Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence The Price of Neutrality for the Ranked Pairs Method Markus Brill Felix Fischer Institut¨ fur¨ Informatik Statistical Laboratory Technische Universitat¨ Munchen¨ University of Cambridge 85748 Garching, Germany Cambridge CB3 0WB, UK Abstract To remedy this situation, a large number of SPFs have been suggested, together with a variety of criteria that a The complexity of the winner determination problem has been studied for almost all common voting rules. A notable reasonable SPF should satisfy (see Arrow, Sen, and Suzu- exception, possibly caused by some confusion regarding its mura 2002 for an overview). Neutrality and anonymity, for exact definition, is the method of ranked pairs. The original instance, are basic fairness criteria which require, loosely version of the method, due to Tideman, yields a social pref- speaking, that all alternatives and all voters are treated erence function that is irresolute and neutral. A variant intro- equally. Another criterion we will be interested in is the duced subsequently uses an exogenously given tie-breaking computational effort required to evaluate an SPF. Compu- rule and therefore fails neutrality. The latter variant is the tational tractability of the winner determination problem is one most commonly studied in the area of computational so- obviously a significant property of any SPF: the inability cial choice, and it is easy to see that its winner determina- to efficiently compute social preferences would render the tion problem is computationally tractable. We show that by method virtually useless, at least for large problem instances contrast, computing the set of winners selected by Tideman’s original ranked pairs method is NP-complete, thus revealing that do not exhibit additional structure. As a consequence, a trade-off between tractability and neutrality. In addition, computational aspects of preference aggregation have re- several known results concerning the hardness of manipula- ceived tremendous interest in recent years (see, e.g., Fal- tion and the complexity of computing possible and necessary iszewski et al. 2009; Conitzer 2010). winners are shown to follow as corollaries from our findings. In this paper, we study the computational complexity of the ranked pairs method (Tideman 1987). To the best of our 1 Introduction knowledge, this question has not been considered before, which is particularly surprising given the extensive litera- The fundamental problem of social choice theory can be ture that is concerned with computational aspects of ranked concisely described as follows: given a number of individ- pairs.2 A possible reason for this gap might be the confu- uals, or voters, each having a preference ordering over a set sion of two variants of the method, only one of which sat- of alternatives, how can we aggregate these preferences into isfies neutrality. In Section 2, we address this confusion a collective, or social, preference ordering that is in some and describe both variants as well as a closely related SPF sense faithful to the individual preferences? By a preference known as Kemeny’s rule. After introducing the necessary ordering we here understand a (transitive) ranking of all al- notation in Section 3, we show in Section 4 that deciding ternatives, and a function aggregating individual preferences whether a given alternative is a ranked pairs winner for the orderings into social preference orderings is called a social 1 neutral variant is NP-complete. Section 5 shows that a num- preference function (SPF). ber of known results follow as corollaries from our results, A natural idea to construct an SPF is by letting an al- and Section 6 discusses variants of the ranked pairs method ternative a be socially preferred to another alternative b if that are not anonymous. Finally, Section 7 concludes with a and only if a majority of voters prefers a to b. However, complexity-theoretic comparison of the ranked pairs method it was observed as early as the 18th century that this ap- and Kemeny’s rule. proach might lead to paradoxical situations: the collective preference relation may be cyclic even when all individual preferences are transitive (de Condorcet 1785). 2 Kemeny’s Rule and Two Variants of the Ranked Pairs Method Copyright c 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. In this section we address the difference between two vari- 1In contrast to a social welfare function as studied by Ar- ants of the ranked pairs method that are commonly studied row (1951), an SPF can output multiple social preference order- ings with the interpretation that all those rankings are tied for win- 2Typical problems include the hardness of manipulation (Bet- ner. The rationale behind this is to allow for a symmetric outcome zler, Hemmann, and Niedermeier 2009; Xia and Conitzer 2011) when individual preferences are symmetric, like in the case of two and the complexity of computing possible and necessary winners individuals with diametrically opposed preferences. (Xia et al. 2009; Obraztsova and Elkind 2011). 