Notes Voting

8/12/2019 Notes Voting

1/8

Ch. 17.1: Voting Methods

The mathematical/sociological study of voting methods is more properly referred to asSocial Choice Theory. It is a very important and hotly-debated subject, for obvious

reasons! Methods of voting have been developed and debated since democracy in an-cient Athens (probably earlier). It was in 1785 that a French philosopher named JeanAntoine Nicolas Caritat (the Marquis of Condorcet) published two foundational math-ematical results, now known as Condorcets Jury Theorem, and Condorcets Paradox.The term Social Choice Theory, however, was not coined until after 1951, with thepublishing of Social Choice and Individual Values by American economist KennethArrow. These two men are often credited as the founders of the modern theory.

In this section, we will look at seven different methods of casting and evaluating votes:

i. Majority Ruleii. Plurality Rule

iii. Borda Count

iv. Hare Method

v. Pairwise Comparison Method

vi. Approval Method

vii. Tournament Method

Here on, we consider the general problem in which a group ofn people must chooseexactly 1 option(e.g. C) from a list of distinct options (A,B,C, etc.). These optionscould be candidates running for office, bills, legal decisions, etc. The one that is chosenwe shall call the winner.

1


2/8

Plurality Voting

In plurality voting (also called first-past-the-postvoting in reference to horse races),each voter casts exactly 1 vote in favour of just 1 option. The winner is the option

that earns a certain number of votes.

Majority Rule:

The winner is the option that earns more than 50% of all votes (asimple majority). If the number of votes, n, is even, then a simplemajority is at least

n

2+ 1.

Ifn is odd, then a simple majority is at least

n+ 12 .

Plurality Rule:

The winner is the option that earns more votes than any other candi-date individually (a plurality).

Example 1: Imagine we have a group of 15 people that are voting on taking 3 differentcourses of action, A,B,C. Suppose the results of the vote are as follows:

Preference: ABC ACB BAC BCA CAB CBANo. of Votes: 3 1 5 1 0 5

Here, each person casts a vote by writing their order of preference (top line). For themajority/plurality methods, we will assume that for each vote, the 1st choice (left-mostletter) gets the entire vote, and the others we ignore (for instance, 5 votes for the orderCBA, means 5 votes for C alone; A and B are not counted). Which alternative wonby plurality? Majority?

ANSWER: We start by tallying up the total number of votes for each option:

Option No. of Votes

A 4B 6C 5

2


3/8

From this, we immediately see that option B wins by plurality, since B got the morevotes than any other option. Since the total number of votes, 15, is odd, we need atleast (15 + 1)/2 = 8 votes for a simple majority. However, noneof the options gotmore than 6 votes, so there is no majority winner.

This example shows us that an election does not always produce a majority winner. Itis even possible for there to be no plurality winner, if, say, two options get the exactsame number of votes. However, if there is a simple majority winner, then that optionwill also be a plurality winner.

Preferential Voting

In a prefential voting system, voters are required to rank candidates in order of rela-tive preference (like we did in the above example). The winner can then be determinedwith a variety of methods. One method is called the Borda Count, invented in 1770by the French mathematician and physicist Jean-Charles, chevalier de Borda.

The Borda Count:

Each voter ranks all candidates in order of preference. If there arem candidates, then for each vote, m points are assigned to the 1stchoice, (m 1) points are assigned to the 2nd choice, and so on.

The candidate that receives more points in total than any other isdeclared the winner.

3


4/8

Example 2: Suppose 6 people vote on 4 different candidates (A,B,C,D) by rankingthem in order of preference (4=highest preference, 1=lowest preference). If the resultsof the vote are the following:

BallotsCandidates #1 #2 #3 #4 #5 #6 Total Points:

A 2 3 1 1 4 4 15B 3 2 4 3 1 1 14C 1 1 3 4 2 3 14D 4 4 2 2 3 2 17

then we would conclude that candidate D won the election by the Borda Count, becauseshe earned the most total points (17). Notice that if we only counted peoples 1st choice

votes, then we would have A and D with 2 votes each, and B and C with 1 vote each then there would be no plurality winner.

Since the Borda Count is more sensitive to voters preferences, it is technically possiblefor an option to win a majority of 1st-choice votes, and yet lose in a Borda Count.

Example 3: Suppose 3 candidates (A,B,C) are running for an election, and there are100 voters. Suppose we list the results as we did in Example 1:

Choices: ABC ACB BAC BCA CAB CBANo. of Votes: 25 26 2 8 4 35

Who won according to the plurality method? Who won by the Borda Count?

ANSWER: As we did before, we can tally all 1st-choice votes for each candidate:

Candidate No. of Votes

A 51B 10C 39

We clearly see that A wins by simple majority. For the Borda Count, we must tallythe points given to each candidate by all voters:

4


5/8

Points for A: 25(3) + 26(3) + 2(2) + 8(1) + 4(2) + 35(1) = 208 pts

Points for B: 25(2) + 26(1) + 2(3) + 8(3) + 4(1) + 35(2) = 180 pts

Points for C: 25(1) + 26(2) + 2(1) + 8(2) + 4(3) + 35(3) = 212 pts

Now, C wins over A by a few points. The reasoning here is that while A was supportedby over 50% of voters, a comparably large fraction, 43%, voted A as their last choice.By contrast, a majority of the supporters of A and B chose C as their 2nd-choice.Thus, candidate C is, in some sense, the best compromise.

