MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum.
A voting system contains rules for valid voting, and how votes are counted and aggregated to yield a final result. Common voting systems are majority rule , proportional representation or plurality voting. The study of formally defined voting systems is called voting theory , a subfield of political science, economics or mathematics.
With majority rule, most of us who are unfamiliar with voting theory are often surprised that another voting system exists, or that "majority rule" systems can produce results not supported by a majority. If every election had only two choices, the winner would be determined using majority rule alone. However, when there are three or more options/candidates, there may not be a single option/candidate that is preferred by a majority. Different voting systems may give very different results, particularly in cases where there is no clear majority preference.
A voting system specifies the form of the ballot (the set of allowable votes) and the tallying method
(an algorithm for determining the outcome). This outcome may be a single winner, or may involve multiple winners such as in the election of a legislative body.
Different voting systems have different forms for allowing the individual to express his or her vote.
In ranked ballot or "preference" voting systems, such as Instant-runoff voting, the Borda count, or a
Condorcet method, voters order the list of options from most to least preferred. In range voting , voters rate each option separately on a scale. In plurality voting (also known as "first-past-the-post"), voters select only one option, while in approval voting , they can select as many as they want.
Some voting systems include additional choices on the ballot, such as write-in candidates , a none of the above option, or a no confidence in that candidate option.
Some methods call for a primary election first to determine which candidates will be on the ballot.
Single-winner systems can be classified based on their ballot type. In one vote systems, a voter picks one choice at a time. In ranked voting systems, each voter ranks the candidates in order of preference. In rated voting systems, voters give a score to each candidate.
The most prevalent single-winner voting method, by far, is plurality (also called "first-past-the-post",
"relative majority", or "winner-take-all"), where each voter votes for one candidate, and the candidate that receives the most votes is declared the winner. In an election where there are just two candidates, that system works just fine, and is the same as selecting a winner according to majority rule .
Majority rule has at least 3 desirable properties:
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
1.
All voters are treated equally – if any 2 voters were to exchange ballots prior to submitting them, the outcome of the election would be the same.
2.
Both candidates are treated equally – if the election were re-run and every voter switched side, then the outcome of the election would be reversed as well.
3.
If the election were held again and a single voter switched his/her vote from a vote for the loser to a vote for the winner of the original election and everyone else voted exactly the same as before, then the outcome of the election would not change.
May’s theorem : If the number of voters is odd then the majority rule is the only voting system for 2 candidates that satisfies the 3 conditions just listed and never results in a tie.
It's when there are three or more candidates that problems can arise. Plurality voting can result in the election of a candidate whom almost two-thirds of voters detest when there are more than 2 candidates.
For instance, in 1998, in a three-party (Democrat, Republican, and Reform Party) race, plurality voting resulted in the election of former wrestler Jesse Ventura (Reform Party candidate) as
Governor of Minnesota (a state in USA), despite the fact that only 37% of the electors voted for him.
The almost two-thirds of electors who voted Democrat or Republican had to come to terms with a governor that none of them wanted. In this case, Ventura won not because the majority of voters chose him, but because plurality voting effectively thwarted the will of the people and electing someone whom the majority opposes. Had the voters been able to vote in such a way that, if their preferred candidate were not going to win, their preference between the remaining two could be counted, the outcome could have been quite different.
Plurality voting is also easily subjected to manipulation. A major problem is the so-called " spoiler effect ", which describes the effect a minor party candidate with little chance of winning has upon a close election, when that candidate's presence in the election draws votes from a candidate similar to him, thereby causing a candidate dissimilar to them to win the election. The minor candidate causing this effect is often referred to as a spoiler . Often such spoilers are financially supported by those who despise that candidate's politics, utilizing an "any enemy of my enemy is my friend" strategy.
One often cited example of the "spoiler effect" at work was the 2000 U.S. Presidential election. In that election, George W. Bush and Al Gore had a very close election in many states, with neither candidate winning a majority of the votes. In the state of Florida, the final certified vote totals showed that Bush won just 537 more votes than Gore with more than 6 million votes casted, thus winning the state.
Some Gore supporters believed that many of the 97,421 votes that went to Ralph Nader in that state would likely have been votes for Gore, had Nader not been running in the election. Some Gore supporters contend that Nader's candidacy "spoiled" the election for Gore by "taking away" enough votes from Gore in Florida and many other states (in particular, New Hampshire being the allegation most statistically supportable) to allow Bush to win. Their argument is bolstered by a poll of Nader voters, asking them for whom they would have voted had Nader not run, which said 45 percent of them would have voted for Gore, 27 percent would have voted for Bush, and the rest would not
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang have voted. Nader supporters said he was running, in part, to protest the positions of Bush and
Gore.
Because of the drawbacks of the plurality voting, other types of voting systems have been proposed.
Several countries, among them Australia, the Irish Republic, and Northern Ireland, use a system called single transferable vote (the Hare system) . Introduced by Thomas Hare in England in the
1850s, this system takes account of the entire range of preferences each voter has for the candidates. All electors rank all the candidates in order of preference. When the votes are tallied, the candidates are first ranked based on the number of first-place votes each received. The candidate who comes out last is dropped from the list. This, of course, effectively "disenfranchises" all those voters who picked that candidate. So, their vote is automatically transferred to their second choice of candidate -- which means that their vote still counts. Then the process is repeated: the candidates are ranked a second time, according to the new distribution of votes. Again, the candidate who comes out last is dropped from the list. With just three candidates, this leaves one candidate, who is declared the winner. In a contest with more than three candidates, the process is repeated one or more additional times until only one candidate remains, with that individual winning the election. Since each voter ranks all the candidates in order, this method ensures that at every stage, every voter's preferences among the remaining candidates is taken into account.
