Ballots: what is out there?

A team project for MA279, Fall 2013

Team members: Qianyu Deng, Sui Fang, Weichen Gai, Chenkai Wang, Bolun Zhang

Introduction (Chenkai Wang)

The main purpose of using ballots in an election is to record the opinions of electorates and their preferences of the candidates. The goal is to determine a winner from the candidates. Ballots come in many physical forms, such as a piece of paper or a digital document stored in a computer. The actual format of a ballot is called voting system or voting method. A voting system has several built-in rules in order to ensure fair voting during the election. Another functionality of a voting system is counting the voting from ballots to determine a final winner. So to specify a valid voting system, we have to describe two key ingredients: allowable votes, i.e., ballots and the algorithms of collecting votes. In this study, we will study how various voting systems are designed and reveal the mathematical reasons in designing these voting systems.

Fairness Criteria

In order to minimize biased opinions in a voting system, we use fairness criteria to measure the "fairness" of a particular voting system. A fairness criteria is a mathematical description of the rules a voting systems uses. In a formal mathematical treatment, we can define the mathematical meaning of the word "fairness" according to these criteria. Here we describe three important criteria and end with Arrow's impossibility theorem. First, we have the following definition.

Definition Let C be the finite set of candidates and N be the finite set of voters. Let L be the set of all total (linear) ordering on C, i.e., it's the space of all possible ballots submitted by voters. Note since all underlying sets are finite, there is no difference between total ordering and well ordering. Each total ordering assigns a unique natural number $ 1\leq\mathrm{rank}(a)\leq|C| $ to all candidates $ a\in C $ since a finite well ordering is isomorphic to a unique finite ordinal number. A social welfare function is a function $ f:L^N\rightarrow L $. The domain of f is called the set of preference profiles. A generic element of L^N has the form $ \langle \leq_1, \leq_2,\cdots,\leq_N \rangle $, where $ \leq_i $ are total ordering on C, i.e., one generic element (preference profile) represents a possible outcome of all voters. A social welfare function represents the process of choosing the winner from one generic preference profile, i.e., giving the final total ordering of the candidates. Let's denote $ f(\leq_1,\cdots,\leq_N) $ by the single symbol $ \leq $.

Unanimity

Definition Let $ a,b\in C $, if $ \forall i\in N(a<_ib) $, then a < b. In words, if every voters prefer one candidate to another, this order should be preserved in the final decision.

Independence of Irrelevant Alternatives

Definition Let $ r,s\in L^N $ and $ a\in C $, if rank_r(a) = rank_s(a), then rank_f(r)(a) = rank_f(s)(a). In other words, if one candidate has the same ranks in two preference profiles, the rank should also be the same in two corresponding final decisions.

Non-dictatorship

Definition There is no $ i\in N $ such that if $ \langle\leq_1,\cdots,\leq_N\rangle\in L^N $, we have $ f(\leq_1,\cdots,\leq_N)=\leq_i $. In words, the final decision should be different from all elements in a preference profile.

Arrow's Impossibility Theorem

Theorem There is no social welfare function satisfies the criteria of unanimity, independence of Irrelevant Alternatives, and non-dictatorship for candidates of size greater than three.

This theorem imposes a strong constraint on all possible voting systems. In layman's term, there can not be a perfect fair voting system if we need the above three fairness criteria. In the following study, we introduce some actual voting systems which are not on the textbook and demonstrate how they work and how Arrow's Impossibility Theorem applies on them.

Copeland Method (Sui Fang)

History

Copeland method is a Condorcet method to elect winners by using pairwise comparison that the order of candidates is ranked by the difference between the number of pairwise wins and the number of pairwise loses. Supporters argue that this method is fairly understandable and practical in our daily life. Moreover, it is easy for us to calculate data and get results. However, others believe that this method cannot deal with all cases. “When there is Condorcet winner, Copeland method usually meets ties.”—(Wiki) In addition, opponents think this method pay too much attention to the number of rounds’ victories and defeats instead of quantities of voters for candidates. Then I will use an example from website to show how to apply this method and what criterion it fails.

How it works?

step1 Find preference by voters

step2 Pairwise Comparison

step3 Calculate (The # of wins – The # of loses)

step4 Ranking

Example:

“Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities and that everyone wants to live as near to the capital as possible.”—(Wiki)

Step1:Find preference by voters

Step2:Pairwise Comparison

Step3:Calculate (The # of wins – The # of loses)

Step4:Ranking

According to result we get, we find that the order of candidates is the same as the table column2 (26% of voters close to Nashville) in step1. Hence, it doesn't satisfy the Non-dictator criteria.

