The "Maximize Affirmed Majorities" voting method (MAM) 
satisfies immunity from majority complaints 
and local independence of irrelevant alternatives 
(DRAFT - need to complete some proofs)

Steve Eppley <seppley@alumni.caltech.edu>

Revised:  January 02, 2003

Condorcet's Paradox tells us that, no matter what voting procedure is used, when society 
elects one alternative from a set of three or more alternatives, it is possible a majority 
of the voters will prefer a non-elected alternative over the elected alternative, as in 
the following scenario: 

Scenario 1:  9 voters rank three alternatives {a,b,c}:

The table shows each voter's order of preference:  4 voters prefer a over b over c,  
3 prefer b over c over a, and 2 prefer c over a over b.  

Thus a majority (6 voters) prefer a over b, a majority (7 voters) prefer b over c, 
and a majority (5 voters) prefer c over a.  Therefore, for every alternative there is 
a majority who prefer another alternative over it, so no matter which is elected 
there will be a defeated alternative that is preferred over it by a majority.  

Even voting procedures such as Approval and "Vote for one, Plurality rule," which do 
not permit voters to completely express their preference orders, are not immune from this 
phenomenon.  The condition is determined by voters' preferences, not by their votes.  
Under such procedures, unofficial polling may establish the existence of a majority 
preference for a defeated alternative.

A majority who prefer a defeated alternative over the elected alternative might complain 
that "democratic majority rule" has somehow been violated.  When will this be trouble 
for society?  That may depend on the voting procedure.  For instance, in the scenario 
above, if the voting procedure socially orders the alternatives as "a over b over c" 
then a 5 voter majority may complain that a majority voted for c over the winner.  
They can be rebutted by turning their "majority" argument against them, by showing 
a larger majority (7 voters) rank b over c and a larger majority (6 voters) rank a over b.  
On the other hand, if the voting procedure does not elect a or declares the social 
ordering to be something other than "a over b over c" it may be harder to rebut 
the complainers' argument:  

If b is elected, the 6 voters who prefer a over b cannot be so easily rebutted 
since the only majority that prefers some alternative over a is smaller.

If c is elected, the 7 voters who prefer b over c cannot be so rebutted 
except perhaps with an argument which favors electing a, not c.  

If the voting procedure socially orders the alternatives as "a over c over b" 
then the attempted rebuttal involving "b over c" and "a over b" undermines 
the social ordering by its inconsistency, which undermines the voting 
procedure and undermines the election of a.  

Even if the voting procedure elects a and does not socially order the alternatives, 
it may be hard to rebut the complainers if there exists a natural way to extend the 
procedure to produce a social ordering that differs from "a over b over c."  
This is a question of consistency: If the same principles that allow the elected 
alternative to be declared "better" than the other alternatives cannot also be used to 
find which of any pair of alternatives is "better" than the other, this incompleteness 
of the "better" relation suggests something is amiss with the voting procedure's 
principles, which in turn may undermine confidence in the procedure.

Complaints by a "losing" majority which are hard to rebut could undermine the winner's 
mandate, and could undermine confidence in the voting procedure, particularly if there 
exists another procedure that satisfies the criteria the people consider important, 
would have elected the alternative favored by the complainers, and over the long run 
would elect alternatives preferred by more voters.  We consider it desirable to be 
able to turn the complainers' argument against them in a manner as easy as in the 
"a over b over c" example above since we do not expect people to entirely agree 
about the relative importance of the other criteria used to select the voting procedure 
(and there would certainly be an incentive for cheap talk in favor of some other voting 
procedure in most voting scenarios).   

The remainder of this paper formalizes several criteria which are elaborations of this 
idea, provides examples showing some prominent voting procedures fail some or all 
of these criteria, and provides proofs that the "Maximize Affirmed Majorities" (MAM) 
procedure satisfies all of these criteria.

Under some voting procedures, in scenarios where a majority rank some alternative x  
over the winner, the minority who prefer the winner can always rebut the complainers by 
turning that majority's "larger is better" argument against them by pointing out the existence 
of a majority at least as large who rank over x some alternative a₁ socially ranked over x, 
and a majority at least as large who rank over a₁ some alternative a₂ socially ranked 
over a₁, ..., etc., and a majority at least as large who rank over a_k-1 some a_k socially 
ranked over a_k-1, and a majority at least as large who rank the elected alternative over a_k.  
This seems to be the most we can expect of any voting procedure.  We call voting 
procedures under which such a rebutting sequence is always available immune from 
majority complaints.  Ideally, of course, we would like to always be able to show 
that for any x ranked over the winner by a majority, there exist a strictly larger majority 
who rank over x an alternative a₁ that precedes x in the social order, and a larger 
majority who rank over a₁ an alternative a₂ that precedes a₁, ..., etc., and a larger 
majority who rank over a_k-1 an alternative a_k that precedes a_k-1, and a larger majority 
who rank the winner over a_k.  The following two examples show that is too demanding 
in some "tied" scenarios:

Scenario 2:  3 voters rank three alternatives {a,b,c}:

The cycle in scenario 2 is perfectly symmetric.  Without loss of generality, assume a  
is elected (perhaps by drawing straws to break the tie).  A majority (2 voters) rank c  
over a.  No majority is larger than 2 voters.  Thus the most we can expect here from 
a rebutting sequence of majorities "from a to c" is that the rebutting majorities be 
at least as large (i.e., 2 voters) as the majority who prefer c over a.  It is too much 
to demand the rebutting sequence be comprised of larger majorities in such "tied" 
scenarios; however we trust it will suffice that the majorities be at least as large. 
(Of course, in elections involving many voters it is unlikely there will be same-size 
majorities, so the rebutting sequence would typically be comprised of strictly larger 
majorities.) 

Scenario 3:  15 voters rank four alternatives {a,b,c,d}:

If d were deleted from all the votes then scenario 3 would be similar to scenario 1, 
where only the social ordering "a over b over c" provides an easy rebuttal to majority 
complaints.  Alternative d is presumably similar to a since every voter ranks them 
together (with slightly more than half ranking a over d).  Assume the social ordering 
has a on top as in scenario 3, followed by d since d does as well against b and c  
as a does, followed by b and then by c.  Since d almost ties a, it is impossible to 
construct a sequence to rebut the "c over a" majority (10 voters) that includes a 
majority of at least 10 for the winner a over the alternative that is second in the 
social ordering (d in this case).  We can, however, merely demand that the rebutting 
sequence be a subset of the alternatives that is consistent with the social ordering 
in the sense that each alternative in the sequence is socially ranked over the 
alternatives that follow it in the sequence.  In scenario 3, the rebutting sequence 
"a over b over c" is consistent with the social ordering "a over d over b over c."

