(DRAFT)    The Maximize Affirmed Majorities (MAM) voting method    (DRAFT)
Steve Eppley  <seppley@alumni.caltech.edu>

This paper describes a voting method, MAM, which may come as close as 
possible to satisfying Arrow's criteria:  non-dictatorship, unanimityrationality 
and independence from irrelevant alternatives.  Besides satisfying 
non-dictatorshipunanimityrationality and Peyton Young's local 
independence from  irrelevant alternatives
, MAM also satisfies 
TN Tideman's independence from clone alternatives as well as 
other desirable criteria.


1. Introduction 

    Arrow's "impossibility theorem" [1963] showed that no voting method can, in every voting 
scenario, satisfy a certain set of desirable criteria: non-dictatorship, unanimity, rationality, and 
independence from irrelevant alternatives (IIA).  Thus no voting method is ideal.  Some scholars 
describe Arrow's result as meaning no voting method is "reasonable" but, since society must make 
choices, that is unhelpful.  That is, it is important to discover the best (not necessarily ideal) voting 
method.  Scholars such as Young [__,1995] and Campbell & Kelly [2000] have suggested that 
the best voting method is one that satisfies as much of the force of Arrow's criteria as possible.  

    The Arrow criterion that is too demanding is IIA, which requires the relative social ranking 
of each pair of candidates to depend only on the voters' relative orderings of that pair alone. 
(Some people such as Plott [1976], McKelvey [2000] and Hild [2001] rewrite the criteria so 
that IIA is easy to satisfy and rationality is the one which is too demanding, but the problem 
remains the same: adding or deleting a losing candidate from the set of nominees can change the 
winner.)  Since IIA cannot be entirely satisfied, elites may sometimes--perhaps often, depending 
on the voting method--easily manipulate outcomes by manipulating the set of of nominees, and 
superior candidates may sometimes--perhaps often, depending on the voting method--avoid 
competing to prevent a "greater evil" from defeating a compromise (called "spoiling"). 

    Instead of calling all voting methods unreasonable, we should call a voting method reasonable 
if it satisfies non-dictatorship, unanimity, rationality, and as much of IIA as possible.  When 
Young [__,1995] explored this approach, he proposed local independence from irrelevant 
 (LIIA) and the voting method Maximum Likelihood Estimation (MLE), and 
showed that MLE satisfies LIIA.  

Local independence of irrelevant alternatives (LIIA):  For all pairs of alternatives 
x,y such that x and y are adjacent in the social ordering and a majority prefer x to y
x must be socially ordered over y.   

Maximum Likelihood Estimation (MLE):  From the set of possible social orderings, 
find the ordering that maximizes the sum of sizes of majorities that agree with it, 
and elect the candidate atop that ordering.  

But Young erred significantly by calling LIIA a "slight" weakening of IIA.  Voting methods can 
satisfy not only LIIA but also other, perhaps more important, independence criteria such as 
independence from clone alternatives [Tideman 1987] which is not satisfied by MLE: 

Independence from clone alternatives (ICA):  Let X denote the set of alternatives.  
Call YX a set of exact clones if, for all y,zY, every voter ranks y equal to z.  
Call YX a set of clones if, for all y,zY and all xX\Y, every voter who ranks x 
over y also ranks x over z and every voter who ranks y over x also ranks z over x.  
For all YX which is a set of clones, all xX\Y such that Y ∪ {x} is not a set of 
exact clones, and all Y' which is a strict subset of Y, the probability that x is elected 
must not change if alternatives in Y' are deleted from all votes. 

ICA should be required for informational and anti-manipulation reasons.  It is easy, given 
an alternative x, to find alternatives that are similar or inferior to x, and the extra information 
elicited from the voters regarding such alternatives does not really tell us anything new regarding 
the voters preferences, and thus should not affect the outcomes of elections.  Furthermore, 
this seems a much easier manipulation, assuming the voting method does not satisfy ICA
than the manipulation of voting methods that satisfy ICA but do not satisfy certain other 
anti-manipulation criteria.  For instance, the reinforcement criterion satisfied by the Borda 
voting method requires that if there exists a partitioning of the voters into two groups such that 
the same candidate would be elected by both groups tallied separately, then that candidate must 
be elected when all the votes are tallied together.  A weaker reinforcement criterion satisfied 
by MLE requires that if there exists a partitioning of the voters into two groups such that both 
groups tallied separately would produce the identical social ordering, then that must be the social 
ordering when all the votes are tallied together.  To exploit non-satisfaction of reinforcement
would-be manipulators would need the power to choose whether and how the voters are 
partitioned, but since it is simple for the rules to force no partitioning or to require that only a 
majority vote can partition the voters, we can conclude that satisfaction of reinforcement is 
much less important. 

