The scope is limited relative to what you seem to want, but I thought I might mention the book Theories of Distributive Justice by John Roemer. It walks through several popular philosophical approaches to distributive justice like utilitarianism, maximin principle, etc at a formal level, and discusses relevant results that have been proven about them. I've only leafed through it though, so I can't necessarily vouch for it, but you might want to check it out.
The scope is limited relative to what you seem to want, but I thought I might mention the book Theories of Distributive Justice by John Roemer. It walks through several popular philosophical approaches to distributive justice like utilitarianism, maximin principle, etc at a formal level, and discusses relevant results that have been proven about them. I've only leafed through it though, so I can't necessarily vouch for it, but you might want to check it out.