I should probably provide a corollary to this. It's an interesting question and deserves more than a pithy one-word response.

Logistics:

It is difficult enough to coordinate the work diaries, social calendars, birthdays, anniversaries, dietary requirements, travel plans, in-laws, etc. of two reasonably busy people who live in close proximity to one another. The more people and locations you add, the more it compounds any orchestration problem.

Economics:

I claim romantic relationships do not enjoy the benefits of economies of scale, and the overhead of each additional relationship actually increases logarithmically. I also claim additional partners are subject to diminishing returns. In fairness, if this is accurate, it is less of a case against polyamory and more of a case against an arbitrarily high number of partners. Still, it's not unreasonable to suggest that the optimal number of typical partners for a given person is between 0 and 2.

"Love Anarchy":

Much like the international system, my lovelife has no police force. I am generally quite pleased with this state of affairs. In a monogamous relationship my partner and I each have a single trade partner for our romantic resources. The quantity of those resources may or may not be to our exact liking, but the distribution is not contested. This is a relatively stable system. Once a third (or fourth, or fifth...) party becomes involved, we have a negotiation problem.

There's no feasible method for someone to commit to a set distribution of time/effort/attention between partners. I'm not saying there should be, just pointing out that such things can't realistically be budgeted for or enforced. The absence of such a mechanism makes polyamory highly unstable compared to monogamy, though I suppose this only really sits in the pro-monogamy column if you place a premium on stability.