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

over alternative

that ensures

rank

by Mike Ossipoff under the name

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

of alternatives such that, for each alternative in the subset, say

each alternative outside the subset, say

sincerely prefer

over

alternative, and more than half the voters rank some alternative in

the sincere top set, call it

top set, call it

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

the elected alternative

(1) Majorities at least as large as the majority who rank

rank

(2) The election procedure generates an order of finish—it doesn't

just pick a winner—and the order of finish places

**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.")

alternative

then the election outcome must not change if

and from the the set of nominees.

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.)

elect the same alternative, then that alternative must be elected.

preferred by them over the election outcome when they vote their

sincere preferences.

must not change if they are deleted from the collection of votes.

(Note that

by any procedure that satisfies

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

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

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 excerpt from The Simpsons' 1996 Treehouse of Horror VII (low res, fair use)**