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 6
An election rule is usually thought of as a method that, given a
profile, chooses a corresponding outcome-or, in the event of a tie, chooses
two or more outcomes, one of which must then be selected in some other way
(such as by tossing a coin). However, this description is not quite adequate
to deal with the complexities of ties. Consider Election 1 above, with the
votes counted by STV (or, rather, by the Alternative Vote (AV), which is the
rule to which STV reduces in a single-seat election). No candidate reaches
the quota of 0.5, and there is an initial tie for exclusion between a
and b. If b is excluded then a is immediately elected,
whereas if a is excluded then b and c tie for election.
Thus a is elected with probability ½, and b and c
are elected with probability ¼ each.
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 P0 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 P0 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 1/3
A proper election rule is one that satisfies the above four axioms;
that is, one that is anonymous, neutral, homogeneous and discriminating. The
term "axiom" is used rather freely in the literature as a synonym for
"property", but I shall restrict its use to these four, which I regard as
genuinely axiomatic, in the sense that I am not interested in any rule that
does not satisfy them.
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.
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.
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.
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.
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 Condorcet, and so it certainly does not satisfy either of the above two extensions of it. This can be seen in Election 4 above. Under STV (AV), b is excluded and a is elected. However, b is the Condorcet winner, beating both a and c by the same margin of 0.55 to 0.45. This example highlights a fundamental difference in philosophy between STV and Condorcet-based rules. Loosely speaking, STV tries to keep votes near the tops of the ballots. Thus the preferences of the cba voters for b over a will not even be considered under STV until c is excluded, which means that in this example they are not considered at all, since b is excluded before c. In contrast, Condorcet's principle requires that, right from the outset, the preferences of the cba voters for b over a should be given equal weight with the similar preferences of the bac voters. However, despite this difference in philosophy, Condorcet and majority are not actually incompatible in single-seat elections: if one wishes, one can use AV (or any other system of one's choice) to select a candidate from the Condorcet top tier. Any such rule clearly satisfies Smith-Condorcet, and hence satisfies both majority and Condorcet, although it is a moot point whether it is really any better than AV on its own. In multi-seat elections, Condorcet is undesirable, in my opinion, because it is incompatible with DPC, as shown by Election 5 above. Here the quota is 0.333...., and so DPC requires that a and b should be elected, whereas d is the Condorcet winner.
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.
Monotonicity. A candidate x should not be harmed if:
ab 10
Election 6: bca 8
(1 seat) ca 7
STV satisfies mono-append but none of the other properties, although
in single-seat elections AV satisfies mono-add-plump and
mono-add-top. To see that AV does not satisfy mono-raise,
mono-raise-delete, mono-raise-random, mono-sub-plump,
mono-sub-top or mono-remove-bottom, consider its effect in
Election 6 above. As it stands, c is excluded and a is
elected. But if two of the bca ballots are removed, or replaced by
a or by abc or by anything else starting with a, then
b is excluded and c is elected instead of a.
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 105
To see that STV does not satisfy mono-add-plump or
mono-add-top, consider Election 7. The quota is 300/3 = 100, so that
a is elected with a surplus of 20. This is divided 5 to b, 15
to c, and so b has 64 votes to c's 66, b is
excluded, and d is elected. Suppose now that we add a further 24
ballots with d top. The quota is now 324/3 = 108, so that a's
surplus is now only 12. This is divided 3 to b, 9 to c, and so
b has 62 votes to c's 60, c is excluded, and b
is elected instead of d.
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.
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:
(a) (b) (a)+(b)
ab 6 3 9
Election 9: bc 4 4 8
(1 seat) cb 3 6 9
STV does not satisfy convexity. Again, I cannot do better than to
quote an example of David Hill's (Election 9). In district (a), c is
excluded and b is elected. In district (b), a is excluded and
b is elected. But when the ballots from the two districts are
processed together, b is excluded and c is elected.
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.
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 46
To see that STV does not satisfy symmetric-completion in general,
consider Election 10. The quota is 180/3 = 60, so that a is elected
with a surplus of 60. Under the Newland-Britton rules, the whole of
a's surplus goes to b, who is elected. Under Meek's method,
the transfer of a's surplus ends with the quota reduced to (180 -
36)/3 = 48, with 36 non-transferable votes going to 'excess', and 36 votes
transferred to b. Either way, a and b are elected.
However, if each ballot is replaced by its symmetric completion, then, of
a's surplus of 60 votes, 45 go to b and 15 to c, and
c is elected instead of b.
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 180
David 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 a and b are
elected. In Election 12 the quota is 240/4 = 60, so that e is elected
with a surplus of 120. Under symmetric completion, this would be used to
increase the votes of the remaining candidates by 30 each, so that a
would be elected first, after which d would be excluded and c
would be elected. However, if the quota is reduced to 20 after the election
of e then a and b are elected as in Election 11. To
paraphrase David's comments slightly, "Election 12 has one extra candidate,
one extra seat, and a large number of extra voters whose sole wish
(apparently) is to put that extra candidate into that extra seat. It is
nonsense that the original 60 voters should get a and c
elected in Election 12 instead of the a and b they would have
got from Election 11."
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:
In an attempt to find a property weaker than IIA but expressing a similar idea, I came up with the following.
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.
xx'a 13
x'xa 11
Election 13: abc 10
(2 seats) bc 12
c 14
It is not difficult to see that AV satisfies all the clone
properties. I am fairly sure that STV also satisfies clone-in in
multi-seat elections, although I do not have a formal proof of this. To see
that STV does not satisfy the other two clone properties, consider
Election 13. The quota is 60/3 = 20. Nobody having reached the quota,
a is excluded and b is elected; then x' is excluded and
x is elected. However, if the clones x and x' are
replaced by a single candidate x, then x has 24 votes
initially and so is elected, and the surplus of 4 votes goes to a;
therefore b is excluded, and c is elected instead of b.
So replacing x by a pair of clones helps b and harms
c.
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.