I think you're being downvoted because people consider your original comment poorly constructed and poorly thought out.

That men can participate in polygamy does nothing to encourage birthrates; what is the incentive of the wives to want children? What is the incentive for the man here to want children?

Unless it's implied in your idea that the purpose of marriage is to produce children, polygamy solves nothing here. If that is the presumption that the arrangement requires children, how does this counteract the very common reasons for not having children?

- It's extremely time consuming and life changing; you have to commit substantial time or resources (or both) to ensuring the child is cared for

- A person or persons simply don't want children

- There isn't a strong reason to have children outside of one's own desire to do so (there isn't incentive to maintain a lineage of successors, build out an army of loyal subjects, and so on for the grand majority of people anymore)

- Even within religions, the impetus for having children just isn't emphasized in many sects like it was in the past

- Social mobility, while nowhere near perfect, is far more achievable via other arrangements or even to a degree with personal effort; it won't be the peasant to noble move that existed previously, but it's more possible than it has been in the past (though social mobility is still ridiculously difficult and fraught with discrimination and exploitation, so understand I am not at all saying that it's in a good spot)

Your idea doesn't really address any of the actual reasons a lot of people hold off from having children, it just presents a suggestion to "legalize polygamy for rich men", implying that this somehow addresses the many reasons that people aren't having children.

Your follow-up comment here I think also misses a lot of points as to why people check out and undermines the idea that society can incentivize its way out of these problems. If the problem is a bleak economic future for people, trying to utilize childbirth as a pyramid scheme to overcome this economic uncertainty isn't really a great idea, as that means you need to somehow convince the children of the already checked-out persons that "no, really, it's a great system. Ignore why your parents are so miserable, it will be different for you for reasons."

I don't think you can just economically incentivize your way out of a declining population, not in any meaningful way aside from temporary trends which will fade. If the root cause of the issue is poor economic conditions and unstable comfortable living conditions, you need to first understand why those are happening in the first place. If it turns out that it's because only a select group have the economic stability to support a life that includes children and this select group is not a huge portion of the population, aside from mandating child birth, you haven't actually given incentive to make more children, you just have given special status to the already elite.

> But consider what the future would look like if you don't increase the birth rate:

I have, and I imagine we probably see more adoption of universal income and automation, and far more acceptance of such a system and interest in keeping such a system where you don't need to struggle for basic needs in place by investing time and effort into the system. There's a lot of this which of course is fantastical thoughts right now, but we're already seeing this in many nations, and the quality of life in these places is pretty good while birthrates are just "so-so".