Cage Calculus uses an Elo rating system to develop its fighter rankings and win probabilities. Elo is a system that is typically used to rate chess players. Though, it can be used in any competitive sport with head-to-head results. It judges the teams, players, or (in this case) fighters based on those results. The model creates win probabilities and adjusts ratings after each fight based on how a fighter overperformed or underperformed their expected win probability. Our model, described in more detail below, adds adjustments based on method of victory and championship performance as well.

Generating Probabilities

The Cage Calculus model assigns each fighter a power rating. Win probabilities in fights are based on the difference between the two fighters’ ratings prior to the fight. After each fight, each fighter’s rating changes based on how unexpected their win or loss was and how it happened in the cage.

For any fight between two fighters (we’ll call them Fighter A and Fighter B for this example), the probability of Fighter A winning the fight is as follows:

• 1 / [1 + 10^((B – A)/400)]

So, for example, if Fighter A were rated 1750 prior to the fight while Fighter B was only rated 1500, the probability calculation for Fighter A would be:

• 1 / [1 + 10^((1500 – 1750)/400)]

This works out to roughly 0.808, meaning that the Cage Calculus model expects Fighter A to have an 80.8% chance of winning the upcoming fight. If we were to flip the numbers and do the calculation again, Fighter B’s win probability would come out to roughly 0.192, or a 19.2% win probability.

Rating the Fighters

Fighters’ ratings change based on their wins and losses inside the octagon. But, they need a rating at the very beginning of their UFC careers in order to generate probabilities and to climb or fall in the rankings. To fix this, the model gives each fighter a rating based on the performance in other promotions prior (IOPP) to their UFC debuts. Note: a fighter does not enter the Cage Calculus rankings until the week of their UFC debut.

Historically, the Cage Calculus model has taken two similar, but separate approaches to generating IOPP ratings for incoming UFC fighters. The first method applies only to fighters who made their UFC debuts prior to UFC 23. That system is described below:

• A fighter starts at a base rating of 1500
• + 5 points for every win IOPP
• – 15 points for every loss IOPP (only – 5 if that loss was a title fight)
• + 15 extra points for every title won IOPP
• + 5 extra points for every title defense IOPP

For example, Fighter C made their debut before UFC 23. Fighter C’s record coming in was 7-2. Fighter C also won one championship prior and defended that belt once before ultimately losing that title. We would calculate Fighter C’s rating heading into their UFC debut like this:

• 1500 + 5 (7 wins) – 15 (1 normal loss) – 5 (1 title loss) + 15 (1 title) + 5 (1 defense)
• Fighter C’s rating = 1535

The second method began use after UFC 23 and has been used to rate incoming UFC fighters ever since. The reason for the switch is that prior to UFC 23, the UFC (and other promotions) routinely had tournaments where fighters would fight multiple times in the same night. To combat score inflation where one fighter was essentially guaranteed three wins (and often a title) in the same night, titles were worth less and the relative cost of losing was greater.

The system employed after UFC 23 and still used today is as follows:

• A fighter starts at a base rating of 1500
• + 10 points for every win IOPP
• – 20 points for every loss IOPP (only – 10 if that loss was a title fight)
• + 20 extra points for every title won IOPP
• + 15 extra points for every title defense won IOPP

Say Fighter D made their debut after UFC 23. Fighter D’s record coming in was 17-6. Fighter D won two titles in other promotions and defended each belt once for a total of two defenses. Fighter D lost one title fight before coming to the UFC. We would calculate Fighter D’s rating heading into their UFC debut like this:

• 1500 + 10 (17 wins) – 20 (5 normal losses) – 10 (1 title loss) + 20 (2 titles) + 15 (2 defenses)
• Fighter D’s rating = 1630

These same additions and subtractions are used to alter a fighter’s rating if they go to fight for another promotion between fights in the UFC.

Scoring the Fights

Once each fighter has a rating via their performance prior to joining the UFC and by accumulating wins and losses against other rated opponents, we can evaluate how a fighter’s rating will change at the conclusion of a fight inside the octagon.

The formula for creating the new rating is based on a number of factors:

• The Fighter’s incoming rating
• Their opponent’s incoming rating
• The K-Factor (explained below)
• The expected win probability
• The actual outcome (win = 1, loss = 0, draw = 0.5)
• The method of victory/defeat
• Whether it was a title fight

The K-Factor mentioned above is a multiplier that determines how drastically a rating will change based on the outcome. The higher the K-Factor is, the more drastic score changes will be at the conclusion of fights.

