Yee Pictures

The diagrams on these pages use the visualisation method for single winner election methods invented by Ka-Ping Yee and apply it to the multiwinner case.

Ka-Ping Yee's original page can be see here. His visualisation method has also been used by Warren Smith and Brian Olson.

The basic idea is that each candidate takes up a position on a 2-d plane. The closer a voter is to a candidate, the more the voter likes that candidate. For each pixel on the diagram, an election is held using the election method under test and the winner is determined. The voters for the election are randomly distributed so that the average voter position is at the pixel under test. (This may not be exactly true due to randomness, but is correct as the number of voters tends towards infinity). Each candidate is assigned a colour and each pixel is coloured the colour of the winner.

This visual method cannot be used for handling multiwinner elections due to the fact that there is more than one winner per pixel. However, with a small modification, it is possible. The diagrams on these pages contain 2 elements.

A diagram is created for each candidates. For each pixel where the candidate was one of the winners, the pixel is assigned the colour of the candidate. This means that unlike in Yee's original method, where only one diagram is needed per election method, multiple diagrams are used to display the results for each election method. Each candidate position is shown as a circle of the candidate's colour. The circle for the candidate being shown is larger.

In addition, a java applet is created for each diagram. This java applet has the information for the winners at each location. A circle for each candidate is displayed at the candidate's location. The user can then point the most at a particular pixel. The candidates who won the election when it was centered at that position show up as their colours. The circles for the remaining candidates show up as black. The borders where the the results change from one set of winners to another are highlighted with lines. Approximately, half of the voters lie within the small circle and most of the remaining lie within the larger circle centered on the mouse pointer.

The actual voter distribution used in these simulation is not actually random. The voters are assigned to locations on a square uniform 2-d grid and then transformed using the Box-Muller transformation as described here. This gives a Guassian distribution without randomness and would mean that the average voter location was exactly at the pixel location. In the simulations run on the other sites, random selection is used to pick the voter positions.

The table below shows the results of various arrangements of candidates. The single winner examples would be better visualised using Yee's original method. as giving each candidate his own diagram is not necessary in the 1 winner case.

Candidate Arrangement	Number of Voters	Seats
Equilateral Triangle	10000	1
Triangle with clones	10000	1
4 candidates in a square	10000	1
Skewed Line of 3 candidates	10000	1
Skewed Line of 3 candidates with one offset	10000	1
Equilateral Triangle	10000	2
Triangle with clones	10000	2
4 candidates in a square	10000	2
Skewed Line of 3 candidates	10000	2
Skewed Line of 3 candidates with one offset	10000	2
Triangle with clones	10000	3
4 candidates in a square	10000	3
Skewed Line of 3 candidates with one offset	10000	3