User:Lucasvb/Uncertainty in cardinal voting vs. ranked voting

ToDo:

• Rewrite with a more formal argument, explaining an utility scale is necessary for options to be commensurable
• Point out how interpersonal comparisons of utility are not required. The voting procedure can be understood entirely in terms of optimizing the probability of maximizing utility.

Uncertainties in opinions

Humans are not perfect comparison machines, we are full of uncertainties and indifferences, so opinions are fuzzy.

To model this, assume that a voter's utilities $u_{i}$ for candidates $i$ are not exact numbers, but random variables drawn from distributions over some domain (e.g. issue space). When a voter casts a ballot, they're basing themselves on comparisons of these distributions, so rankings/scores themselves should be seen as representing underlying distributions of opinions/utilities.

Uncertainties under cardinal voting

In this way, scores are random variables drawn from certain (assumed unimodal) scoring distributions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_i(s)} for each candidate $i$ and score $s$ (e.g. $S_{A}(3)$ is the probability that this voter scores A=3). If you repeated the election many times and voters kept drawing scores from their own distributions, each voter's score distribution for each candidate would approximately reproduce their true inner scoring distribution in the limit, and thus, the underlying utility distribution as accurately as possible under the circumstances (we're bound by whatever goes in the ballot).

Now imagine you run a score voting election using those ballot distributions themselves. By linearity of score voting, this would be a perfect representation of the scoring distribution of everyone in aggregate.

Additionally, by linearity of expected values, the winner of this election based on distribution-ballots is the same expected winner from a regular score voting election. Both distributions match, there's no distortion. So score voting elections, on average, are robust under opinion uncertainty and statistical noise.

In other words, the criticisms some people point out that scoring is inconsistent or uncertain are not really adequate. It's what makes it work. Cardinal voting statistically internalizes the natural uncertainty of opinions in a consistent way.

Uncertainties under ranked voting

Rankings are trickier. Saying $A>B$ is saying $u_{A}>u_{B}$. But if these utilities are random variables, then $u_{A}>u_{B}$ is a probabilistic statement: there's a certain degree of certainty that a random utility drawn from $u_{A}(x)$ is larger than $u_{B}(x)$. If the distributions overlap, this certainty decreases.

For example, say there are two Failed to parse (syntax error): {\displaystyle σ=1} normal distributions with means Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle μ_A = +1/2} and Failed to parse (syntax error): {\displaystyle μ_B = -1/2} . The probability $p(u_{A}>u_{B})\approx 75\%$. So the ranking $A>B$ is not just imprecise, as in the cardinal case, it is wrong 25% of the time due to the fuzziness of opinions. Sometimes the voter just feels like putting $B>A$ on the ballot, even though A has a higher mean and would be better most of the time.

Also note that two identical candidates (Failed to parse (syntax error): {\displaystyle μ_A = μ_B} , the distributions are identical) would have $p(u_{A}>u_{B})=50\%$, in this symmetric distribution. Two very different candidates would have $p(u_{A}>u_{B})=100\%$ or $p(u_{A}>u_{B})=0\%$, that is, the distributions are far apart but may be reversed in the 0% case.

But ranking systems assume this is all always 100% certain, as if the $u_{A}(x)andu_{B}(x)$ never have any overlap. (This is why I emphasize that it is rankings, not scores, that are assuming unreasonable infinite precision: they effectively assume the utility distributions $u_{i}(x)$ are perfectly sharp Dirac deltas $u_{i}(x)=\delta (x-\mu _{i})$, entirely defined by their means.)

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.

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 $i$ of the $N$ voters has probability $p(u_{A}>u_{B})=p_{i}$, 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.

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 $B(N,n,p)$, for a population of N, n the number of successes (number of A>B). We then want to compute: $p(\mathrm {Awins} )=\sum _{n>N/2}^{N}B(N,n,p)$. This is just the cumulative values of the upper half of the distribution on the $0\cdots N$ interval, which is well known (involves a hypergeometric function, but we'll be qualitative here).

For large N (many voters), this function is 0% for $p<0.5$ and rises sharply around $p=0.5$ to 100%. This means the probability of A winning is highly sensitive to overlapping distributions. Recall that $p=0.5$ is when the distributions are identical.

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.

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.

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 $u_{A}>u_{B}$ comparisons) with no interpersonal comparisons of cardinal utility whatsoever.

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 $p_{A}$ and $p_{B}$, 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.

Remarks

Now, the assumptions here may seem strong, like normal distributions, overlaps and identical voters, but one can make an argument from the Central Limit Theorem using the means of voters opinions. Basically, if you got utilities at random from voters of each faction, and took the mean of many such set of samples, their means would likely follow a normal distribution, and this would fit the above analysis well. But that would involve aggregating cardinal utilities. However, you could do the same analysis under the $p(A>B)+p(B>A)=1$ ballot distribution alone, instead of $u_{A}(x)$ and $u_{B}(x)$, which wouldn't have this problem.

These overlaps may seem insignificant to the typical ranked voter enthusiast, but they're really not. Indifference plays a large role in elections with multiple candidates, and always forcing perfect distinctions is extremely problematic and exacerbates the above problem. Consider also the role of clone candidates in elections and how we're trying to address that with better voting systems right now. Clones, by definition, have significant overlaps to other candidates. Also note that Condorcet cycles are likely to occur when the Smith set candidates have a lot of overlaps.

So this is a very relevant and realistic scenario.

Also, note that none of this relies on "strategical voting vs honest voting", and applies equally to both scenarios. Strategy can merely reshape the original opinion distributions.