Bayesian inference of the climbing grade scale

TL;DR

This paper applies Bayesian inference and Markov chain Monte Carlo methods to estimate the climbing grade scale, revealing it as a logarithmic measure of difficulty consistent across various grading systems.

Contribution

It introduces a rigorous statistical approach to quantify climbing difficulty scales and estimates the fundamental scale parameter linking different grading systems to difficulty.

Findings

Climbing grade scales are logarithmic in difficulty.

Increment in Ewbank, French, and UIAA grades corresponds to about 2.1 times difficulty increase.

V-grade scale for bouldering corresponds to a 3.17 times difficulty increase.

Abstract

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.