An alternative system is the Borda count , named after Jean-Charles de Borda, who devised it in
1781. Again, the idea is to try to take account of each voter's overall preferences among all the candidates. As with the single transferable vote, in this system, when the poll takes place, each voter ranks all the candidates. If there are n candidates, then when the votes are tallied, the candidate receives n points for each first-place ranking, n-1 points for each second place ranking, n-2 points for each third place ranking, down to just 1 point for each last place ranking. The candidate with the greatest total number of points is then declared the winner.
Yet another system that avoids the Jesse Ventura phenomenon is approval voting . Here the philosophy is to try to ensure that the process does not lead to the election of someone whom the majority opposes. Each voter is allowed to vote for all those candidates of whom he or she approves,
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang and the candidate who gets the most votes wins the election.
This is the method used to elect the officers of both the American Mathematical Society and the Mathematical Association of America.
An example:
To see how these different systems can lead to very different results, let's consider a hypothetical scenario in the state of California, where Green Party candidate Ralph Nader is expected to do well.
Suppose that, on the election day, 15 million Californians go to the polls, and that their preferences between the three main candidates are as follows:
6 million rank Bush first, then Nader, then Gore.
5 million rank Gore first, then Nader, then Bush.
4 million rank Nader first, then Gore, then Bush.
If the votes are tallied by the plurality vote -- the present system -- then Bush's 6 million (first-place) votes make him the clear winner. And yet, 9 million voters (60% of the total) rank him dead last!
That hardly seems fair.
What happens if the votes are counted by the single transferable vote system -- the system used in
Australia and Ireland? The first round of the tally process eliminates Nader, who is only ranked first by 4 million voters. Those 4 million voters all have Gore as their second choice, so in the second round of the tally process their votes are transferred to Gore. The result is that, in the second round,
Bush gets 6 million first place votes while Gore gets 9 million. Thus, Gore wins by a whopping 9 million to 6 million margin.
But wait a minute. Looking at the original rankings, we see that 10 million voters prefer Nader to
Gore -- that's 66% of the total vote. Can it really be fair for such a large majority of the electorate to have their preferences ignored so dramatically?
Thus, both the plurality vote and single transferable vote lead to results that run counter to the overwhelming desires of the electorate. What happens if we use the Borda count ? Well, with this method, Bush gets
6m x 3 + 5m x 1 + 4m x 1 = 27m points,
Gore gets
6m x 1 + 5m x 3 + 4m x 2 = 29m points, and Nader gets
6m x 2 + 5m x 2 + 4m x 3 = 34m points.
The result is a decisive win for Nader , with Gore coming in second and Bush trailing in third place.
What happens with approval voting? Well, we don't have enough information -- we don't know how many electors actively oppose each particular candidate. Let's assume that the Gore supporters and the Nader supporters could live with the others' candidate, but the voters in both groups really don't want to see Bush in the White House. (This is not at all an unreasonable supposition, given the voting preferences we started with, but remember that this is a purely hypothetical example.) In this case, Nader gets 15 million votes, Gore gets 9 million votes, and Bush gets a mere 6 million. All in all,
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang it's beginning to look as though Nader is the one who should win California.
Faced with such confusion in how to count votes in elections with three or more candidates, it's tempting to say that the only fair way to decide the issue is to choose the individual who would beat every other candidate in head-to-head, two-party contests . This approach was suggested by the
Marquis de Condorcet in 1785, and as a result is known today as the Condorcet system .
A little bit of history here can set the record straight. Voting theory became an object of academic study around the time of the French Revolution. Jean-Charles de Borda proposed the Borda count in 1770 as a method for electing members to the French Academy of Sciences. His system was opposed by the Marquis de Condorcet, who proposed instead the method of pairwise comparison that he had devised. Implementations of this method are known as the Condorcet method.
For the scenario in our example, Nader also wins according to the Condorcet system. He gets at least 10 million votes in a straight Nader-Gore contest and at least 9 million votes in a Nader-Bush match-up, in either case a majority of the 15 million voters. Unfortunately, although it works for this example, and despite the fact that it has considerable appeal, the Condorcet method suffers from a major disadvantage: it does not always produce a clear winner !
For example, suppose the Californian voting profile were as follows:
5 million rank Bush first, then Gore, then Nader.
5 million rank Gore first, then Nader, then Bush.
5 million rank Nader first, then Bush, then Gore.
Then 10 million Californian voters prefer Bush to Gore, so Bush would easily win a Bush-Gore battle.
Also, 10 million voters prefer Gore to Nader, so Gore would romp home in a Gore-Nader contest.
The remaining two-party match-up would pit Bush against Nader. But when we look at the preferences, we see that 10 million people prefer Nader to Bush, so Nader comes out on top in that contest. In other words, there is no clear winner. Each candidate wins one of the three possible two-party battles !
This is known as the Condorcet paradox (or the voting paradox ), which was first noted by Condorcet and he called
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang it the intransitivity of majority preferences (i.e. the failure of majority rule to produce transitive preferences for society).
Transitivity is a property of preference rankings: If A is preferred to B, and if B is preferred to C, then A should be preferred to C.
The simplest Condorcet paradox occurs when there are three voters and three political parties.
Suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows
(candidates being listed in decreasing order of preference):
In these circumstances, no matter who is declared the winner, two-thirds of the electorate will have preferred someone else. There is no “Condorcet winner”, i.e. the option, or candidate, in a multicandidate election, which wins a simple majority against each of the others when every pair of candidates is compared.
First, B runs against C. What is the outcome? Voter 1 and voter 2 prefer B over C, so B wins.
Second, A runs against B. voter 1 and voter 3 prefer A over B, so A wins.
However, in a race between A and C, C would win: Voter 2 and voter 3 prefer C to A.
This result violates transitivity!
The lessons we learned from the Condorcet Paradox are:
1.
