Benevolent strategic indifference and group strategy equilibria:
Minimal Defense and Truncation Resistance as criteria for voting rules

Stephen Eppley <seppley@alumni.caltech.edu>

Revised:  February 21, 2003

Abstract 

This paper presents two criteria, minimal defense and truncation resistance, that are 
failed by most voting rules, and argues it is important they be satisfied.  In particular, 
satisfaction of minimal defense makes it easier for voters to reach desirable equilibria 
by making unnecessary "compromising" strategies (which voters are loathe to employ 
since they want to express preferences for their favorite alternatives and they dislike 
compromising unnecessarily).  A benefit is reducing the cost of majority coordination.  
A related benefit in partisan elections is that political parties may at last have a net 
positive incentive to nominate more than one candidate per office since the risk of 
fratricide is reduced; elections can be reduced to a single stage (no primary elections).

Two voting rules which satisfy both criteria (and many other desirable criteria) are 
briefly described, Maximize Affirmed Majorities (MAM) and PathWinner. (A detailed 
description of MAM and proofs of its criteria compliance are provided elsewhere.)

Introduction

Some of the social choice literature has warned that even "nice" voting scenarios having 
a Condorcet winner (Condorcet [1785]) are manipulable.  For instance, Shepsle and 
Bonchek [1997] noted that in the case of three voters and three alternatives, 12 of the 216 
possible combinations of voters' strict orderings produce cyclic majorities.  They went on 
to write [p.68]: 

The Rationality assumption is violated on some occasions.  Perhaps [...] we are 
overreacting to this problem.  The doubtful reader might say, "Sure, coherence and 
fairness in preference aggregation cannot be guaranteed, but perhaps this conflict 
only arises occasionally.  Nothing is perfect, after all."  This is an overly optimistic 
view.  The very fact that some social situations produce either incoherence or 
unfairness means that it will be possible for clever, manipulative, strategic, 
individuals to exploit this fact. 

By exploitation, Shepsle and Bonchek mean voting strategies (and perhaps nomination 
strategies) which produce an outcome more preferred by the strategizers than if they 
straightforwardly vote their sincere preferences, such as in the following scenario:

Scenario 1:  Voters' preferences regarding alternatives x, y and z: 

Alternative y is a "sincere" Condorcet winner given these preferences since a majority 
(54%) prefer y over x and a majority (66%) prefer y over z.  But if the 46% who prefer 
x over y over z are clever and manipulative, they may vote "x over z over y" since then 
x would be elected under most voting rules (assuming the other voters naively vote their 
sincere orders of preference).  The scenario is highly simplified, having single-peaked 
preferences over a linear spectrum in which y is between x and z. (This is evident since 
y is not the least preferred alternative of any voter.)  It will become apparent later that 
generality is not lost by neglecting voters who may have "x over z over y" preferences 
or "z over x over y" preferences. 

In this document, we argue that the existence of voting procedures which satisfy the 
minimal defense and truncation resistance criteria (to be described below) suggests 
their view is overly pessimistic, because satisfaction of these criteria means the other 
voters can deter or thwart such attempts at manipulation without having to compromise.  
(Minimal defense, non-drastic defense and truncation resistance are refinements of 
three criteria proposed by Mike Ossipoff [1995].)  The pessimism of so many writers 
may be due to an erroneous belief that, since it is reasonable to make the simplifying 
assumption that voters' preferences are strict orderings, one may neglect the possibility 
that voters may have incentives to vote non-strict orderings and that such strategies may 
benefit society.  For an example of this oversimplification see Hervé Moulin [1988], 
who defined voting rules as mappings from strict orderings of the alternatives.

To clarify the motivation behind the criteria, suppose scenario 1 describes a large public 
election, and that in advance of the election the voters and candidates have incomplete 
information about what the voters' preferences will be on election day.  Suppose it 
appears likely that y will be the Condorcet winner given voters' sincere preferences, 
but y, being a centrist in a diverse society, will not be the first choice of a majority.  
Additionally suppose it is less clear which of x or z will have more support on election 
day: perhaps more voters will prefer z to x than vice versa, and perhaps not.

The following definitions may help to illuminate the discussion:

Offensive strategy:  a group voting strategy intended to produce an election 
outcome more preferred by every member of that group than the outcome 
which would result if everyone voted sincerely.  Such strategies, when 
successfully employed, are considered manipulations of the outcome.

Defensive strategy:  a group voting strategy intended to deter or thwart 
other groups' offensive strategies.

Compromise:  a voting strategy in which an alternative is voted equal to or 
over a more preferred alternative to avoid the election of an alternative which 
is less preferred than both. (Compromise may be offensive or defensive.)

The outcome of elections may hinge on coordination of group strategy.  Since coordination 
is difficult and expensive in large public elections, particularly if strategy may need to be 
adjusted shortly before the election in response to other groups' strategizing, it is important 
for y's camp to be able to publicize a defensive strategy well in advance of the election 
which will be stable no matter how other groups respond.  If this defensive strategy requires 
voters who prefer another candidate more than y to compromise by ranking y equal to or 
over their more preferred candidate, it will be difficult to coordinate, particularly since 
those voters, lacking complete information, may not be convinced compromise is necessary 
and would dislike compromising unnecessarily.  Thus we seek a group strategy equilibrium 
which does not require compromise. (Ideally the equilibrium should involve minimal 
misrepresentation of preferences.)

Section 1 discusses the minimal defense criterion and shows how its satisfaction can lead 
to desirable equilibria.  Section 1 also defines the non-drastic defense criterion.  Section 2 
discusses the truncation resistance criterion.  Section 3 discusses the larger majority 
criterion, whose satisfaction implies satisfaction of minimal defense, non-drastic defense 
and truncation resistance (assuming voters are permitted to vote non-strict orderings).  
Section 4 describes some voting rules which satisfy these and other desirable criteria.  
Section 5 provides concluding remarks.  

1. Minimal Defense criterion

If the voters rank all alternatives in order of their sincere preferences in scenario 1, most 
voting rules prominent in the literature choose y. (The Instant Runoff rule, also known as 
Hare and Single Transferable Vote and other names, is a notable exception.)

Borda chooses y.
Simpson chooses y.
Copeland (assuming plurality tie-breaker) chooses y.
Instant Runoff (Hare, Single Transferable Vote, Alternative Vote) chooses x. 
Tideman's "ranked pairs" rule chooses y. 

(Definitions of these voting rules are included in Tideman [1987].  
Zavist, Tideman [1989] provides an updated version of Tideman's rule.)

Despite the niceness of the scenario, it is not a group strategy equilibrium under any 
prominent democratic voting rule for all voters to rank the alternatives in their sincere 
orders of preference (abbreviated "sincere ranking").  To be more specific, x would be 
elected if enough supporters of x vote "x over z over y" and the other voters vote sincerely, 
under most voting rules. (Also, sincere ranking is not a group strategy equilibrium under 
Instant Runoff, since y would be chosen instead of x if the 34% having "z over y over x" 
preferences compromise by voting "y over z over x" and they have an incentive to do so 
since they prefer y over x.)

Example 1a:  Offensive reversal strategy employed by some of x's supporters

23%	23%	10%	10%	34%
x	x	y	y	z
y	z	x	z	y
z	y	z	x	x

 

(Many supporters of x misrepresent their "y over z" preference by voting "z over y.")

Borda chooses x.
Simpson chooses x.
Copeland (assuming plurality tie-breaker) chooses x. 
Instant Runoff (Hare, Single Transferable Vote, Alternative Vote) chooses x. 
Tideman's "ranked pairs" rule chooses x.

Clearly there is an incentive for x's supporters to misrepresent their "y over z" 
preference as shown, under the voting rules listed here.