With these considerations in mind, we define several related criteria: 
1.  Weakest resistance to majority complaints (Weakest RMC) 
2.  Weak resistance to majority complaints (Weak RMC) 
3.  Immunity from second-place complaints (I2C) 
4.  Immunity from majority complaints (IMC) 
5.  Local independence of irrelevant alternatives (LIIA).  

Our context is that exactly one alternative must be elected.  Thus we are compelled to 
abandon the simplifying assumption often found in the social choice literature that more 
than one alternative may be selected and then some unspecified "tie-breaking" stage 
of the procedure (e.g., a coin toss) would be used to elect one of those selected.  
To test for satisfaction of our criteria, the overall procedure must be specified since 
an unspecified tie-breaking stage could fall victim to majority complaints.  For instance, 
if a procedure selects all three alternatives in scenario 1, as the Copeland procedure 
does, and does not specify how one of the three is ultimately to be elected, then it 
would not make sense to claim the first stage of the procedure satisfies our criteria.)  
In other words, only resolute voting procedures may satisfy our criteria.  Resolute 
procedures are not necessarily always deterministic, since complete determinism can 
only be gained at the expense of neutrality of the alternatives or anonymity of the 
voters, which are both often desirable criteria. (For instance, given an even number 
of voters electing one of two alternatives, if the voters are equally split then the only 
ways to elect one with nothing left to chance are to privilege an alternative or privilege 
at least one of the voters, contrary to democratic instincts.)  We do not require 
anonymity nor neutrality nor determinism, and have made this point about the 
trade-off between the three properties to explain why the framework of our formal 
definitions will be sufficiently general to encompass deterministic, non-deterministic, 
and "usually deterministic" voting procedures. 

First we define Weakest RMC and Weak RMC and explain why they are too weak:

Let A denote a finite non-empty set of alternatives, from which one must be elected.  
Let N = {1,2,...,n} denote a finite non-empty set of voters.  

For all non-empty B Í A, let O(B) denote the set of all possible orderings of B, 
and let L(B) denote the set of all possible strict orderings of B.  
Abbreviate O = O(A) and L = L(A). 

For all i Î N, let R_i denote i's vote, assumed to be an ordering of A.  

Let O^N denote the set of all possible n-tuples of orderings of A.  
Let R denote the n-tuple of voters' votes (R_i)_iÎN.  (Thus R Î O^N.)

For all x,y Î A, let R(x,y) denote the votes in R that rank x over y.  
For all x,y Î A, let #R(x,y) denote the number of votes in R that rank x over y.  

Weakest Resistance to Majority Complaints (Weakest RMC):  
Let w Î A denote the unique alternative elected from A given votes R.  
For all x Î A, if #R(x,w) > #R(w,x) then there must exist y Î A  
such that #R(y,x) ³ #R(x,w) and #R(y,x) > #R(x,y).  

Clearly, any choice procedure that satisfies Weakest RMC always elects an alternative 
in the subset {w Î A such that #R(w,x) > #R(x,w) for all x Î A} when that subset is 
not empty.  In other words, satisfaction of Weakest RMC implies a "Condorcet winner" 
will be elected if one exists, a property known as Condorcet-consistency.  Thus all 
voting procedures that are not Condorcet-consistent, such as Borda, Plurality Rule, 
Majority Runoff, Instant Runoff (a.k.a Single Transferable Vote) and Minimax, fail 
Weakest RMC. (We include Minimax in this list since it may defeat a Condorcet 
winner when votes may be non-strict orderings.)

Unfortunately, Weakest RMC is too weak for our purposes.  It could allow c to be 
elected in scenario 1, which we have argued above could be troublesome for society.  
It also would permit electing an alternative outside the "top cycle" (the smallest non-empty 
subset B Í A such that #R(x,y) > #R(y,x) for all x Î B and all y Î A\B).  Weakest 
RMC would even permit election of a "Condorcet loser," an alternative x such that, 
for all y Î A\{x}, more than half of the voters voted y over x., as shown by the 
following example:  

Scenario 4:  15 voters rank four alternatives {a,b,c,d}:

A majority of 9 rank a and b and c over d.  Thus d is a Condorcet loser.  
But d would be elected by Minimax since a and b and c each have some larger 
majority, 10 voters, who rank an alternative over them.  That is, 10 voters 
rank c over a, and 10 voters rank a over b, and 10 voters rank b over c.  
This does not violate Weakest RMC, but it would be difficult to rebut a 
complaint by the majority who rank d last.  

With that concern in mind, we can strengthen the criterion somewhat:

Weak Resistance to Majority Complaints (Weak RMC):  
Let w Î A denote the unique alternative elected from A given votes R.  
For all x Î A, if  #R(x,w) > #R(w,x) then there must exist a sequence 
of alternatives a₁,a₂,...,a_k Î A such that a₁ = w and a_k = x and both 
of the following conditions hold: 
        (1)  #R(a_j,a_j+1) ³ #R(x,w) for all j Î {1,2,...,k-1}.
        (2)  #R(a_j,a_j+1) > #R(a_j+1,a_j) for all j Î {1,2,...,k-1}.  

(Condition 1 requires each of the "rebutting" coalitions in the sequence must 
be at least as large as the "complaining" majority.  Condition 2 requires the 
rebutting coalitions must in fact be relative majorities, not minorities or ties.)

Clearly, satisfaction of Weak RMC implies satisfaction of Weakest RMC, and furthermore 
implies the elected alternative will always be in the top cycle. (Thus any voting procedure 
that can elect outside the top cycle fails Weak RMC.)  Typically the set of alternatives 
electable according to Weak RMC will be a strict subset of the top cycle when the top 
cycle includes three or more alternatives, as in scenario 1 where the top cycle includes 
all three alternatives and satisfaction of Weak RMC requires a be elected.

