The FCA concluded in a study on stale pricing in dark pools that it is not... 100% efficient. Not that this is a surprise, but it's good to get the numbers. It's not clear that those numbers are really that reliable, but it's a start.
In particular, they looked at trades that are supposed to be pegged to the mid-point. By pegging dark pool execution to the mid-point (determined by observing lit venues), trades are supposed to remove the market impact and confer no disadvantage to either the side of buyer or the side of the seller. This off course relies on the derived mid-point to be correct (and not out of date). This is what they examined.
The regulator found that around 3.5% of all trades concluded in dark pools in the UK are away from the mid-point of the corresponding lit venue. On the surface that may look like a big number, but when translating it into actually currency terms, the impact shrinks dramatically.
All in all, the monetary disadvantage they put on these "stale transactions" is small: a mere 4.2 million GBP per annum (which is about 17,000 GBP per day). However, if we really look at just the part that happens outside the prevailing BBO at the time this shrinks to 1.3 million GBP per annum (outside of the BBO this represents a "true arbitrage" in the classical sense, although whether this can really be captured is dubious as per the below). Needless to say, these are actually fairly small numbers.
Now, one of the problem when analyzing this sort of dataset is getting isosynchronous timestamps of the message arrival. The FCA does not have that, all they have is timestamps at the various venues matching engines. So inferring what is stale and what isn't is problematic, because you don't know what the latencies are especially when it's busy. Still, it's a good idea to look at it.