**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,yandz:

46% |
10% |
10% |
34% |

x |
y |
y |
z |

y |
x |
z |
y |

z |
z |
x |
x |

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 choosesy.

Copeland (assuming plurality tie-breaker) choosesy.

Instant Runoff (Hare, Single Transferable Vote, Alternative Vote) choosesx.

Tideman's "ranked pairs" rule choosesy.(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 ofx's supporters

23%23%10%10%34%xxyyzyzxzyzyzxx

(Many supporters of

xmisrepresent their "yoverz" preference by voting "zovery.")Borda chooses

x.

Simpson choosesx.

Copeland (assuming plurality tie-breaker) choosesx.

Instant Runoff (Hare, Single Transferable Vote, Alternative Vote) choosesx.

Tideman's "ranked pairs" rule choosesx.Clearly there is an incentive for

x's supporters to misrepresent their "yoverz"

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**:**

For all subsetsMinimal Defense:Xof the alternatives, if there exists an

alternativeythat more than half the voters prefer over every alternative inX,

then there must exist a set of voting strategies for that majority that ensures

no alternative inXwill be elected and does not require any of them to

misrepresent any preferences except possibly lowering alternatives inX.

A nearly equivalent wording ofminimal defense:

Each voter must be permitted to vote any (non-strict) ordering of the

alternatives, and for all alternativesx,xmust not be elected if there is

an alternativeysuch that the number of voters who rank bothyoverx

andxno higher than tied for bottom exceeds the number of voters

who rankxhigher 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**:**

46% |
10% |
10% |
34% |

? | y |
y |
z |

? | x,z |
z |
y |

? | x |
x |

(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

tied with

2) Downranking

supporters of the less preferred alternative, an opportunity which will not become available

if

to downrank

opportunity for

percentages on election day, and thus might not be an equilibrium.

3) It is less misrepresentative to downrank

than to downrank

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

many widely accepted criteria such as

may dodge some questions about whether certain other criteria are more important

than

is desirable. Elsewhere we argue that criteria inconsistent with

such as

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 ofxby some of the "yoverx" majority

23%23%10%10%34%xxyyzyzx,zzyzyxx

(Many of

x's supporters are misrepresenting their "yoverz" preference by voting

"zovery." Some ofy's supporters misrepresent their "xoverz" preference by

voting indifference betweenxandz. More than half the voters (54%) are voting

yoverxandxno better than tied for bottom.)Borda chooses

x.

Simpson choosesx.

Copeland (assuming plurality tie-breaker) choosesx.

Instant Runoff (Hare, STV, Alternative Vote) choosesx.

Tideman's "ranked pairs" rule choosesx.Examples 1a and 1b show that the listed voting rules fail

minimal defensesince

they choosexwith 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 electsyassuming satisfaction ofminimal defense

46%20%34%xyzyx,zyzx

The 20% whose favorite is

yhave voted the others tied for bottom. Thus more

than half (54%) have votedyoverxandxno better than tied for bottom, and

more than half (66%) have votedyoverzandzno better than tied for bottom.

The 46% who favorxcannot vote in a way which electsxgiven the strategies

of the others, so they have no strategy better than their sincere orders of preference.

The 34% who favorzcannot vote in a way which electszgiven 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

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

group strategy equilibrium unattainable. They fail

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

cycle is ranked over

(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

denote the finite set of alternatives.ALet SincereTop denote the minimal non-empty

B⊆such that, for allAb∈B

and alla∈\AB, the number of voters who sincerely preferbtoaexceeds

the number who sincerely preferatob.Say there is no misrepresentation except possibly truncation if and only if,

for alla,b∈, every voter who votesAaoverbsincerely prefersatob.If there is no misrepresentation except possibly truncation, then for all

x∈,A

xmust not be chosen ifx∉ SincereTop and at least one alternative in SincereTop

is ranked overxby more than half of the voters.

An equivalent wording of *truncation resistance* is the
following:

Suppose * R*
and

in

express strict preference regarding those alternatives. Let

cycle given

and

*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 byx's supporters of their "yoverz" preference

46%10%10%34%xyyzxzyzxx

(Assume the "

x" votes of the 46% are treated the same as "xovery~z.")Since

yis the sincere Condorcet winner and there is no misrepresentation

except truncation and more than half of the voters voteyoverx, satisfaction

oftruncation resistancerequiresxmust not be chosen.Borda chooses

x.

Simpson choosesx.

Copeland (assuming plurality tie-breaker) choosesx.

Instant Runoff (Hare, STV, Alternative Vote) choosesx.

Tideman's "ranked pairs" rule choosesx.