It is of course possible for z's supporters to employ a compromise strategy, ranking y  
equal to or over z, to ensure x cannot win.  That would be a group strategy equilibrium 
since no group would have an incentive to deviate given knowledge of others' strategies.  
But z's supporters may lack enough information to be certain z cannot win.  For instance, 
if z is more popular than x on election day, then z could win if z's supporters employ 
some non-compromising strategy.  Also there is a chance z will be the sincere winner 
on election day.  Furthermore, z's supporters may be reluctant to misrepresent their 
preference for z over y due to long term thinking.  Thus it may be difficult to convince 
z's supporters of the need for compromise.  But this does not mean no voting rule has 
desirable group strategy equilibria achievable without compromise.  This motivates the 
following criterion, presented with two nearly equivalent wordings:

Minimal Defense:  For all subsets X of the alternatives, if there exists an 
alternative y that more than half the voters prefer over every alternative in X, 
then there must exist a set of voting strategies for that majority that ensures 
no alternative in X will be elected and does not require any of them to 
misrepresent any preferences except possibly lowering alternatives in X.

A nearly equivalent wording of minimal defense:
Each voter must be permitted to vote any (non-strict) ordering of the 
alternatives, and for all alternatives x, x must not be elected if there is 
an alternative y such that the number of voters who rank both y over x  
and x no higher than tied for bottom exceeds the number of voters 
who rank x higher than tied for bottom.

Assuming the voting procedure satisfies minimal defense, x will be defeated if 
the 54% who prefer y over x in scenario 1 vote as follows:

(The votes of the 46% who favor x are not shown, to indicate that x will be 
defeated no matter how they vote, assuming minimal defense is satisfied.)

Note that only 10% need to misrepresent any preferences (by lowering x to tied with z for 
bottom) to ensure x is defeated, and that z's supporters can express their preference for z 
over y rather than compromising. 

The criterion is named "minimal defense" because it allows the voters to express their 
sincere preferences for their favored alternatives.  They won't be compelled to pretend 
to prefer a compromise alternative, and thus won't accidentally compromise unnecessarily 
when unsure about the support by other voters for their favored alternatives.  

The clause about not lowering x below a tie for bottom has three consequences: 
1) The strategy can simultaneously be employed by more than one majority to defeat more 
than one alternative, which would not be the case if the strategy required a unique bottom 
alternative.  For instance, the other 10%, who prefer y over z over x, can downrank z to 
tied with x for bottom, in order to defeat z.
2) Downranking x below a less preferred alternative could create a strategic opportunity for 
supporters of the less preferred alternative, an opportunity which will not become available 
if x is downranked to tied with the least preferred alterative.  For instance, if the 10% needed 
to downrank x below z instead of to tied with z at the bottom, this could create a strategic 
opportunity for z's supporters to elect z, particularly if there is uncertainty about the expected 
percentages on election day, and thus might not be an equilibrium.
3) It is less misrepresentative to downrank x to tied with one's least preferred alternative 
than to downrank x below one's least preferred alternative.

Ossipoff's wording, which he named the strong defensive strategy criterion (SDSC), 
said that when more than half the voters prefer some alternative y over x, there must exist 
a voting strategy that ensures x will not be chosen and doesn't require them to vote y 
equal to or over any alternative preferred over y. (In other words, they must never 
need to use a compromise strategy to defeat x.)  SDSC unfortunately fails to capture 
the requirement that the voters must not have to lower x below a tie for bottom, 
thereby allowing some voting rules which fail minimal defense and lack the desired 
equilibrium to satisfy SDSC.

Another application of the "minimal defense" voting strategy is by sophisticated legislators 
whose strategic behavior may be constrained by accountability to unsophisticated constituents.  
More drastic misrepresentation (i.e., ranking a compromise alternative equal to or over a 
more preferred alternative) may not be accepted by strategically unsophisticated constituents.

In section 4 it is shown that it is possible to satisfy minimal defense (and non-drastic 
defense, defined below, and truncation resistance, defined in section 2) along with 
many widely accepted criteria such as Condorcet-consistency, top cycle, Pareto, 
monotonicity, resolvability, independence of clone alternatives, etc.  Thus one 
may dodge some questions about whether certain other criteria are more important 
than minimal defense, as long as one is persuaded that satisfaction of minimal defense 
is desirable.  Elsewhere we argue that criteria inconsistent with minimal defense etc., 
such as reinforcement and participation (satisfied by the Borda voting rule), are 
much less important.

In scenario 1, candidates x and z are covered by the premise of minimal defense since 
for each of them there is a candidate (y) preferred by more than half the voters.  
Candidate y is not covered by the premise since no candidate is preferred over y by 
more than half the voters.  Looking first at x, satisfaction of minimal defense requires 
there be a way for the 54% who prefer y over x to ensure x's defeat, without 
misrepresentating except by downranking x, and without having to downrank x  
below their least preferred candidates.  Of that 54%, only the 10% who have 
"y over x over z" preferences do not already consider x least preferred, so only two 
sets of strategies for the 54% are acceptable to the criterion:  (1) sincere ranking, or 
(2) some of those 10% vote "y over x ~ z". (The '~' symbol indicates indifference.)  
However, under many prominent voting rules, x is elected given either of these strategies 
if supporters of x vote "x over z over y" (misrepresenting their "y over z" preference):

Example 1b:  Defensive downranking of x by some of the "y over x" majority

23%	23%	10%	10%	34%
x	x	y	y	z
y	z	x,z	z	y
z	y		x	x

 

(Many of x's supporters are misrepresenting their "y over z" preference by voting 
"z over y."  Some of y's supporters misrepresent their "x over z" preference by 
voting indifference between x and z.  More than half the voters (54%) are voting 
y over x and x no better than tied for bottom.)

Borda chooses x.
Simpson chooses x.
Copeland (assuming plurality tie-breaker) chooses x.
Instant Runoff (Hare, STV, Alternative Vote) chooses x. 
Tideman's "ranked pairs" rule chooses x. 

Examples 1a and 1b show that the listed voting rules fail minimal defense since 
they choose x with either of the strategies allowable by the criterion.

The minimal defense criterion does not require that y be elected whenever the minimal 
defensive strategy is employed.  It suffices for us that y be elected at equilibrium when 
the minimal defensive strategy is employed.  Since x cannot win assuming satisfaction of 
minimal defense, the best response to the minimal defensive strategy for x's supporters is 
to vote sincerely rather than elect their least preferred choice z.  This best response is part 
of a group strategy equilibrium in which y is elected.  

A similar argument can be constructed showing that if the 10% who have "y over z over x" 
preferences downrank z to tied for bottom, the best response for z's supporters is to vote 
sincerely.  Thus the following is a group strategy equilibrium in scenario 1 under a voting 
rule that satisfies minimal defense, and the only misrepresentation of preferences is the 
defensive downranking by y's supporters:

Example 1c:  Equilibrium elects y assuming satisfaction of minimal defense

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

 

The 20% whose favorite is y have voted the others tied for bottom.  Thus more 
than half (54%) have voted y over x and x no better than tied for bottom, and 
more than half (66%) have voted y over z and z no better than tied for bottom.  
The 46% who favor x cannot vote in a way which elects x given the strategies 
of the others, so they have no strategy better than their sincere orders of preference.  
The 34% who favor z cannot vote in a way which elects z given the strategies 
of the others, so they have no strategy better than their sincere orders of preference.)

Since the minimal defensive strategy does not change the outcome from what it would be 
given all sincere voting, it should not be considered manipulative.  Furthermore, it deters 
manipulation of the outcome.  Because this strategy does not require compromise, it is 
relatively easy to coordinate.  Thus one can argue that if the voting rule satisfies minimal 
defense then this scenario should not be considered manipulable, and Shepsle, Bonchek 
et al are overly pessimistic.

