Douglas Woodall is Reader in Pure Mathematics at Nottingham University.

In this article, he argues that more attention should be paid to properties of electoral systems, and less to procedures. He lists many properties that a preferential election rule may or may not have, and discusses them with reference to STV.

Properties of electoral systems can be thought of as "performance indicators", and like any other performance indicators they need to be used with care. If one chooses a set of performance indicators in advance, it may well be possible to manufacture a high score on those indicators in an artificial way, which does not represent good performance in any real sense. Nevertheless, it seems to me that the Electoral Reform Society needs to pay more attention to properties if it is not to be sidelined in the electoral debate. In particular, since different desirable properties often turn out to be mutually incompatible, it is important to discover which sets of properties can hold simultaneously in an electoral system. Only then will it be possible to decide whether there are electoral systems that retain what is essential in STV while avoiding some of the pitfalls.

The purpose of this article is to introduce a long list of technical properties that an election rule may or may not have, to invent snappy descriptive names for them all, and to discuss them with special reference to STV. Except where otherwise indicated, statements made about STV apply equally well to the Newland-Britton and Meek versions of STV. In a later article I hope to address the question of monotonicity in more detail.

The term *outcome* will be used in the sense of "possible outcome"
(assuming there are no ties). Thus in an election to fill two seats from
four candidates *a, b, c, d*, there are six outcomes, corresponding to
the six possible ways of choosing the two candidates to be elected:
{*a*, *b*}, {*a*, *c*}, {*a*, *d*}, {*b*,
*c*}, {*b*, *d*} and {*c*, *d*}.

Election 1 Election 2 (1 seat) (2 seats) ab 0.17 a 9 ea 4 ac 0.16 b 9 eb 4 bac 0.33 c 10 fc 1 cb 0.34 d 10 fd 1 fe 6An

A similar situation arises in Election 2, again under STV. There are 54
votes cast, so the quota is 18, and there is an initial tie for exclusion
between *e* and *f*. If *e* is excluded then *f*,
*c* and *d* must also be excluded, and *a* and *b* are
elected; whereas, if *f* is excluded, then *a* and *b* must
also be excluded, and then *e* is elected and *c* and *d* tie
for second place. Thus the outcome {*a*, *b*} is chosen with
probability 1/4, and the outcomes {*c*, *e*} and {*d,*
*e*} are chosen with probability ½ each.

Because of examples like these, I define a (preferential) *election
rule* to be a procedure that, given a profile, associates a corresponding
non-negative probability with each outcome, in such a way that the
probabilities associated with all possible outcomes add up to 1. The
"normal" situation is that all the outcomes are given probability 0 except
for one, which has probability 1 (meaning that that outcome is chosen
unequivocally). If anything else happens, then we say that the result is a
*tie* between all the outcomes that have non-zero probability.

*Anonymity*. The result should depend only on the number of ballots of
each possible type in the profile (and not, for example, on the order in
which they are cast, or on extraneous information such as the heights of the
candidates).

*Neutrality*. If some permutation is applied to the names of all the
candidates on all the ballots in the profile, then the same permutation
should be applied to the result. For example, since STV is neutral, if
*a* is replaced by *c* and *c* by *a* on every ballot in
Election 2 above, then STV would choose {*b*, *c*} with
probability ½ and {*a*, *e*} and {*d*, *e*} with
probability 1/4 each. One consequence of neutrality is that a tie in a
single-seat election cannot be resolved simply by electing the first in
alphabetical order among the tied candidates.

A rule that is both anonymous and neutral is called *symmetric*.

*Homogeneity*. The result should depend only on the *proportion*
of ballots of each possible type. In particular, if every ballot is
replicated the same number of times, then the result should not change. It
is this property that enables us to describe profiles as in Election 1
above, showing the proportion, rather than the absolute number, of ballots
of each type cast.

