My knee-jerk reaction to Robert Shiller and Eugene Fama getting the Nobel was that it's an example of them being honoured for flatly disagreeing. This reaction is partly true: Fama's thinking is sceptical of the possibility of price bubbles, whilst Shiller's shows that they can happen. But in another sense, their work is actually compatible, because they are talking about different things.
Fama's work is primarily about individual stocks. He has shown that, generally, prices quickly embody all available information so that investors cannot out-perform the market except by taking on more systematic risk. This risk might be market risk (beta), or the sort of cyclical risk that is often associated with small and value stocks - hence the "three factor" model discussed here (pdf). Back in 1970, Fama concluded (pdf):
For the purposes of most investors the efficient markets model seems a good first (and second) approximation to reality.
One piece of evidence for this is the performance of equity neutral hedge funds: according to HFR, these have returned 1.1% a year in the last five years - no better than a risk-free rate. A second piece comes from UK all companies unit trusts. In the last five years, most of these have under-performed the better tracker funds.
Market efficiency is like Newtonian physics. It's not exactly right, but you'll not often go disastrously wrong if you act as if it is.
Shiller's work, by contrast, has focussed more upon the aggregate market. In one famous early paper, for example, he showed that the S&P 500 was far more volatile (pdf) than could be explained by dividends. This implies that - over longish periods at least - aggregate prices might be predictable by the dividend-price ratio (or other things (pdf)).
Whilst this is consistent with the possibility that investors are irrational - contrary to the spirit of Fama's work - it is not proof of irrationality. As David Meenagh has shown in the UK, such excess volatility might be due to investors attaching varying but reasonable probabilities to future scenarios (booms, slumps etc) which might occur but in fact do not.
But here's the thing. Micro efficiency in Fama's sense is quite compatible with excess aggregate volatility in Shiller's. Shiller himself (pdf) has suggested just this.
Why the difference? One big reason lies in how information gets embedded into prices. Imagine you think an individual stock is over-priced. In many cases (not all - the tech bubble being an exception) you can short-sell the stock reasonably easily, as you can diversify the risk of doing so by going long of a comparable stock; this is the idea behind pairs trading. But what if you think the aggregate market is over-priced? It's risky to short-sell it; remember Keynes' famous saying that markets can stay irrational longer than you can stay solvent? And there's less chance of being able to spread this risk by going long of other markets, simply because global stock markets are correlated.
With short-selling of aggregate markets more difficult than of individual stocks, it's likely that aggregate markets will be (occasionally) more over-priced than individual stocks.
In this sense, Shiller and Fama aren't so far apart.