What causes experts at sites like FlightFox to be so much cheaper than search engines?

Lauren (co-founder of Flightfox) here. I can't tell you exactly what's going on in this contest but I can give you a general answer.

Our experts come from many walks of life. Some are travel agents who really know their way around fare rules and ticketing. Some are frequent flyers who have learned from experience. And others are just pure travel hackers who keep abreast of all the deals and technology.

That said, the purpose of Flightfox isn't to beat any particular price. Instead, we give the average person a chance to work "with" experts (not against them) to find the best flights. So our specialities are international and multi-city trips instead of short and/or domestic.

In the example above, which is a relatively simple trip, I suspect the experts were able to cover more airlines and routes by using the right tools. In general, humans clearly can't search quicker than an algorithm, but they can use the right tools with their own knowledge more efficiently. So maybe in this case the experts used low cost airlines, or routed through cheaper airports, or joined flights that the algorithms don't pair together.

The fundamental reason is that airline pricing is just plain ridiculous. Finding the cheapest flights should be a trivial problem that you could assign as homework on an algorithms class. Instead, the airlines insist on having so complicated set of rules and fares that it's logically impossible to have an algorithm that always finds the cheapest flights. In fact, it's logically impossible to even have an algorithm that can always determine correctly, whether it's possible to sell flights from city A to city B on a single ticket.

Or at least that's how it was 10 years ago. There was a nice article in SIAM News, which should be at least partially accessible to people without a mathematical background.

Because of the complexity, even the most sophisticated search engines search only for relatively simple flight combinations. As a consequence, if you travel often enough, you quickly start to learn the strengths and the weaknesses of different search engines on the kinds of routes you fly. With this kind of knowledge, you'll eventually be able to beat any single search engine most of the time.