Dividing the Pie – Visualizing Quantities and Qualities of Majorities in Pie Charts

When designing electronic participation systems, one minor but important challenge is to not just count votes and show the result in form of numbers but to also facilitate an appropriate, intuitionally understandable graphical representation for majorities (and minorities) who voted on an issue.

The easy case: simple majorities and yes/no votings

The results of simple yes/no votings on candidates or proposals usually consist of only 3 numbers to visualize: the count of “Yes” votes, the count of “No” votes, and the count of “Abstention” votes. The most interesting facts that should be instantanously visible by looking at a graphical representation are:

1. 1. Is the candidate/proposal approved or disapproved?
2. 2. How big is the difference between Yes and No votes?
3. 3. How is the relation between Yes/No votes and abstentions? In particular: Is there an absolute majority in favor of the candidate/proposal (i.e. if abstentions were counted as “No”, the candidate/proposal would still win), or is there an absolute majority against the candidate/proposal (i.e. if abstentions were counted as “Yes”, the candidate/proposal would still lose the poll)?

Pie charts allow to immediately recognize the answer to the above stated questions. When, for example, the parts of a pie chart for a voting result are drawn in the sequence 1. Yes, 2. Abstention, 3. No (starting and ending at the top of the circle), it can easily fulfill all three requirements stated before: it is possible to recognize whether the candidate or proposal got a simple majority (see Figures 1 and 3), it can easily be seen how big the difference between Yes and No is (see Figure 2), and the relation between abstentions, Yes, and No votes as well as any existing absolute majorities are visible too.

A pie chart divided in three parts. The biggest part is covering almost half of the pie and is labeled “yes (y)”. The second biggest part is covering a bit more than a quarter and is labeled “no (n)”. The smallest part is covering a bit less than a quarter and is labeled “abstension (o)”. Both the “no” and the “yes” part begin at the top of the pie (at 12'o'clock position). The “no” part spans to the left, the “yes” part spans to the right. Since the “yes” part is bigger than the “no” part, it reaches further down than the “no” part. A predominant part of the “abstension” section is on the left half of the pie chart.

Figure 1: A simple majority

After mirroring the lower edge of the “no” section, the difference between the “yes” and “no” votes is visible between the mirrored edge of the “no” section and the lower edge of the “yes” section.

Figure 2: Difference between Yes and No

Majorities I. Simple majority ( y > n ): The most commonly required majority in democratic decision-making: If there are more “yes” than “no” votes (independently of the abstensions), then a proposal reached a “simple majority”. Absolute majority ( y > n + o ): A more strict variant: the “yes” count needs to be greater than the sum of abstension and “no” votes. In other words: More than half of all valid votes need to be “yes”. Blocking majority ( n >= y + o ): If an absolute majority is against a proposal, we have a blocking majority: even if all abstensions were counted as “yes”, there would still be an (absolute) majority against the proposal or candidate. In case of supermajority requirements, there can also be a blocking minority (see Figure 5).

Figure 3: Visual appearance of simple and absolute majorities as well as blocking majorities

Preferential voting

As already shown in the article “Game of Democracy”[GoD] in this issue of the journal, simple yes/no votings are no suitable means to create a truly democratic process (see also pages 18 through 20). Therefore, when talking about visualization of vote counts, we need to consider preferential voting as well.

In case of preferential voting, it may happen that there is no majority which favors a particular proposal most, but there is a majority which favors a group of proposals to the status quo. There may or may not be a winner which received a simple or absolute majority of first preference votes. In order to display this information, we can split the “Yes” section of a pie chart into two sub-sections: “Yes, first preference” and “Yes, alternative vote”.

