Since most women managed to reproduce, we can assume a winner strategy is having a large number of daughters

But if everyone adopts this strategy, in a few generations women will by far outnumber men, and suddenly having sons is a brilliant strategy instead. You have to think about what strategies are stable in the population of strategies - as you begin to point towards with the comments about game theory. Yes, game theory has of course been used to look at this type of stuff. (I'm certainly not an expert so I won't get into details on how.)

If you haven't read The Selfish Gene by Richard Dawkins, it's a fun read and great for getting into this subject matter. How The Mind Works by Steven Pinker is also a nice readable/popular intro to evolutionary psychology and covers some of the topics you're thinking about here.