On AA I get double miles and free upgrades (unlimited domestic, eight overseas) and lounge access* (I paid to join, but that's a sunk cost). Match that, offer something additional like a better schedule for that particular trip at the same fare or a lower fare for a comparable schedule, and you have a chance. If the only reason is a lower fare, it has to be a whole lot lower.

That's tough to do, so it tends not to happen.

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*In theory I can use an AA Admirals Club lounge no matter what airline I'm flying, but in practice the layouts of most airports big enough to have one in the first place make it difficult or impossible to access them unless one is flying AA.