I'm not an expert, nor have I extensively studied this (feel free to add "I noticed") so I pulled this definition off of wikipedia that pretty much sums up my opinion of it: "Market failures can be viewed as scenarios where individuals' pursuit of pure self-interest leads to results that are not efficient – that can be improved upon from the societal point-of-view." I'll admit that it is not as clean as yours in a mathematical sense.
That's just a different way of phrasing precisely what KH said. "Efficient" in this case pretty much just means "not Pareto optimal".
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