The mathematics of majority


I think this is quite the most fascinating thing I’ve come across for a long time. It was brought to my attention by a reader of the Small Dead Animals blog, where a blogger named Marcus Vitruvius (proprietor of the Sagacious Iconoclast) sometimes guest-posts. It’s called Vitruvius’s Experimental Election Predictor, and offers a quick and dirty way of reckoning whether the Tories are likely to win a majority.

The formula is as follows:


where C is the Conservative vote, and L is the Liberal vote. The notion is that the higher the Ve, the greater the likelihood of a Conservative majority. The reasoning is intuitively appealing. As Vitruvius puts it,

If there were only two parties, C and L, then the (C+L)/100 term is equal to one, and Ve simplifies to: (CL). 

In other words, in a two-party system, any margin of victory suffices, no matter how small.

But the greater the number of other parties, and their share of the vote — and the smaller the combined share of the two leading parties — the larger the gap between the two must be to produce a majority: that is, the lower the Tories’ share of the vote, the more they must depend on a Liberal collapse and consequent splitting of the non-Tory vote to put them over the top.

The historical evidence since 1962, according to Vitruvius, is that a Ve of 8.0 or greater is needed to produce a Conservative majority. A Ve of between 0 and 8 means a Tory minority. Conversely, a Ve of less than -8 produces a Liberal majority, while a Ve of between -4 and -8 means a Liberal minority. 

Simple math shows that for any Conservative vote C, the corresponding Liberal vote L needed to generate a Ve of greater than 8 is the square root of (C2 – 800.) For example, if C=30, then L=10. More plausibly, if the Tories were to get, say, 34% of the vote, the Liberal vote would have to fall all the way to a historically unprecedented 19% — a gap of 15 points — for a Conservative majority. (Even then, the 47% of the vote that remained would have to split just right between the NDP, Greens and Bloc Québécois.)

At higher Conservative votes, however, the required Liberal collapse is not so cataclysmic. At 36%, the Tories would need a 14-point margin of victory (36-22). At 38%, the magic number is 13 points (38-25); at 40%, just 12 points (40-28). We’ve seen these sorts of combinations at times in this race — in the first week, and near the end of the third — but only briefly. For the most part, Vitruvius’s formula suggests a Conservative majority has simply not been in the cards. 

Oh, and what is the Vitruvius predictor’s current reading? The latest polls have the Conservatives at an average of 34%, to the Liberals’ 27 (as of Oct 4), leaving the Ve at just  4.3 — not nearly enough. Either the Tories have to gain 5 points, or the Liberals have to lose 8.

MORE: Following Vitruvius’s lead, I’ve graphed the Predictor’s movements using average daily poll data throughout the race.

You can see the Tories were inching towards majority territory (Ve=8) as of about last weekend, but that since then it seems to have gotten away from them…

MORER: The Vitruvius Predictor also sheds some light on Tory strategy, notably the heavy reliance on attacks on the Liberals, before and after the writ. That is, it wasn’t enough just to raise their own numbers: given a likely ceiling on Tory support of 40%, they also had to depress Liberal support below 28% to have any chance of a majority. Of course, maybe if the Tories weren’t so nasty they’d have a higher ceiling than 40%. Chicken, meet egg.


The mathematics of majority

  1. Reading Coyne is never a break from school work…

  2. Very interesting stuff. Though I wonder if you just wanted an excuse to say “Vitruvius Predictor” a lot.

  3. So.. does adding Greens to your V-8 make it a V-10?

  4. Hit submit too soon.

    On the bright side, this indicates that minority governments are quite likely the new norm. I’m hopeful this will lead the political parties to understand the benefits of either compromise, or proportional representation.

  5. I encourage folks to look at Vitrivirus’ chart, not cut and pasted and out of context,

    like Andrew’s is.

    You’ll see several solid lines, indicating: (at the bottom is a Liberal Majority, then a light red line showing a Liberal minority, then higher is a Conservative minority, then near the top is a Conservative majority.

    You’ll note that all of the activity, including vitrivirus’ mean number is hovering at the “majority bar”.

    Andrew, I’m actually surprised at the level of spin.

    Vit’s chart in its entirety shows precisely the opposite of what Andrew’s trying to say.

