I would use the 10 trillion dollars to give every person on earth access to a system that would be (as close as modern technology allows) equivalent to A Young Lady's Illustrated Primer.
In today's terms, that might manifest itself as giving everyone free access to computers with high bandwidth internet connections, which they could use to access a YouTube-like website housing multimedia explanations of every topic imaginable, tailored to every target audience imaginable, combined with a sophisticated system that would create informational links (and/or allow users to create links) between every piece of media, such that another sophisticated system could propose a 'best route' to achieving deep understanding of any particular piece of knowledge.
After this had been done, all remaining money would be spent on a combination of incentivizing the world's best thinkers and educators to contribute (or continue contributing), research grants, and scholarships.
Does anyone think that this disagreement can be resolved without threat-signalling? I think valuing a particular model of 'fairness' over another (the Xers and Yers from Leif's post) ultimately boils down to the cost/benefit of being accepted/rejected by a particular social group.
So does this disagreement take place in a universe consisting only of the entities Xannon, Yancy, and Zaire, or do they all go back to the same village afterward and reminisce about what happened, or do they each go back to their separate villages?