As the Ontario election goes on and the rise of the NDP was (kinda) confirmed by the Ipsos poll on Wednesday, now is as good of a time as any to talk about vote efficiency. I'm referring to how successful parties are at converting votes into seats into our current electoral system.

I have to admit that when I started writing this post, I thought it'd be very easy. But as I began writing and thinking about it, I realized that defining and measuring vote efficiency isn't as straightforward as I thought. My first instinct was to look at a measure such as votes per seats (or seat per votes). That kinda works but it left me dissatisfied. This works to compare parties within the same election but not across (as the number of votes changes with turnout).

I also quickly realized that the real question wasn't so much which party was more or less efficient at their current vote level, but overall. What I mean here is that a measure such as votes per seat will obviously show that the NDP at 19% is less efficient than the Liberals at 38% with a majority. But that isn't saying much. Our electoral system is such that such result is expected. The better question is more: what if the NDP was to reach 38% and the Liberals fall to 19%? Would the NDP win more or fewer seats than the Liberals had?

So here is what I came up with:

1. Vote efficiency is about getting an optimal distribution of the vote over the map

2. Ideally, a party would want not to waste any vote. That means that as soon as you have one more vote in a riding, you move the rest of your votes to another riding. Winning by 1 vote or 5000 has the same result, except that you wasted 4999 votes in the second case.

Therefore, for every election since 2007, I looked at how each party could have distributed its votes in order to maximize the number of seats won, keeping the distribution of votes of the other parties constant. This isn't very difficult to do. All I did is order the riding in order of the number of votes require to win (after removing the Liberal votes). For instance, imagine one riding where the PC finished second with only 5000 votes, then the OLP would allocate 5001 votes to this riding and move on to the next one. You keep doing this until your party ran out of vote.

With this method, vote efficiency is defined as the ratio of seats actually won to the maximum number of seats that could have been won with an optimal distribution of the votes.

Note that for some years, it does mean a party's optimal distribution would have resulted in this party winning all the seats. This is the case in 2014 for the Liberals for instance. Since they "only" won 58 seats, this gives us an efficiency of 58/107=0.54 (or 54%).

The table below summarizes the findings. It also include the votes per seat measure (which correlates with my more fancy method).

As you can see, the Liberals have been the most efficient party since 2007. This isn't surprising, they won the last 3 elections and got the most votes in all of them.

There is obviously a relationship between your percentage of votes (province-wide) and the efficiency of this vote. Again, this is naturally due to our electoral system.

The graph below shows this relationship. It also illustrates where each party stood for every election. As Ontario didn't experience important swings since 2007 (this is about to change this year it seems), all three parties' data points are close to each other for the 3 elections. This is unfortunate as it would be interesting to see some crossovers.

This graph clearly shows the "money zone" where each extra percentage point turns into many more seats. The slope of the relationship picks up around 25%, which is the usually accepted threshold with FPTP.

Finally this graphs shows that the NDP, while getting overall fewer votes and lower efficiency, is actually doing quite well for the province-wide percentages it got. On the other hand the Conservatives seemed to have slightly under performed for their votes level. This is most likely to the GTA where the Liberals have been incredibly successful over the last decade.

Let's look at other measures (direct or indirect) of vote efficiency.

*1. General distribution of the votes*

The NDP has had a higher standard deviation of its vote compared to the Liberals since 2007 (14 pts versus 12 in 2014 for instance). The standard deviation of the NDP was actually the highest of the top three parties in 2 of the last 3 elections (NDP and PC were very close in 2011). This is particularly visible if you look at the variations across regions. While the Liberals were between 23% and 49% in 2014 in average in the 10 regions of the model, the NDP varied from 13% to 45%.

The NDP therefore has a vote that is fluctuating a lot between regions. Very concentrated in some regions (the North, the Southwest and Hamilton) while being very low in others (Central, Ottawa, the 905). And that could actually explain the more narrow possible distribution of seats for this party: there are regions where the NDP needs to increase substantially before being able to win seats.

In general, we say that small parties want to have a concentrated vote while big parties want it widespread. The NDP, at 20%, has done well with concentrating its votes in some regions. But if Andrea Horwath wants to become Premier, she better hopes her new votes are in the "right" regions.

On that note, the NDP could actually have quite an inefficient vote this time around if its increase is concentrated in the city of Toronto. The Liberals dominated there in 2014 and there is a chance that despite the fall (and the NDP's rise), most seats will still go the Liberals.

*2. Margins of victory in ridings won*

Ideally, you want your margins to be as small as possible otherwise it means you are wasting votes.

In 2014, the NDP won its 21 seats with a margin of 20 points in average (they completely dominated the North for instance as you can see here). The Liberals only needed 17 points while the PC was at 15 points.

*3. Close races (<5points and="" i="" lost="" margin="" of="" victory="" won="">*

If you are well organized and know how to target the right riding (and get the vote out), your efficiency will increase.

No clear pattern here, if only the improvement for the NDP after 2007. The Liberals used to win more races than they lost except in 2014. But they still won a majority that year with less than 40% of the vote.

*4. What if all three parties were around 31%?*

Finally, let's use the model and simulations to see what the predictions would be if there was an almost perfect split between the top 3 parties (the Green are left at around 5%).

In this scenario, the chances of winning would be

OLP: 51.8%

PC: 26.3%

NDP: 18.6%

Similarly, the 95% confidence intervals for the seats would be:

OLP: 29-59

PC: 28-54

NDP: 30-51

This confirms the advantage the Liberals have. After that, it's pretty close between Conservatives and New Democrats. Let's not forget those are fully hypothetical simulations. We noticed again the smaller range for the NDP.

**Conclusion**

Lots of numbers and measures in this post. At the end of the day, I think the main result is the strong vote efficiency of the Liberals in Ontario. But this efficiency was highly dependent on a) being the top party and b) the GTA. Based on the recent polls, it seems fair to say both points aren't valid this year. Both the PC and NDP have okay vote efficiency. They don't waste a ton of votes in one region (like the provincial Liberals used to in the English part of Montreal and therefore needed about 4-5 points more than the PQ to win an election). So as far as winning the most seats, both the PC and NDP can do it. If the race was to become more competitive between PC and NDP, I think the key would simply be: who will win the GTA? And if we observe a massive migration of voters from the Liberals to the NDP, could this party also inherits the high efficiency of the Liberals? Time will tell.