Monotonicity-based consensus states for the monometric rationalisation of ranking rules and how they are affected by ties

Aggregating the preferences of several voters on a set of candidates is a classical problem in several fields of application. In previous work, we have addressed this problem in the case where each voter expresses his/her preferences in the form of a ranking on the set of candidates, by searching for monotonicity of three different types of representation of votes: the scorix, the votrix and the votex. However, this search was addressed differently for each of the three representations of votes. In this paper, we propose a Kemeny-like type of search that will equally deal with the three aforementioned representations of votes. Moreover, in case some voters consider that two or more candidates are equally suitable, the proposed method needs to be adapted. In this setting, the decision is no longer between 'candidate a is better than candidate b' and 'candidate b is better than candidate a', but 'candidates a and b are equally suitable' also needs to be considered. Here, the scorix, the votrix and the votex are extended in order to deal with this new three-way setting.