Voting Theory
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- 1 Voting Theory
Voting Theory
Voting Theory is the study that explores ways in which people can express their preferences among a finite series of options, and through an algorithm decide which choice should be taken as a society. It is in general considered at the base of the Democracy. When people think about voting they often think about plurality rule--often mistakenly called majority rule. For this reason we shall try to be very precise with the terminology.
History of Voting Theory
Although there are more ancient examples of some intuitive studies in Voting Theory, usually the field is considered to have started immediately after the French Revolution. It was in the context of trying to build a perfect democratic society that the first precise mathematical studies were done on the possible different ways to vote.
Ways to Vote
Usually in Voting Theory it is considered that if there are n alternatives, each person would order them from the one he likes the most to the one he likes the least. This is called a preference profile, and is usually represented as a column with all the options, with the one that the person prefers most on top and the one that the person least prefers on the bottom. So, for example, if we have the options a, b, c, the column:
c |
a |
b |
The above would mean that the person in question would rather have the option c, and if this is not possible the option a, and if this is not possible the option b. So if we have several columns, one next to the other, each column would represent a how a person is voting:
c | b |
a | a |
b | c |
In this case we have two people, one which would rather have C over A over B, and the other which would rather have B over A over C. Of course you can have multiple people that have the same preferences. This is represented by adding a row of numbers at the beginning, that represent how many people have that preference. So
5 | 7 |
c | b |
a | a |
b | c |
would mean that there are 5 person that want C over A over B, and 7 that want B over A over C. In passing we note that if there are n options, there are n! ways in which people can have their preferences. So, for example, for n=3 we have 3*2*1=6 possible orderings of the preferences. So independently of the number of people we can always write the matrix in which people vote in the form:
... | ... | ... | ... | ... | ... |
a | a | b | b | c | c |
b | c | a | c | a | b |
c | b | c | a | b | a |
with the first raw indicating how many people share that preference.
Although this is not the only way in which people can give their preferences, many of the others can be extracted as a special case of this. For example, in many modern democracies people are invited to express one preference. This is equivalent to considering only the first two rows of this matrix. The row with the highest preference, and the row above it with the number of people who want that preference.
But then there are other voting systems that cannot be represented in this way. For example Approval Voting divides for each voter the options into two sets: approved, rejected. This is like taking the ordered list, and then establishing a threshold, then dividing the proposal into accepted if they are above the threshold, and rejected if they are below. But then any ordering information between the approved proposal is ignored as is any ordering information between the rejected proposals. Since the threshold is different for each user inevitably this method requires information that is not present in the standard preference profile.
Also the way people vote for approval voting is not a generalization to the more common preference profile, because all the ordering information between options is absent. I.e. if we only know which proposals does a voter approve we still don't know in what order are they liked.
ways to count the votes
Once the preference profiles for the participants in the vote have been defined there are multiple ways in which they can be counted. And thus multiple ways to decide who the winner of an election is. The most common system used today is Plurality Rule.
Plurality Rule
In Plurality Rule only the preferred alternative for each voter is considered. Each proposal counts the votes received, and the proposal that receives more votes is selected. This gives a series of problems, for example let us consider having 4 alternatives, a, b, c, d. And let us suppose that we have the following preference profiles:
3 | 2 | 2 |
a | c | d |
b | b | b |
c | d | c |
d | a | a |
Here with plurality rule, we only consider the first preference and the relative weight, thus we just consider the information in bold:
3 | 2 | 2 |
a | c | d |
b | b | b |
c | d | c |
d | a | a |
A would get 3 votes, b would get 0 votes and c and d would get 2 votes each. So a would win the election, even though 3 people out of 7 like it, and the other 4 hates it (it scores in the lowest position). We can see that in this case, for example, there would be a perfectly good alternative, if we only consider more information. And then we would see how the alternative b, although not being anyones favorite, would be an acceptable compromise for everybody.
Antiplurality Rule
Antiplurality Rule is a similar system than Plurality rule. In Plurality Rule the top proposal takes 1 point, and the other takes 0 points. In Antiplurality rule, all but the bottom proposal takes 1 point. So in the previous example:
3 | 2 | 2 |
a | c | d |
b | b | b |
c | d | c |
d | a | a |
With antiplurality rule, we only consider all except the last preference. thus we just consider the information in bold:
3 | 2 | 2 |
a | c | d |
b | b | b |
c | d | c |
d | a | a |
A would get 3 points, B would get 7 points, C would get 7 points, and D would get 4 points. Thus in this case there would be no simple system. Sometimes the solution to this would be using Runoff systems (see below).
Borda Rule
Condorcet Extensions
Runoff Systems
Fractional Voting
See fractional voting.
paradoxes in voting
requirements from a voting system
IIA, no dictatorship,
IIA: Independence from Irrelevant Alternatives
Suppose you go to a restaurant and the waitress offers you a choice between pasta and pizza. You chose pasta. But then the waitress comes back and says:
-I am sorry, we also have soup.
At which you decide:
-But this changes everything, I will have pizza.
This little story give a sense of what it means to have a decision system that is not Independent of Irrelevant Alternatives. Usually we want our voting system to produce a winning result that is Independent of Irrelevant Alternatives. In other words if we consider two different preference profiles, and in the second an alternative that in not winning in the first is being excluded, the result should be the same. Few voting systems possess this property.