"Maximize Affirmed Majorities" (MAM) satisfies Minimal Defense,
Non-Drastic Defense, Truncation Resistance and Larger Majority.

Revised:  December 31, 2002

and the paragraphs below.

Let A denote the set of (nominated) alternatives, assumed finite and non-empty.

Minimal Defense:  Every (non-strict) ranking of A must be an admissible vote,
and for all xA, x must not be elected if both of the following conditions hold:
(1)  More than half of the voters vote x no better than tied for bottom.
(2)  There exists yA voted over x by more than half of the voters.

Non-Drastic Defense:  Each voter must be allowed to vote as many alternatives
tied for top as she wishes, and for all xA, x must not be elected if there
exists yA such that that more than half of the voters vote y over x and
no worse than tied for top.

Truncation Resistance:  Let SincereTop denote the smallest non-empty BA
such that, for all xA\B and all yB, the number of voters who sincerely prefer y
over x exceeds the number who sincerely prefer x over y.  For all xA\SincereTop,
x must not be elected if both of the following conditions hold:
(1)  For all a,bA, every voter who votes a over b sincerely prefers a over b
(2)  There exists y ∈ SincereTop voted over x by more than half of the voters.

Larger Majority:  For all x,yA, let #R(x,y) denote the number of voters
who vote x over y.  For all x,yA, say that x beats y pairwise if and only if
#R(x,y) > #R(y,x).  For all xA, x must not be elected if there exists yA
such that both of the following conditions hold:
1.  y beats x pairwise.
2.  For all a1,a2,...,akA, if a1 = x and ak = y and aj beats aj+1
pairwise for all j ∈ {1,2,...,k-1}, then there exists j ∈ {1,2,...,k-1}
such that #R(y,x) > #R(aj,aj+1).

Satisfaction of minimal defense is considered more important than satisfaction of
non-drastic defense due to our expectation it will be easier for voters to coordinate
the "minimal defense strategy" (that is, downranking of "greater evil" alternatives)
than the "non-drastic defense strategy" (that is, upranking a compromise alternative
to at least tied for top).  "Defensive" strategies are important in scenarios where
supporters of "greater evil" alternatives would otherwise have an incentive to
strategically misrepresent their preferences (in particular, voting the reverse of
some preferences, such as downranking the "sincere winner" below less preferred
alternatives).  We do not consider defensive strategies "manipulative" since they
serve to defend the election of the alternative that would be elected if every voter
voted her sincere order of preference, by creating an equilibrium in which no other
subset of voters has an incentive to misrepresent preferences.  Satisfaction of
non-drastic defense is better than nothing, however, since the only other defensive
strategy capable of thwarting the "offensive" reversal strategy is the "drastic"
defensive strategy of ranking some compromise alternative over more preferred
alternatives, and we believe this defensive strategy is the most difficult to coordinate.

An example of a voting procedure that satisfies non-drastic defense but fails minimal
defense
is Approval, in which each voter may select as many alternatives as she wishes
and the alternative selected by the most voters is elected.  The non-drastic defensive
strategy under Approval is for a majority to select some compromise alternative they
all prefer over the "greater evil" alternatives, and for each voter in that majority to also
select every alternative she prefers over the compromise.

Two common voting procedures which fail these criteria are "plurality rule" (which
allows each voter to select only one alternative, and elects the alternative selected
by the most voters) and "majority runoff" (which allows each voter to select only one
alternative, and elects the alternative selected by more than half the voters if there
is one, and otherwise conducts a second round of voting to choose between the
two alternatives selected by the most voters).  Some other prominent procedures
that fail all four criteria are Instant Runoff (also known as Hare, Single Transferable
Vote, and the Alternative Vote), Copeland, Simpson and scoring rules such as Borda.
Thus, most voting procedures often force many voters to choose between expressing
their preference for their favorite alternative or pretending to prefer some "compromise"
alternative over their favorite, a strategy that many voters seems to loathe.

