Possible and Necessary Winner Problems in Iterative Elections with Multiple Rules

Abstract

An iterative election eliminates some candidates in each round until the remaining candidates have the same score according to a given voting rule. Prominent iterative voting rules include Hare, Coombs, Baldwin, and Nanson. The Hare/Coombs/Baldwin rules eliminate in each round the candidates with the least plurality/veto/Borda scores, while the Nanson rule eliminates the candidates with below-average Borda scores. Recently, it has been demonstrated that iterative elections admit some desirable properties such as polynomial-time winner determination and NP-hard control/manipulation/bribery. We study new aspects of iterative elections. We suppose that a set R of iterative voting rules is given and each round of the iterative election can choose one rule in R to apply. The question is whether there is a combination of rules, such that a specific candidate p becomes the unique winner (the Possible Winner problem), or whether a specific candidate p wins under all rule combinations (the Necessary Winner problem). The Possible Winner problem can be considered as a special control problem for iterative elections. We prove that for all subsets R of {Hare, Coombs, Baldwin, Nanson} with $$|R|\ge 2$$ , both Possible and Necessary Winner problems are hard to solve, with the only exception of R = {Baldwin, Nanson}. We further provide special cases of the Necessary Winner problem with R = {Baldwin, Nanson}, which are polynomial-time solvable. We also discuss the parameterized complexity of the Possible Winner problems with respect to the number of candidates and the number of votes, and achieve fixed-parameter tractable (FPT) results.