Representations of votes facilitating monotonicity-based ranking rules: From votrix to votex

Monotonicity-based consensus states for the monometric rationalisation of ranking rules with application in decision making

“…Besides the Borda count, all other scoring ranking rules [57,102,104,169] also reduce the profile of rankings given by the voters to the scoring matrix. Based on the ideas of Condorcet [35], another representation of votes gathering a completely different type of information is commonly used in social choice theory: the votrix [124]. The votrix is a matrix where the element at the i-th row and j-th column represents the number of times that the i-th candidate is preferred to the j-th candidate in the preferences given by the voters.…”

“…This type of representation of votes where candidates are head-to-head compared with each other is said to be based on pairwise information. Many other representations of votes based on pairwise information, such as the beatpath matrix [147] and the votex [124], have been used in social choice theory.…”

“…This strength of support defines a matrix V , henceforth called a votrix 2 [124], where the element at the i-th row and the j-th column is the strength of support of the i-th candidate over the j-th candidate.…”

“…, r} 2k−2 . This function will be henceforth called the votex 3 [124] induced by the profile R. Definition 4.13 Let C be a set of k candidates and r be the number of voters. A function W : C 2 = → {0, 1, .…”

“…It has been shown to have applications in decision theory [89], economics [10,72], machine learning [132,163,164] and social choice theory [124,154]. Stochastic dominance establishes a relation between probability distributions that is based on the pairwise comparison of the corresponding cumulative probability distributions.…”