Democratic preferences are not always transitive.
2.
The order on which things are voted can affect the result.
3.
Majority voting does not always reveal what society really wants.
A voting procedure is said to satisfy the Condorcet winner criterion (CWC) if for every possible sequence of preference list, either (1) there is no Condorcet winner (as in the Condorcet paradox), or (2) there is a Condorcet winner and it is the unique winner of the election.
So what do we do next? Faced with such a confusing state of affairs, the obvious thing is to abandon all of the methods we have looked at and search for an alternative approach. After all, there must be a fair way to count the votes in an election, mustn't there?
There are many alternative voting systems, but is there a perfect one? The answer, in a mathematical sense, is no. To see why, let's remind ourselves of what an election is actually trying to do. There is a population of voters, each of whom, if pressed hard enough, can come up with a preference ranking of the candidates: whom they like most, whom they like least, and everything in between, with some candidates possibly occupying equal position. Ideally, a voting system should take these millions of preference rankings as input, and return a single ranking of candidates that makes reasonable democratic sense. The government can then be formed on the basis of this single ranking.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
But what's democratic sense? First of all, we clearly don't want there to be a dictator, so the system should reflect the wishes of more than just one individual ( Non-dictatorship ). Secondly, the system should reflect voters' wishes reasonably accurately. At the very least we should require that if all voters prefer candidate x to candidate y, then x should come above y in the final result (this condition is sometimes called unanimity ). Also, there shouldn't be room for ambiguity: the voting system should always return exactly one clear final ranking (this condition is known as universality ).
In the 1950s the economist Kenneth Arrow asked himself whether there can be a voting system which satisfies these pretty minimal criteria. He added one slightly more subtle condition: in the final result, whether one candidate is ranked above another, say Bush above Gore, should only depend on how individual voters ranked Bush compared to Gore. It shouldn't depend on how they ranked either of the two compared to a third candidate, for example Nader.
For example, suppose that Bush has just won the election, having come top of the final ranking, when Nader becomes embroiled in a scandal, which sees him slide to the bottom of all voters' preference rankings. Then this should not have any effect on the fact that Bush was more popular than Gore: a re-run of the election should still return Bush as the winner, as long as voters didn't change their relative rankings of Bush and Gore. Arrow called this condition independence of irrelevant alternatives (IIA) . In our example, poor Nader has become irrelevant, but this shouldn't affect the contest between Bush and Gore. Independence of irrelevant alternatives is an important condition because non-compliance of a voting system means that the ranking of outside candidates can have an effect on the outcome.
Sadly and surprisingly, Arrow proved mathematically that if there are three or more candidates and two or more voters, no vote-tallying system that works by taking voters' preference rankings as input and returns a single ranking as output can satisfy all the four conditions. Put differently, only a dictatorship (in which one voter's ranking determines the result) can simultaneously satisfy unanimity, universality and independence of irrelevant alternatives. His theorem, called Arrow's
Impossibility Theorem helped to earn him the 1972 Nobel Prize in Economics.
There are many versions of the theorem. In one version, Arrow proved that, no "fair" voting system can be designed to satisfy these four criteria:
1.
Unanimity: If everyone prefers A to B, then A should beat B in the final result.
2.
Universality: There is exactly one clear final ranking
3. Independence of irrelevant alternatives: The ranking between any two outcomes should not depend on whether a third option is available.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
4. Non-dictatorship: There is no person that always gets his way, regardless of everyone else’s preferences.
Arrow proved that no vote-tallying system can satisfy all four properties and this result is known as
Arrow’s Impossibility Theorem .
It should be stressed that Arrow's theorem doesn't just say that none of the tallying systems that have been devised so far is fair. Arrow proved that no fair system can possibly exist.
So does Arrow's theorem mean that democracy is impossible? Not unless you're a mathematical absolutist. The theorem doesn't say that every voting system always returns horribly unfair results, but that there always is a theoretical possibility that one of the four conditions might be breached if voters happen to vote in a particular way. Whether or not these breaches are a terrible thing depends on how often they are likely to happen — you might argue that the cyclical split in the last example is unlikely to occur in a general election — and how important you think the condition is that's being violated. Non-dictatorship is clearly pretty fundamental, but some people argue that we could do without independence of irrelevant alternatives. Other people support voting systems in which candidates aren't ranked, but rated, as happens for example in approval voting, in which people are simply asked to approve or not approve of a candidate, and which escapes Arrow's theorem. It's a question of carefully weighing up pros and cons to choose the best system. And while the first-past-the-post system is undoubtedly crude, there is also an argument for simplicity: the fairest system is of no use if it's so complicated that voters end up not bothering to vote.
Arrow's theorem gives mathematical backing to something we already know from experience: that it's really hard to design a system that produces a reasonably fair result in all circumstances.
Thus, the best we can hope for is to pick the best of a range of imperfect election tallying systems. But how do we make that choice? Things might not be so bad if mathematicians themselves agreed which system is best.
Unfortunately, the only thing everyone does pretty much agree on is that the present system -- plurality voting -- is the worst , and any of the other systems described here would do a better job of representing the preferences of the electorate.
However, the single transferable vote and the Borda count are not without their problems.
One worrying problem with the single transferable vote (Hare) system is that if some voters increase their evaluation of a particular candidate and raise him or her in their rankings, the result can be -- paradoxically -- that the candidate actually does worse!
For example, consider an election in which there are four candidates, A, B, C, D, and 21 electors. Initially, the electors rank the candidates like this:
7 voters rank: A B C D
6 voters rank: B A C D
5 voters rank: C B A D
3 voters rank: D C B A
In the first round of the tally, the candidate with the fewest first-place votes is eliminated, namely D. After D's
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang votes have been redistributed, the following ranking results:
7 voters rank: A B C
6 voters rank: B A C
5 + 3 = 8 voters rank: C B A
Then B is eliminated, leading to the new ranking:
7 + 6 = 13 voters rank: A C
8 voters rank: C A
Thus A wins the election.