Since there are several competing proposals in a preferential voting, such pie charts could be rendered for each competing proposal. Our experiences with LiquidFeedback 1.x and 2.x, however, taught us that displaying the approval rate of several competing alternatives may cause confusion to the user since not the approval rates but the preferences determine which proposal wins if there are multiple eligible winners.[PLF, p.106-108] While each proposal may have its own pie chart, we recommend to not display them concurrently (if applicable to the medium) but to show visualized preference counts instead. In LiquidFeedback 3.0, these preference counts are displayed as bar graphs to be able to distinguish them easily from the pie chart that is displaying the “Yes, first preference”, “Yes, alternative vote”, “Abstention”, and “No” counts.

Majorities II (Preferences). Simple first-preference majority ( y_1 > n ): When using preferential voting, the count of votes where a candidate or a proposal was marked as first preference may be of special interest. Similar to the “simple majority”, we may define a “simple first-preference majority”. Extended first-preference majority ( y_1 > n + o ): Whether the first-preference vote count is greater than the sum of “no” votes and abstensions, can be visually determined by looking at the “yes, alternative” section (y_alt). If that section is predominantly on the left, then y_1 > n + o. Such an “extended first-preference majority” implies both the absolute majority (see Figure 3) and the simple first-preference majority (see above). Absolute first-preference majority ( y_1 > y_alt + n + o ): Similar to the absolute majority as displayed in Figure 3, we can define an “absolute first preference-majority” where the number of first-preference votes must be greater than the sum of alternative votes, abstensions, and “no” votes.

Figure 4: Further qualities of majorities when considering first-preference votes

Supermajorities

We covered only simple majorities yet. But in some situations, decisions require a supermajority, e.g. a 2/3 or 3/4 majority. In these cases, the pie chart as described before looses its ability to fulfill the first requirement (showing if the candidate or proposal was accepted) because it is not easy for a human to visually determine if a 2/3 majority has been reached. The same holds also for a 3/4 majority where a 90° angle could only be used as reference if the number of abstentions is zero.

To solve this problem, we introduced the feature of “supermajority pie rotation” in LiquidFeedback 3.0. Using the supermajority pie rotation formula (see Figure 6) it is possible to rotate the display of a pie chart in such a way, that the lower angles of the Yes and No section are compareable again. If the left (No) part does not reach lower than the right (Yes) part, then the candidate or proposal has reached the supermajority. Otherwise, the proposal is rejected. If the “No” part furthermore reaches into the right side, then a “blocking minority” exists: even if all abstensions were approvals, the candidate may still not win.

Majorities III (Supermajorities). Supermajority ( y/(y+n) >= q ): The ratio of “Yes” votes reaches a quorum q (that is higher than 50%, usually q = 2/3). 40% Yes (beginning 27° left from top, spanning clockwise until 117° right from top), 15% No (beginning 27° left from top, spanning counter-clockwise until 81° left from top). (Non-super)majority ( y > y+n , y/(y+n) < q ): An absolute majority doesn't need to be a supermajority and vice-versa. Here, a 2/3 quorum is failed. 55% Yes (beginning at 55.8° left from top, spanning clockwise until 142.2° right from top), 31% No (beginning at 55.8° left from top, spanning counter-clockwise until 167.4° left from top). Blocking minority ( n/(y+n+o) > 1-q ): A minority greater than (1-q)*(y+n+o) may block any decision. 55% Yes (beginning at 66° left from top, spanning clockwise until 132° right from top), 40% No (beginning at 66° left from top, spanning counter-clockwise until 150° right from top).

Figure 5: Supermajorities

alpha_pie / 360° = min(n/(y+n+o) * (1 / (1/q - 1) - 1) / 2, (1 - n/(y+n+o)) / 2, max(7/12 - n/(y+n+o), 0)) y = number of “yes” votes, n = number of “no” votes, o = number of abstentions, q = required supermajority, e.g. 2/3, 7/12 = an arbitrary value > 0.5 but approx. 0.5 to keep the “no” block mostly left.

Figure 6: The formula to calculate the pie rotation in case of supermajorities
(q is the required supermajority, and 7/12 is an arbitrary value > 1/2 but ≈ 1/2 to keep the “no” block mostly left)