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## Ranking from Stochastic Pairwise Preferences: Recovering Condorcet Winners and Tournament Solution Sets at the Top

Citations: | 1 - 0 self |

### Citations

99 |
Tournament Solutions and Majority Voting
- Laslier
- 1997
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Citation Context ...tournament solution sets such as top cycles and Copeland and Markov sets, which have been used to define ‘winners’ in the computational social choice literature (Moulin, 1986; De Donder et al., 2000; =-=Laslier, 1997-=-; Brandt et al., 2015). We show that when preferences contain cycles, the RC, MB and SVM-RA algorithms can fail to rank such ‘good’ items at the top; we then propose three Ranking from Stochastic Pair... |

75 | Learning mallows models with pairwise preferences
- Lu, Boutilier
- 2011
(Show Context)
Citation Context ...hor(s). 1. Introduction There has been much interest in recent years in designing algorithms for aggregating pairwise preferences to rank a set of items (Fürnkranz & Hüllermeier, 2010; Ailon, 2011; =-=Lu & Boutilier, 2011-=-; Jiang et al., 2011; Jamieson & Nowak, 2011; Negahban et al., 2012; Osting et al., 2013; Wauthier et al., 2013; Busa-Fekete et al., 2014a;b; Rajkumar & Agarwal, 2014). Indeed, the need for aggregatin... |

66 | Comparison of perturbation bounds for the stationary distribution of a Markov chain,” Linear Algebra and Its Applications,
- Cho, Meyer
- 2001
(Show Context)
Citation Context ...̂PM produced by running the PM algorithm on P̂ satisfies σ̂PM(i) < σ̂PM(j) for all i ∈ TC(P), j /∈ TC(P) . The proofs of Theorems 10–11 make use of the Cho-Meyer perturbation bound for Markov chains (=-=Cho & Meyer, 2001-=-) to bound the difference between the stationary vector pic and pi. (see supplementary material for details). 7. Experiments We conducted experiments to compare the performance of different algorithms... |

61 |
Aggregation of preferences with variable electorate
- Smith
- 1973
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Citation Context ... ∈ W, j /∈ W . A Condorcet winner corresponds to a top cycle of size 1, i.e. when |TC(P)| = 1, we have TC(P) = {CW(P)}. The top cycle is also referred to as the Smith set in voting theory literature (=-=Smith, 1973-=-). Copeland and Markov Sets. The top cycle is one form of tournament solution set, which selects a set of elements considered to be ‘winners’ in a tournament (Laslier, 1997; Brandt et al., 2015). One ... |

55 |
Choosing from a tournament,"
- Moulin
- 1986
(Show Context)
Citation Context ...en they exist, and more generally, to tournament solution sets such as top cycles and Copeland and Markov sets, which have been used to define ‘winners’ in the computational social choice literature (=-=Moulin, 1986-=-; De Donder et al., 2000; Laslier, 1997; Brandt et al., 2015). We show that when preferences contain cycles, the RC, MB and SVM-RA algorithms can fail to rank such ‘good’ items at the top; we then pro... |

43 | J.-C.: Mémoire sur les élections au scrutin. Histoire de’Académie Royale Des Sci - Borda |

31 |
A ‘reasonable’ social welfare function.
- Copeland
- 1951
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Citation Context ...items in the Copeland set to appear at the top. The corresponding algorithm is shown in Algorithm 1; we term this the Matrix Copeland (MC) algorithm due to its similarity to the Copeland voting rule (=-=Copeland, 1951-=-). The following result shows this algorithm indeed ranks the Copeland set at the top for any preference matrix and also minimizes the pairwise disagreement error when the preference matrix is acyclic... |

31 | Iterative ranking from pair-wise comparisons.
- Negahban, Oh, et al.
- 2012
(Show Context)
Citation Context ...rs in designing algorithms for aggregating pairwise preferences to rank a set of items (Fürnkranz & Hüllermeier, 2010; Ailon, 2011; Lu & Boutilier, 2011; Jiang et al., 2011; Jamieson & Nowak, 2011; =-=Negahban et al., 2012-=-; Osting et al., 2013; Wauthier et al., 2013; Busa-Fekete et al., 2014a;b; Rajkumar & Agarwal, 2014). Indeed, the need for aggregating pairwise preferences arises in many domains where fully ordered p... |

29 | Active ranking using pairwise comparisons.
- Jamieson, Nowak
- 2011
(Show Context)
Citation Context ...h interest in recent years in designing algorithms for aggregating pairwise preferences to rank a set of items (Fürnkranz & Hüllermeier, 2010; Ailon, 2011; Lu & Boutilier, 2011; Jiang et al., 2011; =-=Jamieson & Nowak, 2011-=-; Negahban et al., 2012; Osting et al., 2013; Wauthier et al., 2013; Busa-Fekete et al., 2014a;b; Rajkumar & Agarwal, 2014). Indeed, the need for aggregating pairwise preferences arises in many domain... |