    So Andrew simply doesn’t show it in its entirety.

    He “recreates” them in his own chart.


  6. kody, I think you need to have another cup of coffee.

  7. Jack,

    take a look at Vit’s chart yourself.

    This is a classic example of the media filter.

    All Andrew had to do is cut and paste the picture with a link/attribution,

    but he “recreates”.

    In this case, vitrivirus’ picture was worth a thousand words,

    just not the words Andrew wanted to hear.

    or wanted you to hear.

  8. kody, if this is “classic” filtering, I’d like to meet the lame-ass ripoff kind.

    Mr. Coyne’s whole post is about the imminence of a Tory majority. He only talks about it in those terms. Just like Vitruvius’s graphs do. What, you want him to add a caveat to every post that an NDP majority remains highly unlikely?

    Perhaps you could try a stronger brand. I get this great Blue Mountain stuff from my local market – very potent. You gotta grind the beans yourself, though. That’s key.

  9. “For the most part, Vitruvius’s formula suggests a Conservative majority has simply not been in the cards”

    along with the out of context graph which seems to show just that.

    Sorry rabbit,

    tricks are for kids.

  10. Kody,

    How do you take math “out of context”? How do you “spin” arithmetic? The chart in my post is simply an average of every single poll taken this election, ground through Vitruvius’s formula. I can send you the spreadsheet if you like.
    The reason I didn’t just cut and paste V.’s chart, though I link to his page, is a) I couldn’t make head or tail of it — he’s got a million squiggly lines, and b) his data set appears to be incomplete — he doesn’t have the latest polls, which as it happens are the worst yet for the Tories.
    If using up-to-date data is “spin,” then I guess I’m guilty.

  11. Another mad mathematician trying to put the science back in PolySci.

    I am reminded of when John Crosbie was ex-finance minister, running in the 1980 election after his gov’t was defeated. He was speaking at Western, and a fellow who would parachute into any number of ridings appeared to heckle. He called himself “the Engineer”, wore some sort of hardhat, and he had some sort of econometric formula he had generated/derived/contrived (I’m not sure what the proper descriptor would be) that he offered to Crosby so that he could better run Canada’s economy. Just plug in the numbers. Crosby said no thanks, he “couldn’t count” or some such dismissive comment when confronted.

    Same thing here. You can come up with any number of formula to fit past data, and, assuming nothing radical changes in the short term, the formula will probably work reasonably well. But, it is worth noting that this is only for Fed Canadian Elections, not provincial (ie it wouldn’t work in BC where I believe both you can get majority gov’ts with less than a plurality of pop support).

    So, interesting model – but it doesn’t take into consideration voting efficiency, or any number of other factors.

    I wouldn’t take it to Vegas.

    But, “the Engineer” would be proud.

  12. As I said, quick and dirty. But it beats the usual journalistic handwaving about “40 per cent” being the usual threshold of majority. In a five party system, with historically low Liberal numbers, Vitruvius’s numbers — and more important, his logic — show it doesn’t have to be that high.
    But neither is it as simple as a “10 point gap.” It’s the combination of the gap and the level. A 12 to 14 point gap, with Tory support in the 36 to 40 point range, seems to be what’s required.

  13. Too many polls, too little discussion of issues.

    Besides Coyne, how many journalists know that if you install a geothermal heat pump in your house, you get a $3,500 cheque from Ottawa? This is one of dozens of ways the ecoACTION program helps individuals make their homes or their vehicles greener. Harper did that.

    I looked it up in Google News. No one talks about it.

    Oh, look, another poll. 300 news articles.

  14. Sorry to go tangentially off topic, but I was also fascinated by MV election thingy. I come from a family of engineers, and I have learned that unsupported hypotheses have a lifespan shorter than snowball in Hades when among them. Here is the OT thing. Their political spectrum ranges from right of Attila the Hun to left of Tommy Douglas, and really only agree on one thing. If you mention the theory of Anthropogenic Global Warming, duck, because depending on the time of day either coffee or beer/wine will be airborne from the snorts of derision which follow.

  15. Andrew:

    *How do you take math “out of context”? How do you “spin” arithmetic?*

    Simple. Take an old-fashioned glass, make a mark about halfway, then add some water up to the mark and put the glass on the table. Then invite some people in to look at the glass and ask them: “Is it half-full, or half-empty?”

