Electoral Math I: Vote Splitting and the NDP+Liberal merger hypothesis
May 3, 2011

Electoral Math I: Vote Splitting and the NDP+Liberal merger hypothesis

This historic election has seen the complete collapse of the Bloc Québécois and the decline in the Liberal party to historic lows while propelling the Conservatives to a majority government and the NDP to the official opposition. With the governing party still only receiving approximately 40% of the popular vote, and with two parties asking themselves existential questions, it is natural to try and measure both the degree to which vote splitting between the NDP and the Liberals is responsible for this election result and what may have happened should a Liberal-NDP merge have occurred. I ran some calculations on this election's data and will present those results and then talk about the consequences.

2011 Election Results
The first and roughest calculation is to simply add up the total vote of the NDP and the Liberals in each riding and compare that with the other parties to see how many ridings were won by each party in a four party scenario of NDP+Liberals vs Conservatives vs Bloc vs Green. Since it is very unlikely that every vote cast to either the NDP or Liberals would add together in votes for the merged NDP+Lib party, I added several scenarios where some proportion of the vote received by the NDP or Liberals would not end up in the combined NDP+Lib party for that riding and instead would be transfered to some combination of the Conservatives candidate or third parties. I assumed a very simply distribution of an even percent of votes leaving the NDP+Lib for other parties across all ridings in the country. Note that results are out of 307 with 1 ignored riding, 155 is the normal majority threshold.

Scenarios with results listed as NDP+Lib/Con/BQ/Green:
-NDP+Lib receive 100% of their respective votes:  186/120/0/1
-NDP+Lib receive 95% of their respective votes, remained transfered to Conservatives: 170/136/0/1
-NDP+Lib receive 90% of their respective votes, remainder transfered to Conservatives: 156/150/0/1
-NDP+Lib receive 85% of their respective votes, remained transfered to Conservatives: 139/167/0/1
-NDP+Lib receive 90% of their respective votes, half of remainder transfered to Conservatives, half to third parties: 167/139/0/1
-NDP+Lib receive 85% of their respective votes, half of remainder transfered to Conservatives, half to third parties: 154/152/0/1

The immediate conclusion from these numbers is that vote splitting clearly occurs and confirms the obvious result that without it the conservatives would be the official opposition and the NDP+Lib party would have a strong majority. That said, in the 100% case, it is still only 186 seats, some 31 over the majority. The conservatives are thus winning 120, or 40%, of the seats in the country regardless of whatever the NDP or Liberals would do and vote splitting cannot be said to be relevant in these seats. Note that this 40% of seats is basically the same as the Conservative popular vote which indicates there isn't too large of an asymmetry between votes and seats with regards to the Conservative party (but an asymmetry remains for the third parties). As votes shift from the NDP+Lib to the Conservatives, it requires only 10% of NDP+Lib votes to make their way to the Conservatives for it to be essentially a tie at 156 vs 150 seats and at 15% transfer from NDP+Lib to Conservatives the Conservatives win a clear majority. That said, if the Conservatives did not receive all of the extra NDP+Lib vote that leaked away and instead went to the Bloc, Greens or other third parties, the NDP+ Lib party would still squeak a majority at 85% with half of the remainder of votes in each riding going to Conservatives.

These numbers reflect to me the fact that there is not a large margin of error on an NDP and Liberal merger. The buffer of seats evaporates with only 10% of votes leaving for the Conservatives. The largest source of uncertainty is Quebec. That the NDP solidly won in Quebec is the single largest change in this election. With 75 seats in Quebec, the reaction of often fickle Quebecers to the NDP and Liberal merger would be more than twice the buffer and hence would essentially determine the outcome of whether it could work. Outside of Quebec there might be slightly more hope simply because a considerable amount of the more moderate Liberals left the Liberal party for the Conservative party in this last election. That body of voters may well overlap the body that would not accept an NDP+Liberal merger but since they have already somewhat walked away in this last election the further dampening would not be overly significant.

Ultimately, the uncertainty here is enormous. How the public in and outside Quebec reacts to a merger is something that cannot be easily approached statistically and could be very strong or very negative. However, we must recognize that the margin of error is not as large as people proposing a merger would probably like and even with completely eliminating vote splitting the conservatives still represent a huge force in the country that only requires a small gain either from itself or a resurgence of the BQ in order to be successful.

(A nicely compiled spreadsheet of the election results by riding with vote numbers for each party is located here: 2011 Canadian Election Results Spreadsheet. Note that only one fringe party is added in each riding, and I have left the computation for one of the cases above in as a simple model.)

Thoughts on this post? Comment below!

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Annie said...

I like the way the Australians vote, election day different than the the Senate. The regular election is called Preferential voting ..that would fix all this FPTP.

bazie said...

Ya I completely agree, there system sounds good. Neither complete FPTP or complete proportional representation are good ideas. One of the various hybrid options with transferable votes is probably the best option.

agmk said...

According Frank Graves (founder of EKOS), there was an 11th hour shift of approximately 3% of Liberal support to the Conservatives, presumably out of anti-NDP sentiment. I wonder how this changes your analysis... did the percentage of voters who'd switch to the Conservatives already switch? Or were those switchers harbingers of further switching.

bazie said...

Ya, how the public reacts to such a merger is of course the dominant question and can't be determined statistically. That shift does compound the amount of vote shifting in this particular election however, as the NDP surge drives liberal voters not just to the NDP but also apparently some 3% to the Conservatives. My simple estimates are that a further 10% beyond the 3% mentioned already could leave from a merged party to the Conservatives before they are out of majority territory. It may be safe to say that given 3% left without them merging, saying another 10% would leave if they actually merged would be possible.

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