On a Bloody Beauty Contest
It is not a case of choosing those which, to the best of one's judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees.
— John Maynard Keynes (The General Theory of Employment, Interest and Money)
The notion that humans are not rational sparked an entire field known as behavioral economics. It appears I am not writing this because a CPU in my brain optimized a utility function. Of course, that is not to say I do not optimize anything at all or that I am irrational. Academics now call me boundedly rational, which might sound insulting, but it is actually a progression from the times they called me perfectly rational. A famous example of this now infamous assumption is the Keynesian beauty contest. A twisted variant of this game featured in Haro Aso's Alice in Borderland as the King of Diamonds. A game-theoretic analysis of it reveals why Keynes did not get it completely wrong.
Five players enter a courtroom and shackle themselves to a seat right beneath scales holding aqua regia. You can imagine a gory conclusion as, eventually, metaphorically speaking, enough evidence would mount against players to prove guilt, the acid would tilt the scales, and punishments would be meted out in the courtroom itself. Amid this fearful atmosphere, the game itself is simple: each round, contestants choose a number between 0 and 100; the average of the selected numbers is multiplied by 0.8; and the person whose number is the closest to that result wins the round, while the others each lose a point. Game is over for players who reach -10 points.
A naive initial guess is 40: the average of 0–100 would be 50, so multiply it by 0.8. This is merely Keynes' second degree. The third degree would assume everyone chooses ~40 with similar reasoning, so choose 40 * 0.8 = 32. Clearly, this converges to 0, so why not just pick 0 from the outset? Anticipate what the average opinion expects the average opinion to expect the average opinion to expect the average opinion... no. That's impossible. Indeed, those who overthink, bearing the curse of knowledge, spiral downward the moment an external shock disturbs their calculations. "When shrouded in a blanket of confusion, their thought processes will strand them." Why? Perfect information does not entail unlimited cognitive capacity. You could think 10 steps ahead, but perhaps others do not. You gave them too much credit. Then where should I stop? At my expectation of the average cognitive capacity of the group? So Keynes was right on a higher plane of reasoning?
As the principle goes, كلموا الناس على قدر عقولهم. How good are humans, then, at reading someone's rationality? The normal distribution comforts me, depicting how most people swirl around the average, but Aristotle warned me about the extremes centuries ago. Before reading someone's rationality, it seems I must read how good my readings are... Not again. I must avoid this cognitive version of Zeno's paradox. Perhaps I know some things that I cannot articulate. Intuition? Come on. After all that I have said, you cannot expect me to be able to answer that.