A brief definition of the Maximize Affirmed Majorities (MAM) voting procedure

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

    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 (LIIA) and the voting method Maximum Likelihood Estimation (MLE), and
showed that MLE satisfies LIIA.

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:

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.

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, Condorcet-consistency, top cycle, resolvability, homogeneity, computability
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 x ∈ X, x must not be elected if there exists y ∈ 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).

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 ∈ X, let support(x,y) denote the number of voters who
ranked x over y. For all x ∈ X such that support(x,w) > support(w,x), there must exist
alternatives y₁,y₂,...,y_n ∈ X (n ≥ 1) where y₁=w and y_n=x such that, for all j ∈ {1,2,...,n-1},
all three of the following conditions hold:
    (1) support(y_j,y_j₊₁) > support(y_j₊₁,y_j).
    (2) support(y_j,y_j₊₁) ≥ support(x,w).
    (3) y_j is socially ordered over y_j₊₁.

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     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

For all x,y ∈ X, 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 p ∈ X²,
let p₁ denote the first candidate in p and let p₂ denote the second candidate in p.

Let M denote {p ∈ X² such that support(p₁,p₂) > support(p₂,p₁)}.
Call M the "majorities."

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

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

Let M* denote {p ∈ M 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 x ∈ X such that, for all p ∈ M*, x ≠ p₂. (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.

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.

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.

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:

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:

    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.

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.

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.