Technically, the larger majority criterion is not failed by plurality rule, majority
runoff, nor Approval since those procedures do not permit voters to rank the
alternatives.  Larger majority serves as a tool for comparing voting procedures
that allow each voter to non-strictly rank the alternatives, since it turns out that
its satisfaction by such procedures implies they also satisfy the other three criteria.
Also, satisfaction of larger majority by a procedure that allows non-strict rankings
implies that a majority seeking to defeat some "greater evil" alternative x do not
necessarily need to rank x no better than tied for bottom (the strategy suggested
by minimal defense), since they can also defeat x by ranking x no better than
the alternatives that may cycle with their "compromise" alternative y.  Thus the
misrepresentation of preferences implied by downranking of x can be even
less than with the blunt strategy suggested by minimal defense

Example 1:  The minimal defense voting strategy.

Suppose there are 3 alternatives x, y and z.  Suppose the voters'
sincere preferences regarding the alternatives are as follows:

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

That is, 46% of the voters prefer x over y and y over z, 10% prefer y over x
and x over z, etc.  Suppose the voters vote the following rankings:

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

The votes of the 46% who voted x over z over y misrepresent their preferences
since they prefer y over z. (This is the classic "reversal" strategy, burying y
in their rankings hoping to defeat y and elect their favored alternative x.)
The votes of the 10% who ranked only y are interpreted as having ranked x
and z tied for bottom, below y.  Thus 54% rank x no better than tied for
bottom and rank y over x, so satisfaction of minimal defense requires
that x not be elected given these votes.  Note that the 34% who prefer z
over y do not need to rank y equal to or over z to ensure x is not elected,
assuming satisfaction of minimal defense.

Now suppose instead that the voters vote the following rankings:

 46% 20% 34% x y z y y z x

The votes of the 20% who ranked only y are interpreted as having ranked x and z
tied for bottom, below y.  Thus 66% rank z no better than tied for bottom and
rank y over z.  Also, 54% rank x no better than tied for bottom and rank y over x.
Therefore, satisfaction of minimal defense requires that neither x nor z be elected
given these votes, which means y must be elected.  Note that this set of votes
constitutes a "strategic equilibrium" since no voter nor coalition of voters has
an incentive to vote differently. (That is, no strategy of the 46% can elect x
the alternative they prefer over y, and no strategy of the 34% can elect z
the alternative they prefer over y.)  Thus no voter outside the 20% has an
incentive to misrepresent preferences, and no voter has an incentive to
misrepresent preferences regarding alternatives preferred over y.

Example 2:  The non-drastic defense voting strategy.

Suppose there are 3 alternatives x, y and z.  Suppose the voters'
preferences regarding the alternatives are as follows:

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

That is, 46% of the voters sincerely prefer x over y and y over z, 10% prefer y
over x and x over z, etc.  Suppose the voters vote the following rankings:

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

The votes of the 46% who voted x over z over y misrepresent their preferences
since they prefer y over z. (As above, this is the classic "reversal" strategy.)
The 34% who ranked both z and y tied for best are part of a 54% coalition
who voted y over x and no worse than tied for best.  Satisfaction of non-drastic
defense
requires that x must not be elected given these votes.  Note that the
34% who prefer z over y do not need to "drastically" rank y over z to ensure
that x is not elected, assuming satisfaction of non-drastic defense.

Example 3:  Resistance to truncation.

Suppose there are 3 alternatives x, y and z.  Suppose the voters'
preferences regarding the alternatives are as follows:

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

That is, 46% of the voters sincerely prefer x over y and y over z, 10% prefer y
over x and x over z, etc.  The sincere top cycle is {y} since more voters
prefer y over x than vice versa and more voters prefer y over z than vice
versa.  Suppose the voters vote the following rankings:

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

The 46% who ranked only x have not expressed their preference for y over z.
Such an expression of indifference that fails to express a voter's sincere preference
is called a "truncation of preference."  Those votes are interpreted as having ranked
y and z tied for bottom, below x.  However, note that all of the strict preferences
expressed by all of the voters accurately represent sincere preferences.  Since 54%
rank y over x, satisfaction of truncation resistance requires x must not be elected.
Thus, satisfaction of truncation resistance implies truncation is ineffective as an
"offensive" voting strategy, so no "defensive" misrepresentation of preferences by
the 54% is needed to ensure x will not be elected.  This property is important
when few voters are sophisticated enough to attempt a "reversal" strategy, and
when voters are representatives accountable to unsophisticated constituents
who will not appreciate reversal strategies.