Now suppose that the 3 voters who originally ranked the candidates D C B A change their mind about A, moving him from their last place choice to their first place: A D C B. These voters do not change their evaluation of the other three candidates, nor do any of the other voters change their rankings of any of the candidates. But when the votes are tallied this time, the end result is that B wins. (If you don't believe this, just work through the tally process one round at a time. The first round eliminates D, the second round eliminates C, and the final result is that 10 voters prefer A to B and 11 voters prefer B to A.)
For all the advantages offered by the single transferable vote system, the fact that a candidate can actually harm his/her chances by increasing his/her voter appeal -- to the point of losing an election that she would otherwise have won -- leads some mathematicians to conclude that the method should not be used.
This example shows that the single transferable vote (Hare) system does not satisfy a mathematical property called monotonicity , which means a candidate’s chance of winning the election should not be hurt by having more voters ranking him/her higher in their preference. Monotonicity is a reasonable property that any fair voting sytem should have.
The Borda count has at least two weaknesses. First, it is easy for blocks of voters to manipulate the outcome. For example, suppose there are 3 candidates A, B, C and 5 electors, who initially rank the candidates:
3 voters rank: A B C
2 voters rank: B C A
The Borda count for this ranking is as follows:
A: 3x3 + 2x1 = 11
B: 3x2 + 2x3 = 12
C: 3x1 + 2x2 = 7
Thus, B wins. Suppose now that A's supporters realize what is likely to happen and deliberately change their ranking from A B C to A C B. The Borda count then changes to:
A: 11; B: 9; C: 10.
This time, A wins. By putting B lower on their lists, A's supporters are able to deprive him of the victory he would otherwise have had.
Of course, almost any method is subject to strategic voting by a sophisticated electorate, and Borda himself
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang acknowledged that his system was particularly vulnerable, commenting: "My scheme is intended only for honest men." Somewhat more worrying to the student of electoral math is the fact that the entry of an additional candidate into the race can dramatically alter the final rankings, even if that additional candidate has no chance of winning, and even if none of the voters changes their rankings of the original candidates. For example, suppose that there are 3 candidates, A, B, C, in an election with 7 voters. The voters rank the candidates as follows:
3 voters rank: C B A
2 voters rank: A C B
2 voters rank: B A C
The Borda count for this ranking is:
A: 13; B: 14; C: 15.
Thus, the candidates final ranking is C B A. Now a new candidate X, who has no hope of winning, enters the race, and no voter has changed his/her ranking of the original candidates (A, B, C) so that the voters' ranking becomes:
3 voters rank: C B A X
2 voters rank: A X C B
2 voters rank: B A X C
The new Borda count is:
A: 20; B: 19; C: 18; X: 13.
Thus, the entry of the losing candidate X into the race has completely reversed the ranking of A, B, and C, giving the result A B C X. This example shows that the Borda count system does not satisfy the condition known as
“independence of irrelevant alternatives” (IIA).
With even seemingly "sophisticated" vote-tallying methods having such drawbacks, how are we to decide which is the best method? Of course, the democratic way to settle the matter would be to vote on the available systems.
But then, how do we tally the votes of that election?
When it comes to elections, it seems that even the math used to count the votes is subject to debate!
Many elections are held to the ideal of "one person, one vote," meaning that every voter's votes should be counted with equal weight. This is not true of all elections, however. Corporate elections, for instance, usually weight votes according to the amount of stock each voter holds in the company, changing the mechanism to "one share, one vote". Votes can also be weighted unequally for other reasons, such as increasing the voting weight of higher-ranked members of an organization, such as the United States Electoral College, the Security Council in the United Nations, and the Council of
Ministers of the of the European Union.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
Voting weight is not the same thing as voting power. In situations where certain groups of voters will all cast the same vote (for example, political parties in a parliament), voting power measures the ability of a group to change the outcome of a vote. Groups may form coalitions to maximize voting power.
A weighted voting system is one in which the preferences of some voters carry more weight than the preferences of other voters.
When a vote deals with only two alternatives, all reasonable voting methods have the same outcome as "majority rule."
For this reason, our major interest here will not be comparing voting systems but rather, the concept of POWER : Who has it and how much do they have?
Following is a list of frequently encountered terminologies in the discussion of weighted voting systems.
Motion
A motion is any vote involving only two alternatives.
Player
A player is a voter in a weighted voting system.
 The N players are: P
1
, P
2
, P
3
,…, P
N
.
Weight
The weight of each player is the number of votes he/she controls.
 The weights of the N players are (in order): w
1
, w
2
, w
3
,…, w
N
.
Quota
The quota is the minimum number votes needed to pass a motion.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
 We will use the letter q to stand for quota.
 q must be at least a majority but not more than the total number of votes.
 ½ ( w
1
+ w
2
+ … + w n
) < q  w
1
+ w
2
+ … + w n
Dictator
A dictator is a player who has enough votes to pass any motion single-handedly.
 A dictator’s weight is always greater than or equal to the quota .
 Whenever there is a dictator, all other players have NO power.

Consider [11:12, 5, 4]
-- A
owns enough votes to carry a motion single handedly.
Dummy
A dummy is a player with no power.
Consider [30: 10, 10, 10, 9] -- A turns out to be a dummy! There is never going to be a time when A is going to make a difference in the outcome of the voting.
Veto Power
A player that is not a dictator but can single-handedly prevent any group of players from passing a motion is said to have veto power .
Consider [12: 9, 5, 4, 2] – A has the power to obstruct by preventing any motion from passing.
Coalition
A coalition is any group of players that join forces to vote together. It is a form of cooperative games through which players can coordinate their strategies and share the payoff.


