This is a really interesting topic.
I think that there are two ideas that combine to create a requirement for traditional monogamous structures - 1) Land Ownership and 2) The family as a power-wielding unit (not just immediate family but extended family)
Land is power, in Asia and everywhere else. It's advantageous for a family to get more land. Also, it needs people to take care of/improve the land. The best way to do this is to have a son who can earn more land, bringing it into the family, and then marry a young girl who can create more members of the family to work and repeat the process.
Now, here's where competition for brides comes into play - it is in the girl's best interests that she picks a mate with lots of land/power, since that means her offspring will be better off, and she will also probably have a better life. So potential suitors will compete to show they have better resources. This is really in the girl's interests - she has the power here.
So why doesn't a girl work herself? Well, in much of Asia, hard labor is still dominant. Men are much better at physical labor than women. With the advent of more IT-related jobs this is starting to change, but men are still in higher demand for jobs than women.