A duopoly is a situation in which two firms control nearly all of the market for a product or service.
Duopolies can be surprisingly competitive. If you remember that the price of a product or service is determined solely by the highest losing bid price and the lowest losing ask price, you’ll realize why a duopoly can be so competitive. A large number of inefficient competitors will have almost no affect on prices in the long run unless someone (either a government or a group of idiotic investors) is willing to continually finance unprofitable operations in an unprofitable industry (think airlines).
Of course, there is always the fear of a price fixing scheme in a duopoly. Generally, however, that fear is unfounded. Human nature suggests a price fixing scheme is far more likely to occur in an oligopoly than a duopoly. Humans weight the fear of loss far more heavily than the greed of gain when making calculations about the future. In a duopoly, mistrust increases the fear of loss inherent to any price fixing scheme (namely, the other guy will stab you in the back). In an oligopoly, the diffusion of power and the lack of excess capacity at any one firm makes price fixing very attractive. Price fixing in an oligopoly is a much safer bet than price fixing in a duopoly.
There are, of course, other reasons why a duopoly is very unlikely to result in a price fixing scheme. In addition to a healthy does of fear, there is an often unhealthy does of hate in duopolies. There is always just one scapegoat in a duopoly. Hatred is a personal emotion; if spread over too many objects it tends to wane away. Finally, there’s the simple fact that both competitors in a duopoly are likely really big, really agile, really cutthroat players. The process leading up to a duopoly tends to be a sort of wolfing run, in which two pups are separated from the runts.
Having said all that, price fixing is possible in a duopoly. Some duopolies are not the result of competition but of nationalization and privatization, although this is relatively rare since a nationalized monopoly won’t often result in a lasting duopoly (it will either remain a monopoly once privatized or get crushed by new, private competitors).
Finally, a price fixing scheme always makes more sense in a commodity business. After all, any product differentiation limits the degree to which general demand is applicable to specific competitors’ products. For example, Coke and Pepsi are highly differentiated products, at least when purchased in their specific packaging (physical differences or similarities are immaterial here; it is only the buyer’s belief that matters). I drink Pepsi, and I can assure you (however irrational it sounds) that no drop in the price of Coke would be sufficient to get me to stop buying Pepsi. There is almost no other tangible good about which I could say the same. So, clearly Coke and Pepsi are differentiated products, and there’s very little chance of an effective price fixing scheme between them.