[EM] Introducing "Insanity Voting" - the only voting system which meets the Insanity Criterion

[Scott Ritchie]

Summary:

Each voter casts a standard, preferential ranked ballot.  It is not
necessary for the voter to give a complete list of preferences, however
like most ranked systems doing so increases the effect the voter has on
the election results.

Insanity Voting is meant to capitalize on both strategic voting and
strategic nomination.  While other systems attempt to minimize the
strategy considerations allowed to voters and potential candidates,
Insanity Voting instead realizes that strategic elements are an
advantage and instead attempts to maximize it.


The method:

Assign each candidate a number from 1 to n - do this in a way such that
the voter knows the candidate's number before voting, perhaps by sorting
them alphabetically.

To determine the first winner, convert each voter's first preference
into a number from 1 to n based on the candidate he chose.  Sum them up,
divide by n, and take the remainder - the winner is the candidate whose
number matches (or the last candidate if remainder 0).

To determine the next winner, examine each voter's second preference
with respect to his first, and determine their difference.  If a voter
voted for candidate 5 first and candidate 8 second, the difference would
be 3.  Add n if the result is negative.  This in turn generates a second
number from that voter's ballot ranging from 1 to n - 1.  Remove the
previous winning candidate from the selection pool and assign each
candidate a new number (original number - 1) if they had a higher number
than the previous round's winner.  Sum up the numbers from the ballots
as we did before, divide by n - 1, and determine a new winner.

This process is then repeated until you have as many winners as you
want.  In fact, you can even generate a complete ordering of all
possible candidates this way, and you're guaranteed to never have a tie,
something that isn't possible even in plurality methods.


Strategic Voting:

Insanity Voting, as stated before, is completely vulnerable to strategic
voting, even when done by a single voter.  This is by design, as it
greatly increases the impact an individual voter will have on the
election results, and therefore increases the incentive for turnout.
Similar arguments have been used to support the electoral college, for
instance, however insanity voting increases the odds an individual will
have an effect on the election outcome to their absolute maximum
possible.

To vote strategically in Insanity Voting, you must dishonestly change
your list of preferences until your preferred order of candidates wins.
Note that it is possible to change your preferences such that _any_
selection of candidates wins, regardless of the other voters.

Strategic nomination is similarly egalitarian - unlike most other
methods, which tend to ignore the fringe candidates that don't receive
any of the vote, in Insanity Voting every candidate matters equally.
Insanity Voting allows for both teaming, spoiler candidates, and
crowding.


Criterion:


Neutrality of Spoiled Ballots: Passed.  Each unmarked ballot (or
preference for an incomplete ballot) will have no effect on the vote.

Arrow's unrestricted domain or universality Criterion: Passed with
flying colors.  The winners are all ranked, in order, and can be from
any of the candidates listed.  Moreover, the outcome is completely
deterministic, not random.

Arrow's non-dictatorship Criterion: Passed.  Arrow defined dictatorship
to be only counting one ballot and ignoring the others.  In Insanity
Voting, every ballot is counted, and every ballot determines the winner.
One might call this dictatorship of the entire electorate, and that
seems far more preferable than tyranny of a mere majority.

Arrow's non-imposition or citizen sovereignty Criterion: Passed as well.
Every candidate can win - in fact, it's quite likely they will.

Summability: Passed.  Insanity voting is first order summable, the best
kind.  We can cut it up into however many districts, counties, or
polling places we want, and we'll be able to break each voter's ballot
down into a series of n numbers on the first try.

Plurality criterion: "If the number of ballots ranking A as the first
preference is greater than the number of ballots on which another
candidate B is given any preference, then A's probability of election
must be no less than B's.": Passed.  Since all candidates always have an
equal chance of winning, this criterion holds.

Favorite Betrayal Criterion: "For any voter who has a unique favorite,
there should be no possible set of votes cast by the other voters such
that the voter can optimize the outcome (from his own perspective) only
by voting someone over his favorite.":  Passed, for two reasons.  If we
assume the voter is voting strategically, then for any and all sets of
votes by others, he can elect his favored candidate.  If, on the other
hand, the voter isn't voting strategically, it should be noted that he
cannot alter the probability of his candidate winning anyway, and thus
can't possibly betray him.

Pareto criterion: "If every voter ranks candidate A above candidate B,
then B must not be elected": Failed, except when there are two
candidates and an even number of voters.  And that's a pretty common
exception, really - the best way of passing.

Participation Criterion: Depending on the definition used, this one is
either passed or failed.  Mike Ossipoff's overly restrictive definition,
"Adding one or more ballots that vote X over Y should never change the
winner from X to Y" is not met. However, Douglas Woodall's definition,
"The addition of a further ballot should not, for any positive whole
number k, reduce the probability that at least one candidate is elected
out of the first k candidates listed on that ballot" is met.  It should
also be noted that an inverse of the participation criterion, that
choosing to vote and rank your candidate can cause him to win, is always
true regardless of the makeup of the electorate - in other words, one
always has a complete incentive to vote, no matter how unpopular his
candidate is.

Independence of Irrelevant Alternatives: Failed.  However, many fairly
good methods fail this criterion too.

Later-no-harm Criterion: Passed.  All candidates have the same
probability of winning regardless of what you put on your ballot, so
adding him later will do him no harm.

Mutual Majority Criterion: Failed.  However, any subset of a majority of
voters can vote strategically and select their preferred candidate.  In
fact, it only takes a very small subset of just one person.

Condorcet and Generalized Condorcet Criterion: Failed, however we could
modify the method to select a Condorcet winner if one exists, then
default to insanity voting.  That seems to be all the rage these days.

Consistency Criterion: Failed, unless the selected winners are the last
candidates.  But that only seems fair, since they had to be listed last
on the ballot.  

Monotonicity Criterion: Failed, but that's not a problem since Instant
Runoff and similar methods fail it too, and most insanity voters will
probably want to vote strategically anyway.

Strategy-Free Criterion: Failed, but that's ok because strategic voting
is important to have in an election system.

Additionally, another Criterion that I believe is important enough to be
listed that I couldn't find anywhere:

Impossibility of Ties Criterion:  The voting system should never rank
two candidates equally or resort to nondeterministic methods to resolve
ambiguity.


And, of course, the criterion specially made up for the Insanity Voting
System:

Insanity Criterion: Every voter must be capable of affecting the outcome
regardless of other voters preferences.  Strangely, this is a common
conception of democracy we hold - the idea that "every vote should
count" - yet as far as I can tell only the Insanity Voting method
adheres to this principle.



It seems as though Insanity Voting meets many, if not most, of the
important Criteria for a good voting system.  I recommend we implement
it immediately for all official disputes.

My vote counts,
Scott Ritchie