The "Maximize Affirmed Majorities" (MAM) voting procedure

Revised:  February 11, 2004

Introduction

Whenever there are more than two alternatives, there will typically be more than one 
majority preference, since each preference is a relative preference regarding a pair of 
alternatives.  For example, suppose there are three alteratives x, y and z, and suppose 
40% of the voters prefer x over y and y over z,  35% prefer y over z and z over x, and 
25% prefer z over x and x over y.  It can be seen that a majority (65%) prefer x over y
another majority (75%) prefer y over z, and another majority (60%) prefer z over x.  
The majority preference for x over y can be interpreted as majority support for the 
proposition that "x shall be socially ranked over y."  Similarly, the majority preference 
for y over z can be interpreted as majority support for the proposition that "y shall be 
socially ranked over z" etc.

In 1785 the Marquis de Condorcet demonstrated in his seminal essay on election theory 
that the majority preferences can be inconsistent, or "cyclic," as they are in the example 
above.  In that essay, Condorcet appears to advocate more than one voting procedure.  
One is tersely described in his introduction.  The following excerpt is translated from 
his introduction:

"... take successively all the propositions that have a majority, beginning 
with those possessing the largest.  As soon as these first propositions 
produce a result, it should be taken as the decision, without regard 
for the less probable decisions that follow.

     -- Marquis de Condorcet, "Essay on the Application of Mathematics to 
         the Theory of Decision-Making" [1785], page lxviii (as translated by 
         Keith Michael Baker in "Condorcet: From Natural Philosophy to Social 
         Mathematics" [1975], p.240, Chicago University Press)

Thus in the example Condorcet would begin by affirming "y over z" since it has the 
largest majority (75%).  Then he would affirm "x over y" since it has the next largest 
majority (65%).  Those two propositions taken together produce the result "x over y 
over z."  The proposition "z over x" would not be affirmed despite its majority support 
since it conflicts with the affirmed propositions.  The social ordering "x over y over z
implies x would be elected.

Condorcet's wording is ambiguous when two or more majorities are the same size and 
when pairings are "ties" (i.e., neither alternative in the pairing has a majority over the other).  
When there are many voters, as in large public elections, neither of these conditions is 
likely to occur.  The "Maximize Affirmed Majorities" (MAM) procedure is based on 
Condorcet's rule, but is defined unambiguously.  

MAM treats each majority preference as evidence that the majority's more preferred 
alternative should finish over the majority's less preferred alternative, the weight of 
the evidence depending on the size of the majority.  Since majority preferences can 
sometimes be inconsistent, it may be impossible to honor all of them.  Therefore, 
MAM considers the majority preferences one at a time, from largest majority to 
smallest majority, affirming only the majority preferences that are consistent with 
those already affirmed.  Since the resulting set of affirmed majority preferences is 
self-consistent, there will always be at least one alternative such that no alternative 
is ranked over it by an affirmed majority.  MAM chooses such an alterative. 
(Typically there will be exactly one such alternative; MAM is usually deterministic.) 

Section 1 below provides two simple examples to illustrate MAM's basic operation.  
Section 2 discusses MAM's tiebreaking mechanism.  Section 3 provides a detailed 
description of the MAM procedure.  Section 4 provides two examples to illustrate 
MAM's operation when tiebreaking is necessary.  A concluding section provides 
additional comments, including a list of desirable criteria satisfied by MAM.

Separate documents provide a mathematically formal definition completely equivalent 
to MAM
(useful in formal proofs that MAM satisfies the criteria it satisfies) and proofs 
that MAM satisfies several important criteria.  Computer software which implements 
MAM will be available for downloading over the internet.

 

1. Two examples to illustrate MAM's basic operation

Example 1.1:  Suppose there are three alternatives, represented here by the letters 
x, y, and z.  The voters are asked to rank the alternatives from most preferred to 
least preferred. (Note that sometimes some voters may have strategic incentives to 
misrepresent their orders of preference.  This is not peculiar to MAM; such incentives 
may exist no matter what voting method is used.)  Suppose the voters submit the 
following votes:

34% 10% 10% 46%
x y y z
y x z y
z z x x

 

By inspection of the votes, three majorities can be identified:  
        66% of the voters ranked y over x.  
        56% of the voters ranked z over x.  
        54% of the voters ranked y over z.  

MAM considers the majorities from largest to smallest:  
        First MAM affirms the "y over x" majority, which is largest.  
        Next, since "z over x" is consistent with "y over x", MAM affirms 
                the "z over x" majority.  
        Next, since "y over z" is consistent with "y over x and z over x", 
                MAM affirms the "y over z" majority.  

Alternative x is less preferred by two affirmed majorities, "y over x" and "z over x."  
Alternative y is not less preferred by any affirmed majority.  
Alternative z is less preferred by one affirmed majority, "y over z."
Since y is not less preferred by any affirmed majority, MAM elects y.