Weak RMC is again too weak for our purposes since it does not require the "rebutting" 
sequence of alternatives to be consistent with a social ordering.  For instance, if the 
social ordering in scenario 1 is "a over c over b" then the sequence "a over b" and 
"b over c" offered in rebuttal to the "c over a" majority is flawed since it is inconsistent 
with the social ordering.  The inconsistency undermines the effectiveness of the rebuttal 
by undermining the social ordering itself, thereby undermining the voting procedure 
that underlies the social ordering.  Voting procedures that do not construct a social 
ordering, and merely elect a winner, do not escape this concern since there is always 
at least one natural way to extend such procedures into social ordering procedures, 
and the complainers may construct a natural social ordering "unofficially."  They have 
a clear incentive to do so if that undermines the rebuttal.  Here are some natural ways 
to construct a social ordering given a voting procedure that elects one alternative:  

1. The most general technique to construct a strict social ordering is to use 
recursion, declaring the alternative in k^th place in the social ordering is the one 
that would be elected if the 1^st through k-1^th alternatives were deleted from 
the votes. (Thus the social ordering can be computed iteratively, beginning 
with the winner, then calculating the second place finisher, etc.)  

2. Another technique, which applies only to procedures that assign a score to 
each alternative and then elect the one with the best score (such as Borda, 
Copeland and Minimax), is to order the alternatives by their scores.  

3. A third technique, which applies only to procedures that eliminate one 
alternative at a time until only one remains (such as Borda Elimination and 
Instant Runoff), is to reverse the order of elimination.  

4. A fourth technique, which applies only to procedures that compute some 
ordering of the alternatives and elects an alternative that tops that relation 
(such as PathWinner, illustrated in an example below), is to call that ordering 
the social ordering. 

5. A fifth technique, which applies only to procedures that "affirm" an acyclic 
subset of the majorities and elect an alternative that is not the loser of any 
majority in that subset (such as MAM), is to construct the social ordering by 
finding an ordering consistent with that subset. (For instance, in scenario 1 
MAM affirms the "a over b" and "b over c" majorities and does not affirm 
the "c over a" majority.  The only social ordering consistent with the two 
affirmed majorities is "a over b over c.") 

Since the first technique listed above is general, typically at least two techniques are 
natural for any resolute voting procedure.  In MAM's case, for instance, both the first 
and fifth techniques can be used to construct a social ordering.  If the two natural 
techniques produce different social orderings, the discrepancy could undermine 
confidence in the voting procedure and undermine the rebuttal of the complainers' 
argument.  Thus we could define an additional criterion that demands that when two 
(or more) techniques are natural, they must produce the same social ordering.  But it 
would be difficult to rigorously state such a criterion. (In MAM's case, both natural 
techniques do produce the same social ordering.  See theorem "MAM₂ = MAM.")  
We will take a different tack here and simply specify the first, general, technique in 
our next two criteria (I2C and IMC), and we will note in the examples that show 
well-known procedures fail I2C where partial credit may be given for satisfying I2C's 
"spirit" if the procedure's other natural technique had instead been specified in the 
wording of I2C.

First, here is an informal definition of I2C:  

Let "second-place finisher" denote the alternative that would be elected 
if the elected alternative were deleted from the votes.  

The number of voters who rank the second-place finisher over the elected 
alternative must not exceed the number who rank the elected alternative 
over the second-place finisher.  

Here is the formal definition of I2C:

For all non-empty B Í A, let R|_B denote the restriction of R to 
the alternatives in B. (In other words, R|_B is the same as R except 
all alternatives not in B are lowered to the bottom in each vote.)  

Call a function f a decision scheme if and only if, for all finite non-empty A, 
all finite N, all R Î O^N and all non-empty B Í A, all four of the following 
conditions hold: 
        (1)  For all x Ï B, f(x,B,R) = 0.  
        (2)  For all x Î B, f(x,B,R) = f(x,B,R|_B).  
        (3)  For all x Î B, 0 £ f(x,B,R) £ 1. 
        (4)  S_{(x Î B)}f(x,B,R) = 1. 

(The interpretation of f(x,B,R) is the probability that alternative x will be elected 
from a "nominated" subset of alternatives B given votes R.  Condition 1 implies 
no alternative outside B may be elected.  Condition 2 implies the parts of votes 
regarding alternatives not in B must be ignored.  Condition 3 reflects the fact that 
a probability is always a real number between 0 and 1.  Condition 4 reflects the 
fact that the sum of probabilities over an exhaustive set of mutually exclusive 
events is always 1, which means a decision scheme corresponds to a voting 
procedure that elects exactly one alternative, in a manner that is not necessarily 
deterministic.  Thus this framework encompasses deterministic, non-deterministic, 
and "usually deterministic" voting procedures.) 

Immunity from Second-Place Complaints (I2C):  A decision scheme f is 
immune from second-place complaints if and only if #R(w,x) ³ #R(x,w) 
for all finite non-empty A, all finite N, all R Î O^N and all w,x Î A such that 
f(w,A,R) > 0 and f(x,A\{w},R) = 1.

Though the wording of I2C does not mention a complete social ordering, it can be seen 
that the second-place finisher is computed in a manner consistent with the first of the five 
natural techniques listed above.  If we were to rewrite I2C so the social ordering must 
be produced (not necessarily deterministically) using any of the five natural techniques 
(assuming the technique is natural for the voting procedure in question), it would 
strengthen the criterion slightly:  Let g(r,A,R) denote the probability that r is the social 
ordering, let r₁ denote the alternative atop r, and let r₂ denote the alternative ranked by r 
immediately below r₁.  Require #R(r₁,r₂) ³ #R(r₂,r₁) for all r such that g(r,A,R) > 0.  
We call this slightly stronger since the "f(x,A\{w},R) = 1" clause of I2C means 
"#R(w,x) ³ #R(x,w)" is only required for alternatives x elected with certainty 
when w is deleted.  To explain why we write "f(x,A\{w},R) = 1" instead of 
"f(x,A\{w},R) > 0" consider the following example: 

Scenario 5:  3 voters rank four alternatives {a,b,c,d}:

If d were deleted, the perfect symmetry amongst {a,b,c} would presumably give a  
a non-zero chance of being elected.  Since every voter ranks a over d, it is thus 
reasonable for a voting procedure to give a a non-zero chance when d is included.  
If a is indeed elected, then when calculating which alternative is the second-place 
finisher the perfect symmetry amongst {b,c,d} when a is deleted implies any one 
of {b,c,d} could be the second-place finisher.  If c does not finish second with 
certainty, I2C will not be violated.  

