Local maxima and the fallacy of jumping to fixed-points

An economist and a computer scientist are walking through the University of Chicago campus discussing the efficient markets hypothesis. The computer scientist spots something on the pavement and exclaims: “look at that $20 on the ground — seems we’ll be getting a free lunch today!”

The economist turns to her without looking down and replies: “Don’t be silly, that’s impossible. If there was a $20 bill there then it would have been picked up already.”

This is the fallacy of jumping to fixed-points.

In this post I want to discuss both the importance and power of local maxima, and the dangers of simply assuming that our system is at a local maximum.

So before we dismiss the economist’s remark with laughter, let’s look at a more convincing discussion of local maxima that falls prey to the same fallacy. I’ll pick on one of my favourite YouTubers, THUNK:

In his video, THUNK discusses a wide range of local maxima and contrasts them with the intended global maximum (or more desired local maxima). He first considers a Roomba vacuum cleaner that is trying to maximize the area that it cleans but gets stuck in the local maximum of his chair’s legs. And then he goes on to discuss similar cases in physics, chemisty, evolution, psychology, and culture.

It is a wonderful set of examples and a nice illustration of the power of fixed-points.

But given that I write so much about algorithmic biology, let’s focus on his discussion of evolution. THUNK describes evolution as follows:

Evolution is a sort of hill-climbing algorithm. One that has identified local maxima of survival and replication.

This is a common characterization of evolution. And it seems much less silly than the economist passing up $20. But it is still an example of the fallacy of jumping to fixed-points.

My goal in this post is to convince you that THUNK describing evolution and the economist passing up $20 are actually using the same kind of argument. Sometimes this is a very useful argument, but sometimes it is just a starting point that without further elaboration becomes a fallacy.

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