Example 1.2:  Suppose the voters submit the following votes:

34% 10% 10% 46%
x y y z
y x z
z z x

 

This example illustrates that voters are not required to vote complete strict orderings 
of the alternatives.  They are allowed to express indifference, and as a shortcut they 
may leave some alternatives unranked.  When a voter leaves alternatives unranked, 
the ballot is interpreted as though the voter had ranked them as equal to each other 
and less preferred than the explicitly ranked alternatives.  Thus the ballots of the 46% 
who voted just "z" are treated the same as if they had voted z as their first choice 
and x & y equally second.

By inspection of the votes, three majorities can be identified:  
        34% of the voters ranked x over y.  
        56% of the voters ranked z over x.  
        54% of the voters ranked y over z.  

MAM considers the majorities from largest to smallest:  
        First MAM affirms the "z over x" majority.  
        Next, since "y over z" is consistent with "z over x", 
                MAM affirms the "y over z" majority.  
        Next, since "x over y" is inconsistent with "y over z and z over x", MAM 
                does not affirm the "x over y" majority.  (The reason "x over y" is 
                inconsistent is that "y over z" and "z over x" together imply "y over x".)  

Alternative x is less preferred by one affirmed majority, "z over x."  
Alternative y is not less preferred by any affirmed majority.  
Alternative z is less preferred by one affirmed majority, "y over z."
Since y is not less preferred by any affirmed majority, MAM elects y.

 

2. MAM's tiebreaking mechanism

Any voting procedure which satisfies the criteria of anonymity and neutrality must 
in some cases depend on chance. (Anonymity requires that no voter be privileged 
and neutrality requires no alternative be privileged.)  For instance, if there are two 
alternatives and the voters are evenly split between them, each alternative must have 
a chance of being elected. (Presumably a fair coin toss could be used to pick one 
of them randomly.)  This is more likely to occur in small committee voting than in 
large public elections, where chance would rarely play a role.  

Though MAM could be modified to privilege someone as a tiebreaking voter (e.g., 
a committee's chairperson) or to privilege one of the alternatives (e.g., the status quo), 
the version of MAM defined here satisfies anonymity and neutrality.  Thus there are 
some possible scenarios in which chance would play a role.  When MAM considers 
the majorities from largest to smallest, chance could play a role if two or more majorities 
are the same size, since it may be the case that the order in which they are considered 
for affirmation determines which are affirmed, which may ultimately affect which 
alternative is not less preferred by any affirmed majority.  Clearly, in elections with 
many voters it would be very rare for two majorities to be exactly the same size 
and also be relevant to each other's affirmation.  

Chance can also play a role in MAM if two or more alternatives are not less preferred 
by any affirmed majority, but this cannot happen unless the voters are evenly divided 
between two alternatives.  This too would be rare in elections having many voters, 
and also in committees comprised of an odd number of voters.

In order to ensure that MAM completely satisfies the independence of clone alternatives 
criterion, we have defined the tiebreaking mechanism in MAM (used when two or more 
majorities are the same size or when two or more alternatives are not less preferred by 
any affirmed majority) to be more sophisticated than blind chance.  The first step after 
encountering such a tie is to construct a strict ordering of the alternatives, call it Tiebreak, 
by picking one of the voters' ballots at random and adopting its preferences. (If this ballot 
is indifferent regarding any pair of alternatives, we would continue randomly picking from 
the remaining ballots, one at a time, incorporating into Tiebreak their strict preferences 
that don't conflict with those already incorporated, until either Tiebreak is a strict ordering 
of the alternatives or every ballot has been picked.  If Tiebreak is still not a strict ordering 
after picking every ballot--which could only happen if all voters are indifferent regarding a 
pair of alternatives--then any remaining indifferences in Tiebreak are resolved randomly.)  

In large public elections the tiebreaking mechanism isn't important since such ties would be 
extremely unlikely, and a simpler (purely random) tiebreaking mechanism could be used.  
But the tiebreaking mechanism could make a difference when voting within legislatures 
or committees, and since we anticipate MAM will be used by such small groups before 
it is used for public elections, it makes sense to advocate the version of MAM presented 
here which is completely independent of clones, in spite of the slight increase in complexity.  
(We could further argue that all anonymous, neutral voting methods should use this 
tiebreaking mechanism instead of blind chance to improve their resistance to clones.)

Here is a detailed description of the mechanism by which MAM constructs a strict 
ordering of the alternatives to be used if necessary for tiebreaking.  We call this the 
Random Voter Hierarchy procedure.  It is a straight-forward generalization of the 
"Random Dictator" procedure (which simply picks one ballot at random and adopts 
its preferences, and resolves any remaining indifferences randomly).

procedure:  Random Voter Hierarchy

Initialize Tiebreak to be the utterly indifferent "ordering" of the alternatives.

