What is the best solution for the Judgment of Paris?
If you pick one goddess, the other two get angry with you. Granted, you could ask Aphrodite for the love of the world's most beautiful *available* woman, but they'd still find a way to get you.
If you waffle and refuse to choose, perhaps they all get angry with you.
So, perhaps I would pick Athena and explain, "Wisdom is the best bribe, because touching this dispute demonstrates I need all the wisdom I can get, and that lack also means you shouldn't take my opinion of your beauty as worth all that much."
Better yet, if the question were in English, I could say, "Athena is the *fairest* because she acts the most justly towards humans instead of making them suffer for things that aren't their fault. (OK, Athena turning that boasting woman into a spider was disproportionate, but at least she got a fair weaving competition.) And if either of you, Hera or Aphrodite, retaliate against me for answering your question, that would be *unfair* and would prove my point." (Granted, they would be so egocentric that they likely would consider punishment for miffing them to be justified.)
For that matter, which goddess would be most willing & able to protect me from the other two?