Implicit in our argument is the assumption that if x's supporters are sufficiently clever and 
manipulative that they would consider attempting an offensive reversal strategy, then enough 
of the other voters will also be sufficiently sophisticated to consider the defensive strategy.  
Part of y's overall campaign strategy could be to publicize the defensive voting strategy.

As example 1c shows, multiple majority coalitions can simultaneously employ the minimal 
defensive strategy since no candidate need be downranked to strictly last.  The minimal 
defensive strategy simultaneously deters reversal by both the x and z camps, and can be 
coordinated publicly before collecting precise polling data regarding the sizes of the 
respective wings.  The strategy need not be kept secret from other camps (which would 
presumably be difficult anyway).  On the contrary, the strategy benefits from being widely 
publicized since it deters others from attempting manipulation.

There is potentially a game of chicken since the x faction may threaten to go ahead with 
their reversal come what may, which would elect z, the least preferred alternative of both 
the x faction and the "y over x over z" faction.  Though it is possible the "y over x over z" 
faction will concede this game, settling for x rather than their least preferred alternative, 
it seems reasonable to expect in practice that enough factors will be on y's side and the 
x faction will concede the game.  For one thing, since y is the "sincere" winner, y has the 
moral high ground.  Second, analysis of utility differences suggests that if y is "between" 
x and z, the x faction has more to lose than the y faction if z is elected, so their threat in 
the chicken game is not much more credible than the threat of a compact car versus an 
SUV in the classic chicken game. Third, there are other more drastic defensive strategies 
that can defeat x if the "y over x over z" faction chickens out:  z's supporters may 
compromise in y's favor, or z may withdraw herself from contention.  So one suspects 
that efforts to organize reversal strategies would not become prevalent in society and 
could eventually die out.  If threats of reversal do indeed die out, then satisfaction of 
truncation resistance (see section 3) means defensive strategy would rarely need to 
be employed. 

Another possible defensive strategy to examine is for the 10% who have "y over x over z" 
preferences to downrank x below their least preferred alternative z (i.e., voting "y  over z  
over x").  This strategy is considered less minimal than the minimal defensive strategy since 
it entails greater misrepresentation of preferences.  Also, this would create an opportunity 
for the z faction to reverse their "y over x" preference and steal the election for z, the least 
preferred alternative of the "less minimal" defenders.  It would exacerbate the coordination 
problem for y's supporters if they need to change strategies at the eleventh hour to defend 
against reversal by z's supporters instead of reversal by x's supporters.  Satisfaction of 
minimal defense means the "y over z over x" defensive strategy is unnecessary, so these 
problems can be avoided.

Note that z's supporters need not misrepresent any preferences to help ensure x's defeat.  
However, this may not be true in the more general case having more than 3 alternatives.  
Given 4 or more alternatives, x might not be the least preferred choice of z's supporters, 
so z's supporters who prefer y over x might also be called upon to downrank x to tied 
with some less preferred alternative w.  But this still leaves them free to express their 
preference for their favorite z over the compromise y, which seems to be important for 
voters.  Thus z's supporters never need to employ a "compromise" strategy -- they do 
not need to insincerely rank their "compromise" candidate (y) equal to or over a favorite 
candidate (z).  In case voters are mistaken about others' preferences and z is actually 
more popular on election day than expected, z's supporters would not be accidentally 
compromising further than necessary.  (Fear of compromising more than necessary 
makes compromise strategies difficult to coordinate, so voting rules that force voters to 
use compromise strategies may not work as well as those that satisfy minimal defense.)

Since z's supporters do not need to compromise, the number of voters who prefer z to y  
can be more accurately counted.  But if the voting rule does not satisfy minimal defense 
(which means y's supporters cannot defeat x using the minimal defense strategy) and y's 
supporters either cannot defeat x with the "less minimal" strategy (ranking x below z, 
strictly last) or cannot coordinate to this less minimal strategy due to a concern about 
creating an opportunity for reversal by z's supporters, then z's supporters, to ensure x's 
defeat, may feel compelled to rank y equal to z ("non-drastic" compromise), or, worse, 
rank y over z or vote only for y ("drastic" compromise), depending on the particular 
voting rule.  Understatement of z's support may harm society, since it is possible z is 
actually best for society and that someday people will realize this after reconsideration 
of z's merits, but reconsideration may be aborted due to the mistaken impression that z  
is very unpopular.  Furthermore, if z cannot rely on her supporters to compromise 
then z may be deterred from competing, which could be a problem if voters' preferences 
on election day are not as estimated and z is really more popular than y and x, or if not 
competing makes z seem less popular than in reality.  The flip side of this unfortunate coin 
is that if z can rely on her supporters to compromise, her incentive to compete may be 
reduced since she expects the voting will make it seem she is less popular than in reality.  
And of course there is the obvious problem that if z competes and her supporters do not 
properly compromise, then x will be elected, a problem for society if x is inferior to y  
or if y is widely perceived as having a sincere mandate. (For instance, in the presidential 
election of 2000, Gore would have won had some Nader voters compromised.)  A mirror 
argument shows that the number who prefer x to y can also be more accurately counted 
under a voting rule that satisfies minimal defense...

In the more general case where there are also some voters having "x over z over y" 
preferences and some having "z over x over y" preferences, there would still be a group 
strategy equilibrium which elects y when y's supporters adopt the minimal defensive 
strategy against both x and z, assuming y is the sincere Condorcet winner and the voting 
rule satisfies minimal defense.  

The second, nearly equivalent, wording of minimal defense is a combination of a universal 
domain criterion ("any ordering, strict or non-strict, of the alternatives must be an admissible 
vote") and a criterion that does not refer to strategies or sincere preferences.  Clearly, any 
voting procedure that satisfies the second wording satisfies the first wording.  The first 
wording more clearly expresses the motivation of the criterion, indicating voters may freely 
express their preferences for favorite alternatives over compromise alternatives.  The second 
wording more clearly describes a defensive strategy which deters offensive strategies, and it 
is straightforward to adapt the second wording into a filter for any voting rule which admits 
non-strict orderings to attain satisfaction of minimal defense, as described in section 4.4 
(but not as robustly as the procedures described in sections 4.2 and 4.3).  

The Approval voting rule (Brams, Fishburn (1978)) satisfies the second part of the second 
wording and allows expression of indifference, but it does not satisfy minimal defense.  
By violating the universal domain part of the criterion, Approval often requires some of 
a majority coalition to employ a (non-drastic) compromise strategy by not expressing 
their preferences for favorite alternatives over compromise alternatives in order to defeat 
less preferred alternatives.  Approval satisfies the following weaker criterion:

Non-Drastic Defense:  Each voter must be allowed to vote as many 
alternatives as s/he wishes tied for top, and if more than half of the voters 
vote some alternative y (tied for) top, then no alternative voted below y  
by more than half of the voters may be chosen.

Some voting procedures accept voters' strict orderings of the alternatives but do not allow 
voters to express indifference between any alternatives.  Such procedures fail minimal 
defense because they do not allow voters to vote x tied for bottom, making the desired 
group strategy equilibrium unattainable.  They fail non-drastic defense because they 
do not allow voters to vote a compromise alternative equal to their favorite alternative.  
Thus they may require a "drastic compromise" defensive strategy in which some voters 
must rank a compromise alternative over their favorite alternative in order to defeat 
an alternative less preferred than both.  The drastic compromise is presumably the most
difficult defensive strategy to coordinate, which explains why political parties avoid 
nominating more than one candidate per office, yet under most voting rules the drastic 
compromise is the only reliable defensive strategy.  Even the non-drastic compromise 
strategy available under Approval may be difficult to coordinate, which is why groups 
placing propositions (initiatives) onto public ballots tend to avoid placing competing 
propositions onto the ballot. 