Repeat the following until Tiebreak is a strict ordering of the alternatives:
{

If at least one voter's ballot has not yet been marked "picked"
{

Randomly pick a voter's ballot not yet marked "picked" and 
let r denote this ballot.  Mark r "picked" (so it won't be picked again).

For all pairs of alternatives x and y such that Tiebreak is indifferent 
between x and y and r is not indifferent between x and y
amend Tiebreak (preserving its strict portions) so it ranks x and
relative to each other the same as r does.

}

Else amend Tiebreak (preserving its already strict portions) into a 
strict ordering by randomly resolving all its remaining indifferences.

}

Note that the Random Voter Hierarchy procedure needs to be performed at most once 
(per election), and not at all by MAM if no pairings are ties and no two majorities are the 
same size, which means the tiebreaker will rarely be needed in elections that have many 
voters.  For simplicity, however, we define MAM here so it performs the Random Voter 
Hierarchy procedure exactly once (per election), prior to sorting the list of majorities from 
largest to smallest.  There are some small timesaving efficiencies that could be implemented 
without affecting outcomes, such as calculating only as much of the strict Tiebreak ordering 
as needed to break ties actually encountered.  In the worst case (where every voter is 
totally indifferent between some pair of alternatives) the time to execute Random Voter 
Hierarchy increases as the number of ballots and increases as the square of the number 
of alternatives.  Thus it is practical to compute on inexpensive computers even in a worst 
case involving hundreds of millions of voters .  In scenarios needing tiebreaking, typically 
most voters will express strict preferences regarding most alternatives, so few ballots 
(possibly only one) will need to be picked.  

Assuming it were time-consuming to randomly pick even a few of the ballots (which 
could conceivably be the case if there are hundreds of millions of ballots occupying 
several billion bytes of computer mass storage, another way time might be saved is 
to recognize that in many cases the precise relative ordering of precedence of two 
or more majorities that are the same size will not affect the outcome.  For instance, 
if two majorities that are the same size are not also the smallest majorities in some 
cycle that includes both, then it will not matter which is considered for affirmation 
before the other.  So, before spending time constructing the portion of the Tiebreak 
ordering "needed" to determine the relative precedence of two or more same size 
majorities, we could first check whether their order of precedence matters.  This 
can be accomplished fairly quickly by first neglecting the ones that already conflict 
with those already affirmed, and then checking whether the other ones could all 
be affirmed.  Often the answer will be yes, in which case it will not matter which 
of them is affirmed before the rest. 

In cases where there are hundreds of millions of voters, planning for the worst case 
would mean keeping track of the ballots that have been (randomly) picked, so that 
the time needed to pick an unpicked ballot won't grow as the number of unpicked 
ballots dwindles.  Keeping track means maintaining a growing list of the ballots that 
have been picked so far.  Since in large public elections this list could conceivably grow 
larger than the amount of RAM memory in a small computer, it might be necessary to 
write the list to mass storage, which is slower than RAM.  In many cases, time can be 
saved by not bothering to keep track of the first few ballots picked, trading the extra 
time wasted if the same ballot is accidentally picked more than once for the time saved 
by not having to write any data to mass storage.  There is no harm in accidentally 
picking the same ballot again, other than the time wasted.  So, if the total number 
of voters is n, it would probably be more efficient not to bother tracking the first 
n/20 ballots picked.  And it would make sense to check how much of the list can 
be stored in RAM before writing any of it to mass storage.  

If additional small amounts of time are to be saved, a limit on the number of ballots 
which Random Voter Hierarchy may pick could be imposed.  If the limit is reached 
before the tiebreaking ranking being constructed is strict, remaining indifferences 
would be resolved randomly. (If the limit is set to one, this is the Random Dictator 
procedure.)  Choosing a limit smaller than the number of voters would make it possible 
to violate the Strong Pareto criterion, and since any time saved would be relatively 
small even in the worst case of massive indifference, Random Voter Hierarchy is 
defined with no such limit.  Random Voter Hierarchy, by having no limit, allows 
satisfaction of an independence of clone alternatives criterion that is slightly stronger 
than the version of independence of clone alternatives defined by TN Tideman 
in 1987, but a limit of one is all that is needed to satisfy Tideman's version.  In a 
large public election, little would be lost by setting the limit to zero (meaning the 
tiebreaking strict ranking of the alternatives, if needed, would be completely random) 
since in large elections it is so unlikely there would be any ties needing breaking 
that no would-be manipulator would expect to gain by exploiting properties of 
the tiebreaking procedure. 

The two examples in section 4 illustrate the operation of MAM when tiebreaking 
is necessary.  But first, section 3 provides a detailed description of MAM.

