User:Lucasvb/Uncertainty in cardinal voting vs. ranked voting: Difference between revisions
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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 p(<math>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. |
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 p(<math>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. 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). |
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 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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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 more much complex scenario to analyze, but similar effects will happen for each of the two factions. |
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== Remarks == |
== Remarks == |
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