[EM] IIAC and Condorcet

Forest Simmons forest.simmons21 at gmail.com
Sat Nov 6 23:54:20 PDT 2021

Nothing new, but under-appreciated....

If an election method M elects some candidate X that is not a Condorcet
candidate, then the ballot set from that election can be used to show that
method M is not IIAC compliant.

Suppose that M is IIAC compliant and elects X. Let Y be any candidate other
than X.

One by one eliminate every candidate other than X or Y, until X and Y are
the only remaining candidates.

The IIAC compliance of M guarantees that X continues to be the winner after
each removal.

So, in the head-to-head contest between X and Y, candidate X must win.

Since the choice of Y was arbitrary, we have shown that X beats every Y
head to head, i.e. X is the ballot head to head winner.

Suppose, now that X cannot beat Y head to head under method M unless X is
ranked ahead of Y on more ballots than not.

This additional supposition completes the proof that X had to be a ballot
Condorcet Winner, after all.

In summary, we have shown that if X wins under an IIAC compliant method,
then X must be a CW for the ballot set in question.

This is the contrapositive of  ...

If an election method M elects some candidate X that is not a (ballot)
Condorcet candidate, then method M is not IIAC compliant.

There are two ways the conclusion can be satisfied... 1. The ballot set had
no CW candidate.  2. The method did not satisfy the Condorcet Criterion.

What are the implications for IIAC compliance simulations?

1. Any CC compliant method will have the same maximal IIA compliance as any
other CC compliant method.

2. Any non-Condorcet method will have IIA violations that no Condorcet
method has.

3. But Condorcet methods violate the IIA only for ballot sets for which no
method can satisfy the IIA.

4. Although all Condorcet methods satisfy the IIA for the same ballot sets
(those for which the Smith set is a singleton), their finish orders may
differ on the question of compliance with Local Independence of Irrelevant

In sum, it is not necessary to test IIA compliance of Condorcet methods by
simulation ... they pass if and only if there is a ballot CW.

It is not necessary to  check IIA compliance on any ballot set for which
there is no CW ... every method fails on every such ballot set ... 100
percent failure. If the simulation doesn't show this, then the simulation
is incomplete and misleading.

There are two things you can learn from a simulation about IIA ...1. how
frequently do various non-condorcet methods satisfy the IIA when possible,
i.e. when there is a ballot CW.... and 2. how frequently does any method,
Condorcet or not yield an LIIA finish order.

Every method whose final step is to sort the finish order pairwise will
have 100 percent compliance with LIIAC, so you don't have to check those.
(But of course you can use them to help debug your simulation.)
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