 

3. A complete definition of MAM 

Here is the detailed definition of the MAM procedure:

procedure:  Maximize Affirmed Majorities

Each voter is allowed to rank (that is, order) the alternatives.  (Voters' rankings 
need not be strict, meaning that voters may express indifferences as well as 
preferences.  To be practical when there are many alternatives, as a shortcut 
each voter may leave some alternatives unranked, which shall be interpreted 
as though the voter had voted the unranked alternatives at the bottom, indifferent 
to each other and below all the explicitly ranked alternatives.  To be practical 
in committees, which typically intermingle voting on alternatives with proposing 
of new alternatives, it is recommended that the technology automatically append 
newly proposed alternatives at the bottom of each voter's previously voted ranking 
and allow each voter to easily edit his/her ranking.)

Construct the vote count table V by counting the votes:  
        For each of the possible pairs of alternatives, for instance x and y
                let Vxy be the number of votes that rank x over y
                and let Vyx be the number of votes that rank y over x

Construct the list of Majorities:  
        Begin with an empty list.  
        For each pair of alternatives, for instance x and y
                if Vxy > Vyx 
                        then append "x over y" to the list. 
                else if Vyx > Vxy  
                        then append "y over x" to the list. 

Construct a strict ordering of the alternatives, called Tiebreak, from the voters' 
rankings by using the Random Voter Hierarchy procedure (defined above). 
(This ordering will be used to break ties, if necessary, in the steps which follow.)

Sort the list of Majorities, primarily from largest majority to smallest majority:  
To be more specific, a majority for x over y precedes a majority for z over w 
if and only if at least one of the following conditions holds:

1.  Vxy > Vzw.

2.  Vxy = Vzw and Vwz > Vyx.

3.  Vxy = Vzw and Vwz = Vyx and Tiebreak ranks w over y.

4.  Vxy = Vzw and Vwz = Vyx and w = y and Tiebreak ranks x over z.

(In other words, the majority having more "support" for its preferred alternative 
has greater precedence, as shown in condition 1.  If two majorities have the 
same amount of support, then the one having less support for its less preferred 
alternative (smaller minority "opposition")  has greater precedence, as shown 
in condition 2.  If two majorities have the same amount of support and have 
the same amount of opposition, then their relative precedence is determined 
by appealing to the Tiebreak ordering as shown in condition 3 or 4.)

Initialize the table FinishOver so that no alternative finishes over any other.  
That is, for all pairs of alternatives x and y, set FinishOverxy to false 
and set FinishOveryx to false.

Conditionally affirm majorities in the Majorities list, considering one majority 
at a time in order of precedence, as follows:  
        Letting "x over y" denote the majority under consideration, 
        if FinishOveryx is still false       //   (which means "x over y" doesn't conflict 
                                                      //   with those already affirmed) 
        and FinishOverxy is still false         //   (which means x finishing over y has not 
                                                            //   already been implied by x finishing 
                                                            //   over some z and z finishing over y
                then affirm("x finishes over y")     //  defined below

Construct the list of Top Alternatives, which is the alternative(s) x such that 
FinishOveryx is not true for any alternative y.  

If the list of Top Alternatives contains only one alternative, elect it.  Otherwise, 
elect the alternative in the Top Alternatives list that Tiebreak ranks over all other 
alternatives in that list.

The affirm procedure (invoked repeatedly within the MAM procedure), when given 
a social preference (such as "x finishes over y") to affirm, sets to true the corresponding 
element of the FinishOver table.  Then it affirms the social preferences not yet affirmed 
but which are logically implied by "transitivity" given all the social preferences affirmed thus 
far.  For example, if it has already been affirmed that x finishes over y and that y finishes 
over z, then together these logically imply that x finishes over z.  Furthermore, it means 
a (smaller) majority for z over x, if there is one, conflicts with the (larger) majorities 
already affirmed.  The affirm procedure is "recursive" but not circular, and its maximum 
depth of recursion is always less than the number of alternatives.  The affirm procedure 
is defined here:

procedure:  affirm("x finishes over y")

Set FinishOverxy to true.
Repeat the following once for every alternative, letting the label "a" denote 
the alternative under consideration:

If FinishOverax is true            //  a over x and x over y together imply a over y.
and FinishOveray is not yet true,             //  (Don't waste time unnecessarily.) 
        then affirm("a finishes over y").                       //  (recursion) 
If FinishOverya is true            //  (x over y and y over a together imply x over a.
and FinishOverxa is not yet true,             //  (Don't waste time unnecessarily.) 
        then affirm("x finishes over a").                       //  (recursion) 