2. Truncation Resistance criterion

Depending on the institution, voters may not be strategically-minded, or they may feel 
constrained in their choice of strategy due to  accountability to constituents.  This section 
considers scenarios of this class, specifically the scenarios where voters are expected 
not to vote any insincere strict preferences.  This is not necessarily the same as sincere 
voting since it also allows expressions of indifference by voters who have sincere strict 
preferences, a voting behavior which might not be based on strategic calculation.

Truncation: a vote of indifference by a voter who has a strict preference. 
(In other words, abstention in one or more pairings of alternatives.)

Some examples of truncation:

1. Given a tediously long list of alternatives to order, a voter might rank only some of them.

2. There may be societies (ours included) where some voters can be induced to express 
indifference by exhortations from their leaders that two less favored alternatives are like 
"Tweedledee and Tweedledum" (i.e., equally bad) when those voters have a preference 
for dee over dum but pay attention to their leaders.  Perhaps the leaders would rather 
have their supporters vote "dum over dee" as part of a reversal strategy, but they find 
the "dum over dee" reversal strategy much harder to sell to non-strategic supporters 
than the "dum indifferent to dee" truncation strategy.

3. A legislator who represents constituents who have strong preferences on some issue 
may fear her non-strategic constituents would not tolerate a vote which reverses their 
preferences, particularly if she fears a challenger will trumpet this alleged "betrayal" 
during her re-election campaign, even when reversal is strategically optimal for her 
constituents' interests.  Legislators may be less fearful of being punished for voting 
indifference or abstention than for voting reversal of constituents' preferences.

The following criterion requires, for a broad class of scenarios, that when no offensive 
strategy (except perhaps truncation) is used by any group, then no defensive strategy 
shall be needed:

Truncation Resistance:  If no voter votes any insincere strict preferences, 
alternative x is not in the sincere top cycle, and an alternative in the sincere top 
cycle is ranked over x by more than half of the voters, then x must not be chosen. 
(A more formal wording is provided below.) 

The clause about no voter voting any insincere strict preferences means no voter ranks 
any alternative over any that is not sincerely less preferred.  Thus if some voter does not 
strictly prefer a over b, yet ranks a over b, the premise does not hold.  If some voter 
does prefer a over b, yet ranks a indifferent to b, she has truncated her preference for a  
over b and this does not violate the premise. 

Here is a more formal wording of the truncation resistance criterion:  

Let A denote the finite set of alternatives.  

Let SincereTop denote the minimal non-empty B ⊆ A such that, for all b ∈ B 
and all a ∈ A\B, the number of voters who sincerely prefer b to a exceeds 
the number who sincerely prefer a to b.  

Say there is no misrepresentation except possibly truncation if and only if, 
for all a,b ∈ A, every voter who votes a over b sincerely prefers a to b. 

If there is no misrepresentation except possibly truncation, then for all x ∈ A, 
x must not be chosen if x ∉ SincereTop and at least one alternative in SincereTop 
is ranked over x by more than half of the voters. 

An equivalent wording of truncation resistance is the following: 

Suppose R and R' are two sets of votes that are the same except some votes 
in R' that express indifference regarding some pairs of alternatives may in R  
express strict preference regarding those alternatives.  Let tc denote the top 
cycle given R.  Alternative x must not be chosen given either R or R' if  x ∉ tc  
and y ∈ tc and more than half the votes in R' rank y over x.  

Truncation resistance is, in a sense, a flip side of minimal defense.  Minimal defense 
says that certain expressions of indifference must be effective as a defensive strategy, 
and truncation resistance says that indifference must be so ineffective as an offensive 
strategy that no defensive strategy would be needed to deter or thwart it.  

Another value of satisfaction of truncation resistance is that, even when there are 
so many "centrist" candidates (i.e., those in the sincere top cycle) that many of them 
are left unranked by voters who hadn't the time to learn about them all or by voters 
put off by the tediousness of ordering them all, a centrist candidate will be elected if 
at least one of them is ranked by a majority over the non-centrist candidates (assuming 
the supporters of the non-centrists do not attempt a reversal strategy, in which case 
the supporters of the centrist would need to employ a defensive strategy such as the 
minimal defensive strategy described in section 1).

Ossipoff's wording of the truncation resistance criterion was less general, saying that 
if no voter votes any insincere strict preferences and a sincere Condorcet winner is voted 
over x by more than half of the voters, then x must not be chosen.  It is straightforward to 
strengthen Ossipoff's criterion by replacing the sincere Condorcet winner with a member 
of the sincere top cycle in the wording.  

Satisfaction of minimal defense does not guarantee satisfaction of truncation resistance.  
Satisfaction of minimal defense may not be enough to mitigate failures of truncation 
resistance in the subclass of scenarios being considered.  Since the voters in these 
scenarios are not strategically minded or are constrained not to downrank so far that 
this would reverse some preferences, they may not be able to employ the minimal 
defensive strategy.

Most voting rules prominent in the literature fail truncation resistance.  Using scenario 1 
as an example, suppose x's supporters truncate their preference for y over z and suppose 
the majority who prefer y to x accurately represent their preferences.  Then the votes are 
as in the following example:

Example 2:  Truncation by x's supporters of their "y over z" preference

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

 

(Assume the "x" votes of the 46% are treated the same as "x over y ~ z.")

Since y is the sincere Condorcet winner and there is no misrepresentation 
except truncation and more than half of the voters vote y over x, satisfaction 
of truncation resistance requires x must not be chosen.

Borda chooses x.
Simpson chooses x.
Copeland (assuming plurality tie-breaker) chooses x.
Instant Runoff (Hare, STV, Alternative Vote) chooses x. 
Tideman's "ranked pairs" rule chooses x.

3. Larger Majority criterion

The larger majority criterion is not being promoted as a normative criterion in its own 
right, but it is useful for purposes of analysis since any voting procedure that satisfies 
larger majority and a universal domain criterion also satisfies minimal defense, 
non-drastic defense and truncation resistance.  

Before defining two equivalent formal definitions of larger majority, here is a brief 
informal description:

If a majority (not necessarily more than half the voters, due to indifference) 
rank some alternative, say y, over alternative x and this majority is not the 
smallest majority in any cycle involving y and x, then x must not be chosen.

Two formal definitions of larger majority are now presented, and then their equivalence 
is demonstrated.  The first definition corresponds to the informal description above.  
The second is defined in terms of "paths" rather than "cycles."

Let A denote a finite set of two or more alternatives.  Let N denote a finite set of 
voters {1,2,...,n}.  Each voter i ∈ N submits a ranking (non-strict ordering) R_i of 
the alternatives in A.  Let R denote the collection of voters' orderings.

For all x,y ∈ A, let R(x,y) denote the orderings in R which rank x over y.  That is, 
R(x,y) = {r ∈ R such that r ranks x over y}.  Thus #R(x,y) denotes the number 
of voters who rank x over y. (Due to the possibility of indifference, the larger 
of #R(x,y) and #R(y,x) is not necessarily more than half the voters.)

Let Pairs(A) denote the set of all possible ordered pairs of alternatives 
{(x,y) such that x ∈ A and y ∈ A}.  Where A is clear from the context, 
abbreviate Pairs. 

Let Majorities(A,R) denote {(x,y) ∈ Pairs(A) such that #R(x,y) > #R(y,x)}.  
Where A and R are clear from the context, abbreviate Majorities.

For all a₁,a₂,...,a_k ∈ A, the sequence a₁a₂...a_k is a majority cycle of (A,R) 
if and only if a_k = a₁ and (a_j,a_j+1) ∈ Majorities(A,R) for all j ∈ {1,2,...,k-1}.  
Let MajorityCycles(A,R) denote the set of  all sequences of alternatives in A 
which are majority cycles of (A,R).  Where A and R are clear from the context, 
abbreviate MajorityCycles.

For all C ∈ MajorityCycles(A,R) and all (x,y) ∈ Pairs(A), (x,y) is a C-majority 
if and only if x immediately precedes y in the sequence C.  

