  The Binder ratios exhibit discrepancy from the Gaussian behavior of the
magnetic cumulants, and their size independence at the critical point has been
widely utilized in numerical studies of critical phenomena. In the present
article we reformulate the nonequilibrium relaxation (NER) analysis in cluster
algorithms using the $(2,1)$-Binder ratio, and apply this scheme to the two-
and three-dimensional Ising models. Although the stretched-exponential
relaxation behavior at the critical point is not explicitly observed in this
quantity, we find that there exists a logarithmic finite-size scaling formula
which can be related with a similar formula recently derived in cluster NER of
the correlation length, and that the formula enables precise evaluation of the
critical point and the stretched-exponential relaxation exponent $\sigma$.
Physical background of this novel behavior is explained by the simulation-time
dependence of the distribution function of magnetization in two dimensions and
temperature dependence of $\sigma$ obtained from magnetization in three
dimensions.