The total number of votes controlled by a coalition is called the weight of the coalition .
A winning coalition is one with enough votes to win (i.e. The total numbers of votes of a winning coalition

quota) .
A losing coalition is one without enough votes to win.

 A blocking coalition is one who gets enough votes to block the motion. (i.e. The total numbers of votes of a blocking coalition > the total number of votes – quota)
 A coalition consisting of all players is often called a grand coalition.
 With N players, there are 2
N
-1 possible coalitions.
Critical Player
A player whose desertion of a winning coalition turns it into a losing one is called a c ritical player .
Pivotal Player
A pivotal player is the player in a sequential coalition who changes the coalition from a losing to a winning one.
A sequential coalition is one in which the players are listed in the order that they entered the coalition.
There are N! sequential coalitions containing all N players.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
MATHEMATICAL NOTATION
If player P
1 has w
1 votes, player P
2 has w
2 votes, P
3 has w
3 votes,…, and player
P
N has w
N votes and the quota is q , then we will write { q : w
1
, w
2
, w
3
,…, w
N
}.
The weights are always listed in numerical order, starting with the highest.
Example:
The three stockholders in a small company form a Board of Directors to oversee the company. John
(P
1
) has 5 votes as the largest stockholder, Ginny (P
2
) has 3 votes, and Ann (P
3
) has 2 votes. The quota is 7; that is, it takes 7 or more votes to pass a motion. This weighted voting system is represented mathematically as { 7: 5, 3, 2 }.
When considering motions, all reasonable voting methods will have the same outcome as majority rules. Thus, the mathematics of weighted voting systems looks at the notion of power: who has it and how much do they have? A player's power is defined as that player's ability to influence decisions.
An important point here is the weight of a player is not a good measure of a player’s POWER .
It is possible for a weighted voting system to actually reduce to a one-person, one-vote situation in which case all players have the same POWER even though they don’t all have the same weight.
For example, for the weighted voting system {5: 4, 3, 2}, none of the players can pass a motion alone and any two players can join together to pass a motion; so, although the players have different weights, each player has the same amount of power.
The Banzhaf power index , named after John F. Banzhaf III (though originally invented by Penrose
(1946) and sometimes called Penrose–Banzhaf index), is a power index defined by the probability of changing an outcome of a vote where voting rights are not necessarily equally divided among the voters or shareholders.
To calculate the power of a voter using the Banzhaf index, list all the winning coalitions, then count the critical voters. A critical voter is a voter who, if he changed his vote from yes to no, would cause the measure to fail. A voter's power is measured as the fraction of all swing votes that he could cast, or is proportional to the number of times the voter is critical.
A simple voting example: [6; 4, 3, 2, 1]
There are 10 total votes, and a simple majority of 6 votes was required for a measure to pass.
The numbers in the brackets mean a measure requires 6 votes to pass, and voter A can cast four
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang votes, B three votes, C two, and D one. The winning coalitions, with underlined swing voters, are as follows:
AB, AC, ABC, ABD, ACD, BCD, ABCD
There are 12 total swing votes, so by the Banzhaf index, power is divided thus.
A = 5/12, B = 3/12, C = 3/12, D = 1/12
Sometimes the Banzhaf power index may lead to very surprising result: the voting weight and voting power are totally different concepts.
The discrepancy between voting weight and voting power is more dramatic in the voting body {51:
50, 49, 1} where, again, a simple majority wins. The 50-vote member is critical in all three winning coalitions—(50, 1), (50, 49), and (50, 49, 1), giving him a veto because his presence is necessary for a coalition to be winning—whereas the 49-vote member is critical in only (50, 49) and the 1-vote member in only (50, 1). Thus, the Banzhaf index for (50, 49, 1) is (3/5, 1/5, 1/5), making the 49-vote member indistinguishable from the 1-vote member; the 50-vote member, with just one more vote than the 49-vote member, has three times as much voting power.
What is known today as the Banzhaf power index was originally introduced by Penrose (1946) and went largely forgotten. It was reinvented by Banzhaf (1965), but it had to be reinvented once more by Coleman (1971) before it became part of the mainstream literature.
Using his index, Banzhaf successfully challenged the constitutionality of the weighted-voting system used in Nassau county, New York, showing that three of the County Board's six members were dummies. Banzhaf wanted to prove objectively the unfairness of the Nassau County Board's voting system in which votes were allocated as follows:
* Hempstead #1: 9
* Hempstead #2: 9
* North Hempstead: 7
* Oyster Bay: 3
* Glen Cove: 1
* Long Beach: 1
There are 30 total votes, and a simple majority of 16 votes was required for a measure to pass.
In Banzhaf's notation, [Hempstead #1, Hempstead #2, North Hempstead, Oyster Bay, Glen Cove,
Long Beach] are A-F in [16; 9, 9, 7, 3, 1, 1]
There are 32 winning coalitions, and 48 swing votes:
AB AC BC ABC ABD ABE ABF ACD ACE ACF BCD BCE BCF ABCD ABCE ABCF ABDE ABDF ABEF ACDE
ACDF ACEF BCDE BCDF BCEF ABCDE ABCDF ABCEF ABDEF ACDEF BCDEF ABCDEF
The Banzhaf index gives these values:
* Hempstead #1 = 16/48
* Hempstead #2 = 16/48
* North Hempstead = 16/48
* Oyster Bay = 0/48
* Glen Cove = 0/48
* Long Beach = 0/48
Banzhaf argued that a voting arrangement that gives 0% of the power to 16% of the population is
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang unfair.
Today, the Banzhaf power index is an accepted way to measure voting power, along with the alternative Shapley–Shubik power index .