    You could predict that half of them will say either half-full or half-empty, but maybe some of them have science training and will argue that the glass is *less* than half-empty because the shape of the glass decreases the volume of water compared with what’s available on top. Some will say that, since they saw you through a window filling the glass, that it influenced them to say full. And some, being cheeky, will drink the water and then declare the glass to be all the way empty.

    *That* is how math (in this case, volume measurement) can be taken out of context and spun. You — with the glass and the water — provided the data. Everyone else provided their education, background, judgement and prejudices — including their own interpretation of what you did.

  16. “I think this is quite the most fascinating thing I’ve come across for a long time. It was brought to my attention by a reader of the Small Dead Animals blog”

    Andrew! You need to get out from behind your computer more often if you’re finding SDA content “fascinating”. It’s a clinically recognized sign of depression, just check the Diagnostic and Statistical Manual of Mental Disorders (DSM IV – Canadian Election Revision). :-)

  17. Phantom,

    But the math remains the math. Whether the glass is half full or half empty, it contains the same 250ml of water.

  18. At the least, it gives some insight into the Tory strategy on hammering the Liberals long, and hard.

    It also makes the prospects of the Tories ever achieving majority somewhat dismal, as it relies on severely weakened Liberals and somewhat broadened support toward the centre.

  19. Hey Andrew Coyne, don’t let yourself get worked up by Kody. Anything that doesn’t predict a Tory majority with a seat count of 200 is Liberal Media Spin, according to them.

    I like this formula, it tends to be, like you said, a better predictor than the 40% number usually quoted in the media as the number necessary for a majority.

    What it doesn’t take into account is the turnout in this election from the right and left. In the last election, the right was angry at the Liberals and motivated. I think a couple of years of Harper rule has made them complacent. Contrast that with the “Anyone But Conservative” panic on the left – I think a higher turnout on the left side of the spectrum may mitigate a bit of the vote splitting.

    Full disclosure to Kody: I voted Conservative last election I’m leaning Liberal this time.

  20. Pete: I think Kody’s downgraded their prediction to 175 now.

    I find it amusing that in a Nanos thread a day or two back, Kody was talking about how one should use the G&M “Poll of Polls” — which is merely an average of the available polls.

    Then, when Andrew posted some observations drawn from the average of available polls — essentially the same thing as the “Poll of Polls” — all of a sudden one shouldn’t be looking at those polls, but rather the longer term trends. (No word on how long term though, and given there’s only a week left in the campaign…)

  21. Kody does need another cup of coffee.

    With a really good lay on top.

  22. The probability that financial turmoil plus climate change hysteria will provide the excuse for a drastic increase in government intervention at all levels of the economy – no matter which party pops out of the Vitruvius formula – is approaching 1.

    The probability that these interventions will make things worse is precisely 1.

    The probability that politicians will push for a world war as part of an even more drastic attempt to force round economic pegs into square holes is, I estimate, approximately one-half, or 0.5.

  23. Whether the glass is half full or half empty, it contains the same 250ml of water.

    Good point, but the real question might be: how many glasses should one be looking at? MV claims one. Others claim four (Maritimes, Quebec, Ontario, Western Canada). Yet others suggest the swing 42 or whatever ridings. But, as a quick and dirty proxy, as it appears was the full extent of your intended use, it has some merit in providing context for more in-depth analysis and discussion.

    Again, relying on past personal experience, I do recall in the 70’s and 80’s, London Ont (and to a similar but lesser degree Peterborough) was deemed to have the ideal demographics for market trials and product introductions. So, as a kid, you’d find yourself one morning eating an early form of Count Chocula for breakfast, or trying out some Dutch Cleanser in a tube toothpaste, and reporting back to P&G in a questionnaire what you thought of the product (if your Mother was adventurous in trying out new things – or just liked free samples.)

    But, in marketing, as in politics, it seems that market segmentation has become much more refined in terms of introducing a new consumer product, or in pandering to niches of the voting electorate through targeted tax credits (ie at soccer Moms etc.)

    The MV approach, treating Canada as a whole, does seem to be a bit simplistic – I’m not sure the earlier data back to ’62 would be relevant to today’s situation, but better than 40% majority/minority threshold, I suppose.

