### Keynesian beauty contest

Problem:
Pick a number from 0 to 100. The winner is the person who chooses the number closest to 2/3rds of the group's average response. What is the rational answer?

Related thought:
Once you get the solution, you would be surprised to see the implications. A Keynesian beauty contest is a concept developed by John Maynard Keynes and introduced in Chapter 12 of his work, "General Theory of Employment Interest and Money" (1936), to explain price fluctuations in equity markets. Read wikipedia article to appreciate it: http://en.wikipedia.org/wiki/Keynesian_beauty_contest

Update: (23 March 2011):
Solution posted by Gautam Kamath (EE, Senior Undergraduate, IIT Bombay) in comments!

1. Rational answer would be all players playing 0. This is because of the following reasoning:
All players have the same information, and are assumed to be equally rational. So if a player plays x, then every player would want to play x. But now the first player thinks that if I play ~2/3*x, I would win, and so he plays 2/3*x.
This applies to every player, and so the iterated value of x keeps reducing till we reach zero, where if every other player plays zero, even I would want to play zero.
This is referred to as a Nash equilibrium in Game theory.

2. Correct! Thanks

3. I guess we r picking natural nos from 1 to 100 and not rationals. In that case shouldn't we be considering closest integer to 2/3x. It creates a problem when we approach the limit(x=1)