20 | Perturbation of the stationary distribution measured by ergodicity coefficients - Seneta - 1988 |

14 | Efficient ranking from pairwise comparisons.
- Wauthier, Jordan, et al.
- 2013
(Show Context)
Citation Context ...airwise preferences to rank a set of items (Fürnkranz & Hüllermeier, 2010; Ailon, 2011; Lu & Boutilier, 2011; Jiang et al., 2011; Jamieson & Nowak, 2011; Negahban et al., 2012; Osting et al., 2013; =-=Wauthier et al., 2013-=-; Busa-Fekete et al., 2014a;b; Rajkumar & Agarwal, 2014). Indeed, the need for aggregating pairwise preferences arises in many domains where fully ordered preferences may be hard to obtain, e.g. in cu... |

13 |
A statistical convergence perspective of algorithms for rank aggregation from pairwise data.
- Rajkumar, Agarwal
- 2014
(Show Context)
Citation Context ...gorithms including the Rank Centrality algorithm, the Matrix Borda algorithm, and the SVMRankAggregation algorithm succeed in recovering a ranking that minimizes a global pairwise disagreement error (=-=Rajkumar and Agarwal, 2014-=-). In this paper, we consider settings where pairwise preferences can contain cycles. In such settings, one may still like to be able to recover ‘good’ items at the top of the ranking. For example, if... |

11 | Choosing from a weighted tournament - Donder, Breton, et al. - 2000 |

10 | Active Learning Ranking from Pairwise Preferences with Almost Optimal Query Complexity
- Ailon
- 2011
(Show Context)
Citation Context ...15 by the author(s). 1. Introduction There has been much interest in recent years in designing algorithms for aggregating pairwise preferences to rank a set of items (Fürnkranz & Hüllermeier, 2010; =-=Ailon, 2011-=-; Lu & Boutilier, 2011; Jiang et al., 2011; Jamieson & Nowak, 2011; Negahban et al., 2012; Osting et al., 2013; Wauthier et al., 2013; Busa-Fekete et al., 2014a;b; Rajkumar & Agarwal, 2014). Indeed, t... |

5 | Tournament solutions
- Brandt, Brill, et al.
- 2016
(Show Context)
Citation Context ...tion sets such as top cycles and Copeland and Markov sets, which have been used to define ‘winners’ in the computational social choice literature (Moulin, 1986; De Donder et al., 2000; Laslier, 1997; =-=Brandt et al., 2015-=-). We show that when preferences contain cycles, the RC, MB and SVM-RA algorithms can fail to rank such ‘good’ items at the top; we then propose three Ranking from Stochastic Pairwise Preferences: Rec... |

4 | Pagerank as a weak tournament solution.
- Brandt, Fischer
- 2007
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Citation Context ...rnament Solution Sets at the Top pairwise preferences; below we discuss three specific algorithms that form the backdrop to our work. Before doing so, we point out that our work differs from that of (=-=Brandt & Fischer, 2007-=-), where any ranking algorithm that given P ∈ Pn produces a score vector f ∈ Rn is considered to induce a corresponding tournament solution TS(P) = argmaxi∈[n] fi; in particular, Brandt & Fischer stud... |

3 |
Sorting from noisy information. arXiv preprint arXiv:0910.1191,
- Braverman, Mossel
- 2009
(Show Context)
Citation Context ...sponding tournament solution TS(P) = argmaxi∈[n] fi; in particular, Brandt & Fischer study properties of the tournament solution induced in this manner by PageRank scores. Our work also differs from (=-=Braverman & Mossel, 2009-=-; Wauthier et al., 2013), where pairs are sampled only once and the conditions on P are significantly stronger than those assumed here. Three representative ranking algorithms studied in recent years ... |

2 | Preference-based rank elicitation using statistical models: The case of mallows - Busa-Fekete, Hüllermeier, et al. - 2014 |

1 | rank elicitation through adaptive sampling of stochastic pairwise preferences - PAC |

1 | to Section 3 (Related Work and Existing Results) For completeness, here we include descriptions of the Rank Centrality (RC), Matrix Borda (MB) and SVMRankAggregation (SVM-RA) algorithms that form the backdrop to our work. For the SMV-RA algorithm, we will - Supplement |

1 | the stationary probability vector of P̆ Output: Permutation σRC ∈ argsort(pi) Algorithm 5 Matrix Borda (MB) Algorithm ((Rajkumar & Agarwal, 2014)) Input: Pairwise comparison matrix P ∈ [0, 1]n×n satisfying the following conditions: (i) for all i 6= j: pij - pi |

1 | of Results in Sections 5–6 The overall strategy followed by our proofs is largely similar to that of (Rajkumar & Agarwal, 2014): namely, that the empirical pairwise comparison matrix P̂ concentrates around the true pairwise preference matrix P, and that w - Proofs |