In terms of a natural social ordering, we would require the social ordering r not 
have r₁ = a and r₂ = c.  That still leaves room under the slightly strengthened I2C 
for the social ordering r to have r₁ = a and r₂ Î {b,d} (or perhaps r₁ ¹ a), 
as long as #R(a,d) ³ #R(d,a) is required whenever r₁ = a and r₂ = d and 
#R(a,b) ³ #R(b,a) is required whenever r₁ = a and r₂ = b (even though 
neither d nor b is second with certainty in the social ordering when a is first).  
This is a minor technical point which will not be relevant when we define 
the strongest criterion IMC below.

Many prominent voting procedures fail I2C.  The following example shows that 
Plurality Rule fails I2C:

Scenario 6:  9 voters rank three alternatives {a,w,x} as follows:

Plurality Rule ignores all but the top of each voter's ranking.  Since more voters (4) 
ranked w top than ranked any other alternative top, Plurality Rule would elect w.  
The second-place finisher is x since 7 voters rank x top when w is dropped. 
(Note that x is also the second-place finisher using the other natural way of 
constructing the social order, since x's 3 tops exceed a's 2 tops.)  Since a 
majority (5 voters) rank x over w, Plurality Rule fails I2C. 

It is easy to construct examples that show that Borda, Copeland, Minimax, and many 
other prominent voting rules fail I2C.  The following example shows Borda fails I2C:

Scenario 7:  5 voters rank three alternatives {a,w,x} as follows:

Borda counts for each alternative a point from each vote for each alternative ranked 
below it.  Thus Borda counts 6 points (3´2 + 2´0) for x, 7 points (3´1 + 2´2) 
for w, and 2 points (3´0 + 2´1) for a.  Thus Borda elects w.  Alternative x is 
the second-place finisher since Borda would elect x if w were dropped. (Note 
that x would also be second using the other natural social ordering technique, 
technique 2, since x has the second largest number of points.)  Since a majority 
(3 voters) prefer x over w, Borda fails I2C. (This example also shows that 
Borda fails Weakest RMC and Weak RMC.)

The following example shows the Copeland procedure fails I2C:

Scenario 8:  15 voters rank eight alternatives {a,b,c,d,e,f,w,x} as follows:

Copeland computes the number of pairings won (by relative majority) 
by each alternative and then elects the alternative that won the most pairings.  
Thus Copeland computes these scores: 

w wins 6 pairings (over a,b,c,d,e,f). 
x wins 5 pairings (over c,d,e,f,w). 
a wins 3 pairings (over e,f,x). 
b wins 3 pairings (over a,b,x). 
c wins 3 pairings (over a,b,f). 
d wins 3 pairings (over a,b,c). 
e wins 3 pairings (over b,c,d). 
f wins 2 pairings (over d,e). 

Thus Copeland elects w.  Alternative x is the second-place finisher since Copeland 
would elect x if w were deleted. (Note that x is also second in the other natural 
Copeland social ordering, having the second-largest score.)  Since a majority 
(13 voters) rank x over w, Copeland fails I2C.  In fact, no other alternative is 
opposed by a majority as large as 13 voters, and x is the only alternative whose 
largest opposing majorities are as small as 8 voters, so this example seems an 
egregious failure by Copeland.  (This example also shows that Copeland fails 
Weakest RMC, Weak RMC and IMC.  But it is easy to construct less complex 
examples that show Copeland fails those criteria.)  Eight alternatives were used 
in this example since the Copeland correspondence is often irresolute and its 
tie-breaker is unspecified; thus the example needed to give the elected alternative 
at least one more pairwin than the second-place alternative and give the second-
place alternative at least two pairwins more than the third-place alternative, 
so the second-place alternative would be elected with certainty when the 
elected alternative is deleted.)

The following example shows Majority Runoff and Instant Runoff fail I2C but may be 
given partial credit for satisfying the spirit of I2C:

Scenario 9:  9 voters rank three alternatives {a,w,x} as follows:

Instant Runoff is an iterative elimination procedure which counts each vote for its 
highest-ranked non-eliminated alternative and eliminates the alternative having the 
smallest count, repeating until one alternative remains.  Given the votes in scenario 9, 
Instant Runoff first counts 3 for w, 2 for x and 4 for a and therefore eliminates x.  
Then Instant Runoff counts 5 for w and 4 for a and eliminates a, electing w.  

Alternative x is the second-place finisher since if w were dropped from the votes 
Instant Runoff would elect x (with a count of 5) over a (with a count of 4).  
Since a majority (6 voters) rank x over w, Instant Runoff fails I2C.  

On the other hand, one may argue that Instant Runoff satisfies I2C in spirit since 
the third natural technique of constructing a social order, reversing the order of 
elimination, is also natural for Instant Runoff.  It is impossible for a majority to 
rank the alternative eliminated last over the elected alternative.  However, 
Instant Runoff fails Weakest RMC and Weak RMC (as do Borda and all 
voting procedures that are not Condorcet-consistent), and we shall see 
below that Instant Runoff also fails the stronger criterion IMC, even in spirit.

The following example shows Simpson (and other minimax and maximin procedures) 
fails I2C:

Scenario 10:  15 voters rank four alternatives {a,b,c,w} as follows:

Given these votes, here are the pairwise counts of the votes: 

     #R(a,b) = 10,  #R(b,a) = 5 
     #R(b,c) = 10,  #R(c,b) = 5 
     #R(c,a) = 10,  #R(a,c) = 5 
     #R(a,w) =  9,  #R(w,a) = 6 
     #R(b,w) =  9,  #R(w,b) = 6 
     #R(c,w) =  9,  #R(w,c) = 6 

Simpson elects w since w's three pairwise defeats (by a and by b and by c) 
have size 9, whereas each of the other alternatives has a larger defeat (size 10).  
Clearly the second-place finisher must be one of {a,b,c}.  Since for each 
alternative in {a,b,c} a majority (9 voters) rank it over w, Simpson fails I2C. 

