The new mathematics of voting?

The reform challenge

Electoral reform is a hot topic. If it is true (as I believe) that mathematics has something useful to say about almost every important topic (because most of our problems are due to misplaced rationalism) then mathematics ought to have something useful to say about first past the post (FPTP) versus the alternative (AV).

The arguments

Many mathematicians prefer AV because it has many seemingly desirable properties that FPTP lacks, but such considerations are notable by their absence from the debate. Today (18/2/2011) Cameron and Clegg have made their cases (FPTP and AV, respectively). Clegg makes lots of assertions , such as that under AV ‘every vote is worth the same’, but with no attempt at justifications. The general standard of the debate is exemplified by “When it comes to our democracy, Britain shouldn’t have to settle for anyone’s second choice.” This sounds good, but what it seems to be saying is that if your first choice isn’t on the ballot then you shouldn’t vote. There may be some important principle here, but it needs to be explained. 

On a more substantive point, Cameron leads with the argument that:

“[AV] won’t make every vote count. The reality is it will make some votes count more than others. There’s an inherent unfairness under AV.”

He provides an example that could hardly be clearer. It seems obvious that ‘arithmetically’, FPTP is better.

A mathematical approach

The Condorcet method is the typical mathematician’s method of choice for referenda, but for general elections there may be other concerns. We can’t expect mathematics to give a definitive answer to every question, but at least we should be able to distinguish a mathematical argument from common-sense reasoning dressed up as mathematics.

A mathematical approach typical starts by considering criteria, and then establishing which methods meet which criteria. Here neither side has any explicit criteria, but simply makes some observations and then says ‘isn’t this bad – so we can’t have this method’. But many things in life are compromises, so it is not enough to identify a single failing: one needs to think about which are the key criteria and trade-offs.

Historically, democracy was intended to avoid rule by a person or party who was the last choice of most people. FPTP does not satisfy this basic ‘majoritarian’ requirement (due to vote-splitting), whereas almost all the alternatives, including AV, do. Thus under FPTP it is not enough if most people are against the status quo: they have to agree on a replacement before they vote. Thus people end up voting ‘tactically’. This means that one can’t tell from the ballots what people’s actual preferences were. It could happen (especially before the Internet) that the media misled people into voting tactically (so as not to ‘waste’ their vote) when they would have preferred the outcome that they would have got by voting for their true preference. Mathematicians tend to prefer AV and PR because they are simpler in this respect.

Cameron argues against majoritarianism thus:

“It could mean that those who are courageous and brave and may not believe in or say things that everyone agrees with are pushed out of politics and those who are boring and the least controversial limping to victory. It could mean a Parliament of second choices. We wouldn’t accept this in any other walk of life.”

Thus, we have to think  of a situation where united oligarchs have 30% support but are hated by the other 70%. Perhaps using the media, they could ‘divide and rule’ so that no opposition party obtains more than 40% of the opposition vote, thus keeping the oligarchs in power indefinitely. Do we want an electoral system that could allow in ‘courageous and brave’ oligarchs?

Cameron makes a big point of:

“If the last election was under AV, there would be the chance, right now, that Gordon Brown would still be Prime Minister. Ok, the last election was not decisive in terms of who won. But it was certainly decisive in terms of who lost. And I think any system that keeps dead governments living on life support is a massive backward step for accountability and trust in our politics.”

One problem with many of the FPTP supporters is ‘anchoring’: they presumably see a government as ‘dead’ when it would lose under FPTP. But this means that the health of the government is heavily dependent on how many opposition parties there are, what there relative strengths are and how they are geographically spread. From a majoritarian viewpoint a government is ‘dead’ when a majority prefer some other party. Similarly a sitting member might be thought dead when a majority prefer some other candidate. Under alternatives to FPTP like AV a dead candidate will never be elected. Under FPTP dead candidates are often elected: in recent decades the opposition vote has always been split, so that a candidate with only 30%  support can win when the opposition vote is split. There is no way to tell from the ballots to tell when this has happened. If Cameron really wants to get rid of dead governments he will need something like AV+ or PR . Interestingly, this seems a very sensible requirement yet I have not seen any analysis of it. At least if anyone came up with a suitable method and we had a referendum, AV would  allow select the most preferred option, whereas under FPTP it would be difficult to predict waht would happen: FPTP would seem to have the advantage.

