One paradox I've found interesting involves the stock market. Over the years I've noticed that stock market crashes tend to occur in the September/October time period. Now, if I noticed this, then actual stock traders have certainly noticed it. But if they know it's going to happen (or if it's more likely than pure randomness would allow) then the rational choice would be to short sell your portfolio just prior to this time period (say, August). But if all the big players in the market do this, then the market would crash in August. But they also know this, so the market will actually crash in August. So they should they should really short sell their portfolios in July. But everyone knows this, so the market should crash in July. They should really be short selling in June. But they all know this... on and on.

It seems as though the rational choice is to always short sell immediately. So there are two paradoxes: why does the market go up over time if the players are rational, and why do the crashes actually occur in September/October, even though everyone knows it's going to happen?