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Computing Slater Rankings Using Similarities Among Candidates∗

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Computing Slater Rankings Using Similarities Among Candidates∗ Computing Slater Rankings Using Similarities Among Candidates∗ Vincent Conitzer Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213, USA [email protected] Abstract where the candidates vote over each other by linking to each other (as in the context of the World Wide Web) [4; Voting (or rank aggregation) is a general method for aggre- gating the preferences of multiple agents. One important vot- 3]; and interpreting common voting rules as maximum like- ing rule is the Slater rule. It selects a ranking of the alter- lihood estimators of a “correct” ranking [10]. natives (or candidates) to minimize the number of pairs of The design of voting rules has been guided by the ax- candidates such that the ranking disagrees with the pairwise iomatic approach: decide on a set of criteria that a good majority vote on these two candidates. The use of the Slater voting rule should satisfy, and determine which (if any) vot- rule has been hindered by a lack of techniques to compute ing rules satisfy all of these criteria. One well-known cri- Slater rankings. In this paper, we show how we can decom- terion is that of independence of irrelevant alternatives.In pose the Slater problem into smaller subproblems if there is a its strongest form, this criterion states that the relative rank- set of similar candidates. We show that this technique suffices to compute a Slater ranking in linear time if the pairwise ma- ing of two candidates by a voting rule should not be affected jority graph is hierarchically structured. For the general case, by the presence or absence of other candidates. That is, if we also give an efficient algorithm for finding a set of similar a is ranked higher than b by a voting rule that satisfies in- candidates. We provide experimental results that show that dependence of irrelevant alternatives, then the voting rule this technique significantly (sometimes drastically) speeds up will still rank a higher than b after the introduction of an- search algorithms. Finally, we also use the technique of sim- other candidate c. Arrow’s impossibility result [5] precludes ilar sets to show that computing an optimal Slater ranking is the existence of any reasonable voting rule satisfying this NP-hard, even in the absence of pairwise ties. criterion. Intuitively, when independence of irrelevant al- ternatives is satisfied, whether candidate a or b is preferred Introduction should depend only on whether more votes prefer a to b than In multiagent systems with self-interested agents, often the b to a—that is, the winner of the pairwise election should be agents need to arrive at a joint decision in spite of differ- ranked higher. Unfortunately, as noted by Condorcet [13], ent preferences over the available alternatives. Voting (or there can be cycles in this relationship: for example, it can rank aggregation) is a general method for doing so. In be the case that a defeats b, b defeats c, and c defeats a in a rank aggregation setting, each voter ranks all the differ- pairwise elections. If so, no ranking of the candidates will ent alternatives (or candidates), and a voting rule maps the be consistent with the outcomes of all pairwise elections. votes to a single ranking of all the candidates. Rank ag- The Slater voting rule is arguably the most straightfor- gregation has applications outside the space of preference ward resolution to this problem: it simply chooses a ranking aggregation as well: for example, we can take the rank- of the candidates that is inconsistent with the outcomes of as ings that different search engines provide over a set of web- few pairwise elections as possible. Unfortunately, as we will pages and produce an aggregate ranking from this. Other discuss later in this paper, computing a Slater ranking is NP- applications include collaborative filtering [22] and planning hard. This suggests that we need a search-based algorithm 1 among automated agents [17; 18]. Recent work in artifi- to compute Slater rankings. cial intelligence and related areas has studied the complex- In this paper, we introduce a powerful preprocessing tech- ity of executing voting rules [19; 7; 14; 12; 1]; the com- nique that can reduce the size of instances of the Slater prob- plexity of manipulating elections [9; 16; 15]; eliciting the lem before the search is started. We say that a subset of the votes efficiently [8]; adapting voting theory to the setting candidates consists of similar candidates if for every candi- date outside of the subset, all candidates inside the subset ∗This material is based on work done while the author was achieve the same result in the pairwise election against that visiting IBM’s T.J. Watson Research Center in the context of an candidate. Given a set of similar candidates, we can recur- IBM Ph.D. Fellowship. The author thanks Andrew Davenport and Jayant Kalagnanam for a significant amount of valuable direction and feedback. 