*Discrimination*. If a particular profile *P*0 gives rise to a
tie, then it should be possible to find a profile *P* that does not
give rise to a tie and in which the proportion of ballots of each type
differs from its value in *P*0 by an arbitrarily small amount. This
rules out, for example, the following method of electing one candidate from
three: elect the candidate who beats both of the others in pairwise
comparisons, if there is such a candidate, and otherwise declare the result
a three-way tie. For in that case, not only would the profile in Election 3
below give rise to a tie, but anything at all close to it would also give a
tie, contrary to the axiom of discrimination.

abc 1/3 Election 3: bca 1/3 (1 seat) cab 1/3A

A word of warning is needed about homogeneity. In any practical election where the count is carried out by computer, there will be a limit to the number of decimal places that the computer can hold accurately. Thus there are bound to be situations in which two numbers that are not really equal are regarded as equal by the computer program, because they become equal when rounded to the appropriate number of decimal places. In this case, if every ballot were replicated the same, sufficiently large, number of times, then the difference between the two numbers of votes would become significant, and the computer might give a different result. However, this is a minor problem, introduced by the practical need to round numbers; the axiom of homogeneity should be applied to the underlying theoretical rule, with no rounding.

With this interpretation, STV is a proper election rule.

The most important single property of STV is what I call the *Droop
proportionality criterion* or *DPC*. Recall that if *v* votes
are cast in an election to fill *s* seats, then the quantity
*v*/(*s* + 1) is called the *Droop quota*.

**DPC.**If, for some whole numbers*k*and*m*satisfying 0 <*k*<=*m*, more than*k*Droop quotas of voters put the same*m*candidates (not necessarily in the same order) as the top*m*candidates in their preference listings, then at least*k*of those*m*candidates should be elected. (In the event of a tie, this should be interpreted as saying that every outcome that is chosen with non-zero probability should include at least*k*of these*m*candidates.)

In statements of properties, the word "should" indicates that the property
says that something should happen, not necessarily that I personally agree.
However, in this case I certainly do: DPC seems to me to be a *sine qua
non* for a fair election rule. I suggest that any system that satisfies
DPC deserves to be called a *quota-preferential* system and to be
regarded as a system of proportional representation (within each
constituency)-an STV-lookalike. Conversely, I assume that no member of the
Electoral Reform Society will be satisfied with anything that does not
satisfy DPC.

The property to which DPC reduces in a single-seat election should hold (as a consequence of DPC) even in a multi-seat election, and it deserves a special name.

**Majority.**If more than half the voters put the same set of candidates (not necessarily in the same order) at the top of their preference listings, then at least one of those candidates should be elected.

The following rather weak property was formulated with single-seat elections in mind, but it makes sense also for multi-seat elections and, again, it clearly holds for STV.

**Plurality.**If some candidate*a*has strictly fewer votes in total than some other candidate*b*has first-preference votes, then*a*should not have greater probability than*b*of being elected.

**No-support.**A candidate who receives no support at all (that is, who is not listed by any voters in their preference listings) should not be elected unless every candidates who receives some support is also elected.

The remaining three global properties consist of *Condorcet's
principle*, which was proposed by M. J. A. N. Caritat, Marquis de
Condorcet (1743-1794), and two modern strengthenings of it. We say that a
voter, ballot or preference listing *prefers* *a* to *b* if
he, she or it lists *a* above (before) *b*, or lists *a* but
not *b*. Let *p*(*a*, *b*) denote the number of voters
who prefer *a* to *b*. We say that *a* *beats* *b*
(in pairwise comparisons) if *p*(*a*, *b*) >
*p*(*b*,* a*); that is, if the number of voters who prefer
*a* to *b* is greater than the number who prefer *b* to
*a*. We say that *a* *ties with* *b* (in pairwise
comparisons) if *p*(*a*, *b*) = *p*(*b*, *a*).
A *Condorcet winner* is a candidate who beats every other candidate in
pairwise comparisons. A *Condorcet non-loser* is a candidate who beats
or ties with every other candidate in pairwise comparisons; note that if
there is more than one Condorcet non-loser then all the Condorcet non-losers
must tie with each other.

