Limitations of spatial models of voting

"Spatial model of voting"are ubiquitous in theoretical study and simulations of voting methods. This article describes many limitations of spatial models of voting.

In spatial models of agent behavior, agents (e.g. voters, candidates) are placed in an abstract geometric space, usually Euclidean, in which each dimension denotes some ideological alignment or opinion on an issue. The behavior of agents is modeled by how "close" (under some appropriate metric) they are to other agents in this space. In the context of voting, voters are modelled as ranking candidates depending on their proximity within this space.

However, models based too strictly on geometric representations have challenges representing both voters and candidates. This article describes some of the challenges.

Number of dimensions

The number of dimensions chosen for this geometric embedding imposes fundamental restrictions on the allowed number of candidates. There is a limited number of dimensions that may be effectively distinguished by the voters using ballots, as there is only a finite number of regions possible for each possible ranking assignment of candidates. Conversely, an insufficient number of candidates in a ballot (either by a small number of candidates or arbitrarily restricting the ballot) will also fundamentally restrict the effective opinion space voters can express, as the effective dimensionality is inherently reduced.

The specific numerical results below assume an Euclidean space and Euclidean distances, but similar qualitative arguments apply to any spatial model and chosen metric, as well as the actual real-life behavior of voters (although quantifying it is impossible).

How many ballots could voters actually cast?

With ${\displaystyle n}$ candidates in an election, be it rated or ranked, there are ${\displaystyle n!}$ possible rankings between candidates. These ${\displaystyle n!}$ possible preferences indicate all the possible distinctions voters could ever possibly make between alternatives, no matter how those distinctions are made.

This is true independently of any abstract mathematical model of reality of human behavior, as it is a constraint of the ballots themselves and their information content. In reality, these distinctions are based on some internal attributes and judgements voters have about the world and the candidates, and this is the information voters want ballots to convey. This is what voting methods attempt to represent from voters.

But due to several limitations, not all these ballots can actually get cast in an election. In practice, we only observe a few preference orders, indicating that there's a lot of correlation between voters and between candidates, or putting it in another way, that the "space" of attributes relevant in the election is smaller than the one expressible by the ballots. This is important to consider when developing a mathematical structure to abstractly discuss real voting methods and voter behavior.

In other words, while a spatial model attempts to reverse engineer real-life behaviors and construct a model of the information underlying an election, the ballots themselves, be it from real life elections or computer simulations, can only capture some of the information.

Mathematics of a spatial model

For d=2 dimensions and n=3 candidates (ABC), there is a region in the space for each of the 3! = 6 possible rankings between the candidates, so no information is lost: all possible opinion distributions and ballots can exist.

In a ${\displaystyle d}$-dimensional spatial model for voter behavior, in which voters judge candidates in terms of proximity using ${\displaystyle d}$ separate attributes (no matter how such attributes are used), there is a fundamental mathematical limit for how many ballots can possibly occur, in any arbitrary distribution of voters and candidates. (Equalities or partial rankings do not matter in this analysis, as they can be included in the same space with minimal adjustment.)

With a fourth candidate there are 4! = 24 possible rankings, but it's impossible to partition the space (under Euclidean metric) into more than 18 regions, one example as shown here. Therefore, many of the rankings cannot occur under this 2-dimensional model, e.g., any ballot with D ranked last, in the image. For 3 dimensions, we can construct all of the 24 required regions for the ballots.

This restriction is less about the existence of an actual "Euclidean space of opinions" in the abstract (i.e. the accuracy of our chosen models), but instead, about how candidates could ever be classified in terms of a finite set of attributes by voters. Any comparison voters are actually doing between any two candidates must occur in at least one attribute between them, that can be used to classify the voter's preference one way or another. This dimension ${\displaystyle d}$ quantifies how many such attributes must exist in order for us to observe a given set of ballots. Thus, this is a very real and fundamental limitation of any realistic and operational description of voter behavior.

Note, however, that opinion space is distinct from ballot space. Opinion space is what contains the actual distribution of voters and candidates, and this may have any number of dimensions, voters and candidates. In contrast, ballot space is the space of possible ballots that voters can cast, which confines them to express their opinions in a particular way. One can think of ballot-casting as a function that takes a voter's opinion and that of the candidates (plus additional external factors), and produces a ballot: ${\displaystyle \text{ballot} = f(\text{voter opinion}| \text{distribution of candidates}, \text{external factors})}$. This article refers to the limitations of this function, that is, how much information about opinion space can in principle survive inside ballot space.

