lembarrasduchoix asked:

thank you for the introduction to Newcomb’s paradox! Could you do a post on your favorite paradoxes? 

The decision theory paradoxes I’m familiar with are:

• Ellsberg Paradox— Theorists encode both
  1. situations with unknown probabilities, such as the chance of extraterrestrial intelligence in the Drake Equation, and
  2. situations that are known to have a “completely random” outcome, like fair dice or the runif function in R,
  the same way. However the two differ materially and so do behavioural responses to the types of situations. 
• Allais Paradox — The difference between 100% chance and 99% chance in people’s minds is not the same as the difference between 56% chance and 55% chance in people’s minds. (In other words, the difference is nonlinear.) At least when those numbers are written on paper.
  
  Prospect theory proposes the following [0,1]→[0,1] function describing how “we” perceive probabilities
  (Remember that it shouldn’t be taken for granted that everybody thinks the same, or that it’s possible to simnply re-map a person’s probability judgment onto another probability. Perhaps the codomain needs to change to something other than [0,1], for example a poset or a von Neumann algebra.)
• Newcomb’s Paradox — This one has a self-referential feel to it. At least as of today, the story is well told on Wikipedia. The Newcomb paradox seems to undercut the notion that “more is always preferred to less” — a central tenet of microeconomics. However, I believe it’s really undercutting the way we reason about counterfactuals. I actually don’t like this one as much as the Ellsberg and Allais paradoxes, which teach an unambiguous lesson.

Despite the name, they’re not really paradoxes. They are just evidence that probability + utility theory ≠ what’s going on inside our 10^10 neurons. I don’t think Herb Simon would be surprised at that. (Simon is famous for arguing to economists that “economic agents” — both people and firms — have a finite computational capacity, so we shouldn’t put too much faith in the optimisation paradigm.)

You can find out a lot more about each of these paradoxes by googling. As is my way, I’ve tried to provide the shortest-possible intro on the subject. Twenty-two slides opening the door for you.

I also think it’s interesting how the calculus disproves Zeno’s paradox and how a proper measure-theory-conscious theory of martingales disproves the St. Petersburg paradox. I also think Vitali sets and the Banach-Tarski paradox are compelling arguments against the real numbers. Particularly since everything practical is accomplished with (finite) floats, I’m not sure why people hold on to ℝ in the face of those results.

But personally, I’m more interested in decision theory / choice theory than those pure-maths clarifications.

I know I am forgetting several interesting paradoxes which have revolutionised the way people think. (Zeno thought his reasoning was so revolutionary that he concluded, via modus tollens, that the world didn’t actually exist. One of many religions that has come to such a belief, not to mention Neo and Morpheus thought so.)

If I’ve neglected one of your favourite paradoxes, please leave a comment below telling us about it.