Life as a Sigmoid Function
Exponential growth is one of the most important mathematical tools used to model the world, and we are surrounded by examples. Human population growth over the last several millennia, compound interest, or the rate of user acquisition for popular startups are all good demonstrations.
While exponential growth can generate lots of new value, it cannot last forever in a finite universe. The Earth will not be able to hold an ever increasing population, we will run out of money, and Facebook won’t be able to find new users to hook. In reality, many exponential like trends are better approximated by a close cousin of exponential functions — sigmoid functions.
The sigmoid function is a bounded, positive function for all x values (It also happens to be the integral of the normal distribution!). At negative x values, the exponential and sigmoid curves look identical, and both functions start to grow exponentially at their critical points. However, a sigmoid will level off at some peak amplitude, which is more accurate for growth models in a finite world. In addition to modeling growth, sigmoid functions have found use as activation functions for neurons in neural nets.
Let’s look at the relevant variables of the sigmoid function.
A sigmoid function has three fundamental components — amplitude, rise rate, and horizontal shift. The amplitude determines the height at which the sigmoid levels off, while the rise rate dictates how quickly the curve goes from zero to that amplitude. Lastly, the horizontal shift determines where on the x-axis the rise will occur. As shown below, adjusting each parameter will cause the curve to vary accordingly.
One of the crucial features of exponential and sigmoid growth is that for a long time, it appears that nothing is happening. Once the critical point is reached though, the subject seems to explode over night. However, this is an illusion — the “overnight” explosion would not have been possible if not for the preceding, undetectable growth earlier in the function.
While finances and populations offer great examples for sigmoid growth, it is harder to identify it on an individual level. Self observation difficult. What you choose to measure about yourself and how you do it can be a murky, qualitative task. However, I think that there are several individual characteristics that can display the sigmoid growth pattern. Our own capacity for producing value on a personal level can be modeled by one (or several) sigmoid functions. Whether it is our ability to coach a sports team, publish papers, earn money, or do any other skill, each can be modeled by a sigmoid function that evaluates our capacity to fulfill that role.
Education is one of the best examples. We start in kindergarten, and for the next thirteen years are given more and more advanced training while seeing little benefit from it. But at the end of high school, we are capable of holding a certain number of jobs that would have been impossible if not for the educational investment. If students continue on through college, they spend more and more time at the bottom part of the curve, extending their takeoff time and seeing very few results for their effort. It can be frustrating to invest this time learning without seeing any tangible results, but the completion of a degree marks a significantly higher peak amplitude of production capacity than a high school diploma.
When a person reaches the “ignition point” of the sigmoid curve, it marks a turning point in their capabilities. Finally mastering one lab technique can allow you to run two new experiments, which turns into 4 published papers. Those published papers lead to introductions with 8 new people, and one of those relationships evolves into a collaborative project that is 16 times more impactful than your previous one. Sports offer another example — months and months in the batting cage during the offseason will offer no results for a long time, until you get into the first game of the new season and a 90 MPH fastball is much easier to hit than it was last year.
As mentioned earlier, sigmoids best model our production capacity, not our actual value produced. The lifetime value we produce is just the integral over time of our capacity sigmoid function, and is approximately linear. Thus, there are different paths to achieve the same lifetime production values. An early takeoff to a moderate amplitude can produce the same value as a later takeoff to a high amplitude. This is demonstrated by the classic plumber vs. doctor story — a plumber can have a higher net worth than a doctor into their 40s or 50s, because the plumber’s monetary sigmoid curve hit its critical point years before the doctor’s did.
Because there are so many pathways to value, everyone’s sigmoid curves will be different. The function parameters will certainly be affected by quality of training, resources invested, and any inherent bias such as racism or sexism. However, I also believe that the level of effort expended, ability to work creatively, and plain old grit can affect a personal sigmoid curve. These are all things that are within our power to control. In particular, the amplitude you rise to is limited only by your willingness to work hard and by when you choose to settle for “good enough”. Your rise rate is also affected by work ethic, and by your ability to combine new ideas and practices together — such as integrating stretching with running, or doing two new lab techniques together. Lastly, your right shift will be affected simply by how long you have been plodding along your sigmoid function.
Ultimately, what aspects of your personal life matter depends on your own value system. I offer no answers to that question, but I’m sure you can find a missionary or philosopher who can help you clarify it. A person who values money will have a salary sigmoid curve with a large amplitude, while a person who values friendships and family time may have a lower net worth peak, but a higher emotional satisfaction sigmoid.
What can we take away from this? In order to grow, you have to be patient. Days, weeks, and months can be spend on the left hand side of a sigmoid function, working hard to hone your craft but seeing nothing in return. But at some point, if effort keeps being steadily invested, you will reach that critical point where everything starts to come together, and you will realize that you are exponentially more capable than you once were. So be patient, and keep at it!