Let SmallestMajorities(A,R) denote {(x,y) ∈ Majorities(A,R) such that 
there exists C ∈ MajorityCycles(A,R) such that (x,y) is a C-majority and 
#R(x,y) ≤ #R(z,w) for all C-majorities (z,w)}.  Where A and R are clear 
from the context, abbreviate SmallestMajorities.

Larger Majority:  For all x ∈ A, x must not be chosen if there exists y ∈ A  
such that (y,x) ∈ Majorities(A,R)\SmallestMajorities(A,R).  In other 
words, the chosen alternative must be in the set {a ∈ A such that, for all b ∈ A, 
(b,a) ∉ Majorities(A,R) or (b,a) ∈ SmallestMajorities(A,R)}.  Let LM(A,R) 
denote this set.  Where A and R are clear from the context, abbreviate LM.

Here is a second, equivalent definition of the LM set and the larger majority criterion, 
expressed in terms of "paths" rather than cycles:

Refer to the definitions above.

Paths:  For all a₁,a₂,...,a_k ∈ A, the sequence a₁a₂...a_k is a path from a₁ to a_k 
if and only if (a_j,a_j+1) ∈ Majorities(A,R) for all j ∈ {1,2,...,k-1}.

Path Strength:  The strength of a path a₁a₂...a_k is the minimum of #R(a_j,a_j+1) 
over j ∈ {1,2,...,k-1}.  (In other words, the strength of a path is the size of its 
smallest majority, where the size of any majority is measured by the size of its 
supporting coalition.)

Strongest Path matrix:  Let SPM(A,R) denote the matrix such that, 
for all x,y ∈ A, SPM_xy(A,R) is the strength of the strongest path from x  
to y if there is at least one path from x to y, and SPM_xy(A,R) = 0 if there 
is no path from x to y.  Where A  and R are clear from the context, 
abbreviate SPM and SPM_xy.

Larger Majority (2):  For all x ∈ A, x must not be chosen if there exists 
y ∈ A such that [#R(y,x) > #R(x,y) and #R(y,x) > SPM_xy].  In other words, 
the chosen alternative must be in the set {a ∈ A such that, for all b ∈ A, 
#R(b,a) ≤ max(#R(a,b),SPM_ab)}.  Let LM(A,R) denote this set.  
Where A and R are clear from the context, abbreviate LM.

The equivalence of the two formal wordings of larger majority can be easily shown.  
Note that if #R(y,x) > #R(x,y), then the concatenation of x to the end of a path from x to y  
is a majority cycle in which y immediately precedes x.  By inspection of the definitions, 
it follows that (y,x) ∈ SmallestMajorities if and only if #R(y,x) > #R(x,y) and 
#R(y,x) ≤ SPM_xy.  From this it follows that the two definitions of the LM set are 
equivalent and the two wordings of larger majority are equivalent.

The LM set is always a non-empty subset of the top cycle, denoted here by τ:

Top Cycle:  τ(A,R) is the minimal non-empty B ⊆ A such that #R(b,a) > #R(a,b) 
for all b ∈ B and all a ∈ A\B.)

(The sincere top cycle is defined similarly, but depends upon the voters' sincere 
orders of preference instead of upon their votes.)

The Appendix provides a proof that LM is a non-empty subset of the top cycle.

It is easy to show that if a voting procedure satisfies larger majority and admits all 
non-strict orderings of the alternatives, then it satisfies minimal defense, non-drastic 
defense and truncation resistance:  

To show satisfaction of minimal defense, assume more than half of the voters rank x  
no better than tied for bottom and more than half of the voters rank y over x.  Clearly 
#R(y,x) > #R(x,y).  Since x is ranked better than tied for bottom by fewer than half 
of the voters, all paths from x have strengths less than half of the voters.  Since y is 
ranked over x by more than half the voters, #R(y,x) is greater than half the voters.  
Therefore #R(y,x) > SPM_xy.  Since #R(y,x) > #R(x,y) and #R(y,x) > SPM_xy, 
x cannot be in LM(A,R) and thus minimal defense is satisfied.  

To show satisfaction of non-drastic defense, assume more than half the voters rank 
y no worse than tied for top and more than half the voters rank y over x.  Clearly 
#R(y,x) > #R(x,y).  Since y is ranked worse than tied for top by fewer than half 
the voters, all paths to y have strengths less than half of the voters.  Since y is ranked 
over x by more than half the voters, #R(y,x) is greater than half of the voters.  
Thus #R(y,x) > SPM_xy.  Since #R(y,x) > #R(x,y) and #R(y,x) > SPM_xy, 
x cannot be in LM(A,R) and non-drastic defense is satisfied.  

To show satisfaction of truncation resistance, let S denote the alternatives in the sincere 
top cycle and let X denote the alternatives outside the sincere top cycle. (X = A\S.)  
Assume x ∈ X, that at least one alternative y ∈ S is ranked over x by more than half 
of the voters, and that no voter ranks any alternative over one that isn't less preferred.  
Clearly #R(y,x) > #R(x,y).  By the definition of the sincere top cycle, fewer than half 
of the voters prefer any alternative in X over any alternative in S.  Thus for all b ∈ X  
and all a ∈ S, SPM_ba must be less than half the voters.  Thus #R(y,x) > SPM_xy.  
Since #R(y,x) > #R(x,y) and #R(y,x) > SPM_xy(A,R), x cannot be in LM(A,R) and 
truncation resistance is satisfied.

If a voting rule satisfies larger majority, a majority desiring to defeat x may not need to 
downrank x as far as tied for bottom.  They only need to downrank x far enough that they 
do not rank x over any alternative that may cycle with an alternative they rank over x.  
A 3-candidate example has too few candidates to illustrate this distinction, but we can 
imagine some voters having "y over x over z over w" preferences who perceive that z  
but not w may cycle with y, and can defeat x by voting "y over x ~ z  over w".  If in doubt 
about whether w may cycle with y, however, due either to uncertain information about 
others' preferences or out of concern that some group of voters may attempt an offensive 
strategy causing w to cycle with y, the safer defensive strategy to ensure defeat of x is 
"y over x ~ z ~ w".

4. Voting rules that satisfy the criteria

Sections 4.2 and 4.3 define two voting rules that satisfy minimal defense, non-drastic 
defense, truncation resistance, larger majority, and other desirable criteria such as 
anonymity, neutrality, strong Pareto, monotonicity, resolvability, Condorcet-
consistency, top cycle, independence of clone alternatives, etc.

4.1 The "Minimax(Defeat)" voting rule

Before describing voting rules satisfying all the criteria listed above, it is useful to first 
describe a variation of the Minimax voting rule in order to more clearly illustrate the 
principles involved in satisfaction of minimal defense, non-drastic defense, truncation 
resistance and larger majority when there are at most three alternatives:

Refer to the definitions above.

For all x ∈ A such that #R(y,x) > #R(x,y) for at least one y ∈ A, 
let LargestDefeat(x,A,R) denote the maximum of #R(y,x) over {y ∈ A  
such that #R(y,x) > #R(x,y)}.  For all x ∈ A such that #R(y,x) ≤ #R(x,y) 
for all y ∈ A, let LargestDefeat(x,A,R) = 0.

Minimax(Defeat) voting rule:  Allow the voters to order the alternatives and 
to express indifference in their orderings.  Choose the alternative(s) {a ∈ A such 
that LargestDefeat(a,A,R) ≤ LargestDefeat(b,A,R) for all b ∈ A}.