**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, sayy, over alternativexand this majority is not the

smallest majority in any cycle involvingyandx, thenxmust 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

denote a finite set of two or more alternatives. LetAdenote a finite set ofN

voters {1,2,...,n}. Each voteri∈submits a ranking (non-strict ordering)NR_{i}of

the alternatives in. LetAdenote the collection of voters' orderings.RFor all

x,y∈, letA(Rx,y) denote the orderings inwhich rankRxovery. That is,

(Rx,y) = {r∈such thatRrranksxovery}. Thus #(Rx,y) denotes the number

of voters who rankxovery. (Due to the possibility of indifference, the larger

of #(Rx,y) and #(Ry,x) is not necessarily more than half the voters.)Let Pairs(

) denote the set of all possible ordered pairs of alternativesA

{(x,y)such thatx∈andAy∈}. WhereAis clear from the context,A

abbreviate Pairs.Let Majorities(

) denote {A,R(x,y)∈ Pairs() such that #A(Rx,y) > #(Ry,x)}.

WhereandAare clear from the context, abbreviate Majorities.RFor all

a_{1},a_{2},...,a∈_{k}, the sequenceAa_{1}a_{2}...ais a majority cycle of_{k}(A,R)

if and only ifa=_{k}a_{1}and(a_{j},a_{j}_{+1})∈ Majorities() for allA,Rj∈ {1,2,...,k-1}.

Let MajorityCycles(A,) denote the set of all sequences of alternatives inRA

which are majority cycles of(. WhereA,R)andAare clear from the context,R

abbreviate MajorityCycles.For all

C∈ MajorityCycles() and allA,R(x,y)∈ Pairs(),A(x,y)is aC-majority

if and only ifximmediately precedesyin the sequenceC.Let SmallestMajorities(

A,) denote {R(x,y)∈ Majorities() such thatA,R

there existsC∈ MajorityCycles() such thatA,R(x,y)is aC-majority and

#(Rx,y) ≤ #(Rz,w) for allC-majorities(z,w)}. WhereandAare clearR

from the context, abbreviate SmallestMajorities.

For allLarger Majority:x∈,Axmust not be chosen if there existsy∈A

such that(y,x)∈ Majorities()\SmallestMajorities(A,R). In otherA,R

words, the chosen alternative must be in the set {a∈such that, for allAb∈,A

(b,a)∉ Majorities() orA,R(b,a)∈ SmallestMajorities()}. Let LM(A,RA,)R

denote this set. WhereandAare clear from the context, abbreviate LM.R

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 alla_{1},a_{2},...,a∈_{k}, the sequenceAa_{1}a_{2}...ais a path from_{k}a_{1}toa_{k}

if and only if(a_{j},a_{j}_{+1})∈ Majorities() for allA,Rj∈ {1,2,...,k-1}.

Path Strength:The strength of a patha_{1}a_{2}...ais the minimum of #_{k}(Ra_{j},a_{j}_{+1})

overj∈ {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:LetSPM(A,) denote the matrix such that,R

for allx,y∈,ASPM(_{xy}A,) is the strength of the strongest path fromRx

toyif there is at least one path fromxtoy, andSPM(_{xy}A,) = 0 if thereR

is no path fromxtoy. WhereandAare clear from the context,R

abbreviateSPMandSPM._{xy}

For allLarger Majority(2):x∈,Axmust not be chosen if there exists

y∈such that [#A(Ry,x) > #(Rx,y) and #(Ry,x) >SPM]. In other words,_{xy}

the chosen alternative must be in the set {a∈such that, for allAb∈,A

#(Rb,a) ≤ max(#(Ra,b),SPM)}. Let LM(_{ab}A,) denote this set.R

WhereandAare clear from the context, abbreviate LM.R

The equivalence of the two formal wordings of *larger majority* can be easily shown.

Note that if #** R**(

is a majority cycle in which

it follows that

#

equivalent and the two wordings of

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

Top Cycle:τ(A,) is the minimal non-emptyRB⊆such that #A(Rb,a) > #(Ra,b)

for allb∈Band alla∈\AB.)(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

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**(

of the voters, all paths from

ranked over

Therefore #

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**(

the voters, all paths to

over

Thus #

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*\

Assume

of the voters, and that no voter ranks any alternative over one that isn't less preferred.

Clearly #

of the voters prefer any alternative in

and all

Since #

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*,

consistency

**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

Refer to the definitions above.

