Schemes to assign to each state in the USA, a number of congressmen (based on state populations);
also mathematically related or equivalent are schemes (such as d'Hondt or Sainte-Laguë) to
assign to each political party, a number of seats in parliament (based on party vote counts
in an election).
Voting system where a voter approves
or disapproves of each candidate, and the most-approved candidate wins.
This is a degenerate form of range voting arising when only two numerical
values (the range-endpoints) are allowed scores.
It is also the same as plurality voting with "overvoting" allowed.
(We could also consider "trinary" approval voting also permitting "intentional blanks";
then an approval counts 1, a disapproval 0, and the candidate with the highest average
score is elected.) Invented by Guy Ottewell in 1968.
(See the linked page.) There are also many other impossibility theorems, some of which I
consider more important (even though it was Arrow's which won the Nobel Prize)
such as the Gibbard-Satterthwaite
The "Bayesian regret" of an election method E is the "expected avoidable
human unhappiness" caused by using E
(within a certain mathematical/probabilistic model – no actual humans are involved).
Better voting systems have smaller regret values.
The regret value of any given election system in any given probabilistic
scenario can be measured. This gives a quantitative way to compare
the quality of two election systems.
"Binary" election or referendum:
Means there are only two candidates (or options), e.g. "yes" versus "no."
Black's Voting Scheme:
Elect a Condorcet Winner if one exists, otherwise use the Borda system.
(A simple way to handle Condorcet cycles due to Duncan Black. This particular
Condorcet method is generally looked down on by modern Condorcetists, but it might
still have some good points.)
Voting system where a vote is a rank-ordering of all the N candidates.
The kth-ranked candidate gets score N-k. The candidate with
the highest score-sum is elected. (It is also possible in various ways to
allow "truncated ballots" in Borda voting where you only
rank some of the candidates; but it is unclear how best to do that
and there are several competing approaches. One is: a candidate ranked S spots above the
bottom-ranked candidate on that ballot gets Borda score S.)
Jean-Charles de Borda
This is a weighted positional voting system.
Attempt to improve IRV voting system to cause the Condorcet winner (if one exists)
always to win.
Each vote is a rank-ordering of all the N candidates, for example
"Nader>Gore>Bush>Buchanan" would be a possible vote (with N=4).
After collecting the votes, the N-candidate election proceeds in a sequence
of N-1 "rounds."
In each round one candidate is eliminated
and he is erased from all votes. For example, if "Bush" were eliminated,
then the above vote would become
The one to eliminate is
found as follows.
Find the two candidates A and B whom the fewest voters top-rank.
Now, ignoring all candidates except A and B in all the votes,
i.e. based solely on the A>B and B>A relations in those votes,
perform a 2-candidate majority election among A and B only.
The loser of that "election" is the one we eliminate.
"Cloning" a candidate is the hypothetical operation of creating a candidate
the same as the original one in every important way (in particular, all clones
are ranked adjacent by every rank-order voter, and within ε by every range-style voter
in the limit ε→0)
and adding him as a new
contender to the race. Of course this rarely or never exactly happens, but it often
approximately happens. Many voting systems react very badly to cloning. For
example, in the plurality system, a candidate who is clearly the best may spawn
imitators ("clones") with mild deviations, exactly
because his stances are so clearly the best. Then all the clones lose (due to vote-splitting)
causing a bad candidate to win. Range voting, in contrast, has no problem with cloning
and vote-splitting simply does not exist with range voting.
A wide class of voting systems are called "Condorcet systems" if they
always elect the "Condorcet winner" if he exists (but do various other things,
depending on which system it is, when and if he does not exist). BTR-IRV is a
recent example of a Condorcet system; new ones seem to be proposed about once a year.
The original concept and the first Condorcet voting system (the "least reversal"
system) both were invented by
Marie Jean Antoine Nicolas de Caritat, the
Marquis de Condorcet
Other Condorcet systems include
Tideman Ranked Pairs,
and several other methods, such as
and Smith-prefaced methods, are described here.
or "preference cycles"
are a common way in which Condorcet winners can fail to exist, i.e. in
which each candidate would lose to some other in a pairwise contest.