The Hare Method is named after a 19th cen. English political scholar, Thomas

Hare. It seems to be more commonly referred to as Instant-Runoff Voting, or thePlurality with Elimination Method. This last name describes the process nicely:basically, it involves several rounds of plurality voting. At the end of each round, theoption with the fewest votes is eliminated from the race, and the remaining options are

voted on in the next round. This continues until some option obtains a majority.

Instead of several rounds of plurality voting, the same theoretical result can be obtainedwith just one round of preferential voting (and multiple rounds of vote counting), asindicated here:

Hare Method

Each voter ranks the candidates in order of preference on a ballot;each ballot counts as exactly 1 vote for the highest-ranking candidateon that ballot that has not yet been eliminated from the race.

If at the end of a count, no candidate has a majority, then the can-didate with the fewest number of votes is eliminated, and the voteis recounted. This process continues until some candidate receives amajority, and they are declared winner.

5


6/8

Example 4: Consider the following election results:

Choices: ABC ACB BAC BCA CAB CBANo. of Votes: 0 7 5 3 4 1

Determine a winner using Plurality with Elimination counting.

ANSWER: There are 20 votes in total, so a majority requires at least 11 votes. Webegin with the same counting method that we used in Example 1: we tally the numberof 1st-choice votes for each candidate, and get the following results:

Candidate No. of Votes

A 0 + 7 = 7B 5 + 3 = 8C 4 + 1 = 5

For the 1st round, no one wins a majority (though B does win a plurality). Thus, weeliminate the contender with the fewest number of votes, C:

2nd Round Choices: AB AB BA BA AB BANo. of Votes: 0 7 5 3 4 1

A recount of these new preferences gives us:

Candidate No. of VotesA 0 + 7 + 4 = 11B 5 + 3 + 1 = 9

For the 2nd round, A has a majority, and is therefore declared the winner.

Note that even though B won a plurality in the beginning, he still lost the electionbecause the C supporters were allowed to change their vote. One of the great benefitsof the Hare Method is that a vote is hardly ever wastedon an unlikely candidate i.e,

strategic voting is not as effective as with the plurality method or Borda Count.

The Pairwise Comparison Method is a system of voting specifically designed tosatisfy the Condorcet Criterionthat well see in the next section.

6


7/8

Pairwise Comparison Method

Each voter ranks the candidates in order of preference. Each can-

didate is then given points depending how they rank against everyother candidate.

If A is more often voted for ahead of B, then A gets 1 point, and Bgets 0 points. If A and B are tied, then they each receive 1/2 point.The candidate with the most points in total is the winner.

Example 5: A club of 30 members needs to elect a president. There are 4 candidates:Kenneth (K), Linda (L), Mike (M), and Nora (N). Assume all 30 members vote, andthe results are:

Choices: KLMN KNLM LNKM LNMK NMLK MKLN MLKNNo. of Votes: 9 1 5 3 2 8 2

Who wins according to the Pairwise Comparison Method?

ANSWER: Lets start with Kenneth and Linda: we see from the table above that Kis voted over L in 9 + 1 + 8 = 18 ballots. Thus L is voted over K in 30 18 = 12

ballots (since there are 30 votes total). Hence K gets 1 point, and L gets 0 points.We do this for each possible pairing of the 4 candidates. An easy way to keep track ofthis is with a table:

K L M N

K (12) 0 (15) 1/2 (10) 0

L (18) 1 (12) 0 (3) 0

M (15) 1/2 (18) 1 (11) 0

N (20) 1 (27) 1 (19) 1

Total: 2.5 2.0 1.5 0.0

(Each column lists the votes and points earned against each candidate on the left).We see that Kenneth is the winner with a total of 2.5 points, more than any othercandidate.

7


8/8

Approval Voting

In contrast to preferential methods, approval voting is more similar to the pluralitymethod, except that voters can vote for not just one, but manycandidates at once, if

they so desire.

Approval Voting Method

Each voter may give up to 1 vote for each option that meets his/herapproval. The option earning more total votes than any other optionis declared the winner.

Example 6: Imagine a ski club with 9 people. Suppose they need to choose the des-tination for their winter ski trip. Members of the club nominate 5 different differentresorts, and they all hold an approval vote. The results are:

BallotsResorts #1 #2 #3 #4 #5 #6 #7 #8 #9 Total:

Tussey x x 2Montage x x x x 4

Snowshoe x x x x x 5Lake Tahoe x x x x x x x 7

Beaver x x x x 4

We see that Lake Tahoe wins the vote, having the most approvals.

8

Documents

Notes Voting