Note that there need not be a Condorcet winner, or even a Condorcet
non-loser. In the profile shown in Election 3 above, *a* beats
*b*, *b* beats *c* and *c* beats *a*, all by the
same margin of 2/3 to 1/3. This is the so-called *Condorcet paradox* or
*paradox of voting*: even though each voter provides a linear ordering
of the candidates, the result when the votes are totalled can be a cyclical
ordering. The *Condorcet top tier* is the smallest nonempty set of
candidates such that every candidate in that set beats every candidate (if
any) outside that set. In Election 3, the Condorcet top tier consists of all
three candidates. If there is a Condorcet winner, then the Condorcet top
tier consists just of the Condorcet winner. If there is a Condorcet
non-loser, then the Condorcet top tier contains all the Condorcet
non-losers, but it may possibly contain other candidates as well.

Condorcet's principle and the two strengthenings of it given below were formulated originally for single-seat elections in which every voter provides a complete preference listing; but I have reworded them here so that they make sense (although they are not necessarily sensible) for all preferential elections.

**Condorcet**.[1] If there is a Condorcet winner, then the Condorcet winner should be elected.**Smith-Condorcet**. [4] At least one candidate from the Condorcet top tier should be elected.**Exclusive-Condorcet**(see Fishburn[2]). If there is a Condorcet non-loser, then at least one Condorcet non-loser should be elected.

Election 4 Election 5 (1 seat) (2 seats) abc 0.30 ad 0.36 bac 0.25 bd 0.34 cab 0.15 cd 0.30 cba 0.30STV does not satisfy

As we saw in Election 4, under STV the later preferences on a ballot are not even considered until the fates of all candidates of earlier preference have been decided. Thus a voter can be certain that adding extra preferences to his or her preference listing can neither help nor harm any candidate already listed. Supporters of STV usually regard this as a very important property, although it has to be said that not everyone agrees; the property has been described (by Michael Dummett, in a letter to Robert Newland) as "quite unreasonable", and (by an anonymous referee) as "unpalatable". There are really two properties here, which we can state as follows.

**Later-no-help.**Adding a later preference to a ballot should not help any candidate already listed.**Later-no-harm.**Adding a later preference to a ballot should not harm any candidate already listed.

**Monotonicity.** A candidate *x* should not be harmed if:

- (
**mono-raise**)*x*is raised on some ballots without changing the orders of the other candidates; - (
**mono-raise-delete**)*x*is raised on some ballots and all candidates now below*x*on those ballots are deleted from them; - (
**mono-raise-random**)*x*is raised on some ballots and the positions now below*x*on those ballots are filled (or left vacant) in any way that results in a valid ballot; - (
**mono-append**)*x*is added at the end of some ballots that did not previously contain*x*; - (
**mono-sub-plump**) some ballots that do not have*x*top are replaced by ballots that have*x*top with no second choice; - (
**mono-sub-top**) some ballots that do not have*x*top are replaced by ballots that have*x*top (and are otherwise arbitrary); - (
**mono-add-plump**) further ballots are added that have*x*top with no second choice; - (
**mono-add-top**) further ballots are added that have*x*top (and are otherwise arbitrary); - (
**mono-remove-bottom**) some ballots are removed, all of which have*x*bottom, below all other candidates.

**Participation**. The addition of a further ballot should not, for any positive whole number*k*, reduce the probability that at least one candidate is elected out of the first*k*candidates listed on that ballot.

**mono-raise-random**implies both**mono-raise**and**mono-raise-delete**;**mono-raise**and**later-no-help**together imply**mono-raise-delete**;**mono-raise-delete**and**later-no-harm**together imply**mono-raise-random**;**mono-sub-top**implies**mono-sub-plump**;**mono-sub-plump**and**later-no-harm**together imply**mono-sub-top**;**mono-append**and**mono-raise-delete**together imply**mono-sub-plump**;**mono-append**and**mono-raise-random**together imply**mono-sub-top**;**mono-add-top**implies**mono-add-plump**;**mono-add-plump**and**later-no-harm**together imply**mono-add-top**;**participation**implies**mono-add-top**.