  24. I’m just happy that you’re using metric half full glasses.

  25. Once again Coyne is putting his spin on things though. Did he measure using imperial units and then convert to metric? If so, how many decimal places did he use in the conversion?

  26. You bet Coyne is spinning. Assuming a traditionally sized glass, 250 ml is about 1 imperial cup; way more than 1/2 a glass.

    You are shameless Andrew.

  27. But I like the fact that after 4 half full cups consisting of 250mL (when half filled), that I’ll have a Litre, and will have 2¢ more if it’s filled with diesel fuel. But only if we elect Harper. If not, you can kiss those two pennies goodbye!

  28. Well.

    The mean, median and mode is Green. So Elizabeth May will say things like “The Greens will enforce the Canada Health Act” to distract you from that wavelength and make you think about the Range. After she says it :: what does Green mean? What are the Liberals learning by this leaning? Your glass is tipped so according to my perspective a half-full glass looks like a majority. Why is Stephane Dion surfing the surface tension of a half-full glass??

    Andrew. I don’t care about stupid maths and polls. Tell me something I don’t already know.

  29. Hello Andrew, Vitruvius here. I’m glad you like my little experiment. To be strictly correct, though, +8 is not required for a CPC majority, a linear interpolation between the 2006 and 1998 results says that +7 or a little less will work too. Still, there isn’t all that much data since ’62, so we’re still doing a lot of guessing here, albeit with better numbers. The key to interpreting the current value of Ve is the final table in my essay at tinyurl.com/53ofes

    I’m not sure where you got the idea that I don’t have the “latest polls”, each SDA update is current to all published polls at 20:00 mountain time every day. Also, I don’t agree with your conjecture that the CPC has generally not been in majority territory, I think Ve is saying they generally have been in majority territory, if one counts all five pollsters.

    Today, however, for the first time in three weeks, Ve is predicting a 3 seat minority. As for me, if anyone wants my personal opinion, ask me next Tuesday ;-)

  30. “I think Ve is saying they generally have been in majority territory, if one counts all five pollsters.”

    I think it’s wrong to assume that averaging all 5 pollsters will give an accurate representation. When you think about it, only one number is the right percentage, and there’s no guarantee that more wrong numbers put together will get closer to the right one. 5 of 5 pollsters could be out to lunch.

  31. No, *all* the advance polls are wrong. Only the election counts. Vitruvius’s Experimental Election Predictor is at best an interesting toy. Anyone who bothers to actually read my essay ~ tinyurl.com/53ofes ~ will find that it is full of provisos, including that the advance polls ended up being two Ve points higher than the final election Ve in 2004 and 2006, and that near the breakpoints (somewhere around +8, 0, and -8) the per-riding splits dominate which side of the breakpoints the final result falls on. Finally, my ΣVe/n mean value plot, shown as the black curve in the graph at the top of the essay, is generally higher than Andrew’s; it’s probably the effect of the weekly tracking polls (which Andrew isn’t holding through) being generally higher than the dailies, and if I cared enough I’d look into it further, but I don’t.

  32. …and I’m sure, once this election is complete, with the benefit of more hindsight data, further provisos and revisions will be added, and the model will be able to more accurately predict the past.

  33. But how many ml of water it will take for the Conservatives to get a majority?

    kidding… the conclusion from the analysis is what it is, but more importantly I simply find the analysis to be interesting and offers a cool way to look at what it takes to get a majority. Good stuff!

  34. But how many ml of water it will take for the Conservatives to get a majority?

    Depends what state they are in. Freezing adds roughly 10%, I thawed.

  35. Now I understand what playwright John Mighton meant when he said: “Failed mathematicians become economists.”

  36. Seeing as there have only been two Conservative majorities since 1962, and that one of those majorities was an absolute whopper, I don’t know if this formula can lay claim to validity.

    I agree that in Canada there likely needs to be a big gap between the first two, but Labour won a majority in the 2005 UK election with only 35.2% of the vote, as compared to the Conservatives 32.4% (so a gap of 2.8%). There was a better formula out there, that came out after that UK election, that factored in total number of seats to be contested (the more there were, the easier to gain a majority with a smaller gap) and the popular vote tallies of the 3rd and 4th parties. I tried googling for it, but couldn’t find it.

