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)