**participation**implies**mono-remove-bottom**.

ab 10 Election 6: bca 8 (1 seat) ca 7STV satisfies

Election 7 Election 8 (2 seats) (2 seats) ab 30 ac 207 ac 90 bd 198 bd 59 bdac 12 cb 51 cd 105 d 70 dc 105To see that STV does not satisfy

Although all the monotonicity properties look attractive, I do not think
that **mono-remove-bottom** is desirable in multi-seat elections.
Consider Election 8. The quota is 627/3 = 209, and so DPC requires that we
elect *b* and either *c* or *d*. It seems to me that
{*b*, *c*} is clearly the better result (although STV gives
{*b*, *d*}). But if we now remove the 12 *bdac* ballots, then
the quota drops to 205, so that we must elect *a* and either *c*
or *d*. It seems to me that now {*a*, *d*} is the better
result (although STV gives {*a*, *c*}). Thus the removal of the 12
ballots that have *c* bottom *should*, in my opinion, harm
*c*.

All the monotonicity properties seem desirable in single-seat elections.
However, I proved[7] that no rule simultaneously
satisfies **mono-sub-plump**, **later-no-help**, **later-no-harm**,
**majority** and **plurality**. Since I do not think anyone would
seriously consider a rule that did not satisfy both **majority** and
**plurality**, this shows that in order to have **mono-sub-plump** one
must sacrifice either **later-no-help** or **later-no-harm** (or
both). Whether or not this is desirable may depend on what other properties
one can gain at the same time.

**Mono-raise-random**, **mono-sub-top** and **participation** are
very strong properties, and it is possible that they are incompatible with
DPC. If one could find a reasonable-looking "STV-lookalike" rule that
satisfied all the other monotonicity properties (except for
**mono-remove-bottom** when there is more than one seat), then I
personally might well prefer it to STV itself. But we are a long way from
finding such a rule at the moment.

While on the subject of monotonicity, I should mention one other monotonicity property, if only to dismiss it immediately.

**House-monotonicity**. No candidate should be harmed by an increase in the number of seats to be filled, with no change to the profile.

Another property that is related to monotonicity is known in the literature
as *consistency*[8] or *reinforcement*[3], but I prefer to call it by its mathematical name:

**Convexity**. If the voters are divided into two districts and the ballots from each district are processed separately and the results in the two districts are the same, then processing the ballots of all voters together should give the same result.

(a) (b) (a)+(b) ab 6 3 9 Election 9: bc 4 4 8 (1 seat) cb 3 6 9STV does not satisfy

**Convexity** is one of the best-understood of all properties. Young[8] proved that a symmetric preferential election rule for
single-seat elections satisfies **convexity** if and only if it is
equivalent to a point scoring rule (in which one gives each candidate so
many points for every voter who puts them first, so many for every voter who
puts them second, and so on, and elects the candidate with the largest
number of points). Since no point scoring rule can possibly satisfy DPC, it
follows that **convexity** and DPC are mutually incompatible. This is a
pity, because **convexity** implies several of the monotonicity
properties; but, sadly, it is of no use to us.

Of course, the absence of convexity will hardly ever be noticed in practice, since elections are not counted both in separate districts and together as a whole. But it is worrying inasmuch as it may suggest that something odd is going on.

- the symmetric completion of a ballot marked
*abcd*is a single ballot marked*abcde*, with weight 1; - the symmetric completion of a ballot marked
*abc*consists of two ballots, each with weight ½, one marked*abcde*and the other marked*abced*; - the symmetric completion of a ballot marked
*ab*consists of six ballots, each with weight 1/6, completed in the six different possible ways: that is,*abcde, abced, abdce, abdec, abecd*and*abedc*; - the symmetric completion of a ballot marked
*a*consists of 24 ballots, each with weight 1/24, completed in the 24 different possible ways; and so on. **Symmetric-completion.**A truncated preference listing should be treated in the same way as its symmetric completion.

