Okay, here goes on the issue of "predatory pricing." Predatory pricing is one of a variety of things which is illegal because it constitutes an attempt to monopolize, which is outlawed by Section 2 of the Sherman Antitrust Act. That is the easy part. What, however, is "predatory pricing."
Courts have struggled with this issue for many years. To understand this discussion, it will help to have some understanding of the concepts of fixed costs and variable costs. Fixed costs are those which a company is going to incur, whether it makes another marginal unit of production or not (or, in the case of airlines, whether they will fill another seat or not). Variable, or marginal costs, are the additional costs incurred in selling that additional or marginal unit of production. As examples, let's look at airline seats and gasoline. The fixed costs involved in Exxon's production of gasoline are the costs of building the refineries, the oil wells, etc. The marginal costs are the costs of the crude oil (if they buy it), or the energy and labor to pull it out of the ground, to refine it , and to transport it to the Exxon station, where you can buy it. So, there are a lot of variable or additional costs to produce that extra gallon of gasoline. As to the additional airline seat, however, there is a big difference. The airplane will already be flying. The pilots, gate agents, flight attendants, baggage handlers will all be paid, regardless of whether the plane takes off with 100 or 120 people on board. Sso, the incremental, marginal or variable costs involved in a few extra tickets is extremely low. The airplane will use a little more fuel, and they will use up some more soda and a meal (and we all know that is going to cost very little!!).
Some proponents have pushed the idea that a company is engaged in predatory pricing anytime it sells its product for less than average TOTAL cost. That has been pretty much discredited, and clearly could not be applied to airlines. Half or more of the people on board are probably paying less than average total cost.
The generally accepted measure these days is that a competitor is engaged in predatory pricing if 1) it has the potential to monopolize the "market" and 2) if it is selling at below its "marginal" or "variable" cost for the product.
For airlines, most antitrust practitioners and economists agree that the "market" is defined on a route by route basis, as "city pairs." For example, there is a market for transportation between Philadelphia and Pittsburgh, which is virtually monopolized by US Air. There is a market for transportation between New York and Los Angeles, which is flown by a dozen airlines (some direct and some through other hubs).
So, an airline probably couldn't be charged with predatory pricing unless it could be shown that it priced below its variable costs (for the extra seats that is decreased the price of to drive the upstart out of the market), and it had a legitimate potential of monopolizing that "city pair" market.
As you can see, some of these explanations get kind of technical and long-winded. I am sorry that I am not able to distill these answers into shorter versions. Hope that this is understandable. (Of course, I am being presumptuous--that anyone is still reading this thread by the time they get to this end of these answers!)