Condorcet Winner (traditional definition):
Consider an election system in which votes can be used (at least if we assume "honest" non-strategic voting)
to deduce the voter's
preference within each candidate pair. If a candidate A is preferred over
each other candidate B by a majority of the voters, then A is a Condorcet Winner
or "beats-all winner." (There is no necessity
that a Condorcet winner exist, however.
One appears to exist somewhere between 80
and 100% of the time in practical elections.)
It is also possible to define Condorcet winner in
inequivalent nontraditional ways. Range voting is a Condorcet voting method under
the nontraditional definition, but not with the traditional one. (The two definitions
are in fact equivalent on every voting method that Condorcet himself ever considered,
and indeed that had ever been considered in the political science literature before the 1990s,
i.e. all "ranked ballot" methods. Hence, it is not possible for us to tell which
definition-version Condorcet himself would have preferred, nor is it possible for us to tell whether
Condorcet himself would have agreed that Range Voting meets his criterion.)
The Coombs system is like IRV
except that the candidate bottom-ranked by the
most voters is eliminated each round.
Unfortunately, strategic voters are tempted to bottom-rank the candidate they
like least among the frontrunners in the pre-election polls
(as opposed to the one they truly like least),
to try to defeat him by getting him eliminated. This will cause
surefire elimination of all the frontrunners and the surefire election
of a "dark horse" every time. Not very useful.
In the Copeland system,
each vote is a rank-ordering of the candidates and
the candidate with the greatest "Copeland score" (i.e. defeating the most
rivals pairwise) wins. Copeland has an unfortunate propensity for
and it is highly vulnerable to candidate-cloning.
But it enjoys some good properties of "resistance to strategic voting," see
Like plurality voting, except each voter can vote some fixed number (for example 5)
of times in the same election, and her 5 votes need not be all the same.
However, strategically speaking, it is best if all her votes are the same,
in which case cumulative voting just becomes the same thing as the
(but more complicated) – which is stupid. (Cumulative voting has also
been suggested for multiwinner elections – which also is not a very good idea –
but we shall not discuss that here.)
A probabilistic model for elections in which, if there are N possible allowed votes a
voter can cast, then the N vote-totals in the election form an N-vector uniformly distributed
on the (N-1)-dimensional simplex
V1≥0, V2≥0, V3≥0, ..., VN≥0.
in N-dimensional space. This is different from the "random elections model."
Our name for a devastatingly common and severe problem that arises in many voting systems whenever
there are three "main rival" good candidates and 1 or more "dark horse" bad candidates who
would initially appear to have no hope to win because all voters unanimously mentally agree they
all are worse quality than all three main rivals. In many voting systems
voters feel strategically forced to
"exaggerate" the differences between the 3 main rivals in their votes, which then
guarantees the election of a bad "dark horse."
We later found out that very similar ideas had been proposed by political
science professor Burt L. Monroe under the name "NIA" in work which apparently was never
The empirical fact (supported by a vast amount of data) that plurality voting systems
tend over time to lead to self-reinforcing 2-party domination. The same is
true (various post-Duverger authors pointed out) with instant runoff voting (IRV).
In contrast (as Duverger
also pointed out) proportional representation party-list systems
and the (French) plurality-with-separate-top-two-runoff system (T2R) both
tend to lead to many political parties.
Our name for the
phenomenon, common to many kinds of voting systems, that it often is strategically
advantageous for a voter to rank her favorite candidate below top
in his vote. This can often be useful to help a "lesser evil" defeat
a greater one. It is a very serious threat that strikes at the very core of democracy, if
a voting system motivates people not to vote their favorites top.
Range and Approval voting both
avoid this flaw, but IRV, Borda, all Condorcet systems, and Plurality
all suffer from it.
In the USA at present (2013) the
majority of voters are female, while the majority of
candidates for office (and office holders) are male.
It therefore is convenient when talking about voting systems to use words like "she" and "her"
when discussing voters, versus "he" and "his" when discussing candidates.
We suggest this convention purely for the purpose of making your writing clearer –
accusations of sexism be damned.
