Yee diagram

A Yee diagram of IRV with four candidates, showing that the Yellow candidate has been squeezed out and cannot win.

A Yee diagram (named after Ka-Ping Yee) is used to illustrate the behavior of election methods, given a fixed set of candidates in a two-dimensional preference space.

Production

Ka-Ping Yee at a 2021 event hosted by The Center for Election Science. In this video, Yee explains several of the diagrams named after Yee, and discusses a broad range of electoral-reform topics.

Each candidate is assigned a color and shown as a point, and every other point in the space is colored according to which candidate would win under a given voting method, if the center of public opinion were at that point. Typically, this forms large win regions of the same color. In other words, the candidates stay fixed, while the collective opinions of the voters move to every point in the space, testing who would win in each case.

The voters are usually modeled using a Gaussian ("bell curve") distribution, though their number, dispersion, and strategy can vary from one diagram to the next. These properties do affect the output, but cannot be known from the image itself.

The ideal case

The ideal single-voter case with the same four candidates as above. The candidate most similar to the voter always wins.

The ideal Yee diagram for a given set of candidates is given by the single-voter scenario: whichever candidate is ideologically most similar to the single voter wins. (This produces a Voronoi diagram of the candidates, with each win region defined by the candidate that minimizes Euclidean distance to that point.)

Any discrepancy from this ideal diagram means that a voting method is unfairly biased in favor of or against some candidates, purely as a consequence of where they are located relative to other candidates (how ideologically similar they are).

For example, a voting method that suffers from center squeeze might not show any win region at all for a candidate who has been "squeezed out" by the others. This candidate can never win under that method, even if their ideology is the best match for the average voter.

This discrepancy from the ideal can be shown as a second heat map diagram alongside the Yee diagram.

Variations

Screenshot of "Nonmonotonicity City" section of http://zesty.ca/voting/sim/

The diagrams can also be animated, quickly illustrating how the voting method would perform under many different scenarios (if the candidates held different sets of positions).

While originally intended for displaying single-winner methods, they can be adapted to multi-winner methods by producing multiple diagrams for a given scenario.

Software

A video featuring Mark Frohnmayer describing how Yee diagrams are created, then showing animated versions that model different sets of candidates, for FPTP, IRV, Score, and STAR, then their divergence from the ideal single-voter case.

• Warren D. Smith's IEVS
• Mark Frohnmayer's animated diagrams
• Brian Olson's sim_one_seat (Archive: https://archive.ph/8YwuP )
  (probably similar to https://github.com/brianolson/election_simulator/blob/master/spacegraph.cpp or https://bolson.org/voting/sim_one_seat/ )