User:Lucasvb/Uncertainty in cardinal voting vs. ranked voting: Difference between revisions
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Repeating the election many times you'll get the same ballot distribution for each voter (e.g. in the example, B>A 25% of the time), just as in the score case. The problem arises from the aggregation. |
Repeating the election many times you'll get the same ballot distribution for each voter (e.g. in the example, B>A 25% of the time), just as in the score case. The problem arises from the aggregation. |
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Let's suppose a simple 2 candidate election between A and B, majority preference wins. (This can be a pairwise match in a Condorcet system, for example). Let's assume each <math>i</math> of the <math>N</math> voters has probability |
Let's suppose a simple 2 candidate election between A and B, majority preference wins. (This can be a pairwise match in a Condorcet system, for example). Let's assume each <math>i</math> of the <math>N</math> voters has probability <math>p(u_A > u_B) = p_i</math>, due to whatever overlap (more overlap = p closer to 50%, if distributions are symmetric). We want to compute the probability that A wins the election, that is, the probability that a majority M > N/2 of voters has A>B against the minority's B>A. |
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For the sake of simplicity and understanding of the problem, and of a closed-form solution, we can assume all ps are the same (this is an unrealistic situation where everyone has exactly the same beliefs). Since all voters are exchangeable, this is a |
For the sake of simplicity and understanding of the problem, and of a closed-form solution, we can assume all ps are the same (this is an unrealistic situation where everyone has exactly the same beliefs, and in a perfectly precise world decision would be unanimous). Since all voters are exchangeable, this is a binomial distribution <math>B(N,n,p)</math>, for a population of N, n the number of successes (number of A>B). We then want to compute: <math>p(\mathrm{A wins}) = \sum^{N}_{n > N/2} B(N, n, p)</math>. This is just the cumulative values of the upper half of the distribution on the <math>0 \cdots N</math> interval, which is well known (involves a hypergeometric function, but we'll be qualitative here). |
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For large N (many voters), this function is 0% for <math>p<0.5</math> and rises sharply around <math>p=0.5</math> to 100%. This means the probability of A winning is ''highly'' sensitive to overlapping distributions. Recall that <math>p=0.5</math> is when the distributions are identical. |
For large N (many voters), this function is 0% for <math>p<0.5</math> and rises sharply around <math>p=0.5</math> to 100%. This means the probability of A winning is ''highly'' sensitive to overlapping distributions. Recall that <math>p=0.5</math> is when the distributions are identical. |
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This means that under ranked voting for large N, any small variation from indifference is immensely amplified on average. So even if there's a 10-30% chance voters actually prefer B>A due to similarity, statistically the ranked election will elect |
This means that under ranked voting for large N, any small variation from indifference is immensely amplified on average. So even if there's a 10-30% chance voters actually prefer B>A due to their similarity, statistically the ranked election will elect A>B 100% of the time due to the way the individual probabilities are combined. |
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The ranked system is incapable of reproducing the underlying distributions of uncertainty on the average election. Therefore, small individual biases are amplified, and voters will on average betray their own interests significantly if there are similar candidates running and they are slightly uncertain about the candidates. |
The ranked system is incapable of reproducing the underlying distributions of uncertainty on the average election. Therefore, small individual biases are amplified, and voters will on average betray their own interests significantly if there are similar candidates running and they are slightly uncertain about the candidates. |
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Note that this has nothing to do with aggregating the utility of the voters in a ranked system. Each voter is betraying their own interests by the aggregation of the rankings itself. This is a strictly ordinal model (beyond the internal cardinal utility, only used to create the <math>u_A > u_B</math> comparisons) with no interpersonal comparisons of cardinal utility whatsoever. |
Note that this has nothing to do with aggregating the utility of the voters in a ranked system. Each voter is betraying their own interests by the aggregation of the rankings itself. This is a strictly ordinal model (beyond the internal cardinal utility, only used to create the <math>u_A > u_B</math> comparisons) with no interpersonal comparisons of cardinal utility whatsoever. |
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The above example assumed every voter had the same beliefs. A similar but more complex reasoning could be used for the case of two factions with <math>p_A</math> and <math>p_B</math>, by assuming a multinomial distribution and the probability that a given population split produces a majority. This is |
The above example assumed every voter had the same beliefs, which is very unrealistic, but it illustrates the internal microcosm of any faction that operates under similar beliefs. A similar but more complex reasoning could be used for the case of two factions with <math>p_A</math> and <math>p_B</math>, by assuming a multinomial distribution and the probability that a given population split produces a majority. This is a much more complex scenario to analyze generally, but similar effects will happen for each of the two factions. |
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== Remarks == |
== Remarks == |
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