A feature of a genuine logical argument is that it still seems sensible when you change the example, subject to the explicit assumptions. In this case, suppose that we had a referendum with three choices: FPTP, AV and PR. If we used FPTP to count the ballots I suspect that the non-FPTP would be split between AV and PR. But suppose that those who put PR first put AV second, and vice-versa. Then under any majoritarian method FPTP could only win by getting 50% of the votes. Otherwise it would be ranked last by most voters, and hence has to be rejected by our majoritarian criterion. Is this fair? The Cameron argument is that the FPTP ballots are only counted once, while the others are counted twice. But which is more important, respecting the wishes of the majority, or arithmetic?

The ‘counting’ argument is also rather spurious, in that under AV we could simply mix up all the ballots before each round and do a full re-count, but ignoring the options that have been deleted. Each ballot would then be counted equally. Do we think it sensible to choose a method based on something that is a feature of how it is implemented, and not inherent in the method itself?

[Someone has since pointed out that while each ballot is counted as many times as their are rounds, for some ballots the first preference will be counted multiple times while for others a different preference could be counted in each round, as their earlier preferences are eliminated. If this is what the NO campaign is trying to say, it is saying it very badly. If we take this objection seriously then we must vote tactically. Is this subtle ‘inequality’ worse than tactical voting?]

The usual interpretation of ‘one person one vote’ admits AV, so this argument of the NO camp seems to be special pleading.

Mathematical speculations

In fact, there are significant differences between a referendum and a general election. The traditional argument in favour of FPTP is that it necessitates politicking and hence favours strong politicians, who can then apply their dark arts to our benefit in dealing with foreigners. PR (and perhaps AV) would lead to straightforward debates on the issues, risking us having leaders who are unpracticed at the dark arts. FPTP seems ideally suited in this role, but does lead the majority vulnerable to rule by the best at the dark arts, who may not always have the majority’s interests at heart.


Once one has settled on one’s criteria and adequately characterised the problem, one can evaluate a range of methods against them, not just FPTP, AV and PR. One might even develop a tailored method. Practically, though, FPTP, AV and PR seem to be the options, and if one does move away from FPTP then at least one will be able to have sensible referenda, to refine the method.

If one thought majoritarianism paramount, with a wish for ‘strong’ politicians second, then a reasonable method might be to reject all those options/candidates that the majority rank below some acceptable core, and to apply FPTP (if necessary) as a tie-break to the core. This would only differ from FPTP where FPTP would elect a candidate for which a clear majority preferred some other candidate. Similarly, the intention of AV is to avoid wasted votes and respect the majority wishes. This slightly different method can be seen as having the same intent, plus breaking ties using FPTP (to give stronger parties). To mathematicians, this is Condorcet modified to take account of the value of electing a candidate with strong support.


A feature of FPTP, with its tactical voting, is that has a raised barrier of entry to new parties compared with most other methods. AV also has a significant entry barrier in that a party with the fewest first place rankings is eliminated, even if it was a clear overall favourite (e.g., would have won under Condorcet). To go from a high to no barrier in one step could lead to an undesirable disruption to political life. Maybe AV is the best compromise? The Jenkins Commission thought so.

Political views

There are clearly political aspects to the choice. While we can say for sure that the FPTP arguments are wrong the AV arguments seem to thin too justify a change and the mathematical arguments too limited. What do politicians think? Cameron and Clegg (and many others) seem to agree that in most constituencies there will be no change. Cameron’s arguments seem mostly spurious, apart from the one that most voters will simply rank the minor parties the way their first choice tells them to. I have not seen any analysis of the impact of this, but it seems minor compared to the majoritarian criterion. Cleggs’ main argument is that candidates and MPs will need to ‘work harder’, which may offset Cameron’s points. But how do they know? What we can say is that:

  • AV would seem not to make any radical short-term difference.
  • AV reduces the need for tactical voting, so that we can better judge how fair the system is.
  • AV allows us to recognize up and coming candidates and parties, independently of media hype.
  • AV is less biased against the formation of new parties, or local independents.
  • AV would provide information on preferences that could inform choices on further reforms, if necessary.
  • AV, used in a referendum, would open the way to sensible further reforms, if needed.

