Manipulation and Control Complexity of Schulze Voting

Abstract

Schulze voting is a recently introduced voting system enjoying unusual popularity and a high degree of real-world use, with users including the Wikimedia foundation, several branches of the Pirate Party, and MTV. It is a Condorcet voting system that determines the winners of an election using information about paths in a graph representation of the election. We resolve the complexity of many electoral control cases for Schulze voting. We find that it falls short of the best known voting systems in terms of control resistance, demonstrating vulnerabilities of concern to some prospective users of the system.

Schulze and ranked-pairs voting are fixed-parameter tractable to bribe, manipulate, and control

“…Clearly, this immediately implies the following result (still keeping in mind that the values of all edge weights are of the same parity). Corollary 6.4 (Corollary to the proofs of Menton and Singh [25]) Even when restricted to instances having all pairwise contests so equal that the total cardinality of the set of absolute values of WMG edges (in the WMG involving all candidates in the instance) is at most 2, Schulze elections are NP-complete (in the nonunique-winner model) for constructive control by deleting candidates. Even when restricted to instances having all pairwise contests so equal that the total cardinality of the set of values of WMG edges is at most 3, Schulze elections are NP-complete (in the nonunique-winner model) for constructive control by deleting candidates.…”

“…That, along with that fact that all weights of edges in the WMG have the same parity, gives us Corollaries 6.3 and 6.4. [25]) Even when restricted to instances having all pairwise contests so equal that each WMG edge 5 has absolute value at most 2, Schulze elections are NP-complete (in the nonunique-winner model) for constructive control by deleting candidates. The same holds for constructive control by adding candidates, unlimited adding of candidates, partition of candidates in the ties-eliminate model, and runoff partition of candidates in 5 In Corollaries 6.3 and 6.4, when we speak of restricting a WMG, the WMG we are speaking of as having to obey the restriction is the WMG involving all candidates involved in the problem.…”

“…We observe that, with respect to parameterizing on the internal addition/deletion bound, for Schulze elections it holds that constructive control by adding voters, constructive control by deleting voters, and constructive control by adding candidates are all W[2]-hard. These problems have already been shown to be NP-complete by Parkes and Xia [27] and Menton and Singh [23] through reductions from exact cover by 3-sets (X3C). In order to prove W [2]-hardness, we need to reduce from a parameterized problem that is known to be W[2]-hard.…”

“…But the "all can vote the same" insight used by Gaspers et al [15] seems potentially fragile, and is currently known only for the constructive, nonunique-winner case. Indeed, Menton and Singh [25] state that the analogue fails for the constructive, unique-winner case. It thus remains an interesting open issue whether the weighted destructive coalitional manipulation problem for Schulze elections is in FPT when parameterized only on the number of candidates.…”

“…Briefly, the reason their algorithm for the weighted constructive coalitional manipulation problem for Schulze elections is clearly even an FPT algorithm is as follows. Their algorithm is using the fact, observed independently by Menton and Singh[23] and Gaspers et al[15], that for the nonunique-winner model, weighted constructive coalitional manipulation problem for Schulze elections, if one can make a given candidate a winner then there is a set of manipulative votes in which all manipulators vote the same way and that candidate is selected as a winner. Once one has this, an FPT algorithm is obtained simply by cycling over all possible preference orders, for each seeing whether, if that is what all the manipulators cast as their vote, the given candidate becomes a winner.…”