A Problem with Strong EMH

Strong versions of the efficient market hypothesis may have a problem with internal consistency.  

The theory requires and assumes successful securities trading strategies will quickly be mimicked, and hence moot.  Thus, an implication of the theory is that trying to pick individual securities to beat the market cannot be done consistently  It then follows, as many adherents to strong EMH claim, that only index funds can offer consistent positive returns.

Here’s the problem: If the most successful strategy is to invest in broad indexes, then why is this strategy not universal?  And if it were universal, then significant arbitrage opportunities would open up and many for some time.

This seems to suggest a more restrictive limit on the strength of EMH than adherents to strong versions claim.