 If it is true that an informed public would choose FPTP, adopting AV now would give them that choice, with no obvious down-side (scares on costs aside).

Tactical voting

The No campaign’s objections to AV would seem to apply equally to any method that did not encourage tactical voting, i.e. voting for someone who isn’t your actual first preference, but who you think has a better chance. The classic problems with tactical voting are:

  • Voters expectations can be manipulated, e.g. by the media.
  • The support for new parties (e.g. the Greens, a while back) is suppressed, and the giving the current main parties have an advantage.
  • It requires some degree of co-ordination to be able to vote a tyrant out ‘tactically’, to avoid vote-splitting.
  • It disenfranchises honest voters and those who are so clued-up in politics. (Especially if there are some political shocks just before the election.)


While the FPTP arguments appears more mathematical than AV’s its notion of ‘counting’ is spurious. There would seem more merit in the AV argument that under FPTP wasted votes do not count at all, and that AV remedies this defect. But – looking at it afresh – it is not enough to ‘count’ the votes: one may also want the results to respect the wishes of a clear majority, in order to be truly ‘democratic’. But this does not uniquely determine a method: a tie-break may be needed, and there may be some merit in FPTP. Indeed, while the No campaign arguments seem largely spurious, there does seem to lie behind it a genuine concern for the ‘health’ of politics and the strength of government.

It seems to me that mathematics can provide some useful insights, but some greater understanding is required to reach a definitive verdict.  Both Cameron and Clegg make the short-term effects seem rather minor. It is unfortunate that there isn’t the option of a pilot scheme, but AV has a clear edge for referenda and if the majority came to view AV as a ‘dead method’ we could easily return to FPTP. Indeed, we could routinely publish both FPTP and AV results, to inform the public. The arguments against such a tentative view seem unconvincing.



The No2AV campaign gives these reasons to vote NO:

  1. AV is costly …
  2. AV is complex and unfair
    The winner should be the candidate that comes first, but under AV the candidate who comes second or third can actually be elected. That’s why it is used by just three countries in the world – Fiji, Australia and Papua New Guinea. Voters should decide who the best candidate is, not the voting system. We can’t afford to let the politicians off the hook by introducing a loser’s charter.
  3. AV is a politician’s fix …

The second point seems to be an insult to the readers’ intelligence: one could equally well say “The winner should be the candidate that comes first, but under FPTP the candidate who comes second or third can actually be elected.” It all depends on who you think ‘should’ come first. E.g., should someone who is ranked last on 70% of the ballots be elected simply because the other candidates’ votes are split? The No campaign’s points seem all spin and no substance.


Lord Reid has given an interview in which he says

  •  [A] cornerstone of our democratic system has been ‘one person one vote’ …
  • [My vote] has the same weight as everyone elses.
  • [AV] completely undermines and corrupts that; some people will have one vote, others … will be counted again and again.
  • [AV] is a theat to the .. basis of our democratic system.

When the interviewer notes that under AV one gets the candidate that ‘most people are happy with’ Reid responds, of AV, that:

  • If you vote for liberal, labour or conservative it is overwhelmingly likely that your vote will be counted once, whereas if I go out and vote for one of the ‘fringe’ candidates [my vote may be counted many times] … how is it fair [?]

 The emerging ‘No’ message seems to be that only ‘fringe’ candidates ‘such as the BNP’ would benefit from AV. I note:

  • The Green party is also a fringe party, as would be a ‘reform democracy’ or ‘reform expenses’ party.
  • In many constituencies (like mine) one of the three main parties is ‘fringe’.
  • Independents, such as doctors standing to save a local hospital, are ultra- fringe. They may also revitalise democracy.
  • If the three main parties have candidates, it is mathematically certain that at least one of them will have their votes counted at least twice.

The example seems bogus, both mathematically and practically. ‘One vote’ and ‘equal weight’ could mean any of:

  1. One ballot paper each.
  2. One mark each (X).
  3. The ballots are counted by a process which only takes account of each ballot once.
  4. The ballots are counted by a process that only takes account of each vote once.
  5. The ballots could be counted by a process that only takes account of each ballot once.
  6. The same opportunities and rules for everyone.

