Just the Facts, Ma'am

     The Mathematics of Voting Systems.  Criteria listed in order of (generally) decreasing importance.

	S c h u l z e	T i d e m a n	C o p e l a n d	R a y n a u d	B a l d w i n	N a n s o n	D o d g s o n	S i m p s o n	B u c k l i n	H a r e aka I R V	B o r d a	C a r e y	C o o m b s
Pareto-Optimal	YES	YES	YES	YES	YES	YES	YES	YES	YES	YES	YES	YES	YES
Majority	YES	YES	YES	YES	YES	YES	YES	YES	YES	YES	NO	YES	NO
Condorcet	YES	YES	YES	YES	YES	YES	YES	YES	NO	NO	NO	NO	NO
Monotonic	YES	YES	YES	NO	NO	NO	YES	YES	YES	NO	YES	NO	NO
Clone-Independent	YES	YES	NO	YES	NO	NO	NO	NO	NO	YES	NO	NO	NO
Mutual majority	YES	YES	YES	YES	YES	YES	NO	NO	YES	YES	NO	NO	NO
Smith	YES	YES	YES	YES	YES	YES	NO	NO	NO	NO	NO	NO	NO
Reverse-Symmetric	YES	YES	YES	NO	NO	NO	NO	NO	NO	NO	YES	NO	NO
Reinforcing	NO	NO	NO	NO	NO	NO	NO	NO	NO	NO	YES	NO	NO
Schwartz	YES	NO	NO	NO	NO	NO	NO	NO	NO	NO	NO	NO	NO
Nonmanipulable	NO	NO	NO	NO	NO	NO	NO	NO	NO	NO	NO	NO	NO

Shamelessly copied (if somewhat improved) from a voting math amateur enthusiast's webpage here.

NOTE: All of the Libertarian Party, the Green Party, and famous political scientist David Hume like IRV.  The only explanation is that they can not do the math, or that the mathematicians have taken no interest in the matter, or have taken an interest, and lack the explanatory powers.