    The following example shows that MLE fails ICA

Example 1.1:  Suppose 15 voters rank four candidates a, b, c and d as follows:  

6               5               4 
a               b               c
b               c               d 
c               d               a 

d               a               b 

Note that {c,d} is a set of clones and d is Pareto-dominated by c.  MLE scores 60 for 
the ordering b>c>d>a since b>c>d>a agrees with the 11 vote majority for "b over c
plus the 11 for "b over d" plus the 15 for "c over d" plus the 9 for "c over a" plus the 9 
for "d over a".  It can be checked that no other ordering has as large a score, so MLE 
elects b.  To satisfy ICA, MLE must still elect b when d is deleted from all votes, 
but MLE elects a when d is deleted since the maximal ordering without d is a>b>c
(Note that nearly all voting methods socially order a>b>c if d is deleted.)  By also 
nominating d, manipulators change the outcome from a to b, and drop a from top 
to bottom.

    This paper briefly describes a voting method called Maximize Affirmed Majorities (MAM) 
that satisfies non-dictatorship, unanimity, rationality, and the independence criteria discussed 
above (except of course IIA).  Its underlying heuristic is that the larger the number of voters who 
hold a preference, the more respect should be accorded that preference.  It is plausible that no 
voting method satisfies non-dictatorshipunanimity, rationality, and more of IIA than MAM.  

    MAM also satisfies other desirable criteria: anonymity, neutralitymonotonicity, strong 
, Condorcet-consistencytop cycleresolvabilityhomogeneitycomputability 
in small polynomial time
minimal defense, truncation resistance, and immunity from 
majority complaints
.  Most of these criteria are well-known in the social choice literature.  
Minimal defense and truncation resistance are slight generalizations of criteria promoted 
by Ossipoff [1996].  

Minimal Defense:  Let X denote the set of alternatives.  The voting method must allow 
voters to order the candidates and express indifference (at least at the bottom of their 
orderings) and, for all xX, x must not be elected if there exists yX such that the 
number of voters who vote [y over x and x no higher than tied for bottom] exceeds 
the number of voters who vote x higher than tied for bottom. 

    Satisfaction of minimal defense means a majority who prefer y over x won't need to employ 
"compromising" voting strategies to ensure x will be defeated, even if the minority who prefer x 
employ voting strategies of their own (such as order reversal). 

Call a voting strategy compromising if, to defeat a less-preferred alternative x
the voter "insincerely" raises some alternative y (the "compromise") equal to or over a 
more-preferred alternative z (and call it "drastically" compromising if y is raised over z). 

If it is true that voters dislike compromising, hate compromising more than necessary, and may 
not know how far they need to compromise, the task of coordinating a voting strategy that raises 
a compromise candidate figures to be more problematic than the task of coordinating a strategy 
involving lowering of "greater evil" candidates.  The lowering strategy associated with the criterion 
does not require lowering below worse candidates--lowering to indifference suffices--in order 
to achieve an equilibrium instead of creating new strategic opportunities for the worse candidates. 
(Such a benign use of strategic indifference has been neglected in the social choice literature, 
which often simplifies analysis by assuming voters vote strict orderings.  Note that such strategies 
are not manipulative since the equilibrium elects the candidate that would win anyway if all voters 
voted their sincere orders of preference.)  

    Immunity from majority complaints is desirable in case the votes reveal a majority cycle, 
and turns out to be essentially an axiomatic characterization of MAM:

Immunity from majority complaints:  Let X denote the set of alternatives.  Let w denote 
the winning alternative.  For all x,yX, let support(x,y) denote the number of voters who 
ranked x over y.  For all xX such that support(x,w) > support(w,x), there must exist 
alternatives y1,y2,...,ynX (n ≥ 1) where y1=w and yn=x such that, for all j ∈ {1,2,...,n-1}, 
all three of the following conditions hold:  
    (1)  support(yj,yj+1) > support(yj+1,yj).  
    (2)  support(yj,yj+1) ≥ support(x,w). 
    (3)  yj is socially ordered over yj+1.  

