Simulated Electorates

My focus here is collective choice, or how we make decisions as a group as part of our group governance. Various studies over centuries have used different representations of an electorate, voter preferences, or even just pairwise majority decision results to come up with ways to find the best result when there are many possible outcomes. The first four explorables in this section are models that let you play with different ways of looking at voting methods and outcomes. The final two models provide more detailed looks at individual preferences and the aggregation of those preferences.

All of these models were created to explore or illustrate some concepts and ideas about collective choice. It might be useful to think about which models are better suited to what types of questions. It is also reasonable to think about how easy it is to work with these models, and how convenient it would be to work with these models without computers.

These models are ordered by the degree in which voters are incorporated into the voting process, ranging from ignoring them entirely, to tracking them in great detail.

The explorables here are:

Tournaments

Tournaments encode the majority decision results from pairwise races between all pairs of outcomes. They can be presented in tabular form, but are most commonly represented as directed graphs. They have been used to present voting problems such as Condorcet cycles or other confusing preference results, and have more recently been used to look at the results of random tournaments to investigate what rules or scores lead to the best outcomes. From my point of view, they are best used as output formats and easily show the win/tie/lose results but have no information about victory margins, voters or anything at all about what the outcomes mean. Tournaments do not encode any information about either voters or whatever is being decided. Nor do they include any information on the source of the tournament data.

Preference Profiles

Preference profiles are ordered lists of the names of the choices being proposed, along with a count of the voters having each particular ordering of the choices. This is a very traditional way of looking at voting systems. It has been used for centuries and is still common. It is reasonable to use when there are relatively few outcomes, but starts to become unwieldy when you go above \(5\) choices, or allow ties. It also does not allow for consideration of the strength of preferences, or any way to reason about some new option.

Voronoi IRV

The Voronoi IRV explorable is a very simple version of the use of a spatial model to investigate a voting system. In this case the voting system being investigated is Instant Runoff Voting, or IRV. IRV is only one voting method from a group of voting systems called rank choice voting. In this model, the large square is treated as a dense, uniform distribution of voters in a two-dimensional issue space. The choices are labeled points in the space, and each "voter" prefers closer choices over further ones. The lines form a Voronoi diagram, which divides the large square into smaller polygonal regions containing a choice and an area defined as the points closest to the choice point. The straight line segments are the points equidistant from the two nearest choices, and any intersections of the lines are equidistant from three or more nearest choices. When calculating a vote, the votes for a choice correspond to the area in the Voronoi cell defined by the choice, and the area of a cell divided by the area of the large square determines the vote percentage for the choice. This is the first model where it is immediately clear why voters would have preferences. Also, this is the first model where choices can be added, removed, or moved to see how the vote percentages would change.

Spatial Models

This Spatial Models explorable introduces individual voters. Whereas the Voronoi IRV model is a continuous spatial model in which infinitely divisible areas represent fractions of the total vote, this spatial model is discrete, having \(100\) distinct voters and \(3\) labeled choices. In this model, the voters are single points in the issue space, and the first three voters have proposed their ideal locations as the three labeled choices. The three lines across the space are the three perpendicular bisectors of the three possible pairs of choice points, with the points on a line being indifferent to the pair of generating choice points. In this model you start to have all the data you need to reason about voter behavior, and to see how the results might change if you change a voter's preference, or alter a choice.

Voter Metrics

The Voter Metrics explorable looks at an individual voter and three choices. It introduces the idea of metrics that differ from the standard Euclidean metric used in classical geometry. The three metric types differ in how they combine the coordinate differences to generate a single distance value. The taxicab or \(L^1\) distance does a simple sum of the \(\delta{X}\) and \(\delta{Y}\) values. The standard Euclidean or \(L^2\) distance takes the square root of the sum of the squared issue differences. The Chebyshev or \(L^{\infty}\) distance takes the largest coordinate difference and ignores the contribution of any other issue. All three are valid metrics, and apply to various choice situations. When weights are allowed, the same objective distance in a coordinate contributes more or less to the final distance number.

Aggregation

The Aggregation explorable expands on the Voter Metrics explorable and includes several different ways of combining individual voter preferences to a collective ideal. The electorate options here are also used in the SAVE Voters explorable. (If you select an Electorate PRNG that is not random, and set the rest of the electorate parameters accordingly, you can generate the exact same electorate configuration in both explorables.) The goal of this explorable is to consider what is the best possible outcome for a collective choice process, depending only on what the individual voters prefer, and not limited to only the choices provided.

The sequence for the explorables listed above is a progression from simple models to more complicated models. All of them can be done by hand when using the right tools. But it would take a lot of time and be a lot of work. In contrast, all the rolling of dice and drawing the images can be done easily and in a fraction of a second on a reasonable computer, with the right software.

There is a strict hierarchy in the model complexity. A spatial model with voters, motions, and the appropriate metrics can generate a preference order for each voter by measuring the distance between the voter and each of the motions using the given metric and weights. Those orders can then be consolidated to one specific preference profile. A preference profile can be used to determine the majority decision result for each pair of motions to generate one specific tournament. In both mappings, there are infinitely many spatial models that map to exactly the same preference profile. And similarly, there are infinitely many preference profiles that map to the same tournament.

It is also the case that there are constructive proofs that every single possible tournament can be generated by a preference profile ?? (????), and that every possible preference profile can be generated by a spatial model Bogomolnaia, Anna and Laslier, Jean-François (2007) and Eguia, Jon X. (2011).

While it is the case that two-dimensional spatial models are not sufficient to represent all possible preference profiles and tournaments, they are sufficient to show many issues that challenge voting systems. It is also not all that difficult to extend two-dimensional models to higher dimensions. (I have done this in Python, but not yet for a browser running JavaScript. It's on my To-Do list.)

The real reason for using these computer simulations and explorables is to make it easier for people to consider other voting models, and ultimately decide whether a voting system that gets past the difficulties of current voting systems and actually converges on an outcome that is good enough is worth the cost in multiple election rounds.

• Home

.

Bogomolnaia, Anna and Laslier, Jean-François (2007). Euclidean Preferences, Elsevier.

Eguia, Jon X. (2011). Foundations of Spatial Preferences, Elsevier.