An Evidence-Based Approach to Measuring Cumulative Climbing Performance

Many attempts have been made to produce a reliable means of ranking climbing performances, especially in the context of cumulative performances (e.g yearly rankings, gym rankings, single-day competitions). Traditional methods, such as simply counting the number of ascents or the highest grade achieved, often fail to capture the nuances and challenges faced by climbers. A new metric, which we call the “Cumulative Performance Grade” (CPG), offers a more evidence-based approach to measuring cumulative performance in climbing.

The CPG is based on empirical studies that have fitted logistic regressions to ascent data, revealing a consistent pattern: for sport climbing (especially from grade 23/5.11d/7a onwards), the number of failed attempts before success, on average, doubles for each increment of grade (Drummond and Popinga, 2021). This means that for each increment in grade, a climber typically requires twice as many failed attempts before successfully climbing the route. For bouldering in the V-grades, the multiplier for failed attempts needed before success is approximately $e$ (2.71828) for each grade increment.

By taking into account the exponential nature of the relationship between grades and the average number of attempts required for success, the CPG provides a more accurate representation of a climber’s overall performance. The formula for calculating the CPG depends on the grading system used:

For sport climbing (Ewbank, French sport, and Yosemite decimal system):

\[CPG = \log_2(\sum_i 2^{\text{grade}_i})\]

For bouldering V-grades, the natural log is arguably a better scale:

\[CPG = \ln(\sum_i e^{\text{grade}_i})\]

where the i’th “grade” is the numerical representation of the climbing grade (e.g., 23 for 5.11d, 24 for 5.12a, etc. for sport climbing, and 0 for V0, 1 for V1, etc. for bouldering).

Choice of Slopes

While the fundamental empirical feature is that the slope between grades and failed attempts is exponential, the precise multiplicative factors are subject to estimation error.

The actual slopes estimated in the Drummond and Popinga paper are 2.1 and 3.17 for sport and bouldering V-grades, respectively. Rather than using these exact numbers (which may be refined by further research), we chose to use 2 and $e$ mainly for simplicity.

Since the numbers we have chosen are slightly lower than the estimated slopes, this could potentially slightly downweight harder climbs compared to their actual expected difficulty.

However, the data used to estimate these numbers was mostly based on amateur climbers, and there is some evidence that top-level and elite climbers have shallower slopes, meaning they are better able to translate their skills to harder climbs. For example, ascentstats.com estimated that Adam Ondra’s V-grade slope was actually 2.0, so using $e$ might even be too generous for some top-level athletes.

It is widely recognized that the bouldering grade bands are broader than sport climbing grade bands, thus the choice of 2 and $e$ reflects this difference while maintaining a simple and intuitive formula.

The CPG effectively weights each climb based on its difficulty, giving credit to harder ascents proportional to the expected increase in difficulty. This approach acknowledges the empirically measured increments in effort needed to succeed at higher grades and provides a fairer comparison between climbers with different preferences for allocation of effort.

One interesting interpretation of the CPG is that it represents the grade that a climber could have potentially sent if they had focused all their performance effort on a single route. This perspective highlights the cumulative nature of the metric and emphasizes the idea that the CPG captures the overall performance potential of a climber based on a body of work.

Michaela Kiersch’s Performance at the 28th Annual Hueco Rock Rodeo

To illustrate the application of the CPG, let’s consider professional climber Michaela Kiersch’s winning ticklist from the 28th annual Hueco Rock Rodeo outdoor bouldering competition in Feb 2024:

  1. Mr. Serious (V8)
  2. Better Eat Your Wheaties (V9)
  3. Swiss Crisp Mix (V10)
  4. Left Chupacabra (V10)
  5. Landjager (V11)
  6. Sunshine (V11)
  7. Rumble in the Jungle (V12)
  8. Phantom Limb (V12)
  9. Thrilla in Manilla (V12/13)
  10. Crown of Aragorn (V13)

Using the formula for bouldering, we calculate Michaela’s CPG as follows:

\[CPG_{MK} = \ln(e^8 + e^9 + 2e^{10} + 2e^{11} + 2e^{12} + e^{12.5} + e^{13}) \approx V14.01\]

This CPG value suggests that Michaela’s performance over the course of the competition was equivalent to sending a V14 boulder problem (if we had chosen base 2, then it would have been equivalent to V14.83!). A remarkable achievement.

The applications of the CPG are numerous. In competitive settings, such as “Rodeo” style boulder competitions, where climbers attempt to complete as many hard problems as possible within a given time frame, the CPG can be used to determine the winner based on the cumulative difficulty of the problems they have solved. This eliminates the need for subjective cutoffs for number of routes included and arbitrary scoring systems not based on empirical evidence. It ensures that the climber who has demonstrated the highest level of performance across the competition is rightfully recognized, whether they choose to focus on a single hard climb, or many routes they can do in 1-2 goes.

Furthermore, the CPG can be employed in annual rankings, such as those provided by platforms like 8a.nu. By calculating the CPG for each climber based on their yearly ascents, these rankings can offer a more comprehensive and equitable assessment of climbing performance, again without arbitrary limits on the number of climbs considered. The exponential drop off of contribution to score with lower grades guarantees that an appropriately large (but not too large) weight is placed on the climbs with the highest grade. This not only rewards climbers who consistently push their limits across a range of grades but also encourages participation and progression throughout the climbing community.

My performance on a week-long trip to the Blue Mountains

During a week in the Blue Mountains over Easter 2024 I compiled the following list of sends:

  1. Snappy Dresser (24/7a+/5.12a), Bell Crag - flash
  2. Good Big Dog (24), The Freezer - red point
  3. Survival Day (24), Sublime Point - second go
  4. Cutopia (23/7a/5.11d), Porters Pass - onsight
  5. Aroma Gunsmoke (23), The Freezer - flash
  6. Searching the Light (23), Bell Crag - second go
  7. Hairy Horrace (23), Centennial Glen - flash
  8. So, Said the King (21), Porters Pass - flash
  9. Nice Vice Baby (20), Porters Pass - flash
  10. Old Blobby (20), The Freezer - flash
  11. Double Standards (19), The Freezer - flash

The CPG of this tick list is 26.4 (approximately 7b+/c or 5.12c/d), which is to say, that if I had picked an appropriate 26/7b+/5.12c and spent the week on it, I would have had a good chance of sending. And indeed, the four 26’s I have sent have taken me around 5 sessions each on average.

The CPG allows you to quantify a body of climbing work and compare compulsive onsight/flash climbers with those climbers that prefer longer projects.

In conclusion, the Cumulative Performance Grade represents a significant step forward in the way we measure and compare cumulative climbing performances. By leveraging empirical evidence and mathematical modeling, the CPG provides a robust and unbiased metric that better captures the level of performance behind a climber’s accomplishments. As the climbing world continues to evolve, the adoption of the CPG has the potential to provide a universal index for outdoor climbing competitions, rankings, and the way we celebrate the achievements of climbers worldwide.

References

Drummond, A., & Popinga, A. (2021). Bayesian inference of the climbing grade scale. arXiv preprint arXiv:2111.08140.