The following example shows that PathWinner (defined in section 4.3 of the document 
"Strategic Indifference") fails I2C:

Scenario 11:  13 voters rank four alternatives {a,b,w,x} as follows:

Given these votes, here are the pairwise counts of the votes: 

     #R(x,a) = 9,  #R(a,x) = 4 
     #R(w,a) = 9,  #R(a,w) = 4 
     #R(a,b) = 9,  #R(b,a) = 4 
     #R(b,x) = 8,  #R(x,b) = 5 
     #R(x,w) = 7,  #R(w,x) = 6 
     #R(b,w) = 7,  #R(w,b) = 6 

PathWinner calculates the strengths of the "strongest paths" as shown in the 
following table: 

The strength of a path (sequence of majorities) is the size of its smallest majority.  
Since the strongest path from w to any other alternative is stronger than the strongest 
path from that alternative to w, PathWinner elects w.  If w were dropped, then 
PathWinner would elect x.  Thus x is the second-place finisher. (Note that x is also 
the second-place finisher using the fourth natural technique of constructing a social 
ordering, which is natural for PathWinner since the "stronger path" relation is 
an ordering of the alternatives, since x has a stronger path to a than vice versa 
and x has a stronger path to b than vice versa.)  Since a majority rank x over w, 
PathWinner fails I2C.  

The majority who rank x over w may point out the inconsistency of electing w over x 
and that there exists a procedure (e.g., MAM) that satisfies the same desired criteria 
as PathWinner (independence of clone alternatives, independence of Pareto-
dominated alternatives, minimal defense, truncation resistance, top cycle, 
anonymity, neutrality, monotonicity, strong Pareto, resolvability, etc.) and 
would elect x. (Note that wabx, the only path from w to x, includes the "b over x" 
majority.  The "b over x" majority is not affirmed by MAM since the "x over a" 
and "a over b" majorities are larger than the "b over x" majority.  PathWinner too 
finds the xab path is stronger than the bx path, which implies bx is "shaky," 
but ignores this shakiness when incorporating bx into wabx.) 

Next we define IMC, which in spirit is stronger than I2C.  IMC uses a different definition 
of decision schemes, however, replacing the lottery concept with a slightly less general 
"s Î L(R,A)" concept defined here.  

Let L(R,A) denote the set of all possible ordered pairs of strict orderings of the 
votes in R and strict orderings of A. (Since some voting procedures are not always 
deterministic to satisfy anonymity and neutrality, we indicate by some randomly 
picked s = (s_R,s_A) Î L(R,A) the random values that may affect the outcome.  
A randomly picked strict ordering of R, s_R, corresponds to a possible hierarchy 
of the voters, and a randomly picked strict ordering of A, s_A, corresponds to 
a possible hierarchy of the alternatives.  Little generality is lost by considering 
only voting functions that elect a single alternative in a seemingly "deterministic" 
manner given a collection of votes R and some randomly picked s Î L(R,A).) 

Call a function f a voting scheme if and only if, given any non-empty B Í A, 
any R Î O^N and any s Î L(R,A), f(B,R,s) is a singleton subset of B.  

Recursively define the "place of finish" of all alternatives. (This formally defines 
the first of the five natural techniques of extending a resolute voting procedure 
to a social ordering, by defining functions F and B such that F(k) is the 
alternative that finishes in k^th place and B(k) is the subset of alternatives 
that remain when the alternatives that finish in the top k places are deleted.)  

For all R Î O^N and all non-empty B Í A, let R|_B denote the restriction 
of R to B. (That is, R|_B is the same as R except all alternatives not in B 
are deleted from each vote.) 

For all s = (s_R,s_A) Î L(R,A) and all non-empty B Í A, let s|_B denote 
the restriction of s  to B. (That is, s|_B Î L(R|_B,B) and s|_B = (s_R|_B,s_A|_B) 
where s_R|_B orders R|_B the same as s_R orders R, and s_A|_B is the same 
as s_A except all alternatives not in B  are deleted.) 

For all voting schemes f, let B(0,f,A,R,s) = A.  Abbreviate as B(0) 
when A is unambiguous.

For all voting schemes f and all integers k Î {1,2,...,#A), let B(k,f,A,R,s) 
denote B(k-1,f,A,R,s)\{F(k,f,A,R,s)}.  Abbreviate B(k) = B(k,f,A,R,s) 
when f, A, R and s are unambiguous. 

For all voting schemes f and all integers k Î {1,2,...,#A), let F(k,f,A,R,s) 
denote the (unique) alternative in f(B(k-1),R|_B(k-1),s|_B(k-1)).  
Abbreviate F(k) = F(k,f,A,R,s) when f, A, R and s are unambiguous. 

For all voting schemes f and all x Î A, let Place(x,f,A,R,s) denote 
the (unique) integer k Î {1,2,...,#A) such that F(k) = x. 
Abbreviate Place(x) = Place(x,f,A,R,s) when f, A, R and s  
are unambiguous. 

Immunity from Majority Complaints (IMC):  A voting scheme f is immune 
from majority complaints if and only if, for all finite non-empty A, all finite N, 
all R Î O^N, all s Î L(R,A) and all x Î A, if #R(x,F(1)) > #R(F(1),x) 
then there exists a sequence of alternatives a₁,a₂,...,a_k Î A such that 
a₁ = F(1) and a_k = x and all three of the following conditions hold: 
        (1)  #R(a_j,a_j+1) ³ #R(x,F(1)) for all j Î {1,2,...,k-1}.  
        (2)  #R(a_j,a_j+1) > #R(a_j+1,a_j) for all j Î {1,2,...,k-1}.
        (3)  Place(a_j) < Place(a_j+1) for all j Î {1,2,...,k-1}.

Clearly satisfaction of IMC implies satisfaction of Weakest IMC, Weak IMC and I2C.  
IMC is stronger since it requires the "rebutting" sequence of alternatives to be consistent 
with the social ordering.  If some voting procedure has a natural social ordering and 
would satisfy IMC if IMC were amended appropriately to use that natural social ordering 
instead, then it is fair to say the procedure satisfies the spirit of IMC. (However, we are 
not aware of any voting procedure that fails IMC yet satisfies the spirit of IMC.) 

The following example shows Kemeny-Young fails IMC: 

Scenario 12:  9 voters rank five alternatives {a,b,c,w,x} as follows:

Kemeny-Young elects the alternative that tops the ordering of alternatives that 
agrees with the most pairwise votes.  In this scenario, that ordering is "w over a  
over b over c over x" which agrees with 62 pairwise votes (6 "w over a", 
6 "w over b", 6 "w over c", 2 "w over x", 9 "a over b", 9 "a over c", 5 "a over x", 
9 "b over c", 5 "b over x", 5 "c over x").  Thus Kemeny-Young elects w.  
Since the 7 voters who rank x over w cannot be rebutted by any sequence 
of majorities each as large as 7, Kemeny-Young fails IMC. (This scenario 
also shows that Kemeny-Young fails Weakest RMC and Weak RMC.  
However, Kemeny-Young satisfies I2C.)