It is not difficult to see that AV satisfies **symmetric-completion**.
Although AV is usually described in terms of a quota, it can alternatively
be described as follows: repeatedly exclude the candidate with the smallest
number of votes, until there is only one candidate left. The effect of
replacing truncated preference listings by their symmetric completions is
simply that, at each stage in the count, the votes of all non-excluded
candidates are increased by the same amount. It follows that the order of
exclusions is not affected, nor therefore is the eventual winner.

a 60 Election 10: ab 60 (2 seats) b 14 c 46To see that STV does not satisfy

Election 11 Election 12 (2 seat) (3 seats) ab 40 ab 40 ba 2 ba 2 cd 12 cd 12 dc 6 dc 6 e 180David Hill has sent me an example, which I have modified slightly above, to show that quota reduction is preferable to symmetric completion in STV. In Election 11 the quota is 60/3 = 20, and so

The remaining properties are all concerned with the avoidance of "wrecking
candidates". A "wrecking candidate" is a candidate who is not elected but
who, by standing for election and so "splitting the vote", prevents someone
else from being elected. One might naively hope to avoid wrecking candidates
altogether, which would result in the Independence of Irrelevant
Alternatives, or **IIA**:

**IIA**. If a candidate*x*is not elected, then the result of the election should be as if*x*had not stood for election.

In an attempt to find a property weaker than **IIA** but expressing a
similar idea, I came up with the following.

**Weak-IIA**. If*x*is elected, and one adds a new candidate*y*ahead of*x*on some of the ballots on which*x*was first preference (and nowhere else), then either*x*or*y*should be elected.

An alternative weakening of **IIA** has been proposed by Tideman
[5]. In his terminology, a number of candidates form a *set of
clones* if every preference listing that contains one of them contains
all of them, in consecutive positions (but not necessarily always in the
same order). He says that a single-seat election rule is *independent of
clones* if it satisfies the following properties, which I have
reformulated here so that they make sense for multi-seat elections as well.

**Clone-in**. The expected number of candidates elected from any given set of clones should not increase if one member of the set is deleted from every ballot containing it.**Clone-no-help**. Replacing a candidate*x*by a set of clones should not help any other candidate*y*.**Clone-no-harm**. Replacing a candidate*x*by a set of clones should not harm any other candidate*y*.

xx'a 13 x'xa 11 Election 13: abc 10 (2 seats) bc 12 c 14It is not difficult to see that AV satisfies all the

**Clone-no-harm** is actually incompatible with DPC. To see this, note
that if only two candidates stand in a 2-seat election, where the voting is
(say) *x* 70, *y* 30, then both must be elected. But if *x*
is replaced by a pair of clones and the voting is now *xx*' 35,
*x'x* 35, *y* 30, then DPC requires that *x* and *x*'
should both be elected. This suggests that **clone-no-harm** is not a
desirable property for multi-seat elections-and Tideman never suggested that
it was. But **clone-in** and **clone-no-help** both look sensible to
me, even for multi-seat elections.

- Marquis de Condorcet,
*Essai sur l'Application de l'Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix*, Paris, 1785. - P. C. Fishburn, Condorcet social choice functions,
*SIAM Journal on Applied Mathematics*33 (1977), 469-489. - H. Moulin, Condorcet's principle implies the no show
paradox,
*Journal of Economic Theory*45 (1988), 53-64. - J. H. Smith, Aggregation of preferences with variable
electorate,
*Econometrica*41 (1973), 1027-1041. - T. N. Tideman, Independence of clones as a criterion for
voting rules,
*Social Choice and Welfare*4 (1987), 185-206. - B. A. Wichmann, Two STV
Elections,
*Voting matters*(The Electoral Reform Society) 2 (1994), 7-9. - D. R. Woodall, An impossibility theorem for electoral
systems,
*Discrete Mathematics*66 (1987), 209-211. - H. P. Young, Social choice scoring functions,
*SIAM Journal on Applied Mathematics*28 (1975), 824-838.