The Maximize Affirmed Majorities voting procedure (MAM)
A detailed definition
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-
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
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
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
("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
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
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:
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)