The big problem with avoiding competition is that you are also avoiding customers. The existence of a competitor indicates the existence of paying customers. If you can't find anyone who is making money with your idea, you really need to wonder if there is any money to be made there at all.
As an example, suppose you have a cool new business idea. You want to revolutionize parking and car rentals at airports. The idea is simple: Airports are visited by two kinds of people: Some people bring their car and leave by plane, so they pay to leave their car in a parking lot. Other people arrive by plane and need a car, so they pay to rent one from a car rental agency. Why not match these people up? Instead of paying to park their car, travelers can actually make money by allowing it be rented while they're gone on their trip. From the perspective of the renter, everything is the same as it is for existing car rental agencies, except that we can charge really low rates since we don't have to carry the capital costs of owning a fleet of vehicles.
As far as I can tell, nobody is doing this business model right now. If you want to get into a new market which is wide open and free of competition, jump on this exciting opportunity today. You don't even have to give me a piece of the profits for contributing the idea. More specifically, I have no interest in sharing a piece of the losses. :-)
There's a very good reason nobody is running this kind of business: Most people do not want a stranger driving their car. Many people just don't treat rental cars very nicely. The owner of the car is not likely to think the risk and aggravation are worth the trouble for the money they'll get.
If we were still in the bubble, you could spray a coat of Internet on this idea and find a VC clueless enough to fund it. But good investors and good press are just not going to be enough to make this idea work.
The lesson here is that "new" ideas aren't as valuable as people think. Most of the time, when you find a market with no players, it's not really a market. Money is made by beating competition, not by avoiding it. If you want to start a new business, don't look for an idea which has never been tried. Instead look for someone who is serving real customers but not doing it very well. Find a way to do it better.
This approach seems scary, but that's merely because the risks of facing a competitor are easier to see. If you're going to fail competing in an existing market, you probably know how that failure is going to happen. In contrast, the risks of creating a new market from scratch are far less obvious and visible. But even though those risks are harder to see, they're actually huge. An optimistic person can too easily convince himself that those risks aren't really there. Believe me, creating a new market segment is much harder than it looks.