As I consider the question of "the advantages of polyamory," the main thing that comes to mind is that it isn't necessarily the best question to ask. There are advantages to both monogamy and polyamory. But the more urgent question, to me, is what are the advantages of a non-possessive versus a possessive style of loving?
What is it to be possessive? http://dictionary.reference.com/browse/possessive
What is a possession? http://dictionary.reference.com/browse/possession
A possession is a kind (or form) of property.
Now, it is conceivable that a couple could choose a monogamous love-style without choosing to behave as if the other was their property, or possession. But often monogamous people treat one another as property or possession without any thought that that's exactly what they are doing.
A piece of property, or a possession, essentially, is something under our control. To control is "to exercise restraint or direction over; dominate; command."
Such control is usually manifested by a rule or an agreement.: You may do this but not do that."
Well, there's nothing intrinsically unloving about establishing agreements, even rules of acceptable and non-acceptable behavior. But, surely, most of us would agree that some
relationship agreements (rules) amount to a kind of abuse which is unloving. Some partners feel they have a right to limit who (or what sex of person) one may choose as a close friend! How loving is that?!? Some are so possessive and fearful (insecure) that they will disallow even intimate non-sexual relationships. "You must love only me, and no others!" How loving is that? It's certainly controlling, possessive, dominating!
Most of us, if our partners insisted that we can't form other intimate freindships, would refuse to follow such a silly and absurd rule. We'd quit a relationship of that sort. But somehow most folks think it is appropriate to impose limitations on HOW we can express or experience love and intimacy with others. We typically do this, I believe, because we secretly believe that, in order for our relationship to be "special" it must also be unique in how we experience and express affection and warmth. We don't want to lose the "specialness," and so we want to restrict our lovers in their expression and experience of intimacy with others -- by refusing them the freedom to express or experience physical intimacy. Or even, in some cases, emotional bonding!
But this "logic" doesn't work for most polyamorous folk. It seems faulty, erronious, manipulative, controlling, fearful. We'd rather trust that our partners can love others fully, however they will, so long as we are also being loved. So a major advantage, I think, of polyamory, is trust. Our trust in the love of our lovers is larger, more expansive. And we trust our love-bonds well enough to not respond with fear or anxiety if our lover also loves another--however expressed. Or, if we find ourselves feeling insecure or fearful about our lovers' love of another, we inquire within to see--well, Why? And we find that there are no good reasons to fear this additional love, so we learn, often gradually, to open up and fear less. The result, often, is growing, not shrinking love with our partners.