    But for argument’s sake, let’s assume the Greens come in with 10% of the vote on E-day, but say that it is too wide and too thin to elect any MP. Effectively, we could lop off the Green’s score, and pad every other party’s vote % by 11.1%, to then compare.

    So say the Conservatives come in with 36% of the vote, and the Liberals 25%. Removing the Greens, this would instead read CPC 40.0%, LPC 27.8%. I recall Mulroney won big in ’88 with a 43-32 lead. Using the revised figures yields a result of 8.27 …

  37. Does anyone really believe that the GPC will actuallty get 10% which means more than doubling its popular vote from last election?

    I think there’s some heavy overreporting going on, people who say Green Party as a kind of none of the above response so they don’t sound dumb on the phone (or admit they aren’t going to vote at all).

    By the way in the last Ekos poll 4% of respondents said they weren’t going to vote on Election Day. Tempered by reality that number should be closer to 40%!

  38. saskboy:

    Look up “Law of Large Numbers.”

  39. “The Law of Large Numbers”, Garth,
    and the “Central Limit Theorem”.

  40. I agree to some degree with Saskboy.

    The Law of Large Numbers and the Central Limit Theory implicitly assume that the same methodology is being used.

    Say, for argument’s sake, the methodology that Nanos uses is the more correct one (where you ask for party preference), and the Harris-Decima is biased (as it prompts you with the choices), or vice versa. According to AC, Nanos under-reports Green and over reports Libs (or vice versa for Harris-Decima).

    Combining a Nanos poll (unbiased) with a Harris-Decima (biased) won’t give you a more representative sample. It may reduce the Harris-Decima bias, but it will worsen the Nanos results.

    I’d be cautious about combining or averaging results where the methodologies are not identical.

  41. First you make fun of me for being cautious, “further provisos” and all that eh what, and then you advise me to be cautious? Well thanks, I just don’t know what I’d do without you!

  42. My words of caution were not to you- if you’ve been playing with this for five years, doubtful anything I write will influence you.

    The caution is to anyone who uses your methodology, or thinks that by your mentioning of some statistical laws and theories, your methodology is somehow more credible.

  43. “Freezing adds roughly 10%, I thawed.”

    Dot, you are hilarious.

  44. Fair enough, Dot, I certainly wouldn’t “use[ my] methodology” for anything significant, for as you note, after playing with Ve for five years, I’m quite aware of its strengths and weaknesses. As I already wrote above: at best Ve is an interesting toy. I really don’t understand why this is so hard to grok, I mean, it’s not like I’m trying to make some sort of Lyapunov or Lindeberg argument. I can’t reasonably be held responsible for the misconstructions of others.

    Meanwhile, though I fail to understand why this sub-thread is a part of the discussion, it remains the case that: The optimist sees the glass half full, the pessimist sees the glass half empty, and the engineer sees that the vessel is inappropriately sized for the fluid level ;-)

  45. and the pragmatist notices that the fine print says “Poison, do not consume”. :)

  46. Yes I’m a pragmatist, no it’s not poison, and no I wasn’t consuming it; as I’ve made clear: it’s a toy. I’m playing with it. I’m using it as an aide to help me think about the various factors involved. For as Isaac Asimov said, “The most exciting phrase to hear in science, the one that heralds new discoveries, is not ‘Eureka!’ but ‘That’s funny…'”. Your mileage may vary.

  47. Well, my trouble with the Asimov quote is that I believe “Political Science” is an oxymoron. Same, to some degree, with the use of the term “Management Science”.

  48. Oh never mind, Dot. My apologies, Andrew, for taking up bandwidth and storage resources here at your blog trying to be reasonable with Dot.

  49. Back to the drawing board…

  50. One other thing, Andrew: thanks for featuring the Ve measure here on your blog. It’s been interesting to see new readers of the Sagacious Iconoclast come in today, from the House of Commons, the PMO, the PCO, and the governments of Ontario, Manitoba, and Alberta, via this referring URL, over and above the previous regular followers there, at SDA, and elsewhere, of the Ve measure, over the last few weeks and years.

