Presumably, if you know a stock is too highly priced at its IPO (or any time, really), you short it. Unfortunately the hypothesis "some IPO prices may be subject to manipulation by insiders who know about the anchoring bias" doesn't tell you which ones. I suppose you could try systematically shorting every IPO that comes along, but presumably if that were a viable strategy someone would already be exploiting it; it's too simple not to have been spotted. Besides, I think the transaction costs are too large for a small player to make any money with this sort of strategy.
Seems to me that prices on stocks subject to this sort of manipulation aren't going to be distributed similarly to those that aren't. I have no idea of what either distribution would look like in practice, but it seems like you ought to be able to make some money by preferentially picking your shorts based on the points at which you'd expect them to differ -- the OP's price-to-earnings ratio being one candidate.
How much money depends on how distinguishable this is from a normal shorting strategy, how common this kind of insider manipulation is, and how much noise there is in the system, though. I wouldn't expect it to be a spectacular win in most cases, just a minor advantage.
Facebook IPO'd at a price of 38 dollars a share, which apparently gave it a price-to-earnings ratio in the range of 100 - extremely, fantastically high. The price dropped pretty rapidly and is currently somewhere around 20 dollars; which still, presumably, gives it a very high P/E ratio somewhere in the forties. Now, suppose it had IPO'd at a more historically-reasonable P/E of, say, 20 - still high, but not stratospheric. That would put the initial share price somewhere around 10 or 12 dollars. Is there any strong reason to believe that the price would then have *risen* to where it is now? It is not obvious to me that the current price is supported by anything but the historical price - in other words, it's trading around 20 because it has recently traded around 25.
My point: I can't help but wonder if someone connected to the IPO had read Kahneman on anchoring. Somebody, clearly, was buying the stock at 33, just as someone is still buying at 20; I wonder if the chain of thought had that apparently-arbitrary number "38" in it somewhere, making 33 look cheap - fundamentals be damned! And if this happened, who benefited, and what ought we to conclude about the efficiency of markets?