While the FinishOver table is under construction during MAM's affirmation of majorities, 
the table offers a quick test of whether a majority being considered for affirmation conflicts 
with the majority preferences affirmed thus far.  For instance, suppose "x over y" is the 
majority being considered for affirmation, and suppose "y over z" and "z over x" are 
majorities that preceded "x over y" due to greater support, and suppose "y over z
and "z over x" were affirmed before "x over y" is considered.  The affirmed majority 
for y over z is treated as evidence that y should finish over z and the affirmed majority 
for z over x is treated as evidence that z should finish over x, so together they also 
constitute affirmed evidence that logically implies (by transitivity) that y should finish 
over x.  Thus those two affirmed majorities contradict evidence that suggests x should 
finish over y.  In other words, "x over y" conflicts with the majorities already affirmed.  
Suppose in addition, for example, that "y over z" was affirmed before "z over x."  
When "z over x" was affirmed, the affirm procedure would at some point have tested 
whether FinishOveryz is true (letting the label x in the definition of affirm represent the 
z in "z over x," letting the label y  in the definition of affirm represent the x in "z over x," 
and at some point letting the label a represent y) and because FinishOveryz was true 
the affirm procedure would have set FinishOveryx to true.  Thus FinishOveryx would 
be true when MAM considers "x over y," indicating "x over y" is inconsistent with 
the affirmed majorities.  Thus in this instance MAM will not affirm "x over y". 

The overall MAM procedure is easily computable since its execution time increases 
only as a small polynomial function of the number of alternatives and the number of 
voters.  Since most children seem capable of ranking things in order of preference, 
and since MAM offers a couple of shortcuts to the voters (non-strict rankings and 
treatment of unranked alternatives as if the voter had ranked them below the explicitly 
ranked alternatives), we consider MAM practical for many democratic institutions, 
such as large public elections and legislatures, councils and committees.  

There is a more sophisticated MAM algorithm that does not merely select a winner.  
It also constructs a complete strict "social" ordering of the alternatives, with the winner 
on top of course, and this requires little extra time.  The MAM social ordering has 
many nice properties.  For one, the alternative that is second in the MAM social 
ordering is the one that would have been elected by MAM if the winner were deleted 
from all the votes.  In other words, deleting from all votes one or more alternatives 
that top the MAM social ordering will not change the rest of the ordering.  Similarly, 
deleting from all votes one or more alternatives at the bottom of the MAM social 
ordering will not change the rest of the ordering.  For more information, see 
the section on the MAM social ordering in "A formal definition of MAM." 

 

4. Two examples involving tiebreaking

Example 4.1:  Suppose the voters rank three alternatives x, y and z as follows:

32% 34% 34%
x y z
y z x
z x y

 

By inspection of the votes, three majorities can be identified:  
        66% of the voters ranked x over y.  
        68% of the voters ranked z over x.  
        66% of the voters ranked y over z.  

MAM considers the majorities from largest to smallest.  The "z over x" majority is 
considered first since it is uniquely largest, and MAM affirms it.  Next, however, 
since the remaining two majorities are the same size (both having the same amount 
of support for the majority's more preferred alternative and both also having the same 
amount of opposition against the preferred alternative), the tiebreaking mechanism is 
invoked:  the Random Voter Hierarchy procedure is used to construct a tiebreaking 
strict ordering of the alternatives, which we name Tiebreak.  Random Voter Hierarchy 
randomly picks one of the voters' ballots and adopts that ballot's strict preferences.  
Thus in this example there are three cases to consider:  

Case 1:  There is a 32% chance that Tiebreak ranks x first, y second, and z last.  
In this case, the "y over z" majority is considered before the "x over y" majority 
since Tiebreak ranks y (the less preferred alternative of the "x over y" majority) 
over z (the less preferred alternative of the "y over z" majority).  Thus MAM 
affirms "y over z."  MAM does not affirm "x over y" since it is inconsistent with 
"y over z" and "z over x."  Since y is not less preferred by any affirmed majority, 
MAM elects y in case 1.

Case 2:  There is a 34% chance that Tiebreak ranks y first, z second, and x last.  
In this case, as in case 1, the "y over z" majority is considered before the 
"x over y" majority since Tiebreak ranks y (the less preferred alternative of 
the "x over y" majority) over z (the less preferred alternative of the "y over z
majority).  Thus MAM affirms "y over z."  MAM does not affirm "x over y
since it is inconsistent with "y over z" and "z over x."  Since y is not less preferred 
by any affirmed majority, MAM elects y in case 2.

Case 3:  There is a 34% chance that Tiebreak ranks z first, x second, and y last.  
In this case, the "x over y" majority is considered before the "y over z" majority 
since Tiebreak ranks z (the less preferred alternative of the "y over z" majority) 
over y (the less preferred alternative of the "x over y" majority).  Thus MAM 
affirms "x over y."  MAM does not affirm "y over z" since it is inconsistent with 
"z over x" and "x over y."  Since z is not less preferred by any affirmed majority, 
MAM elects z in case 3.

Thus there is a 66% chance that MAM elects y, a 34% chance that MAM elects z
and no chance that MAM elects x.