The Shapley–Shubik power index was formulated by Lloyd Shapley and Martin Shubik in 1954 to measure the powers of players in a voting game. The index often reveals surprising power distribution that is not obvious on the surface.
The constituents of a voting system, such as legislative bodies, executives, shareholders, individual legislators, and so forth, can be viewed as players in an n-player game. Players with the same preferences form coalitions. Any coalition that has enough votes to pass a bill or elect a candidate is called winning, and the others are called losing. Based on Shapley value, Shapley and Shubik concluded that the power of a coalition was not simply proportional to its size.
The power of a coalition (or a player) is measured by the fraction of the possible voting sequences in which that coalition casts the deciding vote, that is, the vote that first guarantees passage or failure.
The power index is normalized between 0 and 1. A power of 0 means that a coalition has no effect at all on the outcome of the game; and a power of 1 means a coalition determines the outcome by its vote. Also the sum of the powers of all the players is always equal to 1.
The Multiplication Rule
If there are m different ways to do X, and n different ways to do Y, then X and Y together can be done in m x n different ways.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
The Number of Sequential Coalitions
The number of sequential coalitions with N players is N! = N x (N-1) x…x 3 x 2 x 1.
Three-Player Sequential Coalitions
Pivotal Player
The player that contributes the votes that turn what was a losing coalition into a winning coalition.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
Computing a Shapley-Shubik Power Distribution
 Step 1. Make a list of all possible sequential coalitions of the N players. Let T be the number of such coalitions.
 Step 2. In each sequential coalition determine the pivotal player.
 Step 3.Count the number of times that P1 is pivotal. Call this number S1.
Repeat for each of the other players to find S2, S3, … SN
 Find the ratio

1
= S1/T. This gives the Shapley Shubik power index of P1.
Repeat for each of the other players to find 
2
, 
3
, …, 
N
. The complete list of  ’s gives the
Shapley-Shubik power distribution of the weighted voting system.
Compute the SSPI for the weighted voting system [6: 4, 3, 2].
Solution:
STEP 1: Make a list of all N! (=3!=6) sequential coalitions containing N players.
{P1 , P2 , P3}
{P1 , P3 , P2}
{P2 , P1 , P3}
{P2 , P3 , P1}
{P3 , P1 , P2}
{P3 , P2 , P1}
STEP 2: In each of these N! sequential coalitions, determine the pivotal player.
(There is always exactly one in each of these coalitions for a total of N!.)
(Remember, P1 has 4 votes, P2 has 3 votes, and P3 has 2 votes.
To determine the pivotal Player in each sequential coalition, add votes from left to right. The Player whose votes cause the coalition's votes to equal or exceed the quota is the pivotal Player. The pivotal Players are underlined below.
{P1 , P2 , P3} 4 (P1) + 3 (P2) = 7
{P1 , P3 , P2} 4 (P1) + 2 (P3) = 6
{P2 , P1 , P3} 3 (P2) + 4 (P1) = 7
{P2 , P3 , P1} 3 (P2) + 2 (P3) + 4 (P1) = 9
{P3 , P1 , P2} 2 (P3) + 4 (P1) = 6
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
{P3 , P2 , P1} 2 (P3) + 3 (P2) + 4 (P1) = 9
Here, N! = 3! = 3(2)(1) = 6.
STEP 3: Determine the number of times Player P is pivotal.
P1 is pivotal 4 times, P2 is pivotal 1 time, P3 is pivotal 1 time.
STEP 4: SSPI(P) is the smaller # (the one from STEP 3) divided by the larger number (the N! from STEP
2). DO THIS FOR EACH PLAYER AND YOU HAVE THE SHAPLEY-SHUBIK POWER DISTRIBUTION.
SSPI(P1) = 4/6 = 2/3 SSPI(P2) = 1/6 SSPI(P3) = 1/6
Example 1
In a weighted voting system: {4: 3, 2, 2}, which player has the most power?
From a Banzaf point of view:
Look at all the winning coalitions.
{P
1
}
{P
2
}
{P
3
}
{P
1
, P
2
} (winning)
{P
1
, P
3
} (winning)
{P
2
, P
3
} (winning)
{P
1
, P
2
, P
3
} (winning)
Find the critical players in each.
winning coalition critical players
{P
1
, P
2
}
{P
1
, P
3
}
P
1
& P
2
P
1
& P
3
{P
2
, P
3
}
{P
1
, P
2
, P
3
}
P
2
& P
3 none
BPI(P) = (# of times player P is critical) / (# times all are critical)
BPI(P
1
) = 2/6 = 1/3 ; BPI(P
2
) = 2/6 = 1/3 ; BPI(P
3
) = 2/6 = 1/3
From a Shapley-Shubik point of view:
Find all sequential coalitions containing all players (N! of them).
{P
1
, P
2
, P
3
}
{P
1
, P
3
, P
2
}
{P
2
, P
1
, P
3
}
{P
2
, P
3
, P
1
}
{P
3
, P
2
, P
2
}
{P
3
, P
2
, P
1
}
Find the pivotal player in each.
sequential coalition
{P
1
, P
2
, P
3
}
{P
1
, P
3
, P
2
}
{P
2
, P
1
, P
3
}
{P
2
, P
3
, P
1
}
{P
3
, P
1
, P
2
}
P
1
P
3
P
1 pivotal player
P
2
P
3
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory --
{P
3
, P
2
, P
1
} P
2
SSPI(P) = (# of times player P is pivotal) / (# times all are pivotal)
SSPI(P
1
) = 2/6 = 1/3 ; SSPI(P
2
) = 2/6 = 1/3 ; SSPI(P
3
) = 2/6 = 1/3
Example 2:
In a weighted voting system: {5: 3, 2, 2}, which player has the most power?
From a Banzaf point of view:
Look at all the winning coalitions.
