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?
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!