Possible and Necessary Winner Problems in Iterative Elections with Multiple Rules.

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| >= 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.