VOTING THEORY AND PROPORTIONAL REPRESENTATION
AS A VOTING SYSTEM
DR DAVID NEWMAN
Associate Professor
Department of Politics & Sociology, Lingnan College
I. Introduction
This paper has a set of very modest goals. First, it examines the question of why we use voting to select office
holders. Second, I demonstrate that the choice of rules for aggregating preferences over candidates can have a
considerable influence over the outcome even when the preferences are held fixed. The third section discusses
the mechanics of proportional representation and the fifth discusses the various arguments advanced by
supporters of proportional representation for its adoption. Section six while rejecting many of the arguments in
favor of proportional representation concludes that as far as electoral systems are concerned generally, one is as
good as another. Section seven modifies the general conclusions of section six taking into account the nature of
Hong Kong's mechanisms for selecting other political officials.
II. What purpose does voting serve?
Voting is viewed as an integral part of the democratic process and ideal. Individuals partially achieve all three
elements of democracy, participation, liberty, and equality, through the act of voting. Citizen participation in the
process of periodically selecting leaders is assumed to serve several purposes including, but not limited to, acting
as a constraint on the exercise of power by the ruler and embedding the modern state with additional latent power
derived from the legitimacy of ruling with the consent of the people. Liberty to pursue one's own goals is also
closely tied to the act of voting. Political liberties allow individuals to organize, publicize, and engage in
individual and collective action to pursue a set of interests often either realized or frustrated at the ballot box.
Equality, as a democratic ideal, is also partially achieved at the ballot box as each vote carries the same weight
and serious differences in voting weights are often characterized as undemocratic popularly and by the courts.
“Democrats of all persuasions would probably agree that participation built on the act of voting is the focus of
democracy" (Riker 1988). For liberals in the Madisonian tradition, periodic elections serve the purpose of
allowing the electorate to "throw the rascals out" if they are doing a bad job. This is certainly what happened
to Lyndon Johnson, Jimmy Carter and George Bush in the United States, Indira Gandhi in India, and John Major
in the United Kingdom. The liberal view does not assume that the people are correct, only that they can seek a
better alternative.
The popularist view of voting takes the view that the general interest of the society can be derived from
consulting the people and that the "public interest" can be ascertained through the process of voting. The rules
made and decisions revealed by the voters must be respected as they encapsulate the will of the people and their
liberty. H. Ross Perot's suggestion during the 1992 American presidential elections of periodic "electronic town
meetings" in which the people debated an issue and voted their preferences over a computer/television network
(now the internet) is perhaps one of the more extreme examples of the popularist view.
The popularist view of voting holds the act of voting and the aggregation of votes in much greater reverence than
the liberal view. Is voting deserving of this venerated position or is more at home in the liberal school of
thought? Should we hold the very act of voting, its selection of an alternative as a social decision, or the
alternative itself as somehow “sacred”? Social choice theory provides our point of departure by asking "is the
outcome suggested by counting votes based on individual preferences meaningful".
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Let us assume that there are three voters, A, B, and C, and three alternatives, X, Y, and Z. The voters'
preferences over the alternatives are presented in table 1. We assume that each voter has an ordered set of
transitive preferences and that each voter casts his or her votes for that alternative which is highest on the voter's
preference ordering.
Table 1
Voters' Preferences over the Set of Alternatives
Voter
Preference Ordering
Voter A
XpYpZ
Voter B
YpZpX
Voter C
(p) indicates "preference".
ZpXpY
If we held an election between alternatives X and Y, voter A prefers alternative X to alternative Y, voter B
prefers alternative Y to X and voter C prefers X to Y. Alternative X would be the alternative chosen by the
society and the supporters of X, satisfied with the outcome, would, in the populous spirit, assert that alternative
X is worthy of the support of all citizens since it expresses the will of the people. Is alternative X worthy?
If we held an election between alternatives Y and Z, alternative Y would be selected. If we held an election
between alternatives Z and X, alternative Z would win. Since any alternative can win, no particular alternative
has a higher moral claim to being representative of the "public will" than any other alternative. Even though
each voter acts consistently, the social outcome is inconsistent. There is no social ordering that is transitive.
Hence, Voting is not guaranteed to reveal the interests of the electorate. The conclusion of this section, which
Arrow (1963) proved generally, is that "there is no method of amalgamating individual judgments that can
simultaneously satisfy some reasonable conditions of fairness on the method and a condition of logicality on the
result" (Riker 1988) and "[o]ne cannot simply say that an elected body is representative in the sense that it is the
result of a true and fair amalgamation of voters' preferences and values" (Riker 1984). Elections therefore serve
the purpose of filling constitutionally mandated offices and cannot be assured of doing much more.