IRV=Instant Runoff Voting
(also called "The alternative vote" in, e.g, Britain):
A bastardized version of the "Hare/Droop reweighted STV"
multiwinner proportional voting system. More precisely, IRV is
the single-winner special
case of this voting system. (Hence IRV no longer has any claims to being
a "proportional" voting system, which was the whole reason Hare and Droop invented STV in
1800s Britain – but it is simpler.)
The main US advocates of IRV
unfortunately were influenced by the British
Electoral Reform Society (ERS),
which advocates proportional representation
election of parliament
via Hare/Droop STV. In fact for over 100 years Hare/Droop STV
was the only known voting system achieving proportionality, so the ERS's
stance was well
justified. However, in a single-winner context, where proportionality is not an issue and
STV degenerates to IRV, this whole stance makes no real sense.
(It is like saying "lye tastes good as a trace ingredient of chocolate, therefore,
we should eat lye.")
Incidentally, there is a multiwinner proportional version of Range Voting,
called "reweighted range voting" (paper #78
which is simpler than, as well as apparently superior to,
Hare/Droop STV; thus even Hare/Droop STV should now probably be regarded as obsolete,
see paper #91 here.
Specifically IRV works as follows:
each "vote" is a rank ordering of all the N candidates.
The election proceeds in N-1 "rounds": each round, the candidate top-ranked by
the fewest votes is eliminated (both from the election, and from all orderings inside votes).
After N-1 rounds only one candidate remains, and is declared the winner.
(It is also possible in various ways to
allow "truncated ballots" in IRV voting where you only
rank some of the candidates; but it is unclear how best to do that
and there are several competing
Means more than half of all voters.
(As opposed, e.g. to a mere "plurality.")
Our name for the empirical
phenomenon that, under range voting, small ("infant") third parties
tend to receive the benefits of honest voter-ratings (since there is little incentive
for range voters to be dishonest-strategic about candidates who have little chance to win)
and therefore experimentally collect far greater vote counts than under either
approval or plurality voting (systems in which strategic exaggeration by voters is
virtually mandated). This "coddles" them in a "protective nursery" giving them a chance
to grow into larger parties.
When, in plurality voting, you vote for more than one candidate.
The Plurality System forbids overvotes (if you try it, your vote will be discarded
but the Approval System allows them.
A pathology is a situation in which a voting system appears to malfunction and
misbehave, delivering a result which hurts society. Instant Runoff Voting (IRV) suffers
many kinds of pathology, quite
while range voting is
well behaved and suffers comparatively few.
Also known as "first-past-the-post," plurality is by far the most common
voting system for single-winner races. (Unfortunately.) Your "vote" is the "name of a single
candidate," and the most-named candidate wins.
Voting systems intended for multiwinner elections which strive to satisfy the ideal
that the election winners "represent" the voters well. For example, if the voters are 51%
female but the winners are 4% women (US Congress, approximate historical average)
then that is highly "disproportional." If 33% of all voters are not registered as either
"Democrats" or "Republicans," but 100% of all winners are, that again is
highly disproportional. (Survey of PR voting systems:
Also known in the literature as the "Impartial Culture," the
random elections model
postulates that each of the V voters casts one of the N allowed votes uniformly
at random (and independently of every other voter), so that every possible election
(where "election" means "table saying the vote cast by each voter")
is equally likely. One then generally is interested in the limit V→∞.
It is better to generalize this to voting systems such as
continuum range voting
in which an infinite set of votes is allowed. To do that, we suppose each voter has
an independent identical normal-random-variate for the "utility value"
of each candidate. The voter than acts based on these utilities in deciding on her vote.
For voters who act "honestly" this is equivalent to the impartial culture for rank-order
ballots. But since it also allows other kinds of ballots, it is a more general model.
(This is different from the Dirichlet Model.)
Excellent voting system in which each voter provides a numerical score within a given
range to each candidate and the candidate with the greatest score-sum wins.
(For example, if the allowed range was 0-99, then a valid
range vote might be "Lincoln=99, Harding=0, Washington=99, McKinley=47.")
A system very much like range voting is used by the Olympics to select gold-medal gymnasts.