Once again, Cage Calculus has historically used two similar, yet distinct, models to determine new fighter ratings that hinge on whether a fight occurred before or after UFC 23. Because fighters fought multiple times in a night, the K-Factor was lowered to avoid the model becoming too jumpy.

The formula for determining the score changes for any fight is as follows:

• Rating + K(Outcome – Probability) x Method x Title

However, what differs are the K-Factors used and what the multipliers are worth. Prior to UFC 23, those variables were:

• K-Factor = 50 for the first 4 fights of a fighter’s UFC career
• K-Factor = 30 beyond that point
• Multiplier for any KO win = 1.33
• Multiplier for a unanimous decision = 1.0
• Multiplier for a split decision = 0.67
• Multiplier for a majority decision = 0.83
• Multiplier for a title win = 1.5

After UFC 23, the variables were shifted slightly to reflect the introduction of more structured rounds, the elimination of tournaments, and the introduction of the three-judge panel to which UFC fans have become accustomed:

• K-Factor = 50 for all fights
• Multiplier for First Round finish = 1.5
• Multiplier for Second Round finish = 1.4
• Multiplier for Third Round finish = 1.3
• Multiplier for Fourth Round finish = 1.2
• Multiplier for Fifth Round finish = 1.1
• Multiplier for a unanimous decision = 1.0
• Multiplier for a split decision = 0.67
• Multiplier for a majority decision = 0.83
• Multiplier for a title win = 1.5
• Multiplier for title in a second weight class = 2.0

So, let’s revisit our first example with Fighter A and Fighter B, rated 1750 and 1500 respectively. We established that the win probability for Fighter A was 0.808 and for Fighter B was 0.192. Let’s say, in a post-UFC 23 fight, Fighter A wins by a knockout in the third round in a non-title fight. The new ratings for each fighter would be:

• Fighter A: 1750 + 50 (1 – 0.808) x 1.3
• New Rating = 1762.5
• Fighter B: 1500 + 50 (0- 0.192) x 1.3
• New Rating = 1487.5

All new fighter ratings are rounded to one decimal place.

Problems of Analysis

This model is without bias. It has no eyes and no feelings. It is operated purely on math and the results of each fight. With that comes other issues that readers ought to keep in mind when assessing the Cage Calculus probabilities.

First, the model does not see what happens inside the cage. For example, when Chris Weidman threw a kick at Uriah Hall and snapped his tibia doing so, Hall was credited with a first round TKO and got a sizable bump in his rating as a result. Hall is not responsible for that freak accident and it does not necessarily prove he was a superior fighter. Nevertheless, the model gives the 1.5 multiplier in such instances.

The only time editorial discretion has been used was in UFC 259. Aljamain Sterling won the bantamweight championship by disqualification after the incumbent champion Petr Yan threw an illegal knee to Sterling’s head. Yan was dominating the fight up to that point. Sterling was given credit by the model for the victory and Yan given the defeat. However, Sterling was not awarded the 1.5 multiplier given to title winners for that fight due to the unprecedented circumstances of his win.

The model also does not take into account other factors such as style. A fighter may be particularly good against wrestlers or grapplers, but not so much against strikers. All the model sees is the rating. The rating is particularly useful and can still speak into the potential outcome of the fight, but it is important to remember it does not tell the whole story.

The same is true for age. The model rewards longevity. If a fighter keeps winning, their rating will continue to climb. However, in a physically demanding sport like MMA, eventually Father Time catches up to all fighters. Ratings will fall as a result of losses (and more quickly, given they are likely to be favorites in these fights). However, given the build up of ratings points in their younger years, older fighters are likely to be overestimated by the model in many cases.

Due to the fact that the UFC cuts its underperforming fighters and seeks to add the best talent, there is a significant degree of score inflation. When analyzing who might be the best all time, it is important to realize the average score has climbed significantly. When the UFC began, the average was roughly 1501. As of this post, the average sits at 1648 and has peaked as high as 1665.8.

Lastly, understand that this model generates probabilities, not hard and fast predictions. For example, if a fight card had ten fights on it and each of the favorites had 60% win probabilities, the model only expects that its projected favorite will win 6 times over the course of that night. An underdog winning does not necessarily mean the model was wrong.

Going Forward

As a result, as part of every fight night preview, the analysis will provide some subjective views of what might be happening outside the octagon to give context to the model’s projections. Happy analyzing!