What is your objective? Survive, grow, or kill him? Different answers will lead to different strategies.
Generally, if he's bigger, you want to survive, and ideally, grow despite his superiority. A price fight is rarely won by the smallest company(unless the big boy is already in danger before the price war begins). The sweet spot in terms of pricing I've found in 1vs1(i.e. the same amount of flights on both sides, with similar planes) is at 47/48% of market share. Yes, slightly less than your opponenent. That's how you'll make the most money. Higher pricing is not yet too much penalized. Plus, your opponent feels stronger than he really is - which is usually a strategic advantage for you.
Of course, you're not going to kill him like that. He'll do good money, even if not as much as you(and the difference is not THAT big anyways). But I do assume he's big and with a big profit margin. You cannot kill him, at least not now. You need to survive, and grow. Add flights where he's not flying, and don't underprice him. Once you've grown enough to be bigger than him, you can begin to annoy him. Not when you're the small fish. It's time to hide, not time to strike.