 
Example 4.2:  Suppose the voters rank three alternatives x, y and z as follows:

20% 30% 40% 10%
x y z z
y z y x
z x x y

 

By inspection of the votes, two majorities can be identified:  
        80% of the voters ranked z over x.  
        70% of the voters ranked y over x.  
The remaining pairing, y versus z, is a tie with 50% of the voters ranking y over z 
and 50% ranking z over y.

MAM considers the majorities from largest to smallest:  
        First MAM affirms the "z over x" majority.  
        Next, since the "y over x" majority is consistent with "z over x", 
                MAM affirms the "y over x" majority.  

Alternative x is less preferred by two affirmed majorities, "z over x" and "y over x."
Alternative y is not less preferred by any affirmed majority.  
Alternative z is not less preferred by any affirmed majority. 

Since both y and z are not less preferred by any affirmed majority, MAM invokes its 
tiebreaking mechanism to choose between them.  The Random Voter Hierarchy 
procedure is used to construct a tiebreaking strict ordering of the alternatives, Tiebreak.  
Random Voter Hierarchy randomly picks one of the voters' ballots and adopts its 
strict preferences.  Thus in this example there are four cases to consider:  

Case 1:  There is a 20% chance that Tiebreak ranks x first, y second, and z last.  
In this case, since Tiebreak ranks y over z, MAM elects y.

Case 2:  There is a 30% chance that Tiebreak ranks y first, z second, and x last.  
In this case, since Tiebreak ranks y over z, MAM elects y.

Case 3:  There is a 40% chance that Tiebreak ranks z first, y second, and x last.  
In this case, since Tiebreak ranks z over y, MAM elects z.

Case 4:  There is a 10% chance that Tiebreak ranks z first, x second, and y last.  
In this case, since Tiebreak ranks z over y, MAM elects z.

Thus there is a 50% chance that MAM elects y and a 50% chance that MAM elects z.

 

Conclusion

Here is a list of some criteria satisfied by MAM.  (Definitions of the criteria and proofs 
that MAM satisfies them are provided below, or in linked documents, or are widely 
known in the social choice literature.)

Feasibility (practicality, computability)
Anonymity 
Neutrality 
Pareto (a.k.a. Unanimity), Strong Pareto
Condorcet-consistency, Top Cycle, Condorcet Loser 
Monotonicity (non-negative responsiveness)
Resolvability, Reasonable Determinism 
Minimal Defense 
Non-Drastic Defense 
Truncation Resistance 
Larger Majority
Independence of Clone Alternatives 
Immunity from Majority Complaints 
Local Independence from Irrelevant Alternatives 
Homogeneity 
Independence of Irrelevant Alternatives (the weak version for 
        social choice procedures, not the strong version for social 
        ordering procedures--see "Arrow's Impossibility Theorem
        for the definition of the weak version.)

Here are some criteria not satisfied by MAM, with descriptions provided below:

Reinforcement 
Participation 
Cancellation 
Independence of Irrelevant Alternatives (the strong version for 
        social ordering procedures) 
Choice Consistency (similar in spirit to the strong version of 
        Independence of Irrelevant Alternatives  but worded for 
        social choice procedures--see "Arrow's Impossibility Theorem
        for its definition.) 
Independence of Pareto-Dominated Alternatives
Uncompromising 

Reinforcement requires that if two distinct groups of voters separately choose the 
same alternative, that alternative must be chosen if their votes are tallied together 
as one large group.  Aside from an aesthetic appeal, failure to satisfy reinforcement 
allows manipulation of the outcome only if some minority has the power to divide 
the voters into separate groups.  But in typical voting institutions no minority has 
that power, so this criterion should usually be considered much less important 
than other criteria.  Reinforcement is a weaker criterion than the criterion that it 
be impossible to gain by gerrymandering, which is too strong for any democratic 
voting procedure to satisfy.

Participation requires that no voter prefers the outcome resulting from abstention 
more than the outcome resulting if s/he votes his/her "sincere" preferences.  This is 
a false dichotomy,  however, since voters may choose to vote strategically, for 
instance by ranking a compromise equal to a more preferred alternative.  When 
a voter has information suggesting abstention is preferable to sincere voting, that 
same information suggests compromising is at least as good as abstention.  
So satisfaction of participation should not be considered important.  

Cancellation requires that the outcome must not change if two completely opposite 
votes are removed from the set of votes. (Note that the votes which would tend to 
be completely opposite are those of voters at opposite extremes.)  This criterion is 
aesthetically appealing, but not satisfying it appears to have no cost to society so 
it seems unimportant.  

Reinforcement and participation are satisfied by the Borda procedure but not 
by any voting procedure that satisfies Condorcet-consistency.  Besides failing 
the Condorcet-consistency,  Borda also fails top cycle, minimal defensenon-
drastic defense
, truncation resistance, independence of clone alternatives
and immunity from majority complaints, which are more important (in our 
opinion)  than reinforcement and participation.

