How to transform all multiple solutions of the Kemeny Ranking Problem into a single solution

Preference aggregation as a problem of a single consensus ranking determination, using Kemeny rule, for m rankings, including ties, of n alternatives is considered in the paper. The Kemeny Ranking Problem (KRP) may have considerably more than one optimal solutions (strict orders or permutations of the alternatives) and, hence, special efforts to deal with this phenomenon are needed. In the paper, there is proposed an efficient formal rule for convolution of the N multiple optimal permutations, the output profile β(N, n), into an exact single final consensus ranking, which can include ties. The convolution rule is as follows: in the final consensus ranking, alternatives are arranged in ascending order of their rank sums (total ranks) calculated for the output profile β; some two alternatives are considered to be tolerant if they have the same rank sums in β. The equivalent convolution rule can be also applied as follows: in the final consensus ranking, alternatives are arranged in descending order of row sums (total scores) calculated for a tournament table built for β; some two alternatives are deemed to be tolerant if they have the same row sums. It is shown that, for any alternative, its total rank and total score are equal in sum to the output profile dimension N×n. The convolution rules are validated using Borda count.


Introduction
In this paper we consider a preference aggregation procedure in the form of Kemeny rule. Being essentially the multiple heterogeneous properties measurement in an ordinal scale, preference aggregation belongs to a circle of problems relevant to measurement science [1]. Additionally, in the recent few years, the preference aggregation and the processing of ranked data have attracted considerable interest within a much broader consideration covered by such a rapidly developing area of artificial intelligence as machine learning [2]. Appropriate methods are used to solve newly emerging problems, such as crowd sensing/labeling, sentiment analysis, meta-search engines, interval data fusion and others [2][3][4][5][6][7].
The preference aggregation is a single consensus ranking determination for m rankings (voters), possibly including ties, of n alternatives (candidates). This is a classical problem that has been intensively investigated firstly as a Voting Problem in the framework of Social Choice Theory [8].
Let we have a set A = {a1, a2, ..., an} of n alternatives and a set / = {O1, O2, ..., Om} of rankings defined over the set A, where each of m rankings (also called preference relation or weak order) has a form of chain of size n, e.g. O = a2 a1 ... as ~ at ... ap ~ aq, and may include , a strict preference relation U, and ~, a tolerance relation (or tie) W, so that O = UW. The set /(m, n) is called a preference profile for the given m rankings of n alternatives.
Let 3n be a set of all n! strict order relations on A; each strict order U 3n is in one-to-one correspondence with a permutation of first n natural numbers fn = {1, 2, …, n}; in this sense, we will treat notions of strict (or linear) order and permutation as synonyms. Preference aggregation goal is to determine a consensus ranking E that is a single preference relation that would give an integrative characterization of the profile /(m, n).
There are many rules to find the consensus ranking [9]. We use the Kemeny rule [10] consisting in determination of such linear order (Kemeny ranking) E 3n of the alternatives of A that the distance D(β, /) (defined in terms of the number of pairwise inconsistencies between rankings) from E to the rankings of the initial profile /(m, n) is minimal for all possible strict orders U (permutations), that is arg min Clearly since the set 3n is finite, the Kemeny ranking always exists, however it is not always unique as the optimal distance value may correspond to not the only transposition of the profile matrix rows and columns, that is where N is a number of Kemeny rankings (optimal permutations) [2,3,11,12]. Furthermore, it was shown in [1] that in certain situations the number of multiple optimal permutations may exceed 10 7 even for small m = 4 and n = 15 and also the solutions may rank the alternatives in significantly different ways what produces much more ambiguity than that inherent in the initial profile /. In order to solve the KRP we use the recursive algorithm of our own design, RECURSALL, implementing the branch & bound technique, allowing to find all possible Kemeny rankings for a given preference profile [4]. It is convenient to term the initial preference profile /(m, n) as input profile and the multiple optimal permutations as output profile %(N, n) = {E1, E2, ..., EN}, % 3n (see figure 2). Clear that special efforts to deal with the phenomenon of the KRP multiple optimal solutions, where Ns !! m, are needed. The rest of the paper is devoted to development of an efficient formal rule for convolution of the output profile into a single final consensus ranking Efin. Then, for all i < j, i, j = 1, …, n, ri < rj ai aj and ri = rj ai ~ aj, where both relations and ~ are in the single final consensus ranking Efin. Thus, the final consensus ranking may include tolerances as well as an arbitrary Ok /, i.e. generally speaking, Efin = UW and Efin 3n. The convolution rule (3) guarantees determination of an exact single optimal permutation Efin for any output profile %.

Convolution rule justification
where both relations and ~ are in the single final consensus ranking Efin.
The convolution rule (4) guarantees determination of an exact single optimal permutation Efin for any output profile %, and the single optimal permutation is the same as that found using the rule (3). In fact, the rule (4) coincides with the preference aggregation rule well known as Borda count [13]. 1 For the indicator function, the following statements are hold: (ai, aj) U ai aj and (ai, aj) U ai % aj.

Initial profile matrix P
The profile matrix re-arranged by consensus ranking E 1 Example 3. Table 2 shows the values of scores and rank sums calculated for the profile alternatives from Example 2. The alternatives are sorted in accordance with rules (3) and (4). It can be seen the fairness of expression (5) for these data. The expression (5) can serve as a basis for justification of the convolution rule (3) validity. Indeed, let a social score function f assigns a nonempty choice set C A to a profile B. It was proved by H.P. Young in [13] that "for any fixed number n of alternatives, there is one and only one social score function that is neutral, consistent, faithful, and has cancellation property -namely, Borda's rule". The properties of the function f listed in the Young's statement are deemed as the set of axioms necessary and sufficient in order to have a fully characterized preference aggregation rule [14]. Thus, the scores zi, implementing Borda rule, guarantee that the rule (4) leads to convolution of the output profile В(N, n) into the exact and single consensus ranking Efin. It follows from (5) that social score functions f coincide for the rules (3) and (4).

Conclusion
It was shown in the paper that the multiple optimal permutations, i.e. the output profile %(N, n) = {E1, E2, ..., EN}, as solutions of the KRP for m rankings, including ties, of n alternatives can be efficiently transformed into an exact single final consensus ranking Efin, which can also include ties. The proposed convolution rule is as follows: in the final consensus ranking Efin, alternatives are arranged in ascending order of their rank sums calculated for the output profile %; in Efin some two alternatives are tolerant to each other if they have the same rank sums in the output profile. The equivalent convolution rule can be also applied as follows: in the final consensus ranking Efin, alternatives are arranged in descending order of row sums (total scores) zi calculated for a tournament table S of the output profile %; in Efin some two alternatives are tolerant to each other if they have the same row sums of the tournament table.
It was also shown that, for any i-th alternative, its total rank ri and total score zi are equal in sum to a constant value Nn, that is the dimension of the output profile %. The convolution rules validity has been shown using such well known preference aggregation rule as Borda count.