For all

x∈such that #A(Ry,x) > #(Rx,y) for at least oney∈,A

let LargestDefeat(x,A,) denote the maximum of #R(Ry,x) over {y∈A

such that #(Ry,x) > #(Rx,y)}. For allx∈such that #A(Ry,x) ≤ #(Rx,y)

for ally∈, let LargestDefeat(Ax,A,) = 0.R

Minimax(Defeat)voting rule:Allow the voters to order the alternatives and

to express indifference in their orderings. Choose the alternative(s) {a∈suchA

that LargestDefeat(a,A,) ≤ LargestDefeat(Rb,A,) for allRb∈}.A

Note that each alternative's "largest defeat" score
depends on the size of an opponent's

supporting coalition (e.g., #** R**(

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)

20% |
20% |
20% |
14% |
13% |
13% |

x |
y |
z |
w |
w |
w |

y |
z |
x |
x |
y |
z |

z |
x |
y |
y |
z |
x |

w |
w |
w |
z |
x |
y |

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

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

but suffices to show MAM satisfies the criteria that concern us here:

**Pair Precedence**:
For all *x,y,z,w*
∈ ** A**, if #

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

that

Let AffirmedMajorities(* A,R*)
denote {

not cycle with {

(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 {

Since

This means there exist

and

majority cycle in which

and

precedes

But this implies

which implies MAM cannot choose

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∈such thatASPM≥_{xy}SPMfor all_{yx}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

non-strict orderings, it follows immediately that PathWinner satisfies

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,) denote {Ra∈such that [Aais not ranked (tied for) bottom by more

than half the voters] or [#(Rb,a) is at most half the voters for allb∈]}.A

Borda with Minimal Defense Filter:Allow the voters to vote any orderings of

the alternatives. Choose the alternative in MD(A,) having the best Borda score.R

Borda with Larger Majority Filter:Allow the voters to vote any orderings of

the alternatives. Choose the alternative in LM(A,) having the best Borda score.R

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:

1 |
1 |
2 |
1 |
1 |
2 |
2 |
1 |
2 |
2 |

a |
a |
a |
b |
b |
c |
c |
c |
d |
d |

e |
c |
e |
c |
e |
a |
b |
d |
a |
b |

b |
b |
d |
d |
d |
e |
d |
a |
b |
c |

d |
e |
b |
a |
c |
b |
a |
b |
e |
a |

c |
d |
c |
e |
a |
d |
e |
e |
c |
e |

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(

this, we first show LM(

and

all

This means there exists

Thus

Let * S*(

Let ψ(**A**,** R**)
denote {

this is the PathWinner set defined in section 4.3.)

Since ** A** and

that LM cannot be empty by showing ψ is a non-empty subset of LM. Clearly

from

hold:

(A1) *S _{xy}* ≥ #

Also, for all *x,y,z* ∈ * A*, the
concatenation of a path

one of the two consecutive

to the minimum of the strengths of

(A2) *S _{xz}* ≥
min(

To show ψ ⊆ LM
we will show *x* ∉ ψ for all
*x* ∈
** A**\LM. Assume

there exists

Combining inequalities we have

ψ ⊆ LM. To show ψ cannot be empty, we will first show

and

(A3) [not *x Sx*]
for all

(A4)

(A5) [

Clearly *S _{xx}* =

A3, meaning

By A2,

we produce the following inequality:

(A6) *S _{xy}* >

which implies *S _{xy}* >

(A7) *S _{yz}* >

By A2, *S _{xz}* ≥ min(

(A8) *S _{xz}* ≥
min(

which implies *x Sz*. Thus

the contrary. This implies that for all

it is possible to construct an arbitrarily long sequence

and

we can construct the sequence so

at least twice in the sequence. Thus we can let

some alternative that appears at least twice in the sequence, where 1 ≤

Thus

Since

constraints on

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(

non-empty subset of τ(

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Black,
Duncan (1958), The Theory of Committees and Elections. Cambridge University

Press, Cambridge.

Brams
SJ, Fishburn PC (1978), "Approval Voting." American Political Science

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Condorcet
(1785), "Essai sur l'application de l'analyse à la probabilité des décisions

rendues à la pluralité des voix." Paris.

Moulin,
Hervé (1988), "Axioms of Cooperative Decision Making" chapter 9
pp.225-226

of the 1991 paperback edition. Cambridge University Press, Cambridge.

Ossipoff,
Mike (Nov 7, 1995), "The best single-winner method." Originally
posted to

internet maillist "elections-reform@igc.org".
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"Mike Ossipoff" <nkklrp@hotmail.com>.

Shepsle
KA, Bonchek MS (1997), Analyzing Politics - Rationality, Behavior and

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TN (1987), "Independence of Clones as a Criterion for Voting Rules."

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TM, Tideman TN (1989), "Complete Independence of Clones in the Ranked

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