III. The Choice of Voting Rules Makes a Difference to the Outcome
Voting theorists have shown repeatedly that the choice of voting system can alter the selection made by the
electorate even though the preferences of the electorate is held fixed (Riker 1988). An example from the 1992
American presidential election demonstrates this finding most vividly.
In the 1992 election there were three major candidates for the office of President: George Bush, the Republican
party incumbent, Bill Clinton, the Democratic party nominee, and H. Ross Perot, an independent candidate.
Table 2 summarizes the voters' complete preference ordering over the set of candidates. As we can see from the
table, 43 percent of the voters preferred Clinton, 41 percent Bush and 16 percent Perot. In the example, all the
Clinton supporters preferred Perot second and Bush third. The Perot supporters preferred Perot to any other
candidate but some preferred Bush to Clinton (ten percent of the electorate) and others preferred Clinton to Bush
(six percent of the electorate).
The United States uses a first past the post plurality voting rule where the candidate with the most votes wins.
With the voter preferences as depicted in table 2, it should be clear that Clinton was elected President with 43
percent of the vote. The first past the post electoral scheme requires voters to reveal only their first preference.
Table 2
Voters Preferences
2
Percentage of
Voters
Clinton
Bush
Perot
Pro-Clinton Voters
43 %
1st
3rd
2nd
Pro-Bush Voters
Pro-Perot Democratic Voters
41 %
6%
3rd
2nd
1st
3rd
2nd
1st
Pro-Perot Republican Voters
10 %
3rd
2nd
1st
An equally "democratic" and "fair" voting scheme is the two stage-run off election system used in a variety of
places, including France. In that system, an initial election is held involving all candidates for office. If no
candidate gets a majority of the vote, a second election is held between the two candidates who received the most
votes. The second election ensures that the candidate chosen will have received a majority of the vote. Again,
using the data from table 2, all three candidates would contest an initial election. No candidate would have
received a majority and a second election would have to be held involving just Clinton and Bush. The Perot
supporters can no longer vote for their first choice but rather vote for their second choice. Under this system,
Clinton receives 43 percent of the vote from his supporters and six percent of the vote from the Perot supporters
who prefer Clinton more than Bush. Bush receives 41 percent of the vote from his supporters and 10 percent of
the vote from the Perot supporters who prefer Bush to Clinton. Bush is elected President. This system allows
voters to sometimes express their second preference but only when their first preference has been defeated.
With two different systems, both democratic and fair, producing different outcomes, it is natural to ask whether
there is an accepted "democratic" and "fair" system which would have selected Perot as President of the United
States. The answer is "yes". Under the Borda system of voting, named after its inventor the French
mathematician Jean Charles Borda, electors are allowed to have their votes reflect more than just their first or
second preference. If there are n candidates, a voter casts n-1 votes for her first choice, n-2 votes for her second
choice, n-3 votes for her third choice and so on. A voter's least preferred candidate gets no votes.
Returning to the 1992 US election, the 43 Clinton supporters each cast two votes for Clinton and one vote for
Perot and no votes for Bush. The Bush supporters each cast two votes for Bush, one vote for Perot of no votes for
Clinton. The Perot supporters each cast two votes for Perot and the Clinton and Bush supporters each cast one
vote for Perot (See Table 3). The net effect is that under a Borda count system of aggregating votes, Mr. Perot
would have been fairly and democratically elected President of the United States in 1992. All three systems are
equally fair and democratic and neither has a morally superior claim to somehow being better at aggregating
individual preferences to reveal what the electors preferred.
Candidate
Clinton
Bush
Perot
Table 3
The Borda Rule for Counting Votes
Votes Received
(43*2) + (41*0) + (6*1) + (10*0) = 92
(43*0) + (41*2) + (10*1) + (6*0) = 92
(43*1) + (41*1) + (16*2) = 116
IV. What is Proportional Representation (PR)
The basic principle underlying proportional representation is that all voters deserve to be represented in the
legislature in proportion to their strength in the electorate. Riker (1988) defines PR as "an electoral system in
which, insofar as is arithmetically possible, seats are assigned to a party in proportion to the votes cast for it.