To address this problem, a specific metric space has to be chosen. For the Euclidean case, this dimensional dependence was already addressed by Tideman in 1977[1] (see also [2], and the similar idea of Vapnik–Chervonenkis dimension[3].). Unfortunately, this result and its fundamental implications to the field of voting theory have gone underappreciated.

With these mathematical results, it is possible to infer the minimum dimensions of any real life ranked election or ballot scenario, and maybe even infer whether enough candidate diversity was present. For ${\displaystyle n}$ candidates and ${\displaystyle d}$ dimensions, the following table shows the absolute maximum number of ballots that any distribution of voters and candidates could possibly generate if voters are using those ${\displaystyle d}$ dimensions to classify the candidates.

The rows of the table end at the dimension that exactly allows for all possible ${\displaystyle n!}$ ballots to occur (the minimum dimension for unrestricted domain to apply).

d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 d=9
n=2 2
n=3 4 6
n=4 7 18 24
n=5 11 46 96 120
n=6 16 101 326 600 720
n=7 22 197 932 2556 4320 5040
n=8 29 351 2311 9080 22212 35280 40320
n=9 37 583 5119 27568 94852 212976 322560 362880
n=10 46 916 10366 73639 342964 1066644 2239344 3265920 3628800

For example, suppose a real-life Condorcet election with 6 candidates is observed. There are 720 possible rankings. If in practice we observe 250 unique ballots, then we can know with certainty that to interpret/simulate this election in terms of an ideological space we would need ${\displaystyle d \ge 3}$ dimensions, because for (n=6, d=2) there are only 101 possible ballots, and we observed more than that. In other words, these candidates are being compared by voters in at least three separate attributes, otherwise these 250 ballots couldn't exist, as there would be not enough attributes to construct these observed preferences.

If we now look at the column ${\displaystyle d = 3}$, we can see that we could (in principle) achieve full representation of all voters on 3 issues with only 4 highly distinguished candidates. Given that we only observed 250/720 = 35% of all possible 6-candidate ballots, and about 250/326 = 77% of the maximum (n=6,d=3) ballots, it seems that we did have a lot of similarity between candidates. This suggests that there's probably significant degrees of indifference in the population that are not coming through the ballots, and are instead being expressed as strict preferences due to the limitations of the voting method.

Dimensional resolution of a ballot

The table also informs us about the limitations of a voting method to convey the information voters are using to classify the candidates. For a given number of candidates, after a given dimension any further dimensions will not add any extra resolution, as the ballot cannot express such information. This is why the rows in the table stop, as we would have maxed out the possible ballots.

The practical effect of this is effectively forcing each voter to "collapse" their ideological space to at most a certain number of dimensions (i.e. political issues), which is the dimension in which their ballots saturate. Furthermore, this "collapse" is entirely determined by the candidates themselves, not the voters, further enhancing the distortion of the information collected and lowering representativity.

To interpret this, we consult the table once more. If there are d=4 important issues voters are using to judge candidates, then we require at least 5 candidates to potentially allow voters to account for all possible political positions in an election. This is how when only n=2 candidates exist, any further dimension or attribute will not lead to more resolution than for d=1. In other words, there is a collapse of the entire ideological space in one dimension for each voter. This is, effectively, the problem of two-party domination and single-issue voting.

These observations also have important implications on specific voting methods. An Instant-runoff Voting election limited to top-three rankings fundamentally limits what sort of ideological distributions can be conveyed, no matter how many candidates are running.

From the table above, we see that if every voter is forced to rank only 3 candidates, then every voter can only express information about at most two relevant issues in their ballot[4], as more issues cannot ever classify the 3 ranked candidates further. Even if they are inherently ranking the candidates based on many other things, this information cannot fit into the ballot and information is fundamentally being lost. It is functionally equivalent to a scenario where voters are forced to use only two attributes to judge their candidates.

