K-Level Reasoning

Imagine a game where participants choose a number between 0-100, with the goal of guessing as close to 2/3 of the average as possible. So if the average guess was 50, the correct guess would be 33.3.

How would you solve it?

I couldn't tell you the answer this morning, but now I can (I think). To solve it, you would use k-level reasoning.

K-level reasoning

There isn't a good one-sentence definition of this, but I'll try my best. Someone who is a k-level 0 player, in a given circumstance, acts completely non-rationally and random. A level-0 player guessing the all-too-common "how many jelly beans are in the jar" problem, just guesses a random number.

A >1 level player assumes all other participants are level (k – 1). So a level-2 player assumes everyone else is a level-1 player; a level 5-player assumes everyone else is a level-4 player.

Now back to our problem.

A level-0 player attempting to solve our problem would just choose randomly as normal. Nothing special there.

But a level-1 person would assume everyone else is level-0, so they're all guessing randomly. This level-1 thinker would assume the average is about 50, so they'd guess 33.3 as their number.

A level-2 person would follow a similar pattern. Since they think everyone else is level-1, and therefore they are all guessing 33.3, this level-2 person would choose 2/3 of that: 22.2.

A level-3 player assumes everyone else is a level-2 player...and so on and so forth until eventually, the rational number to guess is zero.

This sort of infinite thinking is defined by Brilliant, as having an unbounded/infinite depth and in all types of problems like these, it's assumed all actors have infinite depth.