Sigmoid perception

Sometimes my mental vocabulary isn’t very rich for extreme situations. Rather than thinking of temperature like this:


I have different mental “buckets” for various temperature ranges, depending on what I would need to feel comfortable:IMG_1110

Notice how the intervals between different experiences get wider the farther you get from the middle, all the way to “Too hot!” and “Too cold!” which extend forever? My mental thermometer, with a mark for each perceived grade of temperature, isn’t very sensitive to extremes—you could graph my experience as a function of temperature, and see how the curve flattens out near the ends:

IMG_1111A curve with this “S”-shape—steeper in the middle, flattening out toward the ends—is called sigmoid, sigma being the Greek letter for the “S” sound. My perception of temperature isn’t the only thing with a sigmoid shape: my mental buckets for stages of musical ability look something like

  • has no skill
  • can play a few notes nicely
  • can play a few songs nicely
  • has a decent-sized repertoire
  • can learn new music quickly
  • makes money from music
  • makes enough money to live on
  • is world famous for their musical talent

But when I think about how much variety there is within each bucket, and how much time it takes to move from one bucket to the next, it seems like the buckets on the ends are much larger than the ones in the middle. If you know a lot of pieces, it’s not that hard to pick up new ones, but there’s a long way to go between earning a few bucks on the side and quitting your day job. And even farther to rocket from “professional” to “internationally acclaimed”—I just don’t have the mental vocabulary to describe that process with any nuance.

IMG_1112And if you think about how you would feel about an unexpected debt or windfall, that probably follows a sigmoid shape too: I have a good sense of exactly how much a gain of $100 would feel compared to a gain of $150, but if I were suddenly tens or hundreds of million dollars in debt, the exact number of zeroes probably wouldn’t have much of an effect on my psyche.


So what is the upshot of having sigmoid perceptions? Well, in a way it’s good to be more sensitive to the differences between small variations than to differences between large ones: it’s more efficient. With limited mental space, it makes sense to have a larger spread of mental vocabulary for the most common sorts of experiences, because those are the ones we think about, talk about, decide between, etc. Putting equal mental focus on the difference between $0 and $1 and on the difference between $1,000,000 and $1,000,001 would be like having to say “perambulate” and “confabulate” instead of “walk” and “talk”: if everything took up that much space in your brain, you wouldn’t be able to think very many thoughts.

The downside of sigmoid perception is that it leaves you subject to framing effects. A classic experiment in prospect theory is to offer one group of people the following choice of “gambles”:

You are given $1000. In addition, would you now rather:

  • A) Receive $500?
  • B) Or receive $1000 with a 50% probability?

And to offer another group a similar choice of losses:

You are given $2000. Would you now rather:

  • C) Lose $500?
  • D) Or lose $1000 with a 50% probability?

An overwhelming majority of the first group prefer (A) to (B)—why risk a sure thing?—but an overwhelming majority of the second group prefer (D) to (C)—there’s a chance not to lose anything—even though (A) and (C) are the same, and (B) and (D) are the same! Why does that happen?

It’s explainable with sigmoid perceptions. While small gains and losses might be perceived accurately, the larger the windfall (or stroke of misfortune) the less sensitive you are to how big it actually is:

Left: A series of gains and losses. Right: A sigmoid perception of the same gains and losses.

So consider gambles (A) and (B) from the experiment and flip a coin—do you want to receive $500 regardless, or $1000 if it was heads and nothing if it was tails?


From a sigmoid perspective, the choice looks like this:

IMG_1115 (1)

Do you want two chances of money, or just one? The choice is clear! And that’s kind of how my gut feels about the two options: I could have a sure $500—what if I go for the thousand and lose?

But when the question is reframed as a choice between losses, the comparison looks like this:IMG_1117

Now it seems that the 50% chance of losing nothing is clearly better than the sure loss! So this kind of sigmoid perception can make you vulnerable to being manipulated, and highlights the importance of choosing a good baseline.

So how can you extend the middle, linear region of your sigmoid curve so that you can be more sensitive to extremes and less vulnerable to framing? I can think of two approaches:

1. Zoom out! It’s easier to be consistent about choices between gambles (100% chance of $500, or 50% chance of $1000?) if you put them into a larger context, say, a choice between a 100% chance of my bank account balance going from $4680 to $5180 versus a 50% chance of going from $4680 to $5680. (Note: not my real balance.) That makes it easier to decide whether you really want to take the risk, and not just have your sigmoid distortion make the choice for you.

2. Spread out! Another strategy is to get more experience with the extremes to which you’re insensitive, or at least to think about or talk about them more. Since moving to Minnesota, I have a much larger mental vocabulary for cold weather: is it cold enough that I need my flannel-lined jeans? Cold enough to get that tingly snot-freezing feeling in my nose? Too cold to snow? The lower branch of my temperature curve has shifted out accordingly:IMG_1118

I’m curious to know if any of these ideas resonated with you! Notice any sigmoid perceptions in your life? Have any subtler associations with 100+ degree weather than “Too Hot!”? I’d love to know in the comments!

4 thoughts on “Sigmoid perception

  1. Definitely have the vocabulary for the hot end of the spectrum- we are about to be in your “too hot” region for the next three months with very few exceptions. I think my vocabulary for that tends to look something like how long do I have to air condition the car before I can safely put my kids in and/or touch the steering wheel without getting burned…

    This was a great post! SUPER interesting to think about those framing situations/how we think about money!


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