If ranked ballots are constrained to ${\displaystyle k}$ out of ${\displaystyle n}$ candidates, the population, as a whole, can only cast ${\displaystyle \frac{n!}{(n-k)!}}$ ballots, which means the voting method "mixes" the information multiple voters expressed, as each voter is using a different subset of attributes in their ballots. Thus, there are no guarantees all the voters are expressing information about the same issues in their ballots, and the ballots cease to be informationally commensurable, even in principle. In effect, we are left to simply hope that their priorities are, on average, similar, as to restore commensurability.

Simulations

All of these observations can also inform us of how restrictive a voting simulation is. If we are simulating a 5 candidate election, we would need at least d=4 to ensure we account for all possible scenarios (realistic or not, which we can't really rule out with any distribution).

If we only run simulations in d=2, with n=6 candidates we would be ignoring (1-101/720) ~ 86% of the possible opinions at any given time, possibly more in practice. Considering that the space of possible elections depends heavily on the number of ballots, this means such a simulation would be ignoring the vast majority of the possible 6 candidate elections that could exist, and almost certainly would exclude very relevant and realistic scenarios for real elections, as well as very real opinion distributions.

In real life scenarios, there's also noise and uncertainty, which can allow for more ballots to appear in practice. But this does not fundamentally affect the above observations. It is always possible to threshold the ballots to infer the minimum "latent space" voters are using to judge candidates, by ignoring ballots who are too infrequent in the analysis.

Cardinal/rated ballots

The immediate question is how this affects cardinal (rated) methods? The ordinal (ranked) information on the cardinal ballots can still be interpreted as above (provided there are at least ${\displaystyle n}$ intervals in the scale), but there's more to it.

In cardinal voting, a voter can have degrees of preference/indifference between every pair of candidate, which strict preferences based on ${\displaystyle n}$ voter-candidate distances can't fully capture.

For example, three candidates in 2 dimensions, voters couldn't be independently closer or far away from the three candidates. Two voters having (A=9 B=8 C=10) and (A=10 B=9 C=0) ballots couldn't coexist in such a space, not even in principle, as it would violate the triangle inequality of any distance metric: they're both nearly equidistant from A and B, but have vastly different distances to C.

To indicate faction overlaps and degrees of preference, we would need at least three dimensions in this case, so this disagreement has "more room" to exist. Strictly speaking, this extra "overlap parameter" could exist for every pair of candidates, independently. So to simulate an arbitrary cardinal election in its full generality, we would need extra information on the ${\displaystyle n(n-1)/2}$ pairs of candidates, giving the dimension required for full (theoretical) generality:

• Ordinal: ${\displaystyle d = n - 1}$
• Cardinal: ${\displaystyle d = n - 1 + n(n-1)/2}$

These extra "dimensions" may perhaps not necessarily be in the same geometric space, but as additional parameters in the model (i.e. the external factors discussed above, such as voters priorities and external circumstances weighting them).

In practice we can use less dimensions, due to real-life correlations. The above specifies an upper bound of how much cardinal ballots would require in terms of additional parameters. This reveals that most of the analyses so far have fundamentally ignored a vast portion of scenarios, and likely many important ones.

A similar analysis as before for number of ballots is much trickier in the cardinal case, but this upper-bound on dimension should be sufficient. This means cardinal ballots naturally allow for much more of the ideological space to be expressed for the same number of candidates, provided there are enough intervals for the ordinal information to be specified. This was known, but the above analysis quantifies it more precisely.

Note, once more, that here we are interested in the opinion space. A score ballot could be written as a point in a ${\displaystyle n}$-dimensional hypercube, but this space has the "axes" as the candidates, not as the opinions of voters, which misses the point of the process of voting, as it is uncorrelated with the underlying opinion distribution of voters. Real-life or simulated voters are not defined by their ballots, but their opinions, which the ballots are intended to represent. It is the opinion differences that generate the ballots. The fact the minimum dimension for a complete opinion and ballot space for ranked methods are both on dimension ${\displaystyle d=n-1}$ is a coincidence. For the rated case, this is unclear.

References

1. Stirling numbers and a geometric, structure from voting theory (Good and Tideman, 1977)
2. Perpendicular Dissections of Space, Thomas Zaslavsky
3. Vapnik–Chervonenkis dimension
4. There's at least one extra dimension, because a voter has to classify which are the "top three" candidates, so there has to be a "line" separating these three candidates from everyone else.