Choosing the nondominant strategy in a duopoly Would a company ever choose a nondominant strategy in a duopoly?

Let's take this specific example (2007 AP MicroEcon B #2). Two airlines, Airtouch and Windward, are scheduling flights for either morning or evening. There is no collusion, but both airlines know the same profit matrix, as below. (edit: This is also a one-shot game)

                           Windward
           Profit      Morning | Evening
                   ----------------------
           Morning | 1000, 700 | 700, 600
Airtouch           ----------------------
           Evening | 750, 950  | 900, 800


Windward has a dominant strategy of choosing the morning; Airtouch has no dominant strategy.

Airtouch, of course, is dependent on Windward for its decision. If Windward's best decision is morning, Airtouch will choose morning. Same for evening.

If we go with the given solution, Windward would choose the dominant strategy of morning, and Airtouch would then have to go with the morning as well.

But then Windward (and Airtouch, since they have all the same information) realizes that Airtouch is entirely dependent, so if Windward chooses the non-dominant strategy of the evening, it will have a greater profit (800) than it would in the morning (700), even though it is not the dominant strategy.

So then both would go for the evening.

Is there something wrong with this reasoning? Particularly, is Airtouch necessarily dependent if Windward has a sketchy dominant strategy like this? I would think that since Windward begins with the dominant strategy, it holds the initiative, but I'm not certain.

(This, of course, assumes perfect reasoning on both sides, which in real life is definitely not a justifiable thing to presume.)
It seems you interpret the game as a sequential game. If Windward would know that its move was observed by Airtouch it would act as you say. But this is not the case, they move simultaneously, and it is a one-shot game (as defined it is only played once).

Given your reasoning, Windward understands the situation and hence it would choose the evening. Anticipating this Airtouch would do the same. But Windward does not really want to choose the evening. If Airtouch anticipates that Windward will choose the evening that's jolly good for Windward. It is even better if Windward goes on to choose the morning. So what you describe is not an equilibrium outcome and is unlikely to occur in a one-shot situation.

Don't believe me? Let's play. As Windward I understand your reasoning and I make the empty promise of choosing 'Evening'. What will you choose? (And me?)

The situation would be different if the game was repeated, so if reputation was important or if, as I mentioned before, the game was sequential and moves could be observed.