I2C and IMC are closely related to the well-known criterion local independence of 
irrelevant alternatives (LIIA, promoted by Peyton Young), which applies to voting 
procedures that socially order the alternatives.  Informally, LIIA can be expressed 
as follows: 

For any subset of two or more alternatives that are contiguous in the social 
ordering, their ordering relative to each other must not change if all alternatives 
outside this subset are deleted from the votes.  

Here is a formal definition of LIIA, generalized for our framework in which voting 
procedures are not necessarily deterministic.  A lottery framework capable of 
encompassing deterministic, non-deterministic and usually deterministic voting 
procedures is used for maximum generality: 

Call a function g a social ordering scheme if and only if all four of the 
following conditions hold for all finite non-empty A, all finite N, 
all R Î O^N and all non-empty B Í A:

(1)  For all r Ï O(B), g(r,B,R) = 0.  
(2)  For all r Î O(B), g(r,B,R) = g(r,B,R|_B).  
(3)  For all r Î O(B), 0 £ g(r,B,R) £ 1. 
(4)  S_{(r Î O(B))}g(r,B,R) = 1. 

(The interpretation of g(r,B,R) is the probability that r will be the social ordering 
of B given votes R.  Condition 1 means only orderings of B have non-zero 
probability.  Condition 2 means the parts of votes regarding alternatives not 
in B must be ignored.  Condition 3 reflects the fact that a probability is always 
a real number between 0 and 1.  Condition 4 reflects the fact that the sum of 
probabilities over an exhaustive set of mutually exclusive events is always 1, 
which means a social ordering scheme corresponds to a voting procedure that 
selects exactly one social ordering, in a manner that is not necessarily deterministic.  
This framework is general since it encompasses deterministic, non-deterministic, 
and "usually deterministic" social ordering procedures.  

For all r Î O(A) and all non-empty B Í A, let r|_B denote the restriction of r  
to B. (That is, r|_B is the same as r except all alternatives not in B are deleted.)  

For all non-empty B Í A and all r' Î O(B), let Consistent(r') denote 
{r Î O(A) such that r|_B = r'}. 

For all non-empty B Í A and all r Î O(A), call B contiguous within r  
if and only if, for all x Î A\B, r ranks x over all of B or r ranks all of B over x. 

For all social ordering schemes g and all non-empty B Í A, 
call B contiguous with certainty given (g,R) if and only if 
B is contiguous within r for all r Î O(A) such that g(r,A,R) > 0. 

Local Independence of Irrelevant Alternatives (LIIA):  A social ordering 
scheme g satisfies LIIA if and only if, for all finite non-empty A, all finite N, 
all R Î O^N and all non-empty B Í A, if B is contiguous with certainty 
given (g,R) then g(r',B,R) = S_{(r Î Consistent(r'))}g(r,A,R) for all r' Î O(B).

LIIA cannot be directly applied to a voting procedure that merely elects one alternative 
instead of socially ordering the alternatives, but I2C and IMC can.  MAM elects one 
alternative, but since the MAM social ordering has also been defined, LIIA can 
also be applied to MAM.  We prove below that MAM satisfies Weakest IMC, 
Weak IMC, I2C, IMC and LIIA. 

Since the PathWinner voting procedure elects an alternative but does not socially 
order the alternatives, one cannot say whether PathWinner satisfies LIIA without 
extending PathWinner's definition somehow.  However, since PathWinner fails I2C 
and IMC we presume there is no reasonable extension of PathWinner to a social 
ordering procedure that satisfies LIIA.  In particular, the extensions of PathWinner 
produced using the two natural techniques (1 and 4 above) fail LIIA, as demonstrated 
by the example above that showed PathWinner fails I2C:  Pick B = {a,d} in that 
example.  Using either technique natural to PathWinner, B is contiguous with certainty, 
with d ranked over a and both ranked over the rest.  But if the alternatives outside B  
are deleted, PathWinner would elect a.  Thus both naturally extended PathWinner 
social ordering procedures fail LIIA.

Theorem (1):  MAM is immune from second-place complaints.

Proof:  Assume MAM(A,R,s) = w for some s Î L(R,A). (Thus w has a non-zero 
chance of being elected by MAM from A given R.)  Abbreviate A' = A\{w} and 
R' = R|_A\{w}.  Assume MAM(A',R',s') = x for all s' Î L(R',A'). (Thus x is 
elected with certainty from A\{w} given R'.)  We must show #R(w,x) ³ #R(x,w).  
Suppose the contrary.  Let s' denote s|_A'.  Make the following abbreviations:  

Maj = Majorities(A,R).  
Maj' = Majorities(A',R').  
RVH = RVH(A,R,s). 
RVH' = RVH(A',R',s'). 
For all a,b Î A, Precede(a,b) = Precede(a,b,A,R,s). 
For all a,b Î A', Precede'(a,b) = Precede(a,b,A',R',s'). 
Aff = Affirmed(A,R,s). 
Aff' = Affirmed(A',R',s'). 
MAM = MAM(A,R,s). 
MAM' = MAM(A',R',s'). 

Since #R(x,w) > #R(w,x), (x,w) Î Maj and (w,x) Ï Maj.  Since MAM = w, 
by the definition of MAM() w is not second in any pair in Aff.  Thus (x,w) Ï Aff.  
Since (x,w) Î Maj\Aff, (x,w) must be inconsistent with Aff Ç Precede(x,w).  This 
means there exist (a₁,a₂),(a₂,a₃),...,(a_k-1,a_k) Î Aff Ç Precede(x,w) such that a₁ = w  
and a_k = x.  By the definition of Affirmed(), Aff Í Maj.  Thus (a_k-1,x) Î Maj.  
Thus a_k-1 ¹ w, which implies a_k-1 Î A'.  Thus (a_k-1,x) Î Maj'.  Since MAM' = x, 
x is not second in any pair in Aff'.  Thus (a_k-1,x) Ï Aff'.  Since (a_k-1,x) Î Maj'\Aff', 
(a_k-1,x) must be inconsistent with Aff' Ç Precede'(a_k-1,x).  This means there exist 
(b₁,b₂),(b₂,b₃),...,(b_j-1,b_j) Î Aff' Ç Precede'(a_k-1,x) such that b₁ = x and b_j = a_k-1.  
Let P denote {(b₁,b₂),(b₂,b₃),...,(b_j-1,b_j)}.  By theorem "Consistency Maintained (2)", 
Aff is internally consistent.  Thus, since (a_k-1,x) Î Aff there must exist (b,b') Î P  
such that (b,b') Ï Aff.  Let P₂ denote the pairs in Aff' or in Aff Ç Pairs(A') 
but not in both.  Since (b,b') Î Aff'\Aff, (b,b') Î P₂.  Since P₂ is not empty, 
by theorem "Precedence is a Strict Ordering" we can pick (y,y') Î P₂ such 
that the following condition holds: 
        (1.1)  For all (z,z') Î P₂, (z,z') Ï Precede'(y,y'). 
There are two cases to consider: 