For more information about the criteria and proofs of their satisfaction by MAM, follow the links 
in www.alumni.caltech.edu/~seppley/MAM procedure definition.htm.


2. The Maximize Affirmed Majorities voting method (MAM) 

Let X denote the set of candidates.  

Each voter votes by ordering the candidates (that is, sorting them from best to worst).  The 
following illustrates a ballot format that is machine-readable using inexpensive technologies:  

              <---better           worse--->
 Bush     ( )  ( )  ( )  (
)  ( )  ( )  ( )  ( )
 Gore     ( )  (
)  ( )  ( )  ( )  ( )  ( )  ( )
 Nader    ( )  (
)  ( )  ( )  ( )  ( )  ( )  ( )
 Bradley  (
·)  ( )  ( )  ( )  ( )  ( )  ( )  ( )
 McCain   ( )  ( )  (
)  ( )  ( )  ( )  ( )  ( )
 Forbes   ( )  ( )  ( )  ( )  ( )  ( )  ( )  ( )
 Dole     ( )  ( )  ( )  ( )  ( )  ( )  ( )  ( )


We could describe this example vote more compactly by the following, in which 
the candidates have been sorted from top to bottom: 

Gore, Nader
Forbes, Dole 

Note that voters may rank candidates as equal, as the voter in this example 
has done for Gore=Nader and Forbes=Dole.  As a shortcut, candidates left 
unranked are treated as if the voter had ranked them worst, as the voter in 
this example has done for Forbes and Dole. (If ballot width is constrained, 
fewer columns may be offered; even two columns would allow one for 
"favorites" and one for "compromises"--the worst candidates would be left 
unranked--providing a significant improvement over the expression possible 
with traditional voting methods).  Given a computer interface, each voter 
could be offered a list of candidates to drag into the desired order (e.g., 
top to bottom).  Given a touchscreen interface, the voter could drag 
candidates into order using a fingertip.  

For all x,yX, let support(x,y) denote the number of voters who ranked x over y.  

Let X2 denote the set of all possible ordered pairs of candidates.  For all pX2
let p1 denote the first candidate in p and let p2 denote the second candidate in p.  

Let M denote {pX2 such that support(p1,p2) > support(p2,p1)}.  
Call M the "majorities." 

For all p,qM, say that p precedes q if support(p1,p2) > support(q1,q2). (This is 
actually an oversimplified definition of precedence.  The complete definition defines it 
to always be a strict ordering of the majorities even when majorities have the same 
amount of support, but the brief definition here suffices for large public elections, 
where it is very unlikely any two conflicting majorities will have equal support.  
See the note below regarding the case where majorities have equal support.)  

For all pM, let M+(p) denote {qM such that q precedes p}. 
(In other words, the "majorities that precede p.")

For all pM and all SM, say that p cycles with S if and only if there exist candidates 
x1,x2,...,xn (n ≥ 1) such that {(p2,x1),(x1,x2),(x2,x3),...,(xn-1,xn),(xn,p1)} ⊆ S
(In other words, p cycles with S if S contains a "majority path" from p2 to p1.)

Let M* denote {pM such that p does not cycle with M*M+(p)}.  Call M* the 
"affirmed majorities." (We might also call M* the "maximal acyclic majorities."  Note 
that the definition of M* is recursive but not circular; it can be computed quickly by 
examining the majorities one at a time in order of precedence, including into M* 
each majority that does not cycle with those already included.)  

Elect the candidate xX such that, for all pM*, xp2. (In other words, the winner 
is the candidate that is not less preferred by any affirmed majority.  There will always 
be at least one such candidate since, by construction, M* is acyclic, and in large public 
elections it would be very unlikely for there to be more than one.  See the note that 
follows regarding which one is elected when there is more than one such candidate.) 

NOTE:  The web pages at www.alumni.caltech.edu/~seppley provides the complete 
definition of MAM.  It unambiguously specifies strict precedence of majorities even when 
two or more majorities have equal support.  It also specifies which candidate is elected 
when more than one is not less preferred by any affirmed majority.  In large public 
elections such cases would be extremely rare, so the brief definition above is practical. 


