  Instant Runoff Voting (IRV) is used in elections for many political offices
around the world. It allows voters to specify their preferences among
candidates as a ranking. We identify a generalization of the rule, called
Approval-IRV, that allows voters more freedom by allowing them to give equal
preference to several candidates. Such weak orders are a more expressive input
format than linear orders, and they help reduce the cognitive effort of voting.
  Just like standard IRV, Approval-IRV proceeds in rounds by successively
eliminating candidates. It interprets each vote as an approval vote for its
most-preferred candidates among those that have not been eliminated. At each
step, it eliminates the candidate who is approved by the fewest voters. Among
the large class of scoring elimination rules, we prove that Approval-IRV is the
unique way of extending IRV to weak orders that preserves its characteristic
axiomatic properties, in particular independence of clones and respecting a
majority's top choices. We also show that Approval-IRV is the unique extension
of IRV among rules in this class that satisfies a natural monotonicity property
defined for weak orders.
  Prior work has proposed a different generalization of IRV, which we call
Split-IRV, where instead of approving, each vote is interpreted as splitting 1
point equally among its top choices (for example, 0.25 points each if a vote
has 4 top choices), and then eliminating the candidate with the lowest score.
Split-IRV fails independence of clones, may not respect majority wishes, and
fails our monotonicity condition.
  The multi-winner version of IRV is known as Single Transferable Vote (STV).
We prove that Approval-STV continues to satisfy the strong proportional
representation properties of STV, underlining that the approval way is the
right way of extending the IRV/STV idea to weak orders.