A variant permits X, i.e. "intentional blank," votes
for candidates about whom the voter has no opinion. In that case the candidate with
the highest average score (where the Xs for you are not incorporated into your average)
System where a second election is held to determine the final winner.
In the most common version, we have a first round plurality election, then if anybody got over 50% of
the votes, they win. Otherwise, the top two finishers participate in an extra "runoff" election
and the winner among them is the final winner.
This system is used in France and many other countries to elect presidents, and it has been
very common historically that the final winner has differed from
the first round's winner.
Quas "1-dimensional political spectrum" model:
Simple probabilistic model of elections introduced by Anthony Quas,
suitable for use with rank-order-ballot voting systems.
The ∞ "voters" are the uniform distribution on the real interval (0,1).
The C "candidates" are C random points on that interval, i.e. C
independent random samples from the voter-distribution. The voters are assumed to order the
candidates by distance away from them (closest candidate ranked top).
Example theorem (proven by W.D.Smith 2010): in the C-candidate Quas model, C≥2,
the probability that "majority-top winner" exists
(i.e. that there exists a candidate ranked unique-top by over 50% of the voters)
is exactly 22-C.
Quas's model has the flaw (or "property") that Condorcet voting methods are optimal
for it (assuming honest voters and a concave-∩ voter-candidate distance-based
utility function). That is a consequence of
Black's 1D theorem.
That is, with honest voting, any Condorcet method will automatically
elect whichever candidate is located nearest to ½. Meanwhile range voting,
instant runoff, etc, will not always do so (and nor will Condorcet
voting with strategic voters).
If all candidates in a set S pairwise-beat all candidates in the complement subset
(i.e a voter majority prefers each S-member over each candidate not in S)
and S is the smallest nonempty such set, then S is called the "Smith Set."
A "Condorcet winner" is a 1-element Smith set. (Related is the "Schwartz set" in which
"beat" is replaced by "beat or tie.")
Ballots that do not meet the rules of the voting system and hence which are discarded
without using them in the election. For example, "overvotes" are spoiled plurality votes,
and ballots not giving a rank order, such as Gore=2, Bush=2, Nader=1, Buchanan=0, are
spoiled for use in Borda and IRV.
Ballot "spoilage" has historically been heavily used by
election fraudsters and manipulators.
Some election systems like plurality and especially IRV are prone to spoilage, other
systems like Range Voting are comparatively immune to spoilage – e.g, in single-digit
range voting, any way to fill in the slots with either single digits or
intentional (or unintentional) blanks is a valid ballot.
A "spoiler" is a candidate S, such that, if you vote for S, that could cause both S and
your second-favorite candidate Q both to lose, whereas if you had voted for Q (e.g. if
S were to drop out of the race, or if you just voted dishonestly to strategically pretend
Q was your true favorite), then Q would have won.
Spoilers can exist in Plurality, IRV, Borda, and Condorcet voting
but do not exist in Approval and Range voting.
Strategic voting (also called insincere or tactical voting):
The practice of submitting a "dishonest" vote, presumably in order to increase its
chances of causing something good (from that voter's point of view) to happen.
(Actually, some gameplayer-types object to the use of words such as "dishonest" or
"distorted," since in
their view votes have no inherent meaning, therefore, it is impossible to mis-state their
meaning and hence
dishonesty is not possible. In their view voting is just a game, the election system defines
the rules of the game, votes are moves
in the game, and you play to win the most that you can.)
For example, in the USA's single-winner
plurality voting system, it is often strategically-poor gameplaying to vote
for your favorite candidate – if he has no chance to win, it is
certainly strategically pointless to waste your vote on him.
Good voting systems have the property that "best strategic votes" and "honest votes" are
usually the same thing (or at least close to being
the same thing, in some metric); but in bad voting systems
like Plurality and Borda, the two often differ greatly.
System (used in many countries as of 2013, e.g. France's 2012
where the top two candidates reckoned by
a plurality-voting election, get to run in a second, binary, election, whose winner
wins the seat. This partially corrects serious distortions from plain plurality voting
(i.e. the first round by itself) and seems often to overcome Duverger's law
of 2-party domination.