Ideally, for each constituency, the ratio of votes cast for a particular party to the total votes equals the ratio of the
number of seats for that party to the total number of seats in the body." While there are various forms of
proportional representation including party list systems (Norway), mixed member systems (New Zealand), and
single transferable vote systems (Cambridge, Massachusetts), the general goals of all are the same and all
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incorporate some form of multi-member district.
The asserted benefits of a PR system are perhaps best illustrated by juxtaposing it to a plurality and majority
single seat constituency system. In what is certainly an extreme case under a single seat constituency majority
rule system with two parties, one party could gain a majority in a legislature with just over twenty-five percent of
the popular support even though the other party has just under seventy-five percent of the popular support. As
an example, consider a ninety-nine seat legislature in which party A received just more than fifty percent of the
vote in fifty districts and party B received all of the remaining votes. Party A would have a majority in the
legislature with just over twenty-five percent of the popular support. In the contested districts where Party A
won, all of the votes cast for Party B were "wasted". One could even view the seats which Party B won as
wasted since under a system of strict party discipline, Party B could not pass any legislation which was not
acceptable to Party A. Even more extreme examples could be constructed using more parties and a plurality
rule.
Under a proportional representation system involving a single national constituency, the above example would
give Party B just about three quarters of the seats - roughly equal to the proportion of vote it received in the
election.
V. What are the Arguments in Favor of Proportional Representation
The intensity of feeling with respect to proportional representation can easily be found in any search of the
literature or the popular press. The most common arguments advanced in favor of proportional representation
include (Reding 1995) :
a)
b)
c)
d)
majority;
e)
f)
g)
VI.
better representation of minorities and women;
higher voter turn-out and lower voter apathy;
no or fewer "wasted votes" -- each voter's vote counts toward electing someone;
eliminates the swing factor in which a small proportion of votes often determine the
promotes better policy since all views are considered;
reduces pork-barrel (special interest) politics and pet projects; and
it is fairer that all taxpayers/citizens be represented.
What are the Arguments Against Proportional Representation
The most general conclusion one can make regarding the arguments made in favor of proportional representation
is that none of them are necessarily true. Evidence can often be mustered in support of the various propositions
but counter examples and evidence are generally easy to find.
Proponents of PR assert that a two party system encouraged by a single seat plurality system does not do a good
job representing minorities. First, this conclusion seems to be at odds with the other assertion of PR supporters
that "it reduces pork-barrel politics and pet projects". What better indicator of the system's responsiveness to
the needs of minority interests than its willingness to "buy them off" or provide the goodies that a particular
minority interest is demanding. Moreover, it is often the intense minorities within parties, the activists, who are
more influential on party policy and candidates than the "majority" of inactive party registrants. Clearly it was an
intense minority of the Republican Party in the United States which first nominated Ronald Reagan and Barry
Goldwater. The American Democratic Party similarly nominated George McGovern, an extreme liberal, in
1972. Finally, it should be kept in mind that all majorities are comprised of a set of minorities. A party which
lost in the last election by two percent, for instance, will seek to find a small minority set of interests previously
outside of the party and seek to represent them in order to ensure that the party will achieve a majority in the next
election. This small set of interests are highly prized by both political parties and derive disproportionate
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benefits.
In terms of representing woman, it is not clear to me that sex is a basis by which one should cast one’s vote or
assess whether a system or an individual is representative of the interests of an individual or the electorate.
Advocates of proportional representation assert that under a PR system a greater number of eligible voters will
turn out to vote on election day and that voters will be less apathetic about voting since their votes are not wasted.
While voter turnout may increase as a result of the adoption of proportional representation it is often at the
expense of greater ideological extremism on issues. Voters view themselves as having more at stake or at risk if
some party other than their party is elected so they have an incentive to turn-out and vote. It is not clear that the
resulting uncertainty over future policy or the increased likelihood of policy swings is a desirable alternative to
lower turn-out.
Proponents of PR also suggest that voters will be less apathetic. Surveys completed in New Zealand six weeks
to three months after their 1996 election, the first under a PR system, did not suggest that the electorate was more
satisfied than under the previous first past the post system. The large delay and considerable negotiations and
trade-offs between the parties soured the electorate on the multi-member proportionate system. I suspect that
the fall of Mr. Jim Bulger and his replacement by Ms. Jenny Shipley late last year did not further endear the
electorate to their new electoral system. With respect to turnout in New Zealand, public support for the new
MMP system has declined given the protracted inter-party negotiations necessary to form a government and it
may well be the case that voter turn-out and apathy increases in New Zealand over the short run.