Two examples illustrate the operation of MAM: 

Example 2.1:  Suppose Gore, Bush and McCain compete, and suppose the votes are:  

 45%      8%      12%      35% 
 Gore   McCain   McCain    Bush  
McCain   Gore     Bush    McCain 
 Bush    Bush     Gore     Gore 

Three majorities exist:  65% ranked McCain over Bush, 55% ranked McCain over Gore, 
and 53% ranked Gore over Bush.  First "McCain over Bush" is affirmed since it is the 
largest.  Then "McCain over Gore" is affirmed since it is next largest and does not cycle 
with {McCain over Bush}.  Then "Gore over Bush" is affirmed since it does not cycle 
with {McCain over Bush, McCain over Gore}.  Since McCain is the candidate who is 
not second in any affirmed majority, he is elected. 




Example 2.2:  (An example where majorities cycle.)  Suppose the votes are: 

45%      8%      12%      35% 
Gore   McCain   McCain    Bush  
        Gore     Bush    McCain 
        Bush     Gore     Gore 

Three majorities exist:  55% ranked McCain over Gore, 53% ranked Gore over Bush, 
and 35% ranked Bush over McCain.  First "McCain over Gore" is affirmed since it is the 
largest.  Then "Gore over Bush" is affirmed since it is next largest and does not cycle 
with {McCain over Gore}.  Then "Bush over McCain" is NOT affirmed since it cycles 
with {McCain over Gore, Gore over Bush}.  Since McCain is the candidate who is 
not second in any affirmed majority, he is elected.


3. Discussion 

    It can be shown that MAM is equivalent to choosing the "best" possible social ordering, 
where the "better than" relation on the set of possible social orderings is a leximax comparison 
of the majorities that agree with the orderings:  

Let O denote the set of all possible strict orderings of the alternatives.  
Define the majorities M as above.  
For all oO and all pM, say that p agrees with o if o ranks p1 over p2.  
For all o,o'O, let M(o,o') denote {pM such that p agrees with o or with o'  
but not with both}.  
For all o,o'O, call o better than o' if M(o,o') is not empty and the largest majority 
in M(o,o') agrees with o'. (Actually, a more careful definition defines the largest 
majority in M(o,o') in such a way that it is unique when M(o,o') is not empty and the 
"better than" relation is a strict ordering of O.  In large public elections, this brief 
definition suffices since it is very unlikely two or more majorities will have the same 
amount of support.) 
Elect the alternative atop the best oO

Thus the difference between MAM and MLE is the difference in their "better than" relations: 
MAM's comparitor is a leximax of agreed majorities while MLE's is a sum of agreed majorities.  
Use of the sum makes MLE easily manipulable by nominating clone alternatives or Pareto-
dominated alternatives, undermining the claim that the MLE social ordering has the 
maximum likelihood of being the best. 

    The definition of MAM may appear complex, but it is natural once one sees that, since each 
one of a voter's preferences is a relative comparison of a pair of candidates, multiple majorities 
exist when more than two candidates compete. (The examples above illustrate this.)  Indeed, 
MAM may be the procedure proposed in 1785 by the Marquis de Condorcet in his seminal 
essay [6].  In his introduction, Condorcet wrote: 

"... 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.

                    --  translated by Keith Michael Baker [1975]. 

    Condorcet aimed to construct a social ordering of the candidates given the voters' preference 
orders, so the candidate atop the social ordering could be elected.  By "propositions" he meant 
all possible pairwise-relative outcomes (e.g., "candidate x shall be socially ordered over y", 
which is supported by voters who rank x over y and opposed by voters who rank y over x).  
By "result" he meant the relative social ordering of a pair of candidates (e.g., "x finishes over y") 
which can be taken either directly (e.g., by existence of a majority who rank x over y) or by 
inference from results already adopted into the social ordering (e.g., "x finishes over z" and 
"z finishes over y" together imply x finishes over y).  Condorcet discovered the possibility that 
majorities can cycle and reasoned that the larger a majority, the greater the likelihood their 
preference is right, so that if majorities cycle the preferences of the larger majorities should 
be respected.  

    It is straightforward to interpret Condorcet's words to mean, "Consider each majority 
preference, one at a time in order of decreasing support, and adopt into the social ordering 
under construction each majority preference that does not conflict with the partial social 
ordering already constructed."  In other words, MAM.)  