{P
1
}
{P
2
}
{P
3
}
{P
1
, P
2
}(winning)
{P
1
, P
3
} (winning)
{P
2
, P
3
}
{P
1
, P
2
, P
3
} (winning)
Find the critical players in each.
winning coalitions
{P
1
, P
2
} critical players
P
1
& P
2
{P
1
, P
3
}
{P
1
, P
2
, P
3
}
P
P
1
1
& P
3
BPI(P) = (# of times player P is critical) / (# times all are critical)
BPI(P
1
) = 3/5 ; BPI(P
2
) = 1/5 ; BPI(P
3
) = 1/5
From a Shapley-Shubik point of view:
Find all sequential coalitions containing all players (N! of them).
{P
1
, P
2
, P
3
}
{P
1
, P
3
, P
2
}
{P
2
, P
1
, P
3
}
{P
2
, P
3
, P
1
}
{P
3
, P
1
, P
2
}
{P
3
, P
2
, P
1
}
Find the pivotal player in each.
sequential coalitions
{P
1
, P
2
, P
3
}
{P
1
, P
3
, P
2
}
{P
2
, P
1
, P
3
} pivotal player
P
2
P
3
P
1
{P
2
, P
3
, P
1
} P
1
{P
3
, P
1
, P
2
} P
1
{P
3
, P
2
, P
1
} P
1
SSPI(P) = (# of times player P is pivotal) / (# times all are pivotal)
SSPI(P
1
) = 4/6 = 2/3 ; SSPI(P
2
) = 1/6 ; SSPI(P
1
) = 1/6.
Dr. Tsang
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
The Banzhaf and other power indices, rooted in cooperative game theory, have been applied to many voting bodies, not necessarily weighted, sometimes with surprising results. For example, the
Banzhaf index has been used to calculate the power of the 5 permanent and 10 nonpermanent members of the United Nations Security Council . (The permanent members, all with a veto, have
83 percent of the power.) It has also been used to compare the power of representatives, senators, and the president in the U.S. federal system.
In 1958 six West European countries formed the European Economic Community (EEC). The three large countries (West Germany, France, and Italy) each had 4 votes on its Council of Ministers, the two medium-size countries (Belgium and The Netherlands) 2 votes each, and the one small country
(Luxembourg) 1 vote. The decision rule of the Council was a qualified majority of 12 out of 17 votes, giving the large countries Banzhaf values of 5/21 each, the medium-size countries 1/7 each, and—amazingly—Luxembourg no voting power at all. From 1958 to 1973—when the EEC admitted three additional members—Luxembourg was a dummy. Luxembourg might as well not have gone to
Council meetings except to participate in the debate, because its one vote could never change the outcome. To see this without calculating the Banzhaf values of all the members, note that the votes of the five other countries are all even numbers. Therefore, a winning coalition with exactly 12 votes could never include Luxembourg's (odd) 1 vote; while a 13-vote winning coalitions that included
Luxembourg could form, Luxembourg's defection would never render such a winning coalition losing. It is worth noting that as the Council kept expanding with the addition of new countries and the formation of the European Union, Luxembourg never reverted to being a dummy, even though its votes became an ever smaller proportion of the total.
The Banzhaf power index is a measure of power based on the idea that a voter’s power is his/her ability to swing a decision (change the fate of a motion) by changing the way his/her vote is cast.
Unlike the Shapley-Shubik power index, the order of the votes being casted is unimportant. In most decision making organizations, either public or private, votes are either casted simultaneously or are counted after all members have casted their votes. So the order of voting does not matter. From this consideration, the Banzhaf power index seems to be more relevant and a better measure of the true power of a voter. However, to a certain extent the Shapley-Shubik power index also reflects the power of a voter by counting how often the voter becomes pivotal, especially when the voting process is sequential and votes are counted immediately. In many cases the two power indexes give similar results. But in large organizations with many voters, there are cases when they may be very different. In those situations, Banzhaf power index is probably more reliable, unless you have reason to think otherwise.
Spatial Models are tools used by political scientists to analyze and optimize political moves/strategies in democratic elections. The purpose of these models is to analyze how voters’ opinion on political issues affects politicians’ strategies to grab votes.
In spatial modeling we focus on how issue positions of both voters and candidates (or parties) are
20

MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang translated into voter preferences and candidate strategy. In any spatial model of electoral competition, both voters and candidates are located at ideal points in a multidimensional space, each dimension of which represents a substantive issue. For example, the issue dimension of health care might be represented by a scale that ranges from the belief that government should provide universal health care to the opinion that medical expenses should be paid by individuals and private insurance plans.
The spatial modeling approach pioneered by Anthony Downs (1957), and subsequently developed by numerous scholars, permits us to represent the preferences of voters and the strategies of candidates in a structured manner and to develop mathematical models of the relationships among voters, among candidates, and between voters and candidates.
The simplest type of spatial models are the representation of candidate positions on a dominant issue along a one-dimensional left-right continuum in order to determine the equilibrium or optimal positions of the candidates.
 Two-Candidate Elections –
 Most common contests in the general election are between just two contenders.
 Assumptions for the model of two-candidate elections:
 voters primarily respond only to the positions that the candidates take on one overriding issue (i.e. other factors, such as personality, ethnicity, religion, race, etc., have no effect on outcomes).
 each candidate must take a definite stand on this issue.
Voter Distribution （投票人分布）
A graph that shows the number of individual voters with different political stands on an important election issue represented quantitatively along a horizontal line (e.g. ranging from left-liberal to right-conservative). The number (percentage) of voters who share the attitude is represented by the vertical height.
 Unimodal Distribution – A voter distribution that has one peak (mode).
 Median （中位数） , M, is the point on the horizontal axis where half of the voters’ attitudes lie left and half lie right.
 Maximin （最大最小） position for a candidate is one where no other position can guarantee a better outcome.