Case 1.2.1:  (y,y') Î Aff'\Aff.  Since (y,y') Î Aff', (y,y') Î Maj'.  Thus (y,y') Î Maj.  
Since (y,y') Î Maj\Aff, (y,y') must be inconsistent with Aff Ç Precede(y,y').  
This means there exist (c₁,c₂),(c₂,c₃),...,(c_h-1,c_h) Î Aff Ç Precede(y,y') 
such that c₁ = y' and c_h = y.  Let P₃ denote {(c₁,c₂),(c₂,c₃),...,(c_h-1,c_h)}.  
There are two subcases to consider:

Case 1.2.1.1:  w Î {c₁,c₂,...,c_h}.  Since c₁ Î A' and w Ï A', w Î {c₂,...,c_h}.  
Therefore w is second in at least one pair in P₃.  But since P₃ Í Aff 
and w is not second in any pair in Aff, this is a contradiction.  
Therefore case 1.2.1.1 is impossible. 

Case 1.2.1.2:  w Ï {c₁,c₂,...,c_h}.  Thus P₃ Í Pairs(A').  Since P₃ Í Aff, 
P₃ Í Maj.  Since P₃ Í Pairs(A') Ç Maj, P₃ Í Maj'.  By theorem 
"Consistency Maintained (2)" (substituting A' for A, R' for R and s' for s), 
Aff' is internally consistent.  Therefore, since (y,y') Î Aff' there must 
exist (c,c') Î P₃ such that (c,c') Ï Aff'.  Since (c,c') Î Aff\Aff' and 
(c,c') Î Pairs(A'), (c,c') Î P₂.  Since s' = s|_A' and (c,c') Î Precede(y,y'), 
by theorem "Precedence is Independent of Irrelevant Alternatives" 
(c,c') Î Precede'(y,y').  But since (c,c') Î P₂ and (c,c') Î Precede'(y,y'), 
this contradicts 1.1.  Therefore case 1.2.1.2 is impossible.

Since both subcases are impossible, case 1.2.1 is impossible.

Case 1.2.2:  (y,y') Ï Aff'\Aff.  Since (y,y') Î P₂, (y,y') Î Aff Ç Pairs(A') and 
(y,y') Ï Aff'.  Since (y,y') Î Aff, (y,y') Î Maj.  Since (y,y') Î Maj Ç Pairs(A'), 
(y,y') Î Maj'.  Since (y,y') Î Maj'\Aff', by the definition of Affirmed() 
(y,y') must be inconsistent with Aff' Ç Precede'(y,y').  This means there exist 
(c₁,c₂),(c₂,c₃),...,(c_h-1,c_h) Î Aff' Ç Precede'(y,y') such that c₁ = y' and c_h = y.  
Let P₃ denote {(c₁,c₂),(c₂,c₃),...,(c_h-1,c_h)}.  By theorem "Consistency Maintained 
(2)", Aff is internally consistent.  Thus, since (y,y') Î Aff there must exist (c,c') Î P₃  
such that (c,c') Ï Aff.  Since (c,c') Î Aff'\Aff, (c,c') Î P₂.  But since (c,c') Î P₂  
and (c,c') Î Precede'(y,y'), this contradicts 1.1.  Thus case 1.2.2 is impossible. 

Since both cases are impossible, the contrary assumption cannot hold.  
Therefore #R(w,x) ³ #R(x,w), establishing MAM satisfies I2C.     QED 

Theorem (2):  MAM satisfies Immunity from Majority Complaints.  

Proof:  Assume MAM(A,R,s) = w for some s Î L(R,A) and assume 
#R(x,w) > #R(w,x) for some x Î A.  We must show there exists a sequence 
of alternatives a₁,a₂,...,a_k Î A such that a₁ = w and a_k = x and all three 
of the following conditions hold: 
        (2.1)  #R(a_j,a_j+1) ³ #R(x,w) for all j Î {1,2,...,k-1}.  
        (2.2)  #R(a_j,a_j+1) > #R(a_j+1,a_j) for all j Î {1,2,...,k-1}.
        (2.3)  Place(a_j) < Place(a_j+1) for all j Î {1,2,...,k-1}.

Make the following abbreviations: 

Maj = Majorities(A,R).  
For all a,b Î A, Precede(a,b) = Precede(a,b,A,R,s). 
Aff = Affirmed(A,R,s).  
MAM = MAM(A,R,s).  
R_s = R_social(A,R,s).

By the definition of MAM(), since MAM = w it follows that w is not second in any 
ordered pair in Aff.  Thus (x,w) Ï Aff.  Since #R(x,w) > #R(w,x), (x,w) Î Maj.  
Since (x,w) Î Maj\Aff, by the definition of Affirmed() (x,w) must be inconsistent 
with Aff Ç Precede(x,w).  This means there exists a sequence of order pairs of 
alternatives (a₁,a₂),(a₂,a₃),...,(a_k-1,a_k) Î Aff Ç Precede(x,w) such that a₁ = w  
and a_k = x.  Let P denote {(a₁,a₂),(a₂,a₃),...,(a_k-1,a_k)}.  Since P Í Precede(x,w), 
by the definition of Precede() #R(a_j,a_j+1) ³ #R(x,w) for all j Î {1,2,...,k-1}, 
establishing condition 2.1 holds for a₁,a₂,...,a_k.  By the definition of Affirmed(), 
Aff Í Maj.  Thus P Í Maj.  This means (a_j,a_j+1) Î Maj for all j Î {1,2,...,k-1}, 
establishing condition 2.2 holds for a₁,a₂,...,a_k.  Since P Í Aff, by theorem 
"Properties of RSocial (3)" R_s ranks a_j over a_j+1 for all j Î {1,2,...,k-1}.  
By the definition of R_social() (i.e., MAM's social ordering, constructed in 
accordance with the first of the five natural techniques for constructing 
a social ordering), the following condition must hold:  

(2.4)  For all a,b Î A, R_s ranks a over b if and only if Place(a) < Place(b).  

