2006;Ukkonen et al, 2009;Kenkre et al, 2011;Aledo et al, 2017b), i.e., dealing with rank aggregation while allowing ties in the solution, is a recent description of the problem originally stateci by Kemeny and Snell (1962) when defined the median ranking. Within the Kemeny's axiomatic approach, both exact (Emond and Mason, 2002) and heuristic algorithms (Amadio et al, 2016;D'Ambrosia et al, 2017) have been proposed.…”

“…When judges are asked to rank only a subset of the entire set of objects ( or when the rank associated with some items is missing), the resulting ordering is called partial ranking (Heiser and D'Ambrosia, 2013). Sometimes tied rankings are called bucket orders (Gionis et al, 2006;Ukkonen et al, 2009;Kenkre et al, 2011), when a set of items is tied for a given location. Many statistical analyses can be performed with preference rankings and paired comparison rankings; examples include inference on top-k lists (Hall and Schimek, 2012;Sampath and Verducci, 2013), cluster analysis and relateci techniques (Murphy and Martin, 2003;Busse et al, 2007;Heiser and D'Ambrosia, 2013;Brentari et al, 2016), supervised classification methods (D'Ambrosia, 2008;Lee and Yu, 2010;D'Ambrosia and Heiser, 2016;Plaia and Sciandra, 2017), multi variate analysis (Cohen and Mallows, 1980;Busing et al, 2005;Busing, 2006;Yu et al, 2013), and probability models (Bradley and Terry, 1952;Fienberg and Larntz, 1976;Dittrich et al, 2000;Yu et al, 2016).…”

Approaching rank aggregation problems by using evolution strategies: The case of the optimal bucket order problem

“…Although BPA exhibits a good performance when applied to the OBOP [9,26,22], it does suffer from certain drawbacks. In fact, the random selection of the initial pivot has a significant impact on the bucket order obtained.…”

“…Figure 1: The bucket pivot algorithm (BPA) et al [26] propose to reduce this risk by using a two-step approach (BPA-CC): first, the items are clustered into groups by using precedence-based similarity; then, BPA is run over the clusters obtained. To do this, a secondary precedence matrix C , which is defined over the clusters, is computed by collapsing the original one.…”

Rank quantization

“…In addition to practical uses of this problem [10,16], it has also been studied from a theoretical perspective [12,23]. While the positive results known for this problem imply almost linear-time algorithms, they all assume that the input is a tournament, i.e., every pair of elements has a preference relation [23].…”