Kemeny-Young Method (Qianyu Deng)

History

Kemeny-Young method is a voting system first developed by John Kemeny in 1959 and showed as the unique neutral method satisfying reinforcement and the Condorcet Criterion by Peyton Young and Arthur Levenglick in 1978. It uses preferential ballots and pairwise comparison to find the most popular ranking in an election. This method satisfying Condorcet Criterion since if there is a Condorcet winner, then it is always the most popular one.

How it works?

step1 pairwise comparison

step2 Create a tally table of the pairwise comparison

step3 Count ranking score

step4 Find the ranking which gets the highest ranking score

Now,let's look at the previous example of the election on the location of capital of Tennessee we use before to see how it works.

Example

Step1:Find the pairwise comparison in terms of the population percentage

Step 2: Create a tally table of the pairwise comparison

Step3: Count ranking score
Suppose we want to calculate the ranking score of the following ranking:

This ranking satisfies the preferences Memphis>Nashville, Memphis>Chattanooga, Memphis>Knoxville, Nashville>Chattanooga, Nashville> Knoxville, Chattanooga>Knoxville.The respective score, according to the tally table, are:
Memphis>Nashville : 42
Memphis>Chattanooga: 42
Memphis>Knoxville: 42
Nashville>Chattanooga: 68
Nashville> Knoxville: 68
Chattanooga>Knoxville: 83
So, 42+42+42+68+68+83 = 345.
Continuing calculating in this way, we can get the following table of all possible ranking score:
Here, we denote Memphis as M, Nashville as N, Chattanooga as C, and Knoxville as K in abbreviation.

Step4: Find the ranking which gets the highest ranking score.
we see that the highest ranking score is 393 which is the score of the ranking of

According to the ranking result, we can see that this method fails to satisfy non-dictatorship criterion like the previous method does.

Schulze Method (Weichen Gai)

History

The Schulze method, also known as Schwartz Sequential Dropping (SSD), was developed in 1997 by Markus Schulze in order to select a single winner but can also apply to select a list of winners. It was first used in public mailing list in 1997-1998 and then widely adpoted.

How it works?

step 1 Pairwise comparison

step 2 Create a matrix of pairwise preferences. (We can color the cell with green if d[X,Y]>d[Y,X], otherwise color it with red for better visualization.)

step 3 Identify the strongest paths and the strengths of each paths.

step 4 Ranking.

Example  

In the following example 45 voters rank 5 candidates.

• 5 ACBED
• 5 ADECB
• 8 BEDAC
• 3 CABED
• 7 CAEBD
• 2 CBADE
• 7 DCEBA
• 8 EBADC

Step 1. Pairwise comparison

For example, when comparing B and C, there are 8+8=16 voters who prefer B to C. So d[B,C]=16.

Step 2: Create a matrix of pairwise preferences

Step 3: Identify the strongest paths and the strengths of each paths.

Each direction of an arrow is determined the value of d[X,Y] and d[Y,X]. The arrow points at X if d[X,Y]>d[Y,X] and vice versa.

From the picture we can see that, for instance, if we want to determine the strongest path from A to C, we have two options, A-C or A-D-C. The strength of each path is the value of its weakest link. In this case, the strongest path from A to C is A-D-C that has strenth 28> that of A-C, which has strenth 26.

Step 4: Ranking

From step 3 we have the strength of each strongest path.

The winner is X between X and Y if p[X,Y]>p[Y,X]. For example, the winner is C since p[C,D]=29>p[D,C]=28.

As a result, the Schulze ranking is E>A>C>B>D.

Instant-runoff Voting Method (Bolun Zhang)

History

Instant runoff voting was devised in 1871 by American architect William Robert Ware, although it is, in effect, a special case of the single transferable vote system, which emerged independently in the 1850s. Unlike the single transferable vote in multi-seat elections, however, the only ballot transfers are from backers of candidates who have been eliminated.

How it works?

step1 Count the first-place votes for each candidate.

step2 Eliminate the candidate with the fewest first-place votes.

step3 Count ranking score

step4 Repeat the process until there is a candidate with a majority of first-place votes.

Example

Step1: Count the first place votes for each candidate. Memphis 42%
Nashville 26%
Chattanooga 15%
Knoxville 17%

Step2: Eliminate the candidate with the fewest first-place votes.
Chattanooga is eliminated

Rearrange the ballots:

Step3: Repeat the process: Count the first place votes for each candidate.
Memphis 42%
Nashville 26%
Knoxville 32%
Nashville gets the fewest first-place votes and is eliminated.

Rearrange the ballot:

Count the first-place votes for each candidates.
Memphis 42%
Knoxville 58%

Knoxville gets 58% votes over a half, which means Knoxville has a majority of first-place votes.

Knoxville is the winner.

Therefore it does not satisfy Independence of Irrelevant Alternatives criteria.

Bibliography

1. Ballots. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Ballot
2. Copeland. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Copeland's_method
3. Schulze Method. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Schulze_method
4. Instant-runoff voting. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Instant-runoff_voting

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