[EM] voter strategy & 2-party domination under IRV voting

[Warren Smith]

On the probability that insincerely ranking the two frontrunners max and min, is
optimal voter-strategy in an IRV (Instant Runoff Votng) election.
----------Warren D. Smith Aug 2005----------------------------------------------

MATHEMATICAL MODEL: 3-candidate V-voter IRV elections
with random voters (all 3!=6 permutations=votes equally likely).

QUESTION: Is there a subset of identically-voting voters, who, by changing their vote
to rank the two "perceived frontrunners" max and min ("betraying" their 
true favorite "third party" candidate) can make their least-worst frontrunner win
(whereas, their true favorite cannot be made to win no matter what they do)?

THEOREM: The answer to the above question is "yes" with probability
at least  25% = 1/4  in the V=large limit.

PROOF SKETCH:
1. Assume wlog that B is the name of the IRV-winner.

2. Assume wlog that A is the name of the candidate with the fewest top-rank votes,
i.e. the one eliminated in the first IRV round, so it comes down to B versus C.

3. The probability in the V=large limit tends to 1 that all the pairwise
victory margins are of order approximately sqrt(V), and that all the 6 kinds of voters 
occur with counts approximately V/6 each, i.e. much larger than sqrt(V).

4. With probability 50%, candidate B has the most top-rank votes.

5. Given that (4) is true: with probability 50%, the B-C gap in top-rank vote count,
exceeds the C-A gap in top-rank vote count.  9We are now down to probability 25%.)

6. Choose a subset, of cardinality of order sqrt(V), of the voters of type "C>A>B".
(More precisely, we must choose the cardinality*2 to lie above the
previously-mentioned C-A top-rank vote count victory margin, but below the B-C margin.)
If they betray their favorite C by insincerely switching to "A>C>B", 
then A becomes the IRV (and Condorcet) winner,
which from their point of view is a better outcome.
Q.E.D.

STRENGTHENING:
Note our "C>A>B"-type voter subset can argue that obviously, nothing they can do
will elect C, since when they rank C top honestly that fails to do it. Therefore,
their only chance for an improvement is to go for electing A.  And the only
way they can try is to raise A in the rankings.  As we've seen, this reasoning
yields success for them, at least 25% of the time.  However, given their preconception
that C has essentially no chance (or anyway, a chance well below 25%) of victory, 
it actually makes sense for them to rank A top 100%, not 25%, of the time, 
even though we know this will only be successful for them with probability 25%.  
Because given their belief C has no chance, this cannot hurt them - and they know there 
is a 25% chance it will help them.  So we conclude from this that in fact, the "betray C"
strategy is better or at least as good for them as honesty, 100% of the time.

SUMMARY:
This discussion presumably is the underlying theoretical explanation
for the fact that all three IRV countries (Australia, Malta, and Ireland)
historically have been 2-party dominated.

Eveidently the IRV voting method leads to 2-party domination, just like the flawed 
plurality system that method was supposed to "fix."

So anybody who is interested in third parties ever having a chance, would
be advised NOT to foolishly advocate IRV, but instead would be advised to 
advocate RANGE VOTING (which experimentally favors all third parties
far more than either plurality or approval, incidentally,
see the CRV web site).
-wds