It follows that Place(a_j) < Place(a_j+1) for all j Î {1,2,...,k-1}, establishing 
condition 2.3 holds for a₁,a₂,...,a_k.  Since conditions 2.1, 2.2 and 2.3 have been 
established for a₁,a₂,...,a_k and since a₁ = w and a_k = x, it follows that MAM 
satisfies IMC.     QED 

Theorem ():  MAM satisfies Local Independence of Irrelevant Alternatives.  
That is:  

For all finite non-empty sets  of alternatives A, all finite sets of votes R, 
all non-empty B Í A and all r Î L(B), let g_MAM(r,B,R) denote 
#{s Î L(R|_B,B) such that R_social(B,R|_B,s|_B) = r}/#L(R|_B,B). 

g_MAM is a social ordering scheme that satisfies local independence 
of irrelevant alternatives.

Proof:  By inspection of the definition of MAM(), MAM(B,R|_B,s|_B) is a single alternative 
in B for all finite non-empty A, all finite R, all non-empty B Í A and all s Î L(R,A).  
Thus it follows by the definition of R_social() that R_social(A,R,s) is a strict ordering of A 
for all finite non-empty A, all finite R and all s Î L(R,A).  For all non-empty B Í A, 
we can relabel A = B, R = R|_B and s = s|_B and still satisfy the premise that A is finite 
and non-empty and R is finite and s Î L(R,A).  Thus it follows that R_social(B,R|_B,s|_B) 
is a strict ordering of B for all finite non-empty A, all finite R, all non-empty B Í A 
and all s Î L(R,A).  Since L(B) Í O(B), it follows immediately that g_MAM is a 
social ordering scheme.  

Thus it remains only to show g_MAM satisfies LIIA.  Abbreviate R_s(s) = R_social(A,R,s) 
for all s Î L(R,A).  By inspection of the definition of L(), both of the following 
statements must hold:

(2.1)  For all non-empty B Í A, #L(R,A) = (#R)! ´ (#A)!  
(2.2)  For all non-empty B Í A, #L(R|_B,B) = (#R|_B)! ´ (#B)! 

Since the number of ballots does not change when alternatives are deleted from 
the ballots, #R|_B = #R for all non-empty B Í A.  Therefore, by the definition 
of L() the following statement must hold: 

(2.3)  For all non-empty B Í A, #L(R|_B,B) =  #L(R,A) ´ (#B)!/(#A)!

Thus satisfaction of LIIA will follow if we establish the following proposition: 

Proposition 2.4:  R_s(s)|_B = R_social(B,R|_B,s|_B) for all s Î L(R,A) and 
all non-empty B Í A that is contiguous within R_s(s).

Proof of 2.4:  Pick any s Î L(R,A) and any non-empty C Í A that is contiguous 
within R_s(s).  Abbreviate R_s = R_s(s) and r_s = R_social(C,R|_C,s|_C).  We must 
show R_s|_B = r_s.  Make the following abbreviations: 

Maj = Majorities(A,R). 
Maj' = Majorities(C,R|_C). 
RVH = RVH(A,R,s). 
RVH' = RVH(C,R|_C,s|_C).
Aff = Affirmed(A,R,s).
Aff' = Affirmed(C,R|_C,s|_C).  
w = MAM(A,R,s). 
c = MAM(C,R|_C,s|_C). 

Let T denote the alternatives in A that R_s ranks over all of C. (T may be empty.)  
Let B denote the alternatives in A that R_s ranks below all of C. (B may be empty.)  
By theorem "Properties of RSocial (1)", R_s is a strict ordering of A.  Therefore, 
since C is contiguous within R_s, A = T È C È B.  Make the following abbreviations: 

A' = T È C. 
R'_s = R_social(T È C,R|_TÈC,s|_TÈC).  
A" = C È B. 
R"_s = R_social(C È B,R|_CÈB,s|_CÈB).  

Proposition 2.4 will follow easily if we establish the following two propositions: 

Proposition 2.5:  R'_s = R_s|_A'.  
Proposition 2.6:  R"_s = R_s|_A". 

Proof of 2.5:  There are two cases to consider:  

Case 2.5.1:  B is empty.  Since A' = A, R'_s = R_s|_A' in case 2.5.1. 

Case 2.5.2:  B is not empty.  Make the following abbreviations: 

RVH = RVH(A,R,s).
RVH' = RVH(A',R|_A',s|_A').
Aff = Affirmed(A,R,s). 
Aff' = Affirmed(A',R|_A',s|_A'). 
w' = MAM(A',R|_A',s|_A'). 

We first show w = w'.  Suppose the contrary.  Since B is a strict subset 
of A and R_s ranks all of A' over all of B, by theorem "Properties of 
RSocial (7.2)" the following condition holds: 

(2.5.2.1)  Aff' = Aff Ç Pairs(A').  

Since R_s ranks all of A' over all of B, by theorem "Properties of 
RSocial (3)" the following condition holds: 

(2.5.2.2)  (b,a) Ï Aff for all b Î B and all a Î A'.

By the definition of MAM(), w is not second in any pair in Aff.  By 2.5.2.1, 
w is not second in any pair in Aff'.  By the definition of MAM(), w' is not 
second in any pair in Aff'.  By 2.5.2.1 and 2.5.2.2, w' is not second in 
any pair in Aff.  Since w ¹ w' and neither w nor w' is second in any 
pair in Aff and MAM(A,R,s) = w, this implies RVH ranks w over w'.  
Similarly, since neither w nor w' is  second in any pair in Aff' and 
MAM(A',R|_A',s|_A') = w', this implies RVH' ranks w' over w.  But 
by theorem "Random Voter Hierarchy is Independent of Irrelevant 
Alternatives", RVH' ranks w relative to w' the same as RVH does, 
a contradiction.  This means the contrary assumption cannot hold, 
establishing w = w'.  

Since 

Proof of 2.6:  This follows by inspection of the definition of R_social(),  since 
the iterative manner in which the social ranking is constructed implies  the 
relative ranking of "bottom" alternatives (that is, A" = C È B) by R_s neglects 
information about the "non-bottom" alternatives (T).               QED