Structure of a Voting System
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I've been thinking about how to be careful when describing a general voting system in a way that clearly and formally delineates its structure---more or less I want to make sure that we (including myself) are all mostly on the same page about the general, functional anatomy of a voting system. I will list some of the structures/apparatuses that appear to be central to the construct of a voting system with the hope that others will examine it for generality and completeness, and perhaps also for specificity. I don't purport this to be complete, just a starting point.
Firstly there are some obvious possible constructs: namely, the electorate and the candidates. Mathematically these can be seen as unstructured sets V and C, respectively. There is also the set B of all possible ballots. By casting ballots, the voters in V define a function from V to B. But more generally, we could consider the voters to define a probability measure P over the set of all possible ballots, turning it into a probability space (B,P). For example, ordinarily for a ballot b in B, its measure P(b) is simply the fraction of voters that indicated b as their vote, which seems to be a natural requirement in terms of the relationship between V and P.
Next, there is the decision algorithm D. In my conception, D takes as its inputs the probability measure P over B determined by V, and perhaps auxiliary information I, and returns as its output, for generality, a probability measure Q over C. For example, a single-winner system might return a probability measure that is fully concentrated on the single winning candidate, while a multi-winner system might return a measure with full support over the winning subset of the candidates. Notice that the auxiliary information I can be fixed to produce a deterministic system, or the information could in principle be random to produce a nondeterministic system, or perhaps the information could be dynamic.
My reasoning for this measure-theoretic conception is generality. One might argue that there is a missing apparatus here: how a probability measure Q over C actually translates into an electoral decision. However, as I hope I made somewhat clear, the translation can be very straightforward, and the computation of a more arbitrary probability measure can be seen as an intermediate step in the computation of a "final" probability measure that "directly" indicates the election results. For instance, for a single-winner system, one might only allow the final measure to be concentrated on a single candidate as indicated above, but could utilize a less concentrated measure to construct the final measure, for example, by choosing to concentrate all of the probability on the candidate with the highest measure, or some such thing. The translation will be a social interpretation of the results and will constitute the electoral agreement pertaining to the intended correct operation of the voting system.
This conception of the decision algorithm D as taking as its input a distribution P over B and constructing a distribution Q over C is in line with the argument that a voting system is essentially a kind of compression algorithm. The fact that the best compression algorithms used in industry exploit dynamic information methods (machine pseudo-learning) to achieve state of the art results is suggestive, in my opinion. Any thoughts are welcome.
score-stratified-condorcet [10] cardinal-condorcet [9] ranked-condorcet [8] score [7] approval [6] ranked-bucklin [5] star [4] ranked-irv [3] ranked-borda [2] for-against [1] distribute [0] choose-one [0]
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@cfrank said in Structure of a Voting System:
The fact that the best compression algorithms used in industry exploit dynamic information methods (machine pseudo-learning) to achieve state of the art results is suggestive, in my opinion. Any thoughts are welcome.
Voting methods have to deal with the fact that voters actively try to manipulate the decisions made by the method. This can be a challenge to ML models, for example, spammers design their spam to evade getting caught by fillters. However, spam filters can update themselves against the latest strategies, which doesn't really apply to voting methods. So there are unique challenges to voting method design that don't necessarily apply in ML.
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@marylander I don’t understand why a voting system shouldn’t be adaptive, it seems like an arbitrary requirement.
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@cfrank Who gets to decide how it adapts?
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@andy-dienes that would just be included in the rules for the operation of the system.
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@cfrank ok but what would those rules be?
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@andy-dienes I think in principle alternative dynamical rules could vary as much as different voting systems vary from each other already, and the benefits of utilizing adaptive procedures would need to be balanced against the detriments to complexity.
I defined one method before that uses past election data or predefined distributions to measure a formal construct of consensuality among the candidates, and then chooses the highest scoring candidate according to that metric. The distributions could be fixed, or could be updated to include or adapt to newer election data in any prescribed way that seems reasonable.
I think the specific kind of procedure would depend on the kind of information that is being utilized.
As an explicit example that I considered, for a score system I defined a candidate to be "(S,P)-consensual" if that candidate was scored at least an S by at least a P-fraction of the electorate. Then I defined the (S,P)-set of candidate C as the collection of all (S,P) such that C is (S,P)-consensual, and essentially measured the “size” of their (S,P)-sets.
The way of measuring the size used probability distributions for the fraction of the electorate that scored a random candidate at least any fixed score.
