How Long Does the GOAT in Sports Usually Stay the GOAT?
You can’t stay king (or queen) forever. The Great One will eventually just become a great one. The title of GOAT (greatest of all time) will inevitably be passed down to a worthy heir. But how long is someone’s reign likely to last?
If you look at all the major US sports, there is a current player who is knocking on the door to be the GOAT – Lebron, Brady, Ovechkin, and Trout. Would it be weird if all of these guys took the crown? That is, would it strike anyone as out of place that the best player in each of the four major US sports over the last 70+ years actually played in the last month?
In this article, we want to look at the reign of the GOAT from a statistical perspective. How many years is a GOAT’s reign likely to last? How often would we expect the GOAT title to be passed down? This, and other questions, can be answered with the tools of combinatorics and statistics.
To receive email updates when new articles are posted, use the subscription form below!
Assigning a Number for Player Quality
Throughout this entire article we’re going to simplify matters when it comes to measuring “the GOAT”. Usually, determining the GOAT requires debates about efficiency, longevity, peak performance, championship counts, and other factors. We’re going to assume that a player’s career quality can be boiled down to one number (like we did in our article about the top 10 WNBA players ever).
You can think of this number as representing something meaningful if it helps. For example, maybe we measure the GOAT by measuring total wins that player earned. Or maybe the GOAT is measured by estimating the number of points they contributed on offense and subtracting the points they allowed on defense.
No matter how you like to think of it, it is helpful to equate player quality with a single number. If we do this, then determining the GOAT boils down to finding the highest number. It is much easier to talk about mathematically when we write things this way.
A Combinatorics Approach to Changing GOATs
When player quality is represented by a single number, the GOAT is just the highest number in the list. A new GOAT occurs if the newest or last number on the list is also the largest. This corresponds to the newest player being a better player than everyone before them! Can we estimate the probability of this?
We’ll take a straightforward combinatorics approach to this question. Combinatorics is the study of ordering, combining, counting, and doing other operations with finite lists of numbers. The question we want to ask can be formalized in the following way.
Suppose N-1 players have ever played professionally in the league. If the GOAT is going to change, that means that the N^{th} number will be the largest in the list. The probability that this happens is very simple: \frac{1}{N} .
However, we are more interested in the length of time somebody reigns as the GOAT for, not necessarily the number of players. Perhaps a better question is this: In a given year, what is the probability that the title of GOAT changes from one player to another.
Now, we are talking about groups of players instead of individuals. For example, in the NBA 60 players enter into the league at a given time. Depending how you count, the NBA has existed for roughly 50 years. That means that roughly 3000 players have already entered the league.
If the title of The Goat is going to change in the current year, it means that one of the new 60 players is going to be better than the previous 3000. Perhaps the easiest way to think about this problem is to break down the league into years. We want to know the probability that someone in year 51 is better than anyone in the previous 50 years. This probability is going to be \frac{1}{51} or about 2%
This gives us a way to estimate the probability of the GOAT title changing in a given year. Does it work?
Evaluating the Accuracy of our Combinatorial Approach
In the previous section, we argued that if somebody is crowned goat in the N^{th} season, the probability that they are dethroned in the next year is \frac{1}{N+1} . The year after that, the probability they are dethroned is \frac{1}{N+2} , and so on.
That means that if someone is crowned the GOAT in a league’s N^{th} year, then by the 2N^{th} season, there is a 50% chance there is a new GOAT. Let’s look at the NBA to roughly check the accuracy of this model.
A good indication that this model works pretty well is that the length of the reign of the GOAT should roughly double every time the crown is passed.
This website lists the progression of the NBA GOAT via their own metric. We can summarize the change of GOAT status via the following table.
Player | League Age in Crowning Year (from 1950) | Reign Length |
|---|---|---|
George Mikan | 0 | 4 |
Dolph Schayes | 4 | 9 |
Bill Russell | 13 | 3 |
Wilt Chamberlain | 16 | 13 |
Kareem Abdul-Jabar | 29 | 35 |
Lebron James | 64 | – |
Notice that if the Crown had passed from Dolph Schayes to Wilt immediately without passing through Russell, then the length of the reigns roughly double every time! From 4, to 9, to (hypothetically 16), to 35 years. This means that our model is working really well.
What Does this Mean for the Lebron James
We should expect Lebron’s reign as the GOAT NBA player to last around 60-70 years. We might expect somebody to take the crown from him in the 120th-130th NBA season. Because this is measured from 1950, this means that there is a good chance that Lebron James is considered the best NBA player until roughly 2070-2080.
There are a few ways this might not come true. First of all, the media generally has a recency bias. A 25 year old beat writer in 2050 who has never seen Lebron James play a game of basketball would probably favor newer players and transfer the crown prematurely.
A secondary possibility is that player quality is likely to improve over time. Our combinatorial approach only works if a random player in 1950 is as good as a random player in 2020. This probably isn’t the case. A random player today is certainly better than a random player from back then. This effect means that a GOAT’s reign will not be as long.
It is generally impossible to measure either of these effects, but in the next section we run a few experiments numerically to see what other conclusions can be made.
A Simple Experiment for GOAT Longevity
In the last section, we didn’t have to talk at all about probability distributions. In this section, we’re going to use statistical tools instead of combinatorial tools so will need to assume a distribution. In particular, we will study progressive maximums of samples from a standard normal distribution.
We sampled a standard normal distribution 75 times (roughly the age of the NBA) and recorded the max value through the first N samples. This progressive maximum is plotted over two different runs below.
In both cases, even though the patterns are different, there are some similarities. The first plot shows 4 different GOATs over time while the second plot shows 5. We ran 10,000 experiments and computed an average of 4.85 GOATs over a 75 year period. The modern NBA according to the link I’ve used for reference has experienced 6. This is pretty well within reasonable bounds.
Notice also that as time goes on, somebody usually remains the GOAT for a longer period of time. This is what we saw with the roughly doubling of the length of the GOAT’s reign using the real NBA data in the last section.
What happens if we use an older league? Baseball which has been around for closer to 125-150 years, again depending how you count. Here are some sample plots using 150 samples.
Again, we see a very similar pattern. As the number of samples – or the age of the league – increases, we see longer reigns and fewer changes from one GOAT to the next.
In the MLB, we ran 10,000 experiments and have computed an expected value of roughly 5.6 GOATS. It might be kind of fascinating that in a league that is twice as old, we only expect the GOAT list to be one longer. Look at the second example in the picture above, though. The light blue GOAT was on top for nearly 100 years. When you get that good, you get harder to dethrone.
The Harmonic Series
Those of you paying especially close attention might notice that there is an application of the harmonic series floating around here. The harmonic series is the sum of reciprocals of the integers. Specifically, the harmonic series is the sum 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots . This series famously diverges.
In our setting, the probability that the GOAT changes as each year passes is the probability that the last element in the list is the largest, \frac{1}{N} . Therefore, the expected number of GOAT changes in the first M years of the league is \sum_{i=2}^M \frac{1}{i} . This is the harmonic series.
Commentary
There is one peculiarity with the analysis we’ve done. First, all of our analysis assumes that the league quality isn’t getting any better or worse over time. If the quality of players changes dramatically over time, you get weird results.
The most important takeaway from all this analysis is that every time the GOAT title changes, it will likely stick for longer and longer. That is, the GOATs of today will be talked about for much longer than the GOATs of the past.
To receive email updates when new articles are posted, use the subscription form below!
