Group choice using matrix norms

The article describes the approach to the construction of methods of the group choice and ranking of objects in order of preference, based on the minimizing the deviation of the matrix, characterizing objects (of an evaluation matrix) from some peer matrix, the columns of which are the same (the matrix of consistent ranking). To evaluate the deviation is proposed to use matrix norms: $p$-norm, $p-q$ norm, Schatten norm based on the difference of evaluation and peer matrix, on the difference of their covariances matrix, as well as on other forms. It is proved that the ranking, obtained by minimizing the difference between the evaluation matrix of ranks and matrix of consistent ranking by the Frobenius matrix norm coincides with the ranking obtained by evaluation matrix of the ranks by the Borda rule. It is considered the connection between matrix of consistent ranking, obtained by the Frobenius matrix norm with peer matrix in the singular decomposition of an evaluation matrix and related results of ranking by influence method. For matrix $p$-norm is proved that under sufficiently large exponent matrix norm the set of rankings, that give the minimum of this matrix norms from the difference between the evaluation matrix and a matrix consistent ranking, becomes stable — does not change during the subsequent increase in the degree (the results of this a ranking are called balanced ranking). The examples show that balanced ranking gives the minimum losses under non-linear increase of penalties from mismatches ranking with actually realized ranking.