The complexity of the winner determination problem has been studied for almost all common voting rules. A notable exception, possibly caused by some confusion regarding its exact definition, is the method of ranked pairs. The original version of the method, due to Tideman, yields a social preference function that is irresolute and neutral. A variant introduced subsequently uses an exogenously given tie-breaking rule and therefore fails neutrality. The latter variant is the one most commonly studied in the area of computational social choice, and it is easy to see that its winner determination problem is computationally tractable. We show that by contrast, computing the set of winners selected by Tideman's original ranked pairs method is NP-complete, thus revealing a trade-off between tractability and neutrality. In addition, several known results concerning the hardness of manipulation and the complexity of computing possible and necessary winners are shown to follow as corollaries from our findings.

Schulze's rule and ranked pairs are two Condorcet methods that both satisfy many natural axiomatic properties. Schulze's rule is used in the elections of many organizations, including the Wikimedia Foundation, the Pirate Party of Sweden and Germany, the Debian project, and the Gento Project. Both rules are immune to control by cloning alternatives, but little is otherwise known about their strategic robustness, including resistance to manipulation by one or more voters, control by adding or deleting alternatives, adding or deleting votes, and bribery. Considering computational barriers, we show that these types of strategic behavior are NP-hard for ranked pairs (both constructive, in making an alternative a winner, and destructive, in precluding an alternative from being a winner). Schulze's rule, in comparison, remains vulnerable at least to constructive manipulation by a single voter and destructive manipulation by a coalition. As the first such polynomial-time rule known to resist all such manipulations, and considering also the broad axiomatic support, ranked pairs seems worthwhile to consider for practical applications.