Independence of Irrelevant Alternatives (IIA) has been defined in more than 
one way in the political science literature, beginning with Kenneth Arrow in 1953.  
Depending on the formulation, its meaning may be encapsulated by the choice 
consistency
criterion.  It essentially requires that if the voters' preferences regarding 
any non-chosen alternative are deleted from the voters' rankings, then the alternative 
chosen originally must still be chosen. (The term "irrelevant" is misleading, since 
IIA essentially defines as irrelevant every alternative besides the chosen one.)  
However, IIA (or choice consistency) is so demanding that no reasonable voting 
procedure can comply.  IIA is controversial since it is derived from a similar criterion 
which assumes voters' utility valuations of the alternatives are elicited instead of voters' 
ordinal rankings of the alternatives; it is easy to justify an independence criterion for 
voting rules which tally voters' utilities, but it is well-established that voters' utilities 
cannot be reliably elicited, which justifies eliciting ordinal rankings instead but loses 
thereby  the justification for independence.  To the contrary, since voters' ordinal 
rankings  lack information about their preference intensities, preferences regarding 
so-called  "irrelevant" alternatives may provide useful hints. 

Uncompromising requires that if some voters uprank some alternative from the 
bottom (or from having been left unranked)--for example ranking a compromise over 
a "greater evil"-- this must not cause the defeat of an  alternative still ranked over it.  
One voting procedure that satisfies uncompromising is Instant Runoff Voting (IRV, 
also known as the Alternative Vote and as the single-winner versions of Hare and 
Single Transferable Vote).  IRV eliminates one candidate at a time, counting for 
each non-eliminated candidate the number of voters who rank it over all other 
non-eliminated candidates  and eliminating the candidate with the smallest count.  
Unfortunately, IRV can easily neglect voters' expressed preferences for compromise 
alternatives over less preferred alternatives, which means it conflicts with criteria that 
are more important, in our opinion.  For instance, IRV egregiously fails Condorcet-
consistency
, top cycle, monotonicityminimal defense, non-drastic  defense
and immunity from majority complaints (also less important criteria  such as 
reinforcement and participation, and the essentially impossible strong IIA).  
Uncompromising has been promoted  by advocates of IRV such as the Center 
for Voting and Democracy, who are more concerned with promoting proportional 
representation systems for electing  legislatures than with improving elections where 
one alternative is to be elected. (Since the IRV algorithm is the special single-winner 
form  of the Hare Single Transferable Vote proportional representation algorithm, 
teaching the public about IRV (instead of about better procedures) serves the 
interests of proportional  representation advocates by helping to demystify an 
important proportional representation  algorithm.  Arguably their interests would 
be better served by a voting procedure such as MAM that would be better than 
IRV at reducing the "spoiler" problem, thereby allowing voters to express their 
preferences for their most preferred alternatives more often.)  

Another voting procedure that satisfies uncompromising is the minimax variation 
of Simpson-Kramer, which elects the candidate whose largest pairwise opposition 
is the smallest.  Minimax satisfies monotonicity and independence from 
Pareto-dominated alternatives
, but not Condorcet-consistency, top cycle
independence from clone alternatives or immunity from majority complaints
(Its failures of some of these criteria are not egregious, however.  For instance, 
it is Condorcet-consistent if all votes are strict rankings of the alternatives, 
and it can only fail top cycle in the presumably rare cases where the majorities 
within the top cycle are larger than other majorities.) 

Another criterion promoted on behalf of Instant Runoff by the Center for Voting and 
Democracy is not well-defined, but was described in Science News (Nov. 2, 2002). 
Essentially, they asserted that voting procedures need to strike the proper balance of
two conflicting criteria:  rewarding the candidate ranked top by the most voters yet
not ignoring the rest of each voter's preferences.  They claimed IRV strikes this 
balance best.  Their justification for placing so much weight on voters' top choices
(as if each voter cares strongly for her top choice and is nearly indifferent between 
the rest) is that being ranked top by a large minority of voters signifies possession of 
essential leadership skills.  By implication, a candidate ranked second by every voter 
is less qualified than a candidate ranked top by a large minority, and in particular
possesses less leadership ability.  They cited no evidence to support such a claim,
and it stands to reason that, all else being equal, voters would rank candidates
possessing superior leadership ability over other candidates, so that in a competitive
election a candidate ranked second by most voters would indeed possess those skills. 
If their argument is based on the theory that being the sole nominee of a "large" party
indicates possession of those skills, a counter-argument is that the nominee typically
has had little to do with building or leading the party organization; rather, s/he won a
brief "king of the hill" contest over other potential nominees in which special interest
money may be more important than leadership skills.