 Equilibrium occurs if neither candidate has an incentive to depart from either one of his or her positions.
 Median-voter theorem: In a two-candidate election with an odd number of voters, M is the unique equilibrium position.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
In many occasions, the distribution of voters’ opinion is unimodal in a single issue competition. A mathematical result known as the median voter theorem has been proved.
If the following conditions are true:
[1] The issue is unidimensional and voters decide based on their preferences on that issue,
[2] Preferences are single-peaked,
[3] Voting proceeds under pure majority rule, then the median voter’s ideal point will prevail.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
The median voter theory, also known as the median voter theorem or Black's theorem, was first proposed in Duncan Black's 1948 article, "On the Rationale of Group Decision-making" and popularized by Anthony Downs's 1957 book, “An Economic Theory of Democracy”.
It states that in a majority election, if voter policy preferences can be represented as a point along a single dimension, if all voters vote deterministically for the politician who commits to a policy position closest to their own preference, and if there are only two politicians, then a politician/party maximizes their number of votes by committing to the policy position preferred by the median voter.
This strategy is a Nash equilibrium. It results in voters being indifferent between the candidates and
23

MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang casting their votes for either candidate with equal probability. Hence, in expectation each politician receives half of the votes. If either candidate deviates to commit to a different policy position, the deviating candidate receives less than half of the vote.
Generally, single-peaked preferences are a sufficient condition for the theorem to apply. When preferences are not single-peaked or the policy space is multi-dimensional (e.g., individuals vote on both taxation and public expenditure), the median voter theorem yields no prediction.
Approval voting is a single-winner voting system used for elections. Each voter may vote for
(approve of) as many of the candidates as they wish. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most
24

MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang one vote.
Approval voting can be considered to be a form of range voting (also known as score voting, the score system, or the point system), with only two levels: approved (1) and disapproved (0).
A Quaker poll is a variation with three levels: approved (1), neutral (0), and disapproved (-1).
An example ballot for Approval voting:
Approval voting ballot which marks every candidate the same (whether yes or no) has no effect on the outcome of the election. Each ballot can therefore be viewed as a choice which separates candidates into two groups, or a single-pair of ranks (e.g. if a ballot indicates that A & C are approved and B & D are not, the ballot can be considered to convey the ranking [A=C]>[B=D]).
This special form of preference is called dichotomous preference which means that a voter has bi-level preference for the candidates. All of the candidates are divided into two groups such that the voter is indifferent between any two candidates in the same group and any candidate in the top-level group is preferred to any candidate in the bottom-level group. A voter that has strict preferences between three candidates—prefers A to B and B to C—does not have dichotomous preferences.
Historically, several voting methods which incorporate aspects of approval voting have been used:
* Approving voting was used for papal conclaves between 1294 and 1621, with an average of about forty cardinals engaging in repeated rounds of voting until one candidate was listed on at least two-thirds of ballots.
* In the 13th through 18th centuries, the Republic of Venice elected the Doge of Venice using a multi-stage process that featured random selection and voting which allowed approval of multiple candidates and required a supermajority.
* The selection of the Secretary-General of the United Nations has involved rounds of approval polling to help discover and build a consensus before a formal vote is held in the Security Council.
When all voters have dichotomous preferences and vote the sincere, strategy-proof vote, approval
25

MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang voting is guaranteed to elect the Condorcet winner, if one exists. However, having dichotomous preferences when there are three or more candidates would not be typical. It would be an unlikely situation for all voters to have dichotomous preferences when there are more than a few voters.
For example, in an election with 4 candidates (A, B, C, and D) there are 3 groups of voters:
Group I 4 votes: (A B) (C D),
Group II 3 votes: (C) (A B D),
Group III 2 votes: (B C D) (A).
Assuming each group of voters is sincere, B wins with 6 votes to 5 votes for C, 4 votes for A, and 2 votes for D.
References: http://www.maa.org/devlin/devlin_11_00.html
, “The perplexing mathematics of presidential elections” http://en.wikipedia.org/wiki/Voting_system http://plus.maths.org/latestnews/jan-apr10/election/index.html
Extra-reading:
1. http://www.newyorker.com/arts/critics/books/2010/07/26/100726crbo_books_gottlieb?currentPage
=all , “Win or Lose” The New Yorker Magazine, July 2010,
2. http://www.newsweek.com/2007/12/16/the-hot-or-not-solution.html
, “The ‘Hot or Not’ Solution
A mathematical—but controversial—idea for fixing the flaws in voting”, Newsweek Magazine, Dec
2007.
3. http://yoursdp.org/index.php/perspective/special-feature/3692-electoral-dysfunction-why-democra cy-is-always-unfair , “Electoral dysfunction: Why democracy is always unfair”, Sunday, 09 May 2010,
Ian Stewart, New Scientist.
4. http://www.slate.com/id/2073262/ , “Fifty-Fifty Forever--Why we shouldn't expect America's political ‘tie’ to be broken anytime soon”, Slate Magazine, By Mickey Kaus, Monday, Nov. 29, 2004.
This article provides evidence that this phenomenon predicted by the median voter theorem is taking place in the United States. At first sight this article may be a bit difficult for you to understand because you need to have some understanding of the US domestic politics. However, if you can overlook the details in the US domestic political issues and focus on the big picture, this article is a good illustration of how the median voter theorem is the invisible hand guiding the political life in the US.
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MATH1020: Mathematics For Non-science – Chap 2: Voting Theory -- Dr. Tsang
Writing Project:
Form groups of 3 students in your class. Each group should submit a report (about 3 pages) on the reading material listed above. You can choose one of the following topics:
[A] a report on the first 3 articles above.
[B] a report on the last article above: http://www.slate.com/id/2073262/ + your own research on the median voter theorem.
27