The Maximize Affirmed Majorities voting procedure (MAM)
A detailed definition
of MAM
A formal
definition of MAM (useful in proofs about MAM's
properties)
See "Set
Operators and Binary Relations" for definitions of
some
basic mathematical terms.
A
brief definition of MAM (in math notation, as is the formal
definition above).
Some criteria satisfied by MAM:
feasibility: The election outcome must be computable
in small polynomial time,
and no voter may be required to strictly order a (huge) set of
alternatives.
anonymity: Every voter must be
treated equally.
neutrality: Every alternative
must be treated equally.
strong
Pareto: If at least one voter ranks alternative y
over alternative x
and no voters rank x over y, then x must not be
elected.
monotonicity
(non-negative responsiveness): If some voters
raise
an
alternative in their rankings, then its probability of being
elected
must not
decrease.
resolvability,
reasonable determinism (These criteria can be
described
non-technically as "Outcomes must rarely depend on
chance.")
homogeneity: Adding a duplicate
of every vote must not change the outcome.
Condorcet-consistency:
If an alternative, say x, is such that for each other
alternative, say y, some majority of the voters rank x
over y, then x
must be elected.
top cycle: The elected
alternative must be in the smallest non-empty subset
of the alternatives such that, for each alternative in
the subset, say x,
and each alternative not in the subset, say y, the number
of voters who
rank x over y exceeds the number of voters who rank y
over x.
non-dictatorship: For each
voter, say v, there must be a pair of alternatives,
say x and y, and some collection of admissible votes
such that v's vote
ranks x over y and y is elected. (Note
that non-dictatorship is easy
to satisfy; for instance, any election procedure that satisfies
anonymity
or Condorcet-consistency also satisfies
non-dictatorship.)
independence of irrelevant
alternatives (IIA, the weak version for social
choice
procedures): The winner must not change if voters raise or
lower non-
nominated
alternatives in their votes. (See "Arrow's
Impossibility Theorem."
Also see below for the strong version for social ordering procedures,
which has the same
name IIA in the literature of social choice
theory.)
independence
of clone alternatives (ICA, promoted by TN
Tideman):
If
there is a subset of the nominated alternatives such that no voter
ranks
any
alternative outside the subset between any alternatives in the
subset,
then the
subset is called a “set of clones.” The election odds of
every
alternative
that's not one of the clones must not change if a strict subset
of the clones is
deleted from both the votes and the set of nominees.
local
independence of irrelevant alternatives (LIIA): If
the order of finish
ranks some subset of alternatives together (in other words, no
alternative
outside
the subset finishes between any in the subset) then the relative
order of finish of the
alternatives within this subset must not change
if all other alternatives are deleted from the votes and from the
set of
nominees. (This
criterion is promoted by Peyton Young.)
minimal
defense: If more than half of the voters prefer
alternative y over
alternative x, then that majority must have some way of voting
that
ensures x
will not be elected and does not require any of them to
rank y equal to or over any alternatives preferred over y.
(Another
wording
is nearly equivalent: Any ordering of the alternatives must
be
an admissible
vote, and if more than half of the voters rank y over
x
and x
no higher than tied for bottom, then x must not be
elected.
This criterion, in particular the first wording, has been promoted by
Mike Ossipoff
under the name Strong Defensive Strategy Criterion.
Satisfaction means a majority can defeat "greater evil"
alternatives
without having to pretend to prefer some compromise
alternative
as
much as or more than favored alternatives. Since voters in
public
elections
cannot be relied upon to misrepresent their preferences in
this way,
non-satisfaction usually means elites will offer an electoral
system in which
there are only two viable parties, each of which
nominates only one alternative.)
non-drastic
defense: If more than half of the voters prefer
alternative y
over alternative x, then that majority must have some way of
voting
that
ensures x will not be elected and does not require any of
them to
rank y
over any more-preferred alternatives. (This has been promoted
by Mike Ossipoff under
the name Weak Defensive Strategy Criterion.
Non-satisfaction means some members of the majority may need
to
misrepresent
their preferences by voting a compromise alternative
over favored alternatives if they want to ensure the defeat of
less-
preferred
alternatives.)
truncation
resistance: Define the "sincere top set" as
the smallest subset
of alternatives such that, for each alternative in the subset, say x,
and
each
alternative outside the subset, say y, the number of voters
who
sincerely
prefer x over y exceeds the number who sincerely prefer
y
over x.
If no voter votes any alternative over any less-preferred
alternative, and more than half the voters rank some alternative
in
the sincere
top set, call it x, over some alternative not in the sincere
top set, call it y,
then y must not be elected. (This is a strengthening
of a criterion having
the same name promoted by Mike Ossipoff.
