About voting procedures

The aim of this page is to quickly introduce some notions of voting theory, and how they are interpreted in Whale, in order to help you better understanding the results you get when using Whale.

General notions

The problem that the web interface you are currently using (or just quickly visiting) is dedicated to is collective decision making. This problem occurs every time a group of people have to take a decision collectively. The most prominent example of such a problem is a political election, occurring when a collectivity has to elect an individual that will represent them.

The collective decision making problem can be stated has follows:

Given a set C of candidates (also called issues, alternatives, or outcomes), and a set V of voters having some preferences over the set of candidates, find the candidate that best reflects the preferences of the agents.

Of course this definition is a bit informal and fuzzy, both on the notion of preferences, and on what the "best candidate" should be. Fortunately, voting theory gives us some hints on how to compute such a candidate.

Preference representation

Preference is a general term referring to the fact that a human being is not indifferent between several states of the world or outcomes. There are several ways to formalize this notion (the usual way in voting theory being to consider that each voter gives a total ordering over the set of candidates). We will distinguish the way the voters can express their preferences in Whale (the language), and the way it is formally represented and interpreted by Whale. For the latter, we will consider that the preferences are formally represented by natural numbers, called utilities: one for each pair (voter, candidate). The higher the utility is, the more preferred the candidate is by the voter

Representation language for one voter v → utility profile ⟨ u(v, c₁), …, u(v, c_|C|) ⟩

Rankings with no ties: linear orders

The voter is expected to completely rank all the candidates, from the first one to the last one. The candidate labelled "1st" is the most preferred one; the next one is labelled "2nd", and so on. No ties are allowed. If there are |C| candidates,the utility of a given candidate c is |C| + 1 minus the number of candidates that are ranked strictly over c (the first one gets a utility of |C| and the last one a utility of 1).

Rankings with ties: total preorders

You are expected to completely rank all the candidates, from the first one to the last one. The candidate(s) labelled "1st" is (are) the most preferred one(s); the next one is labelled "2nd", and so on. Ties are allowed. If there are |C| candidates, the utility of a given candidate c is |C| + 1 minus the number of candidates that are ranked strictly over c (the first one gets a utility of |C|, and two candidates which have the same rank get exactly the same utility).

NB: with this kind of preference representation, profiles (1st, 1st, 2nd) and (1st, 1st, 3rd) are completely equivalent (and will be translated into the second one for internal representation).

Utilities

The voter is expected to give a utility to each candidate, among a fixed range (e.g. from 0 to 10). The higher the utility is, the better the candidate is. Ties are allowed. This utility is exactly the one that will be used by Whale for computing the candidate scores.

Ballot types

These parameters do not influence the way the best candidate is elected, but only concern the infomation about votes (ballots) themselves, that will be available to the other voters.

Privacy

In Whale, there are three different levels of privacy:

• Public ballots: in this case, the votes are completely visible, for anyone visiting the poll page.
• Private ballots: in this case, the votes are only visible to the poll owner (and hidden to anyone else).
• Private ballots: in this case, noone (including the poll owner) can see the votes. Only the results are visible.

Anonymity

The anonymity property is orthogonal to privacy. If a poll is anonymous, votes might be visible (according to the privacy property), but the voters' names will be anonymized (e.g. Voter #1...).

Voting procedures

Scoring methods

Scoring methods are based on the computation of a score for each candidate, computed from the utilities given by each voter. Each scoring method M is defined by a scoring function g_M which maps every possible utility u to a score g_M(u). The score obtained by a candidate c is simply Σ_v∈V(g_M(u(v, c))).

The most classical scoring methods are the following:

• Borda: g_Borda(u)=u. The Borda score of a candidate is the sum of utilities associated to this candidate.
• Plurality: g_Plurality(u)={1 if u = max, 0 otherwise}. The plurality score of a candidate c is the number of voters that give the maximal possible utility to c.
• Veto: g_Veto(u)={0 if u = min, 1 otherwise}. The veto score of a candidate c is the number of voters that do not give the minimal possible utility to c.

Condorcet-based methods

The Condorcet score N(c, c') of a pair of candidates is the number of voters v such that u(v, c) > u(v, c'). The Condorcet matrix associates to each pair of candidates (one per row, one per column) the corresponding Condorcet score. A Condorcet winner is a candidate c such that N(c, c') > |V| / 2 for each c' ≠ c. Such a candidate may or may not exist. If it exists, it is unique.

The following voting methods are Condorcet-consistent, which means that they elect the Condorcet winner if it exists.

• Maximin (Simpson): The Simpson score of a candidate c is min_{c' ≠ c}N(c, c').
• Copeland: The Copeland score of a candidate c is |{c' ≠ c such that N(c, c') > N (c', c)}| - |{c' ≠ c such that N(c', c) > N (c, c')}|, that is, the number of candidates that c beats in a Condorcet fight minus the number of candidates that beat c in a Condorcet fight.

Bucklin

The Bucklin score of a candidate c is the minimum integer l such that c is ranked among the l top candidates by more than half of the voters. The smaller Bucklin score a candidate has, the better she is.

Run-off voting procedures

Run-off voting procedures are performed in successive rounds. Among those, the two prominent ones are the following.

• Single Transferable Vote (STV): In each round, the candidate with the lowest plurality score is eliminated (ties might be broken by another rule, like the Borda rule for example). Each vote for this candidate, if any, transfers, for the next round, to the candidate coming next in the order given by that vote.
• Plurality with run-off: In each round, we keep, for the next round, only the candidates with the highest plurality scores. Votes are transferred exactly like for the STV rule.

In Whale, the score displayed for these procedures is the round number that eliminates this candidate (the higher the better). It is not very clear how votes can be transferred from round to round when the procedure has to deal with scores, and not ranks. Therefore, these procedures are not used with score-based preferences.

Social Welfare Orderings (SWO)

Social Welfare Ordering are classical aggregation procedures when the agents preferences are given by utilities.

• Leximin SWO: for each candidate c, a vector L(c) is computed. L(c)_k is the number of voters that give the score k to this candidate. Then the vectors are compared component by component (a candidate is eliminated as soon as its current component is the maximal one among the candidates).
• Leximax SWO: same as leximin, except that the vectors are compared in reverse order, and that a candidate is eliminated as soon as its current component is not the maximal one among the candidates.
• Nash SWO: the Nash score of a candidate cis just the product Π_v (u(v, c) + 1) (NB : the "+1" is just to avoid the absorption effect of utility 0).