It is not clear which Lord Reid considers essential to democracy. FPTP satisfies them all, but would allow an oligarchy with enough influence over the media to retain power even if it was most voters last choice, as above. AV meets 1 but fails 2, as would any alternative to FPTP. AV also fails 3, but so what? We could stage a series of rounds (as in France) with trailing candidates being eliminated until one candidate has a majority. The result would be the same, but each vote would be counted once (per round) and counted equally. Do we wish to reject AV on a technicality? With one round AV satisfies 4, if by ‘vote’ one means ballot, and also satisfies 5 and 6. Isn’t it 6 that matters?

Lord Reid also refers to ‘weight’, without defining it. Suppose that two parties traditionally vie for the seat, with the rest being no-hopers. Then a doctor stands on the ticket of supporting the local hospital and otherwise consulting his constituents. Suppose you would prefer to vote for this ‘fringe’ candidate’. Under FPTP you could vote for the doctor, thus recording your support for the hospital but taking no part in the main contest. Or you could vote for a main candidate, failing to record your support for the doctor. Under AV you would simply record your actual preferences, thus recording support for the hospital and taking part in the main contest. And the doctor might even win. Under AV your vote clearly has more ‘weight’, but which is fairer? If we think of a group of people with similar views, then those who support a main candidate will all vote for them, so that their weight of support is not split. Under AV the weight of support for the main candidate is undiminished (it is what it would have been had everyone voted tactically). The support for the fringe is also undiminished (it is what it would have been had they all voted honestly). Under FPTP the weight of support will be divided, unless they all vote tactically. So which is a fairer definition of ‘weight’? Is it obvious that AV undermines democracy?

I think there are some sensible arguments for ‘No’, but the No campaign isn’t using them, and the ability of a party whom most people hate to get elected (due to vote splitting) under FPTP seems much more significant in undermining democracy.

See also

Dave Marsay


About Dave Marsay
Mathematician with an interest in 'good' reasoning.

6 Responses to The new mathematics of voting?

  1. Dear Dave

    An interesting run through the arguments.

    Condorcet with one of the resolution methods when there is no Condorcet winner has real intellectual appeal. However most people will struggle to understand it; it is hard enough to get them to understand AV/IRV. Also, if you consider the “physical” analogues, one can easily visualise a series of elimination ballots the analogue of AV/IRV) but it is hard to imagine any society choosing to hold a large number of pairwise ballots, although if someone won all of the pairwise ballots (ie became the Condorcet winner) then they would have great legitimacy.

    Your article should come to an explicit conclusion as the real referendum is a binary choice. Will you be voting YES or NO?

  2. Pingback: AV or FPTP, according to wikipedia and Jenkins | djmarsay

  3. djmarsay says:

    Mohammed, at the end I’ve added a link to my latest notes comparing FPTP and AV, based on wikipedia and Jenkins. The reasoned arguments seem currently to favour AV, and the more the NO campaign draws attention to particular features of AV, the more it seems reasonable. Right now I would vote AV, but I would prefer the NO campaign to make a clearer case before finally deciding. Dave

  4. Pingback: AV: Yes or No? | djmarsay

  5. sfnhltb says:

    Nice analysis, some of the key points I deduced for myself (partly informed from some vague previous reading on Condorcet and voting methods). It must be said the final decision for me was made by the NO campaign – it was full of such obvious lies and misrepresentations that I wouldn’t vote for it just on principle. The YES campaign was pretty shoddy though – as you picked out the rubbish about “one person one vote” should have been easy to knock down, but wasn’t. I guess the other problem is the same one that persists with local politics – that all votes are treated as referenda on the what is going on in the Commons, whether it be local by-elections, or this AV vote which was turned in many minds to a vote for or against Nick Clegg.

    • djmarsay says:

      Yes. I see the referendum as starkly indicative of the quality of political debate. You have commented on my notes on risk. I don’t see how we could prevent risks from being politicised, in which case the population will be only confused by debates of this ‘quality’.

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