    When Tideman [1987] proposed independence from clone alternatives, he defined a 
voting method closely related to MAM that he called Ranked Pairs. (He did not notice its 
similarity to the Condorcet algorithm, which is forgivable.  This was noticed later by [1995].)  
But Ranked Pairs measures majority size by margin, deducting the size of the opposition from 
the size of the supporting majority.  Tideman also required each vote to be a strict ordering 
(perhaps to simplify his analysis).  Each of these choices, substracting the opposition and 
disallowing expressions of indifference, suffices by itself to prevent Ranked Pairs from satisfying 
minimal defense.  If voters may submit non-strict orderings, Ranked Pairs also fails to satisfy 
truncation resistance and immunity from majority complaints. (Even if one assumes every 
voter's sincere preferences are strict orderings, it is desirable to allow voters to vote non-strict 
orderings for at least two reasons:  (1) to provide a shortcut when many candidates are 
nominated, allowing the voter to leave candidates unranked knowing those will be treated 
as if she had ranked them at the bottom, and (2) to make feasible the voting strategy needed 
to satisfy minimal defense; that is, down-ranking to tied for bottom the candidates whose 
supporters may attempt an order reversal strategy.  Note that the minimal defense voting 
strategy is not manipulative: it does not alter the outcome; rather, it creates an equilibrium 
that defends the sincere winner.) 

    Though it is tempting to focus on the relative complexity of the mechanics of tallying various 
voting methods, for instance pointing out that MAM is more complex to tally than Plurality Rule, 
now that computers are used to tally the votes it is reasonable to conclude that the complexity 
which matters most is how each voter translates her preferences into an optimal vote.  Voters 
need learn at most once how a voting method is tallied (when they are invited to start using 
that method) but they have to learn anew for each election which voting strategy is optimal and 
the difficulty of their strategy coordination varies depending on the voting method.  This may be 
less complex with MAM than with many methods which are simpler to tally--even young children 
are capable of sorting things from most-preferred to least preferred--to the point where parties 
may have incentives to nominate more than one candidate.  For instance, a party might increase 
the turnout of its supporters on election day by nominating a diverse set of candidates.  Thus 
parties could dispense with (expensive) primary elections, and it would arguably be reasonable 
for society to require parties that still want to hold primary elections to finance them themselves.  


4. Variations of MAM 

    Use of MAM in U.S. presidential elections presents a challenge due to the need to win a 
majority of the Electoral College.  To avoid fragmenting the Electoral College, candidates could 
be allowed to withdraw after election day.  For example, suppose Bill Bradley  finished ahead 
of Al Gore, a fellow Democrat, in New York State and suppose that would prevent Gore  from 
winning a majority of the Electoral College.  Bradley could withdraw, unblocking New York 
for Gore--after having given the voters the opportunity to express their preferences for him.  
The withdrawal option could be allowed in non-presidential elections too, if reasonable 
candidates are deterred from competing out of fear of worsening the outcome (which may 
happen since IIA cannot always be satisfied), or to provide a second line of defense against 
order reversal voting strategy (the first line of defense being the minimal defense deterrent 
strategy mentioned above). 

    For elections with more than one winner (multimember districts, city council, school board) 
and assuming proportional representation voting methods are undesired, MAM can be extended 
into a multiwinner method by electing the highest candidates in the social ordering inferable from 
the affirmed majorities.  

    For voting on citizens' initiatives and other public ballot propositions, MAM can replace the 
Yes/No voting method.  Then conflicting initiatives could be placed on the ballot without imposing 
the common strategy dilemma that a more-preferred proposition can be defeated if some of its 
supporters also approve a compromise proposition.  Perhaps more important, society would 
elicit potentially valuable information about voters' preferences. 

    For voting within assemblies on proposals and amendments, MAM can replace the agenda 
voting method (also called successive pairwise elimination) recommended by Robert's Rules of 
Order.  Instead, alternatives could be added to the ballot anytime in any order, and the number 
of rounds of voting would be determined by letting each voter include a special alternative, 
"continue voting," in her ranking which, whenever it wins a round, causes there to be at least 
one more round.  Using MAM instead of agenda voting would eliminate opportunities for 
chairmen to manipulate outcomes by controlling the agenda order, and appropriately sensitize 
outcomes to the sizes of majorities.  

    To meet any desired supermajority requirements that may be desired to protect a status quo, 
such as the common 2/3 or 3/4 requirement to amend a charter or constitution, a rule can be 
added to MAM to keep the status quo when it is not second in any "large" affirmed majority. 



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