In the real world
T2R exhibits some important statistical differences
versus the "instant runoff" (IRV)
system, in particular IRV still yields 2-party domination.
The percentage of people eligible to vote who actually do vote in some election.
Uncovered set (and the notion of "covering"):
In a multicandidate election using rank-order ballots,
candidate X "covers" candidate Y if and only if
every candidate that beats X also beats Y (pairwise),
X beats at least one candidate that Y does not,
or X ties at least one candidate that
Equivalent definition of "X covers Y":
X does at least as well as Y (with regard to pairwise win/tie/loss –
we ignore how severe those losses are, merely taking account of whether they exist)
against each candidate, and better versus at least one.
(Here we assume that any candidate is automatically tied with itself.
Hence "X covers Y" prevents "Y beats X," and in the absence of
nontrivial ties, would force "X beats Y."
If X is tied with Y, it can still cover Y, as long as it does as
well as Y against the other candidates and strictly better against at least one of them.)
Final equivalent definition of "X covers Y": Either
X beats Y and does at least as well as Y versus each rival, or
X ties Y and does at least as well as Y versus each rival and better than Y versus at
least one rival.
The covering relation forms a "partial order" among the candidates.
Hence there always is at least one uncovered candidate.
If and when a condorcet winner
exists, it is the sole uncovered candidate.
The uncovered set is a subset of the Smith set.
A geometrical theorem:
If the voter locations are centro-symmetric around some center in some Euclidean space,
and voter preferences are based on the Euclidean distance
from that voter to each candidate (closer preferred) then X covers
Y if and only if X lies closer to the center than Y.
The covering relation then is a total order and the
uncovered set consists exactly of the candidates at minimum distance to the central point.
But for nonsymmetric voter distributions in Euclidean space,
the situation is more complicated.
The uncovered set can then contain many candidates and have a non-ball, indeed nonconvex,
A theorem about "agenda manipulation" (by Nicholas R. Miller):
Suppose the one winner among a set of candidates is decided by a pairwise
That is, in each round of the tournament, two candidates X and Y compete, and if the voters
by majority prefer Y over X, then X is eliminated. We continue
until only a single candidate remains, and he is the winner.
Suppose the designer of the tournament (i.e. of the schedule of who is paired
with who – we assume designer knows all voters' preference orderings, and
that all voters vote honestly) can force any specified
candidate in some set T to win.
T, and the uncovered set S, are the same.
does not necessarily elect a member of the uncovered set, albeit if
we change the definition of "X covers Y" to incorporate preference strengths
instead of ignoring them – which seems more sensible if using a score-based ballots
– then it does, indeed then the score voting winner
always is the only uncovered candidate.
So with this redefinition and score voting, the uncovered set is trivialized and
agenda manipulation is
made impossible if
voters must score self-consistently in different rounds.
Agenda manipulation unfortunately would remain possible
with no such self-consistency constraint, but by just having a single range voting election,
one round only, there is no "agenda" to be manipulated...
Voting system where a vote is a rank-ordering of all the N candidates.
The kth-ranked candidate gets score Wk for some fixed
set of "weights" W1≥W2≥W3≥...≥WN.
The candidate with
the greatest score-sum is elected.
(For example, Borda is the weighted positional system
with Wk=N-k, and plurality is the
weighted positional system with
W1=1 and Wk=0 for 2≤k≤N.)
demonstrates one very embarrassing flaw in all weighted positional systems.
In Condorcet voting methods in which equal rankings are permitted in votes,
there are two natural ways to reckon the "strength"
of a "pairwise victory":
margins and winning votes.
The "margin" of A over B is the number of votes saying A>B,
minus the number saying B>A.
If more votes say A>B than B>A, then A wins that pairwise contest, and the number
of "winning votes" is precisely the number of votes saying A>B. Otherwise it is zero.
The distinction between these two concepts may sound minor, but it can have some
profound consequences. If equal rankings are forbidden
(i.e. you are allowed to vote "A>B" or "B>A" but never "A=B") then the
distinction between winning votes and margins vanishes.