1Another approach is to look for rankings that are approxi- Copyright c 2006, American Association for Artificial Intelli- mately optimal in the Slater sense [1; 11]. Of course, this is not gence (www.aaai.org). All rights reserved. entirely satisfactory as it is effectively changing the voting rule. 613 sively solve the Slater problem for that subset, and for the pairwise election graph, and hence it is robust to a few voters original set with the entire subset replaced by a single can- who do not perceive the candidates as similar. didate, to obtain a solution to the original Slater problem. In There are a few trivial sets of similar candidates: 1. the addition, we also make the following contributions: set of all candidates, and 2. any set of at most one candidate. We show that if the results of the pairwise elections have We will be interested in nontrivial sets of similar candidates, a particular• hierarchical structure, the preprocessing tech- because, as will become clear shortly, the trivial sets have no nique is sufficient to solve the Slater problem in linear time. algorithmic use. For the general case, we give a polynomial-time algo- The following is the key observation of this paper: rithm• for finding a set of similar candidates (if it exists). This Theorem 1 If S consists of similar candidates, then there algorithm is based on satisfiability techniques. exists a Slater ranking in which the candidates in S form We exhibit the power of the preprocessing technique a (contiguous) block (that is, there do not exist s ,s S experimentally.• 1 2 and c C S such that s c s ). ∈ We use the concept of a set of similar candidates to give ∈ − 1 2 • the first straightforward reduction (that is, not a randomized Proof: Consider any ranking 1 of the candidates in which reduction or a derandomization thereof) showing that the the candidates in S are split into k>1 blocks; we will show Slater problem is NP-hard in the absence of pairwise ties. how to transform this ranking into another ranking 2 with the properties that: Definitions the candidates in S are split into k 1 blocks in , and • − 2 We use C to denote the set of candidates. We say that can- the Slater score of is at least as high as that of . didate a defeats candidate b in their pairwise election, de- • 2 1 noted by a b, if the number of votes ranking a above b By applying this transformation repeatedly, we can trans- is greater than→ the number of votes ranking b above a. The form the original ranking into an ranking in which the can- Slater rule is defined as follows: find a ranking on the didates in S form a single block, and that has at least as high candidates that minimizes the number of ordered pairs (a, b) a Slater score as the original ranking. such that a b but b defeats a in their pairwise election. Consider a subsequence of 1 consisting of two blocks 1 2 (Equivalently, we want to maximize the number of pairs of of candidates in S, si and si , and a block of candidates { } { } 1 1 candidates for which is consistent with the result of the in C S that divides them, ci : s1 1 s2 1 ... 1 1 − { } 2 2 2 pairwise election—we will refer to this number as the Slater s 1 c1 1 c2 1 ... 1 cl 1 s 1 s 1 ... 1 s . l1 1 2 l2 score.) We will refer to the problem of computing a Slater Because S consists of similar candidates, a given candidate ranking as the Slater problem. An instance of the Slater ci has the same relationship in the pairwise election graph to j problem can be represented by a “pairwise election” graph every si . Hence, one of the following two cases must apply: whose vertices are the candidates, and which has a directed 1. For at least half of the candidates c , for every sj, c sj edge from a to b if and only if a defeats b in their pairwise i i i → i election. The goal, then, is to minimize the number of edges j j 2. For at least half of the candidates ci, for every si , si ci. that must be flipped in order to make the graph acyclic. → In case 1, we can replace the subsequence by the subse- Most elections do not have any ties in pairwise elections. 1 1 quence c1 2 c2 2 ... 2 cl 2 s1 2 s2 2 ... 2 For example, if the number of votes is odd, there is no pos- 1 2 2 2 s 1 s 2 s 2 ..
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