** (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,unanimity,rationality

andindependence from irrelevant alternatives. Besides satisfying

non-dictatorship,unanimity,rationalityand Peyton Young'slocal, MAM also satisfies

independence from irrelevant alternatives

TN Tideman'sindependence from clone alternativesas 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
alternatives* (

showed that MLE satisfies

Local independence of irrelevant alternatives(LIIA): For all pairs of alternatives

x,ysuch thatxandyare adjacent in the social ordering and a majority preferxtoy,

xmust be socially ordered overy.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): LetXdenote the set of alternatives.

CallYÍXa set of exact clones if, for ally,zÎY, every voter ranksyequal toz.

CallYÍXa set of clones if, for ally,zÎYand allxÎX\Y, every voter who ranksx

overyalso ranksxoverzand every voter who ranksyoverxalso rankszoverx.

For allYÍXwhich is a set of clones, allxÎX\Ysuch thatYÈ {x} is not a set of

exact clones, and allYwhich is a strict subset of'Y, the probability thatxis elected

must not change if alternatives inYare 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:

654

abc

bcd

cda

da 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-dictatorship*, *unanimity*, *rationality*, and more of
* IIA* than MAM.

MAM also satisfies other desirable criteria:
*anonymity*, *neutrality*, *monotonicity*, * strong
Pareto*,

in small polynomial time

majority complaints

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 * x* Î *X*, * x* must not be elected if
there exists *y* Î *X* 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

compromisingif, to defeat a less-preferred alternativex,

the voter "insincerely" raises some alternativey(the "compromise") equal to or over a

more-preferred alternativez(and call it "drastically" compromising ifyis raised overz).

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,y* Î *X*,
let support(*x,y*) denote the number of voters who

ranked *x* over *y*. For
all * x* Î *X* such that support(*x,w*)
> support(*w,x*), there must exist

alternatives *y*_{1}*,y*_{2}*,...,y _{n}* Î

all three of the following conditions hold:

(1) support(

(2) support(

(3)

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:Bradley

Gore, Nader

McCain

Bush

Forbes, DoleNote 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,y* Î
*X*, let support(*x,y*) denote the number of voters who
ranked *x*
over *y*.

Let *X*^{2}
denote
the set of all possible ordered pairs of candidates. For all *p* Î
*X*^{2},

let *p*_{1} denote
the first candidate in *p* and let *p*_{2} denote the second candidate
in *p*.

Let M denote {*p* Î
*X*^{2}
such that support(*p*_{1}*,p*_{2}) > support(*p*_{2}*,p*_{1})}.

Call M the "majorities."

For all *p,q* Î
M, say that *p* precedes *q* if support(*p*_{1}*,p*_{2})
> support(*q*_{1}*,q*_{2}). (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 *p* Î
M, let M^{+}(*p*)
denote {*q* Î
M such that *q* precedes *p*}.

(In other words, the "majorities that precede *p*.")

For all *p* Î
M and all *S* Í M,
say that *p* cycles with *S* if and only if there
exist candidates

*x*_{1}*,x*_{2}*,...,x _{n}* (

(In other words,

Let M*
denote {*p*
Î M such that *p* does
not cycle with M*
Ç 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 *x* Î
*X*
such that, for all *p*
Î
M*, *x* ¹ *p*_{2}.
(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

Odenote the set of all possible strict orderings of the alternatives.

Define the majorities M as above.

For alloÎOand allpÎ M, say thatpagrees withoiforanksp_{1}overp_{2}.

For allo,o'ÎO, letM(o,o') denote {pÎ M such thatpagrees withoor witho'

but not with both}.

For allo,o'ÎO, callobetter thano'ifM(o,o') is not empty and the largest majority

inM(o,o') agrees witho'. (Actually, a more careful definition defines the largest

majority inM(o,o') in such a way that it is unique whenM(o,o') is not empty and the

"better than" relation is a strict ordering ofO. 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 bestoÎO.

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.

**References**

1. Arrow KJ (1963): Social Choice and Individual Values. Yale University Press.

2. Baker KM
(1975): Condorcet: From Natural Philosophy to Social
Mathematics.

Chicago University Press, p.240.

3. Campbell DE, Kelly J
(2000): Information and Preference Aggregation. Social
Choice

and Welfare, 17: 3-24.

4. Condorcet (1785): Essai
sur l'application de l'analyse à la probabilité des décisions

rendues à la pluralité des voix. Paris.

5. Hild M (2001): Lecture notes on Arrow's
theorem for SES/Pl 169
("Who Gets The

Kidney?"), spring term 2001, California Institute of Technology.

6. McKelvey R (2000): Social Choice (draft). California Institute of Technology.

7. Ossipoff M (1996): various messages in
the email list election-methods-list@eskimo.com

(archived and searchable on the www).

8. Plott CR (1976): Axiomatic Social
Choice Theory: An Overview and Interpretation.

American Journal of Political Science, XX, 3, August 1976, 511-596.

9. Tideman TN (1987): Independence of Clones as a Criterion for Voting
Rules. Social

Choice and Welfare, 4: 185-206.

10. Young HP (__): Equity In Theory and Practice. Princeton University Press, pp.38-40.

11. Young HP (1995): ___. Journal of Economic
Perspectives - Winter1995 Symposium

on Voting Methods.

___ (1995): (interpretation of Condorcet
as Ranked Pairs) ___. Journal of Economic

Perspectives - Winter1995 Symposium on Voting Methods.