Furthermore, it is important to keep in mind the effect of the voting method on the
nomination process.  Just as Plurality Rule induces parties to nominate at most one
candidate per office to avoid electing less preferred candidates, so would IRV. 
If party X nominates more than one candidate for an office under IRV, they risk 
causing their own defeat since IRV can easily eliminate their most electable candidate 
before their less electable candidates.  To see this, suppose "moderate" candidate x  
would win, in part by votes of some of the "swing voters" whose policy preferences 
lie between x and party Y's candidate y.  Now suppose X also nominates x' who is 
less moderate than x but more appealing than x to X's hard-core base and other 
less moderate voters.  Thus x no longer appears to IRV as the top choice of as 
many voters as before, and thus x may be eliminated before x'.  Assuming many 
of the swing voters prefer y over (less moderate) x', y would win.  This risk may 
be great enough and ubiquitous enough that parties would rarely nominate more 
than one candidate per office, and thus would still rely on a flawed nomination
process (for instance, primary elections dominated by special interest money) that
would stifle competition and perhaps prevent the best candidates from competing. 
By similar reasoning, "less moderate" third party candidacies could result in the defeat
of more moderate candidates. (Thus the claim made by many advocates of IRV that 
IRV would eliminate "spoiling" is false.)  Also, "centrist" third parties adopting policies 
between the two large parties would be eliminated early by IRV, and possibly 
perceived as being unpopular. 

Indeed, books on Robert's Rules of Order (including the prestigious Scott, Foreman
editions) clearly recommend against using IRV when better methods are feasible, 
noting that IRV can easily defeat the best compromises. (The books on Robert's 
Rules unfortunately call IRV by the generic name "preferential voting" even though 
many other methods, such as MAM, allow voters to vote orders of preference.)  
In hindsight, the U.S. is fortunate that IRV was not the procedure used at the 
Constitutional Convention of 1787, since the "Great Compromise" (population-
proportional House of Representatives and state-equal Senate) would have been 
deemed the least popular alternative by IRV: big states preferred proportionality 
and small states preferred equality, with the compromise being the favorite of 
almost no one but the second choice of nearly every delegate.  Possibly that 
would have eliminated the compromise from further consideration and prevented 
ratification of a U.S. constitution.

Implied two paragraphs above is the claim that MAM would induce parties to
nominate more than one candidate per office.  Each party would have at least five 
incentives to do so:  (1) By nominating candidates favored by diverse factions, 
voter turnout of their supporters would presumably increase, and while voting 
for their favorites many of those voters would also presumably rank the party's 
other candidates over those of other parties.  (2) The party may not know at 
nomination time which of their potential nominees have the best chances to win 
later in the general election.  (3) Since MAM satisfies about as much of spirit of 
the strong IIA criterion as possible, nominating more than one candidate would 
rarely cause the defeat of all of the party's candidates, and such cases might be 
predictable (and thus avoidable on a case-by-case basis).  (4) The minimal 
defense
voting strategy would be available to deter reversal strategies that 
might be attempted by supporters of other parties.  (5) The party could save 
considerable campaign resources by eliminating its primary elections. 


 

In addition to the Marquis de Condorcet, some credit for the development of the 
MAM procedure should also be given to TN Tideman.  Tideman described a 
procedure he called Ranked Pairs in his paper "Independence of Clones as a 
Criterion for Voting Rules
" (Social Choice and Welfare, 1987).  Ranked Pairs 
is closely related to MAM even though MAM was independently developed.  
Ranked Pairs differs from MAM in three ways.  The two differences that are 
most important in the context of large public elections (where it is unlikely any 
two majorities will be the same size) are: 
        1. Ranked Pairs requires each voter to vote a strict ordering of the 
            alternatives (at least in its 1987 incarnation). 
        2. When Ranked Pairs determines the order of precedence of the majorities, 
             it measures their sizes by subtracting opposition from support.  
Because of these differences, Ranked Pairs fails minimal defense, non-drastic 
defense
truncation resistance and immunity from majority complaints.  
The other difference, unimportant in large public elections but possibly significant 
in small elections such as committees and legislatures, Ranked Pairs uses a 
different tiebreaking scheme when determining the relative precedence of two 
or more majorities that have the same size, and as a result it does not completely 
satisfy independence of clones or strong Pareto-efficiency.  

In a subsequent 1989 paper, TM Zavist and Tideman revised Ranked Pairs so it 
allows voters to express non-strict orderings and uses a different tiebreaking scheme.  
The 1989 tiebreaking scheme provides complete satisfaction of independence of 
clones
and strong Pareto-efficiency, but sacrifices monotonicity by permitting 
the chance that an alternative is elected to fall when voters rank it higher. (This was 
not mentioned in their paper.)  Ranked Pairs 1989 still fails minimal defense
non-drastic defense
truncation resistance and immunity from majority 
complaints
.  The tiebreaking scheme in MAM does not sacrifice monotonicity.  
(The MAM tiebreaker was developed as a refinement of Zavist's tiebreaker, 
so the development of MAM was not entirely independent of Tideman and 
Zavist's work.)