His version applies only when the sincere top set contains only
one
alternative.)
immunity
from majority complaints: If a majority ranked an
alternative x over
the elected alternative w, then there must be a sequence of
alternatives
a1,a2,...,an
such that a1 = w and an
= x and the following conditions hold:
(1) Majorities at least as large as the majority who rank x
over w
rank a1 over a2, a2
over a3, etc., and an-1
over an.
(2) The election procedure generates an order of finish—it
doesn't
just pick a winner—and the order of finish places a1
over a2,
a2 over a3, etc., and an-1
over an.
Some criteria not satisfied by MAM:
independence of irrelevant alternatives (IIA, the
strong version for social
ordering procedures): For all pairs of alternatives, for
instance x and y,
their relative social ordering must not change if voters raise or
lower
other
alternatives in their votes. (This was proposed by Kenneth
Arrow
and is
similar in spirit to his choice consistency criterion for
social choice
procedures, described below. It is too demanding for any
reasonable
social
ordering procedure to satisfy. See "Arrow's
Impossibility Theorem.")
choice
consistency: If y but not x is elected
from some nominated subset of
alternatives that includes both, then x must not be elected
from any set
of
nominees that includes both. (This was proposed by Kenneth
Arrow
for social
choice procedures, and is similar in spirit to his
independence
of irrelevant alternatives criterion for social ordering
procedures. It
is too demanding for any reasonable social choice procedure to
satisfy.
See "Arrow's
Impossibility Theorem.")
independence
of Pareto-dominated alternatives (IPDA): If
no voters rank
alternative x over alternative y and at least one voter ranks y over x
then the election outcome must not change if x is deleted from the votes
and from the the set of nominees.
uncompromising:
The winning alternative must still win if some voters
who
ranked it top
continue to rank it top but rearrange the lower portion of
their votes. (This is promoted by the Center for Voting and
Democracy,
a
non-profit organization who promote proportional
representation
and, recently, the Instant Runoff voting procedure.)
reinforcement:
If the voters can be partitioned so each partition would
elect the same alternative, then that alternative must be
elected.
no-show:
The election outcome when some voters abstain must not be
preferred by them over the election outcome when they vote
their
sincere
preferences.
cancellation:
If two votes are exactly opposite, the election outcome
must not change if they are deleted from the collection of
votes.
(Note
that reinforcement, no-show and cancellation
cannot be satisfied
by any procedure that satisfies
Condorcet-consistency or minimal defense,
and
we consider them far less important.)
People such as Donald
Campbell and Jerry Kelly (2000) and Peyton
Young
("Equity in Theory and Practice")
suggest that since Kenneth Arrow's choice
consistency,
and the strong version of IIA for social ordering
procedures, are
too demanding for reasonable election
procedures to satisfy, they should be
relaxed, but only as much
as necessary. In that spirit, we conjecture that since
MAM satisfies independence criteria such as those
listed above, it comes as
close as possible to satisfying
choice consistency and strong IIA
without
sacrificing satisfaction of Arrow's other
criteria.
For more about the criteria listed above, see the concluding
section of "A detailed
definition of MAM."
Some possible modifications of MAM:
To provide complete determinism (at the expense of
anonymity and/or
neutrality, of course) when
breaking ties by privileging some voters (e.g.,
the
chairperson or senior members) and/or privileging some
alternatives
(e.g., alternatives nominated earlier).
To slightly privilege some alternative(s) (e.g., the status quo, in ties).
To strongly privilege the status quo, so that it wins if the
number of voters
who rank the “pure MAM” winner
over the status quo is not large enough.
(For instance, 2/3
of the voters.)
To strengthen the minimal defense criterion to the
following:
Sincere defense: If more than half the voters
prefer some ("compromise")
candidate x over every candidate in some ("greater evil")
subset of the
candidates, call them Y, that majority must be able to vote
their sincere
orders of preference yet ensure every candidate in Y
loses.
(That is, allow each voter to insert a dividing
line in her ranking. Then, for all
pairs of candidates,
say x and y, such that the number of voters who rank
x
over y exceeds the number who rank y
over x and the number who rank x
over the
line and y under the line exceeds the number who rank y
over the line
and x under the line, give maximal
precedence to the majority for x over y.
Then
MAM's affirmation procedure will ensure that y is not
elected.)
Software that implements MAM:
Online demo <not yet available>
Online
voting servers:
This server requires
someone to enter all voters' ballots into a text field.
(Copy&paste is
recommended when possible.)
This server <not yet available> supports each voter entering
her own vote.
Downloadable software for setting up
your own voting server <not yet available>
Some comparisons of MAM with other voting procedures, based on computer simulations:
MAM
vs. Instant Runoff
MAM
vs. PathWinner (also known as Markus Schulze' method)
“Citizen Kang” 8 minute video (from The Simpsons' 1996 Treehouse of Horror episode)