Theorem "MAM satisfies Minimal Defense."

Proof:  Refer to the definitions in the document that provides the formal definition
of MAM
.  By inspection of the definition of MAM, all strict and non-strict rankings
of A are admissible votes under MAM.  Thus the first clause of minimal defense is
satisfied by MAM.  Assume more than half of the voters rank y over x and more
than half of the voters rank x no better than tied for bottom.  We must show that
MAM does not elect x.  Suppose to the contrary  there is a non-zero chance that
MAM elects x.  This means there exists σ ∈ L(R,A) such  that x = MAM(A,R,σ).
Make the following abbreviations:

Maj = Majorities(A,R).
Aff = Affirmed(A,R,σ).
Precede(y,x) = Precede(y,x,A,R,σ).
Top = Top(A,R,σ).
MAM = MAM(A,R,σ).

Since more than half of the voters rank y over x and x no better than tied for bottom,
clearly both of the following statements hold:

(1.1)  #R(y,x) > #R/2.
(1.2)  For all aA, #R(x,a) < #R/2.

Thus the following statement holds:

(1.3)  For all aA, (y,x) is larger than (x,a).

By 1.3 and the definition of precedence, the following statement must hold:

(1.4)  For all aA, (x,a) ∉ Precede(y,x).

Since MAM = x, x ∈ Top.  This means (a,x) ∉ Aff for all aA.  Thus (y,x) ∉ Aff.
By 1.1, (y,x) ∈ Maj.  Since (y,x) ∈ Maj and (y,x) ∉ Aff, it follows by the
definition of Affirmed()
that (y,x) is inconsistent with Aff ∩ Precede(y,x).
This means there exist (a1,a2),(a2,a3),...,(ak-1,ak) ∈ Aff ∩ Precede(y,x
such that a1 = x and ak = y.  This implies (x,a2) ∈ Precede(y,x).  But this
contradicts 1.4, which means the contrary assumption cannot hold, establishing
MAM cannot elect x.  This implies MAM satisfies minimal defense.        QED

Theorem "MAM satisfies Non-Drastic Defense."

Proof:  Refer to the definitions in the document that provides the formal definition
of MAM
.  By inspection of the definition of MAM, all strict and non-strict rankings
of A are admissible votes under MAM.  Thus each voter may rank as many alternatives
as she wishes at the top of her vote, which means the first clause of non-drastic defense
is satisfied by MAM.  Assume more than half of the voters rank y over x and no worse
than tied for best.  We must show that MAM does not elect x.  Suppose to the contrary
there is a non-zero chance that MAM elects x.  This implies there exists σ ∈ L(R,A
such  that x = MAM(A,R,σ).  Make the same abbreviations as in the proof above
that MAM satisfies minimal defense.  Since more than half of the voters rank y
over x and no worse than tied for top, clearly both of the following statements hold:

(2.1)  #R(y,x) > #R/2.
(2.2)  For all aA, #R(a,y) < #R/2.

Thus the following statement holds:

(2.3)  For all aA, (y,x) is larger than (a,y).

By 2.3 and the definition of precedence, the following statement must hold:

(2.4)  For all aA, (a,y) ∉ Precede(y,x).