With respect to fairness, most advocates of proportional representation assume that political power is somehow
correlated with representation (Johnston 1984). Again, while this can be true, it does not have to be. If a party,
under a strict discipline legislative system, gets more than fifty percent of the vote and receives more than fifty
percent of the seats in the legislature, the other parties may have representation but no seats. Where under a
first past the post system they may have been "locked out" of the legislature by the electoral system, under PR
they are as impotent but in the legislature. For practical purposes, depending on the rules of the legislature
regarding committee assignments and chairmanships, there may be little difference (See, Powell 1989).
As we have just seen, under some circumstances PR, such as one party having a majority or two parties in
coalition being a majority, one cannot be assured that PR will result in better policies as a result of broad
participation. In such circumstances, the minority wishes can be ignored just as they would be under a plurality
single seat system. Moreover, less pork barrel spending and fewer pet projects cannot be assured. As we have
seen in the past several weeks in Israel with its legislative debate over the budget, proportional representation can
require broad based "payoffs" to minor interests in a coalition government in order to keep them in the coalition.
Israel's budgetary expenditures would very likely be much smaller if Prime Minister Netanyahu's Lekud Party
had a majority in the Israeli parliament. Coalition governments under PR systems can be inefficient.
It is important to remember that a public official, once elected, represents all of the people in his or her district so
the interests of all voter’s are represented. It is rare for any individual seeking public office to intentionally
chase potential supporters away as noted above. Unless one adopts a directly democratic system of continuous
referendums al la Perot, representation always assumes some degree of difference between the electors and the
elected. As a proponent of individual rights in the United States, I am rarely able to find a candidate that
mirrors my pro-gun and pro-abortion stance. This should not mean that my elected representative does not
represent me and my interests in a broad sense.
The final assertion noted in the previous section is that PR is somehow "fairer". Given there is no objective
basis for asserting that one system is "fairer" than another and all are democratic, no voting scheme has a better
moral claim to being accepted. In the example generated in section III, neither Clinton, Bush nor Perot has a
better claim to being elected than any of the other two.
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VII. Conclusions and Recommendations
While there may not be much to commend PR to an electorate, there may not be much to commend any particular
system. Returning to our earlier discussion of the role of voting I again note that any outcome, which is
selected by the process of voting, cannot assert that it is the will of the voters. There is no voting system which
can, with certainty, avoid “Arrow’s Paradox”. Therefore, in a general sense, any system which is democratic
and fair and allows the voters to replace one set of rulers with another should be acceptable. 1
Voting thus should be viewed as a signalling device for office bearers on the quality of the service they are
providing to the electorate -- it is a referendum on performance.
If we are interested in protecting individual citizens from the tyranny of their government, we need to look
elsewhere. Generally, there are three constitutional constraints available to preserve individual rights:
separation of powers, political decentralization and parliamentary multipartism (Riker 1984). Unfortunately for
Hong Kong, true separation of powers and political decentralization are not real prospects. In addition,
parliamentary multipartism, where any one party in coalition can bring down the executive, is not possible given
the current mechanism for (s)electing the Chief Executive.
Since I have concluded that voting is a signalling device for the electorate to comment on public office bearers,
the issue is under the current "constitutional" arrangements in Hong Kong will PR be an effective. With a
narrowly (s)elected Chief Executive, an appointed Executive Council, two-thirds of the legislature narrowly
elected, most of our officials are beyond the reach of public accountability.2 Proportional representation will
also tend to reduce the accountability of elected legislators from geographic constituencies since no one political
party is in power or holding office and each elected legislator can claim that they and their party are not in charge.
The future system makes “responsible government” difficult to achieve. An example of this is the current bird
flu crisis in Hong Kong. To whom does the electorate vent its frustration regarding the government's response
to the threat and how is the government and its (s)elected leaders held accountable. Unfortunately, they are not
and this is not likely to change under the new electoral scheme.
- END -
1 In a technical sense what we mean by fair is that the voting system has three properties: monotonicity
(the more votes an alternative gets, the more likely it is to be selected), anonymity (any one vote is not
distinguishable from another), and neutrality (that the system does not favor one alternative over another) (Riker
1988).
2
By narrowly selected, I mean to imply that the functional constituency legislators and Chief Executive
are not returned by a broad based popular vote.
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