# 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.)