A Note on Paired Comparison Rankings

Abstract

If $m$ objects $x_1, x_2, \cdots, x_m$ are compared pairwise, then let $s_{ij}$ denote the number of times $x_i$ beats $x_j$ in $n_{ij}$ independent comparisons. In a ranking, if $x_i$ precedes $x_j$ then one may require that the probability of $x_i$ beating $x_j$ be at least $\frac{1}{2}$. Such a ranking is called weak stochastic ranking. Let $I(R)$ be the set of all pairs $(i, j)$ such that $x_i$ precedes $x_j$ in the ranking $R$ in spite of the paired comparison outcomes resulting in $s_{ij} < s_{ji}$. A statistic $D(R) = \sum_{I(R)} (S_{ij} - S_{ji})^2/n_{ij}$ is derived and proposed for estimating a weak stochastic ranking. Since $D(R)$ is seen to be the sum of a random number of asymptotically distributed chi-square variates, a ranking is called minimum chi-square weak stochastic if $D(R) \leqq D(R_t)$, for $t = 1,2, \cdots, m$! It is proved that minimum chi-square rankings share at least two properties with the maximum likelihood rankings. That is, every minimum chi-square ranking is Hamiltonian ranking and when in particular $n_{ij} = 1$, every minimum chi-square ranking minimizes the violations of observed outcomes. Moreover, the branch and bound algorithm can be used for estimating the minimum chi-square rankings.