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

Revised:  December 31, 2002

For more information about these criteria, see the document "Strategic Indifference" 
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 x Î A, 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 y Î A 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 x Î A, x must not be elected if there 
exists y Î A 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 B Í A  
such that, for all x Î A\B and all y Î B, the number of voters who sincerely prefer y  
over x exceeds the number who sincerely prefer x over y.  For all x Î A\SincereTop, 
x must not be elected if both of the following conditions hold: 
     (1)  For all a,b Î A, every voter who votes a over b sincerely prefers a over b. 
     (2)  There exists y Î SincereTop voted over x by more than half of the voters. 

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

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:

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:

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:

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:

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:

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:

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 s Î L(R,A) such  that x = MAM(A,R,s).  
Make the following abbreviations: 

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

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 a Î A, #R(x,a) < #R/2.  

Thus the following statement holds: 

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

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

(1.4)  For all a Î A, (x,a) Ï Precede(y,x). 

Since MAM = x, x Î Top.  This means (a,x) Ï Aff for all a Î A.  Thus (y,x) Ï Aff.  
By 1.1, (y,x) Î Maj.  Since (y,x) Î Maj and (y,x) Ï Aff, it follows by the 
definition of Affirmed() that (y,x) is inconsistent with Aff Ç Precede(y,x).  
This means there exist (a₁,a₂),(a₂,a₃),...,(a_k-1,a_k) Î Aff Ç Precede(y,x) 
such that a₁ = x and a_k = y.  This implies (x,a₂) Î 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 s Î L(R,A) 
such  that x = MAM(A,R,s).  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 a Î A, #R(a,y) < #R/2.  

Thus the following statement holds: 

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

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

(2.4)  For all a Î A, (a,y) Ï Precede(y,x).

Since MAM = x, x Î Top.  This means (a,x) Ï Aff for all a Î A.  Thus (y,x) Ï Aff.  
By 2.1, (y,x) Î Maj.  Since (y,x) Î Maj and (y,x) Ï Aff, it follows by the 
definition of Affirmed() that (y,x) is inconsistent with Aff Ç Precede(y,x).  
This means there exist (a₁,a₂),(a₂,a₃),...,(a_k-1,a_k) Î Aff Ç Precede(y,x) 
such that a₁ = x and a_k = y.  This implies (a_k-1,y) Î 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 a Î T and all b Î A\T the number of voters who prefer b over a must be 
less than #R/2.  Assume that for all a,b Î A, every voter who votes a over b  
sincerely prefers a over b.  This means the following statement holds: 

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

Assume also that x Î A\T and there exists y Î T 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 s Î L(R,A) such  that 
x = MAM(A,R,s).  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 a Î T and all b Î A\T, #R(y,x) > #R(b,a).

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

(3.3)  For all a Î T, all b Î A\T, (b,a) Ï Precede(y,x). 

Since MAM = x, x Î Top.  This means (a,x) Ï Aff for all a Î A, so (y,x) Ï Aff.  
Since #R(y,x) > #R/2, #R(x,y) < #R(y,x).  Thus (y,x) Î Maj.  Since (y,x) Î Maj and 
(y,x) Ï Aff, by the definition of Affirmed() (y,x) is inconsistent with Aff Ç Precede(y,x).  
This means there exist (a₁,a₂),(a₂,a₃),...,(a_k-1,a_k) Î Aff Ç Precede(y,x) such 
that a₁ = x and a_k = y.  Since a₁ Ï T and a_k Î T, by induction there must exist 
j Î {1,2,...,k-1} such that a_j Ï T and a_j+1 Î T.  But since (a_j,a_j+1) Î 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,y Î A.  Make the following assumptions: 

(4.1)  (y,x) Î Majorities(A,R). (That is, #R(y,x) > #R(x,y).)
(4.2)  For all a₁,a₂,...,a_k Î A, if a₁ = x and a_k = y 
          and (a_j,a_j+1) Î Majorities(A,R) for all j Î {1,2,...,k-1}, 
          then there exists j Î {1,2,...,k-1} such that #R(y,x) > #R(a_j,a_j+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 s Î L(R,A) 
such  that x = MAM(A,R,s).  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 Í Maj, if (y,x) is inconsistent with P then 
          there exists (a,b) Î P such that (a,b) Ï Precede(y,x).

Since MAM = x, x Î Top.  This means (a,x) Ï Aff for all a Î A, so (y,x) Ï Aff.  
By 4.1, (y,x) Î Maj.  Since (y,x) Î Maj\Aff, by the definition of Affirmed() 
(y,x) must be inconsistent with Aff Ç 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