Note that each alternative's "largest defeat" score depends on the size of an opponent's 
supporting coalition (e.g., #R(y,x)), not on a margin of defeat (e.g., #R(y,x) - #R(x,y)).  
This distinction is vital.  Also note that an alternative's largest defeat depends only on 
the pairings in which it is beaten, not on pairings it wins or ties.  This is not as vital but 
adds robustness when the sizes of some of a sincere Condorcet winner's majorities 
are less than half the voters.  Assuming no voter expresses indifference, these distinctions 
would not matter and any Minimax rule would choose the same.  (And given three or fewer 
alternatives, Maximin and Minimax would choose the same.)  But the assumption that no 
voter expresses indifference should not be made.  Even if it is reasonable to assume all 
voters have strict preferences over all alternatives -- a common simplifying assumption 
in the social choice literature -- the arguments in the preceding sections indicate that 
permitting strategic expressions of indifference may be useful for voters and benevolent 
for society.

Since each candidate's largest defeat depends on the size of an opponent's supporting 
coalition, no voting strategy for a candidate's supporters can reduce the sizes of their 
candidate's defeats.  In scenario 1, x's score will be 54% if the voters who prefer y to x  
either vote sincerely or employ the minimal defensive strategy to defeat x.  There are 
two strategies which x's supporters may consider: (1) they can truncate their "y over z" 
preference by voting "x over y ~ z" or (2) they can reverse their "y over z" preference 
by voting "x over z over y".  Under Minimax(Defeat), the first strategy cannot elect x  
since it cannot raise y's largest defeat to be as large as x's largest defeat.  But if the sizes 
of majorities were measured by margin, as is often done in the social choice literature, 
the truncation strategy could elect x.  The second strategy will backfire if the 10% having 
"y over x over z" preferences employ the minimal defensive strategy, because the minimal 
defensive strategy potentially reduces z's largest defeat to less than half the voters, and 
that potential is realized if x's supporters proceed with their reversal scheme.  But if 
defeat size were measured by margin, the reversal strategy could elect x.  Thus, if defeat 
size is measured by the size of the winning coalition and not by margin, the best response 
for x's supporters, facing the minimal defensive strategy, is to vote sincerely.  Similarly, 
the 10% who have "y over z over x" preferences can make sincere voting the best 
response for z's supporters by voting "y over z ~ x."

The Minimax(Defeat) rule fails minimal defense and truncation resistance when there 
are more than three alternatives, for the same reason that it (and other Minimax and 
Maximin rules) fail top cycle, Condorcet loser, and independence from clones: 
adding two alternatives similar to the Condorcet winner can create a "vicious" cycle 
amongst those three similar alternatives that causes the defeat of all three under Minimax 
and Maximin rules:

Example 4.1:  4-alternative failure of Minimax(Defeat)

Alternative w is a Condorcet loser, yet Minimax chooses w.  Since the majority (60%) 
who want to defeat w already rank w bottom, they have no strategy allowed by the 
minimal defense criterion that will defeat w.  Thus Minimax fails minimal defense.  
Assuming the votes are sincere representations of preferences, w is not in the 
sincere top cycle but is not defeated, so Minimax fails truncation resistance.

Minimax(Defeat) might be considered a reasonably practical voting rule even with more 
than three alternatives, since cycles among top candidates would be expected to be less 
vicious than the 66%-67%-67% cycle in the example, but there are rules that are more 
robust, completely satisfying these criteria plus those listed at the beginning of section 4.  
Two such voting rules are presented in the next two sections.

4.2 The "Maximize Affirmed Majorities" (MAM) procedure

This section briefly describes the MAM procedure, which satisfies all the criteria listed 
at the beginning of section 4.  For more information and details, see the documents 
"MAM procedure definition" and "A mathematically formal definition of MAM."  

MAM is an implementation of a terse suggestion written by the Marquis de Condorcet 
in the introduction of his seminal 1785 essay on election theory: 

... take successively all the propositions that have a majority, beginning 
with those possessing the largest.  As soon as these first propositions 
produce a result, it should be taken as the decision, without regard for 
the less probable decisions that follow.  
--  Marquis de Condorcet, "Essay on the Application of Mathematics to 
the Theory of Decision-Making" [1785], page lxviii (as translated by Keith 
Michael Baker  in "Condorcet: From Natural Philosophy to Social Mathematics" 
[1975], p.240, Chicago University Press)

MAM constructs an acyclic subset of Majorities(A,R) by starting with an empty subset and 
considering the majorities one at a time in order of precedence (i.e., from largest to smallest, 
where size is measured by the number of voters who ranked the pairing's winner over the 
pairing's loser):  If a majority under consideration does not cycle with those already included 
into the subset, then it too is included ("affirmed") into the subset.  Since the final subset is 
acyclic by construction, there must exist at least one alternative that is not the pairwise 
loser of any majority in the subset.  MAM chooses such an alternative.  

An omitted detail is how MAM orders the majorities largest to smallest in the ambiguous 
case where two or more majorities have the same size.  Another omitted detail is which 
alternative is chosen in the ambiguous case where two or more alternatives are not the 
pairwise loser of any majority in the acyclic subset.  These details are important for 
complete satisfaction of other criteria (monotonicity, independence of clones, etc.) 
but are not relevant for satisfaction of minimal defense, non-drastic defense, truncation 
resistance or larger majority.  Thus the following is an incomplete definition of MAM 
but suffices to show MAM satisfies the criteria that concern us here:

Pair Precedence:  For all x,y,z,w ∈ A, if #R(x,y) > #R(z,w) then (x,y) precedes 
(z,w) and (z,w) does not precede (x,y). (Note the ambiguity if #R(x,y) = #R(z,w).  
This ambiguity is eliminated in the complete definition of MAM showing precedence 
is a strict ordering of the majorities, but for our purposes that detail is unimportant.) 

Majority Cycles:  For all M ⊆ Majorities(A,R) and all (x,y) ∈ Majorities(A,R), 
(x,y) cycles with M if and only if there exist (a₁,a₂),(a₂,a₃),...,(a_k-1,a_k) ∈ M such 
that a₁ = y and a_k = x. 

Let AffirmedMajorities(A,R) denote {(x,y) ∈ Majorities(A,R) such that (x,y) does 
not cycle with {(z,w) ∈ AffirmedMajorities(A,R) such that (z,w) precedes (x,y)}}. 
(Note that AffirmedMajorities() is defined recursively but not circularly.  It can be 
computed quickly by considering the majorities one at a time in order of precedence.)  

MAM chooses an alternative that is not second in any pair in AffirmedMajorities(A,R). 
(The complete definition of MAM resolves the ambiguity when there are two or more 
such alternatives, but for our purposes that detail is unimportant.) 

For complete proofs that MAM satisfies minimal defense, non-drastic defense, truncation resistance, and larger majority, see the document "Proof MAM satisfies Minimal Defense 
and Truncation Resistance."

Here is a sketch of the proof that MAM satisfies larger majority criterion:  

Assume (y,x) ∈ Majorities\SmallestMajorities.  We must show MAM cannot 
choose x.  Suppose the contrary.  This implies (y,x) ∉ AffirmedMajorities.  
Let Aff_yx+ denote {(z,w) ∈ AffirmedMajorities such that (z,w) precedes (y,x)}.  
Since (y,x) ∈ Majorities\AffirmedMajorities, (y,x) must cycle with Aff_yx+.  
This means there exist (a₁,a₂),(a₂,a₃),...,(a_k-1,a_k) ∈ Aff_yx+ such that a₁ = x 
and a_k = y.  Let s denote the sequence a₁a₂a₃...a_kx.  By inspection, s is a 
majority cycle in which y immediately precedes x.  Thus (y,x) is an s-majority 
and (a_j,a_j+1) is an s-majority for all j ∈ {1,2,...,k-1}.  Since every pair in Aff_yx+ 
precedes (y,x), this implies #R(y,x) ≤ #R(a_j,a_j+1) for all j ∈ {1,2,...,k-1}.  
But this implies (y,x) ∈ SmallestMajorities, contradicting the assumption that 
(y,x) ∉ SmallestMajorities.  This contradiction refutes the contrary assumption, 
which implies MAM cannot choose x.     QED

Since MAM satisfies larger majority and permits voters to vote any orderings of the 
alternatives, it follows that MAM also satisfies minimal defense, non-drastic defense 
and truncation resistance.