1299 in the literature. Both variants are anonymous, i.e., treat all We will henceforth denote this variant by RP. voters equally. Non-anonymous variants of the ranked pairs In a subsequent paper, Zavist and Tideman (1989) showed method have been suggested by Tideman (1987) and Zavist that a tie-breaking rule is in fact necessary in order to achieve and Tideman (1989), and will be discussed in Section 6. the property of independence of clones, which was the main It will be instructive to first consider Kemeny’s rule (Ke- motivation for introducing the ranked pairs method. While meny 1959; Young 1995). The latter, which is also known Zavist and Tideman (1989) proposed a way to define a tie- as the Kemeny-Young method, chooses the ranking or rank- breaking rule based on the preferences of a distinguished ings with maximal Kemeny score, where the Kemeny score voter (see Section 6 for details), the variant that is most of a ranking measures the extent to which it agrees with indi- commonly studied in the literature considers the tie-breaking vidual preferences: for every pair (a, b) of alternatives, each rule to be exogenously given and fixed for all profiles. This voter i contributes one point to the Kemeny score of a rank- variant of ranked pairs will be denoted by RPT. Whereas ing if and only if i ranks a and b in the same order as the RP may output a set of rankings, with the interpretation that ranking does. As the number of possible rankings grows all the rankings in the set are tied for winner, RPT always rapidly with the number of alternatives (for n alternatives outputs a single ranking. In social choice terminology, RP there are n! rankings), computing all Kemeny scores is com- is an irresolute SPF, and RPT is a resolute one. It is straight- putationally demanding. Indeed, it was shown by Bartholdi, forward to see that RP is neutral, i.e., treats all alternatives III, Tovey, and Trick (1989) that finding the so-called Ke- equally, and that RPT is not. An easy example for the latter meny rankings, i.e., the rankings with a maximal Kemeny statement is the case of two alternatives and two voters who score, is NP-hard.3 This is commonly seen as strong evi- each prefer a different alternative. dence that no efficient algorithm exists for this problem. Rather than completely ranking all alternatives, it is often The method of ranked pairs, introduced by Tide- sufficient to identify the socially “best” alternatives. This man (1987) and extended by Zavist and Tideman (1989), can is the purpose of a social choice function (SCF). An SCF be viewed as a heuristic to find a ranking with high—if not has the same input as an SPF, but returns alternatives instead maximal—Kemeny score. This is not to say that the main of rankings. Each SPF gives rise to a corresponding SCF motivation for ranked pairs was the approximation of Ke- that returns the top elements of the rankings instead of the meny’s rule. Actually, the ranked pairs method is interesting rankings themselves, and we will frequently switch between in its own right and satisfies a number of desirable proper- these two settings. Interestingly, deciding whether a given ties, some of which are not satisfied by Kemeny’s rule (see ranking is chosen by an SPF can be considerably easier than Tideman 1987; Lamboray 2009; Parkes and Xia 2012). deciding whether a given alternative is chosen by the corre- The easiest way to describe the ranked pairs method is to sponding SCF (see Table 1). formulate it as a procedure. The procedure first defines a pri- From a computational perspective, RPT is easy: con- ority ordering over the set of all (ordered) pairs (a, b) of al- structing the ranking for a given tie-breaking rule takes ternatives by giving priority to pairs (a, b) with a larger num- time polynomial in the size of the input (see Proposition 1). ber of voters preferring a to b. Then, it constructs a ranking For RP, however, the picture is different: as the number of of the alternatives by starting with the empty ranking and it- tie-breaking rules is exponential, executing the iterative pro- eratively considering pairs in order of priority. When a pair cedure for every single tie-breaking rule is infeasible. Of (a, b) is considered, the ranking is extended by fixing that course, this does not preclude the existence a clever algo- a precedes b—unless fixing this pairwise comparison would rithm that efficiently computes the set of all alternatives that create a cycle together with the previously fixed pairs, in are the top element of some chosen ranking.6 Our main which case the pair (a, b) is discarded.