    Yours in Numeracy,
    Marcus Vitruvius,
    The Sagacious Iconoclast

    PS: I first commented at your original blog on 2004-04-07, Andrew. We sure had some good discussions back then. How time flies. Salut!

  51. “There’s a sucker born every minute”

    P.T. Barnum


  52. Um, no, actually, although it’s not entirely clear, it was probably David Hannum who said “There’s a sucker born every minute”, although it might have been Michael Cassius McDonald. But it almost certainly was not Mr. Barnum. I mean, just for the record, of course, you know, in the name of “caution”.

    Back to the drawing board?

  53. Yeah, I knew, but my wiki link where it said as you report got blocked. Yet, the sentiment remains.

    Besides, as with most urban myths, the more times you allow it to be repeated unchallenged, the more likely it becomes accepted or conventional wisdom.

  54. Will the Leafs win the cup this year………. just asking?

  55. Extrapolating from the 61-62,62-63 and 63-64 seasons – Yes with 100% certainty!

    Looking at this year’s roster, No.

  56. Math, yeah, that’s great, great. Too bad Steven Harper looks like a serial killer.

  57. “Besides, as with most urban myths, the more times you allow it to be repeated unchallenged, the more likely it becomes accepted or conventional wisdom.”

    You mean, like Global Warming….:^)))

  58. Global warming is the ultimate govt spoof, the peak, the epitome, the sum knowledge of everything politicians have learned in the last two centuries about corralling sheep, shearing them, and making them bleat for more.

    Mencken said, “The whole aim of practical politics is to keep the populace alarmed (and hence clamorous to be led to safety) by menacing it with an endless series of hobgoblins, all of them imaginary.”

    AGW is the most imaginary and the most laughable of any political hobgoblin ever invented – yet that is not a flaw, it is its great strength. CO2 is invisible, odorless, colorless, non toxic, almost inert, exists in only minute quantities, and has an insignificant effect on anything in the atmosphere except in extremely dubious and unproven theories about climate “feedback”. The harmlessness of CO2 is what makes it such a scary hobgoblin – pure, invisible CO2 is stealthily creeping around your atmosphere and it’s going to kill you all! The fact that it’s the weather which will be affected, according to the scary stories, is perfect because they can point to any weather event on any day in any place at all and announce that “this is because of AGW!” How can anyone refute it? How can anyone prove it? Better to be safe and let us fight the imaginary hobgoblin for you, before it’s too late.

    Before he got a serious sniff of power Harper tool the sensible approach of denouncing it as a fraud up with he shall not put. But give him the country’s game-boy controller and oh boy, we gotta get on top of this thing and the only way to do that is to place total control of the entire economy – outside of hydro-powered Quebec – into the hands of us politicians.

    That’s why Vitruvius’ formula is no more than a meaningless toy, like a magic 8-ball or a Rubik’s Cube. Play with it all you want, but you’re still going to get tied up, sheared, tatooed and then locked up in a tiny paddock – for your own good, as broadly defined by the dumbest, laziest and most mendacious pack of opportunists in the land.

  59. I find Mr. Vitruvius’s theory interesting but we often neglect the fact that there is not one election but 308. National polls are just part of the equation.

    A site like electionprediction.org provide us with better insights on what the next Parliament will look like because they analyse the election on a riding by riding basis.

  60. Numbers are like people ( ripped from a commentor at Climate Audit , a while ago ) , …. if you torture them long enough , they’ll tell you anything you want to hear .

  61. Does anybody see the Tory trend reminiscent of the Peterson flame out in 1990 in Ontario?

  62. I think that KRB has something here. In trying to debunk the “40% threshold” idea I argued here:http://notweighingourmerits.blogspot.com/2008/09/on-cusp-state-of-canadian-race-2909.html that, for the purpose of predicting whether the Conservatives will win a majority, the vote of the Greens (and all other parties not likely to win more than 1 or 2 seats) could simply be removed from the poll results and the votes recalculated. I did note the importance of the margin between the first and second parties, but couldn’t conceptualize it further.

    Now what I’d love to know whether Vitrivius’ formula would work better if the denominator, instead of 100, was the combined vote percentages of only those parties likely to win seats. the Greens this year may be the only party likely to affect the results significantly, so this may be something to look at after the election.

  63. Andrew,

    Label your axes.

Sign in to comment.