4.3 The "PathWinner" voting rule

Here is another voting rule that satisfies all the criteria listed at the beginning of section 4:

PathWinner:  Refer to the definitions above.  Allow the voters to order the 
alternatives and to express indifference in their orderings.  Choose an alternative 
in the "PathWinner" set {x ∈ A such that SPM_xy ≥ SPM_yx for all y ∈ A}. 
(If there is more than one such alternative, the one ranked over the others 
by a strict ordering constructed by the Random Voter Hierarchy tiebreak 
procedure is chosen, but for our purposes here that detail is unimportant.) 

PathWinner was described in the internet maillist election-methods-list@eskimo.com 
by Markus Schulze.  Schulze did not propose a name for the rule nor credit anyone 
for its invention; presumably it is his invention.  

The PathWinner set is a non-empty subset of LM(A,R). (This is proved in the Appendix.)  
Thus PathWinner satisfies larger majority.  Since PathWinner permits voters to express 
non-strict orderings, it follows immediately that PathWinner satisfies minimal defense, 
non-drastic defense and truncation resistance.  Like MAM, PathWinner has an 
algorithm that executes in small polynomial time, provided elsewhere.  

A nice property of the PathWinner rule is that, for most of the criteria it satisfies, it is 
fairly easy to prove satisfaction.  For instance, monotonicity follows from the fact 
that when an alternative x is raised in some voters' orderings, no path from x is 
weakened and no path to x is strengthened.

Nevertheless, MAM may be preferable to PathWinner for a couple of reasons: 

1. MAM (but not PathWinner) satisfies immunity from majority complaints (IMC), 
immunity from second-place complaints (I2C) and other criteria described in the 
document Immunity from Majority Complaints, and also satisfies Peyton Young's 
criterion local independence of irrelevant alternatives (LIIA).  

2. Computer simulations using randomly generated profiles of voters' orderings suggest 
the alternative chosen by MAM will beat pairwise the alternative chosen by PathWinner 
more often than vice versa, and that over the long run more voters will prefer MAM 
winners over PathWinner winners than vice versa.  For more information, see 
"Comparisons of the MAM and PathWinner voting rules."  

4.4 Filters for other voting rules

The minimal defense criterion or the larger majority criterion could be adapted into 
filters for other voting rules.  For instance:

Let MD(A,R) denote {a ∈ A such that [a is not ranked (tied for) bottom by more 
than half the voters] or [#R(b,a) is at most half the voters for all b ∈ A]}.

Borda with Minimal Defense Filter:  Allow the voters to vote any orderings of 
the alternatives.  Choose the alternative in MD(A,R) having the best Borda score.

Borda with Larger Majority Filter:  Allow the voters to vote any orderings of 
the alternatives.  Choose the alternative in LM(A,R) having the best Borda score.

The "Borda with Minimal Defense Filter" rule fails truncation resistance, choosing x  
in the example in section 3.  The "Borda with Larger Majority Filter" rule satisfies both 
minimal defense and truncation resistance.

Using a filter to shrink the choosable set of alternatives is less robust, as described by 
Tideman [1987] in his discussion of independence of clones, and may cost compliance 
with other desirable criteria such as monotonicity, independence of clones, etc.  
For instance, the following example shows "Borda with Larger Majority Filter" is 
not monotonic:

Example 4.2:  "Borda with Larger Majority Filter" is not monotonic. 
15 voters rank five alternatives {a,b,c,d,e} as follows:

The alternatives' Borda scores are: 
     a:  4×4 + 3×4 + 2×1 + 1×5 = 35 
     b:  4×2 + 3×4 + 2×4 + 1×5 = 33
     c:  4×5 + 3×2 + 2×2 + 1×1 = 31 
     d:  4×4 + 3×1 + 2×6 + 1×1 = 32 
     e:  4×0 + 3×4 + 2×2 + 1×3 = 19

The ten pairwise majorities are: 
     a over e (14 voters) 
     b over e (10 voters) 
     a over b (9 voters) 
     b over c (9 voters) 
     c over a (9 voters) 
     d over a (9 voters) 
     c over e (9 voters) 
     b over d (8 voters) 
     d over c (8 voters) 
     d over e (8 voters) 
Thus Majorities = {(a,e),(b,e),(a,b),(b,c),(c,a),(d,a),(c,e),(b,d),(d,c),(d,e)}. 

It can be checked that SmallestMajorities = {(a,b),(b,c),(c,a),(b,d),(d,c)}. 
Thus Majorities\SmallestMajorities = {(a,e),(b,e),(d,a),(c,e),(d,e)}. 
Since a and e are the only alternatives which are second in any pairs 
in Majorities\SmallestMajorities, it follows that LM = {b,c,d}.  
Since b is the alternative in LM having the largest Borda score, 
"Borda with Larger Majority Filter" chooses b. 

Now suppose one of the four voters who ranked d immediately over b had 
instead ranked b over d.  Then the "b over d" majority would be 9 voters 
instead of 8.  It follows that (d,a) would also be in SmallestMajorities, 
and LM would be {a,b,c,d}, and "Borda with Larger Majority Filter" 
would choose a since b's Borda score would have increased only to 34.  
Since b is not still chosen when b is upranked, this implies "Borda with 
Larger Majority Filter" is not monotonic.

5.  Conclusions

Satisfaction of minimal defense and truncation resistance makes it easier for a voting 
majority to coordinate voting strategies and thus makes nomination strategies (and 
partisan primary elections) less important.  Lacking satisfaction, in order to defeat a less 
preferred "greater evil" alternative, the majority may need to rank a compromise alternative 
equal to or over favored alternatives.  This they are reluctant to do, particularly when 
they lack accurate information about voters' preferences.  Three negative consequences 
of non-satisfaction are:  

1. Understatement of the support for some alternatives when their supporters 
    are forced to compromise, which may deter reconsideration and greater 
    popularity of those alternatives in the future due to misperception of the 
    degree of their unpopularity.

2. Election of "less popular" alternatives when voters compromise further than 
    needed or not far enough due to inaccurate information about other voters' 
    preferences and strategies. 

3. Formation and maintenance of two "big tent" parties (plus "sure-loser" third 
    parties) which each nominate only one alternative per office, since the parties 
    cannot rely on all their supporters to properly compromise. 

Minimal defense and truncation resistance are compatible with many desirable criteria.  
It is possible to simultaneously satisfy Condorcet-consistency, top cycle, strong Pareto, 
monotonicity, anonymity, neutrality, local independence of irrelevant alternatives, 
independence of clone alternatives, immunity from majority complaints, and other 
criteria.  Some exceptions are reinforcement and participation (satisfied by the Borda 
procedure) and uncompromising (satisfied by Instant Runoff and Minimax), which seem 
much less important.  Reinforcement requires that if x is chosen by each partition of some 
partitioning of the voters, then x must be chosen when all votes are tallied together.  But 
it is easy to design the institutional rules to prevent a minority from controlling how or 
whether voters are partitioned, so it is simple to prevent failure of reinforcement from 
being exploited by a minority.  Participation requires that abstention never be a better 
strategy than sincere voting for any voter.  But a voter who knows she may gain by 
abstaining has the information needed to vote strategically, so it is a false dichotomy to 
consider only abstention and sincere voting.  Uncompromising, another form of 
resistance to truncation, requires that if a voter's favorite alternative is elected when 
the voter ranks all other alternatives below it and tied for bottom then that alternative 
must still be elected if instead the voter raises some (compromise) alternative above 
bottom (but still below the favorite).  Occasional violations of uncompromising seems 
a small price to pay for satisfaction of the other desirable criteria.  