Since MAM = x, x ∈ Top.  This means (a,x) ∉ Aff for all aA.  Thus (y,x) ∉ Aff.
By 2.1, (y,x) ∈ Maj.  Since (y,x) ∈ Maj and (y,x) ∉ Aff, it follows by the
definition of Affirmed()
that (y,x) is inconsistent with Aff ∩ Precede(y,x).
This means there exist (a1,a2),(a2,a3),...,(ak-1,ak) ∈ Aff ∩ Precede(y,x
such that a1 = x and ak = y.  This implies (ak-1,y) ∈ Precede(y,x).  But this
contradicts 2.4, which means the contrary assumption cannot hold, establishing
MAM cannot elect x.  This implies MAM satisfies non-drastic defense.      QED

Theorem "MAM satisfies Truncation Resistance."

Proof:  Refer to the definitions in the document that provides the formal definition
of MAM
.  Abbreviate T = SincereTop.  By the definition of SincereTop, for
all aT and all bA\T the number of voters who prefer b over a must be
less than #R/2.  Assume that for all a,bA, every voter who votes a over b
sincerely prefers a over b.  This means the following statement holds:

(3.1)  For all aT and all bA\T, #R(b,a) < #R/2.

Assume also that xA\T and there exists yT such that #R(y,x) > #R/2.  We
must show MAM does not elect x.  Suppose to the contrary there is a non-zero
chance that MAM elects x.  This implies there exists σ ∈ L(R,A) such  that
x = MAM(A,R,σ).  Make the same abbreviations as in the proof above that
MAM satisfies minimal defense.  Since #R(y,x) > #R/2, by 3.1 the following
statement holds:

(3.2)  For all aT and all bA\T, #R(y,x) > #R(b,a).

By 3.2 and the definition of precedence, the following statement must hold:

(3.3)  For all aT, all bA\T(b,a) ∉ Precede(y,x).

Since MAM = x, x ∈ Top.  This means (a,x) ∉ Aff for all aA, so (y,x) ∉ Aff.
Since #R(y,x) > #R/2, #R(x,y) < #R(y,x).  Thus (y,x) ∈ Maj.  Since (y,x) ∈ Maj and
(y,x) ∉ Aff, by the definition of Affirmed() (y,x) is inconsistent with Aff ∩ Precede(y,x).
This means there exist (a1,a2),(a2,a3),...,(ak-1,ak) ∈ Aff ∩ Precede(y,x) such
that a1 = x and ak = y.  Since a1T and akT, by induction there must exist
j ∈ {1,2,...,k-1} such that ajT and aj+1T.  But since (aj,aj+1) ∈ Precede(y,x),
this contradicts 3.3, which means the contrary assumption cannot hold, establishing
MAM cannot elect x.  This implies MAM satisfies truncation resistance.     QED

Theorem "MAM satisfies Larger Majority"

Proof:  Refer to the definitions in the document that provides the formal
definition of MAM
.  Pick any x,yA.  Make the following assumptions:

(4.1)  (y,x)Majorities(A,R). (That is, #R(y,x) > #R(x,y).)
(4.2)  For all a1,a2,...,akA, if a1 = x and ak = y
and (aj,aj+1)Majorities(A,R) for all j ∈ {1,2,...,k-1},
then there exists j ∈ {1,2,...,k-1} such that #R(y,x) > #R(aj,aj+1).

We must show that MAM does not elect x.  Suppose to the contrary  there is
a non-zero chance that MAM elects x.  This implies there exists σ ∈ L(R,A
such  that x = MAM(A,R,σ).  Make the same abbreviations as in the proof
above that MAM satisfies minimal defense.

By 4.2 and the definitions of inconsistency and precedence, the following holds:

(4.3)  For all P ⊆ Maj, if (y,x) is inconsistent with P then
there exists (a,b)P such that (a,b) ∉ Precede(y,x).

Since MAM = x, x ∈ Top.  This means (a,x) ∉ Aff for all aA, so (y,x) ∉ Aff.
By 4.1, (y,x) ∈ Maj.  Since (y,x) ∈ Maj\Aff, by the definition of Affirmed()
(y,x) must be inconsistent with Aff ∩ Precede(y,x).  But this  contradicts 4.3,
so the contrary assumption cannot hold, establishing MAM does not elect x.
This implies MAM satisfies larger majority.        QED