  Climbing grades are used to classify a climbing route based on its perceived
difficulty, and have come to play a central role in the sport of rock climbing.
Recently, the first statistically rigorous method for estimating climbing
grades from whole-history ascent data was described, based on the dynamic
Bradley-Terry model for games between players of time-varying ability. In this
paper, we implement inference under the whole-history rating model using Markov
chain Monte Carlo and apply the method to a curated data set made up of
climbers who climb regularly. We use these data to get an estimate of the
model's fundamental scale parameter m, which defines the proportional increase
in difficulty associated with an increment of grade. We show that the data
conform to assumptions that the climbing grade scale is a logarithmic scale of
difficulty, like decibels or stellar magnitude. We estimate that an increment
in Ewbank, French and UIAA climbing grade systems corresponds to 2.1, 2.09 and
2.13 times increase in difficulty respectively, assuming a logistic model of
probability of success as a function of grade. Whereas we find that the Vermin
scale for bouldering (V-grade scale) corresponds to a 3.17 increase in
difficulty per grade increment. In addition, we highlight potential connections
between the logarithmic properties of climbing grade scales and the
psychophysical laws of Weber and Fechner.