Voting procedures such as MAM that satisfy minimal defense, truncation resistance, 
Condorcet-consistency, etc., are practical now that the technological advances of the past 
few decades permit use of machine-readable ballots or computer interfaces for voting, 
obsoleting old claims of impracticality in the social choice literature.  For instance, optical 
scanning technology is already widely used in large public elections and would permit 
voters to order the candidates.  Even if space on the optically-scanned ballot is insufficient 
to allow the voter to strictly order a large number of candidates, it would nevertheless be 
a significant improvement if there is at least enough space to allow the voter to express 
trichotomous preferences (i.e., indicate which alternatives are best, which are compromises, 
and which are worst).  Even in the worst case, where space only permits the voter to select 
one candidate, each candidate could publish an ordering of all candidates prior to election 
day and each vote could be treated as though it were the ordering published by the voter's 
selected candidate.   

Two more arguments to support practicality are that (1) even children can order alternatives 
from best to worst, given a reasonable interface, and (2) the algorithm to tally the election 
executes in a time which in the worst case is a small polynomial function of the number of 
voters and number of alternatives.  

Voting procedures such as MAM are useful for almost any (democratic) group decision, 
from small committees to large public elections.  In small groups such as committees, 
multistage voting procedures (e.g., the Successive Elimination pairwise agenda procedure 
defined by Robert's Rules of Order) are also feasible, but since agenda control can be 
exploited to manipulate outcomes, an agendaless procedure such as MAM may be more 
desirable.  For instance, a reasonable variation of MAM for committees would be the 
following:  As alternatives are proposed they are automatically appended to the bottom 
of each voter's ordering.  Each voter is allowed to freely edit her ordering.  When no one 
wishes to propose any more alternatives, the tally of the votes is finalized.  

Voting procedures such as MAM may also be compatible with the Electoral College system 
used in the United States' presidential elections.  However, fragmentation of the Electoral 
College delegates' votes among more than two candidates could send the decision to the 
House of Representatives.  Thus, to persuade parties to nominate more than one candidate 
apiece the system would need to allow candidates to withdraw from contention after the 
public's votes are cast and a summary of the votes published.

Voting procedures such as MAM can also enhance proportional representation systems.  
For instance, each voter could be allowed to order the parties, which would allow seats 
to be awarded proportionally (if desired) yet also allow a "best compromise" party to be 
identified and rewarded with agenda control (and possibly also rewarded with extra seats). 

Appendix:  Proof that LM(A,R) is a non-empty subset of the top cycle

In section 3 it was claimed LM(A,R) is a non-empty subset of the top cycle τ(A,R).  To show 
this, we first show LM(A,R) ⊆ τ(A,R) and then show LM(A,R) cannot be empty.  Since A  
and R are held fixed here, abbreviate LM = LM(A,R), τ = τ(A,R) and S_xy = SPM_xy(A,R) for 
all x,y ∈ A.  To show LM ⊆ τ means showing x ∉ LM for for all x ∈ A\τ.  Assume x ∈ A\τ.  
This means there exists y ∈ τ such that #R(y,x) > #R(x,y) and there is no path from x to y.  
Thus S_xy = 0 and #R(y,x) > S_xy.  Since #R(y,x) > #R(x,y) and #R(y,x) > S_xy, this implies 
x ∉ LM.  Thus LM ⊆ τ.  It remains to show LM is not empty.  Make the following definitions:

Let S(A,R) denote the binary relation defined over A such that, for all x,y ∈ A, 
xS(A,R)y if and only if S_xy > S_yx.  

Let ψ(A,R) denote {x ∈ A such that yS(A,R)x for no y ∈ A}. (Note that 
this is the PathWinner set defined in section 4.3.)    

Since A and R are held fixed here, abbreviate S = S(A,R) and ψ = ψ(A,R).  We will show 
that LM cannot be empty by showing ψ is a non-empty subset of LM.  Clearly xy is a path 
from x to y for all x,y ∈ A such that #R(x,y) > #R(y,x).  Thus the following statement must 
hold: 

(A1)  S_xy ≥ #R(x,y) for all x,y ∈ A such that #R(x,y) > #R(y,x).  

Also, for all x,y,z ∈ A, the concatenation of a path y...z to the end of a path x...y (deleting 
one of the two consecutive y's, of course) is itself  a path from x to z having strength equal 
to the minimum of the strengths of x...y and y...z.  Thus the following statement must hold: 

(A2)  S_xz ≥ min(S_xy,S_yz) for all x,y,z ∈ A.  

To show ψ ⊆ LM we will show x ∉ ψ for all x ∈ A\LM.  Assume x ∈ A\LM.  This implies 
there exists y ∈ A such that #R(y,x) > #R(x,y) and #R(y,x) > S_xy.  By A1.1, S_yx ≥ #R(y,x).  
Combining inequalities we have S_yx > S_xy which implies ySx.  Thus x ∉ ψ, establishing 
ψ ⊆ LM.  To show ψ cannot be empty, we will first show S is irreflexive, asymmetric 
and strictly transitive.  That is, we will establish the following three propositions: 

(A3)  [not xSx] for all x ∈ A.                                     (irreflexivity)
(A4)  xSy implies [not ySx] for all x,y ∈ A.                (asymmetry)
(A5)  [xSy and ySz] implies xSz for all x,y,z ∈ A.      (strict transitivity) 

Clearly S_xx = S_xx for all x ∈ A, which implies [not S_xx > S_xx] for all x ∈ A, which implies 
A3, meaning S is irreflexive.  To show S is asymmetric, assume xSy.  This means 
S_xy > S_yx, implying [not S_yx > S_xy], implying [not ySx].  Thus S is asymmetric.  To show 
S is strictly transitive, assume xSy and ySz.  This means S_xy > S_yx and S_yz > S_zy.  
By A2, S_yx ≥ min(S_yz,S_zx) and S_zy ≥ min(S_zx,S_xy).  Combining these four inequalities, 
we produce the following inequality: 

(A6)  S_xy > S_yx ≥ min(S_yz,S_zx) ≥ min(S_zy,S_zx) ≥ min(min(S_zx,S_xy),S_zx) 

which implies S_xy > S_zx.  Combining again, we produce the following inequality: 

(A7)  S_yz > S_zy ≥ min(S_zx,S_xy) ≥ min(S_zx,S_zx) = S_zx

By A2, S_xz ≥ min(S_xy,S_yz).  Combining inequalities, we produce the following inequality: 

(A8)  S_xz ≥ min(S_xy,S_yz) > min(S_zx,S_zx) = S_zx

which implies xSz.  Thus S is strictly transitive.  We now show ψ is not empty.  Suppose 
the contrary.  This implies that for all x ∈ A there exists y ∈ A such that ySx.  This implies 
it is possible to construct an arbitrarily long sequence a₁a₂...a_k such that a₁,a₂,...,a_k ∈ A  
and a_j+1Sa_j for all j ∈ {1,...,k-1}.  Since the sequence is arbitrarily long and A is finite, 
we can construct the sequence so k > #A.  This means at least one alternative must appear 
at least twice in the sequence.  Thus we can let p and q denote two distinct indices of 
some alternative that appears at least twice in the sequence, where 1 ≤ p < q ≤ k.  
Thus a_p = a_q.  Since S is irreflexive, q ≠ p+1.  Since S is asymmetric, q ≠ p+2.  
Since S is strictly transitive and irreflexive, by induction q < p+3.  But these 
constraints on p and q are mutually contradictory, refuting the contrary assumption 
and thereby establishing ψ is not empty.  Having established ψ is a non-empty subset 
of LM, this implies LM is not empty.  Thus it has been established that LM(A,R) is a 
non-empty subset of τ(A,R).  The result can be summarized as φ ≠ ψ ⊆ LM ⊆ τ.     QED  

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