## Can polls be reliable?

Election polls in many countries have seemed unusually unreliable recently. Why? And can they be fixed?

The most basic observation is that if one has a random sample of a population in which x% has some attribute then it is reasonable to estimate that x% of the whole population has that attribute, and that this estimate will tend to be more accurate the larger the sample is. In some polls sample size can be an issue, but not in the main political polls.

A fundamental problem with most polls is that the ‘random’ sample may not be uniformly distributed, with some sub-groups over or under represented. Political polls have some additional issues, that are sometimes blamed:

• People with certain opinions may be reluctant to express them, or may even mislead.
• There may be a shift in opinions with time, due to campaigns or events.
• Different groups may differ in whether they actually vote, for example depending on the weather.

I also think that in the UK the trend to postal voting may have confused things, as postal voters will have missed out on the later stages of campaigns, and on later events. (Which were significant in the UK 2017 general election.)

Pollsters have a lot of experience at compensating for these distortions, and are increasingly using ‘sophisticated mathematical tools’. How is this possible, and is there any residual uncertainty?

Back to mathematics, suppose that we have a science-like situation in which we know which factors (e.g. gender, age, social class ..) are relevant. With a large enough sample we can partition the results by combination of factors, measure the proportions for each combination, and then combine these proportions, weighting by the prevalence of the combinations in the whole population. (More sophisticated approaches are used for smaller samples, but they only reduce the statistical reliability.)

Systematic errors can creep in in two ways:

1. Instead of using just the poll data, some ‘laws of politics’ (such as the effect of rain) or other heuristics (such as that the swing among postal votes will be similar to that for votes in person) may be wrong.
2. An important factor is missed. (For example, people with teenage children or grandchildren may vote differently from their peers when student fees are an issue.)

These issues have analogues in the science lab. In the first place one is using the wrong theory to interpret the data, and so the results are corrupted. In the second case one has some unnoticed ‘uncontrolled variable’ that can really confuse things.

A polling method using fixed factors and laws will only be reliable when they reasonably accurately the attributes of interest, and not when ‘the nature of politics’ is changing, as it often does and as it seems to be right now in North America and Europe. (According to game theory one should expect such changes when coalitions change or are under threat, as they are.) To do better, the polling organisation would need to understand the factors that the parties were bringing into play at least as well as the parties themselves, and possibly better. This seems unlikely, at least in the UK.

What can be done?

It seems to me that polls used to be relatively easy to interpret, possibly because they were simpler. Our more sophisticated contemporary methods make more detailed assumptions. To interpret them we would need to know what these assumptions were. We could then ‘aim off’, based on our own judgment. But this would involve pollsters in publishing some details of their methods, which they are naturally loth to do. So what could be done? Maybe we could have some agreed simple methods and publish findings as ‘extrapolations’ to inform debate, rather than predictions. We could then factor in our own assumptions. (For example, our assumptions about students turnout.)

So, I don’t think that we can expect reliable poll findings that are predictions, but possibly we could have useful poll findings that would inform debate and allow us to take our own views. (A bit like any ‘big data’.)

Dave Marsay

## Is there a reasonable way to count votes?

I have seen a supposedly learned journal which suggests that there is no ‘fair’ way to select a winner from a set of ballots. Although not emphasised by wikipedia or the UN, this is a commonly held view, often relying on abstruse maths.

## A simple example

Suppose that there are two candidates who are the first choice of an equal number of voters: a tie. Then no fair deterministic method can select a winner. But is this a practical problem?

## Typical proofs

Condorcet was at least one of the first to note that a set of ballots could imply a 3-way tie, with A being above B on most ballots, B above C on most, and yet C above A on most ballots. Such ties could be relatively robust, in that the addition of another ballot would not necessarily break the tie. For example, if a voters vote ABC, b voters vote BCA and c vote CAB then there is a 3-way tie (A over B over C over A) whenever (a+b)>c etc. If those who voted ABC wish to eliminate C they can do so by voting BAC, in which case B has a clear majority. This is certainly a problem, but it did not stop Condorcet from devising what he regarded as a ‘reasonable’ method. The difficulties lie in breaking n-way ties. The best known impossibility theorems is Arrow’s. It is implicit that the ‘voting system’ is deterministic. The problem is in what it calls “irrelevant alternatives”. But in Condorcet’s 3-way tie, C is ‘relevant’ to the choice between A and B, although Arrow says that it isn’t. So although Arrow’s approach is interesting, the result could be summarised as saying that ties are a problem. The Gibbard-Satterthwaite Theorem extends Arrow by replacing consideration of ‘irrelevant’ alternatives consideration of tactical voting. This is a much more reasonable criterion. As Condorcet and others appreciated, tactical voting can easily arise in any definite method that seeks to break a 3-way tie. But how significant is this?

## A simple counter-example

The mathematics above is correct, but we may doubt the interpretation. A simple method is to put the ballots into a hat, pull one out ‘at random’, and to select a winner from it.

Irrelevant alternatives:  the winner depends only on the chosen ballot, not on any alternatives.

Tactical voting: how people vote only matters if their ballot is selected, in which case they would do best to vote honestly.

This method is not very attractive, but it does show that the impossibility theorems need careful interpretation.

## A more reasonable counter-example?

I have devised this example to be ‘obviously’ reasonable, without necessarily being the best. First, a candidate is said to have a majority over another if it is ranked higher on more ballots. The method is:

• If we can divide the candidates into two sets, such that all members of the first set have a majority over all members of the second set, then the second set is eliminated.
• This is done to yield a minimal candidate set, with ties between its members.
• A ballot is chosen at random and that member of the candidate set that is ranked highest is selected.

## Independence from Irrelevant Alternatives

All members of the candidate set are relevant in the sense that there is no agreed way to choose between them. In this sense the above method meets the requirement for ‘independence from irrelevant alternatives’. The more conventional requirement seems much too strong.

## Tactical Voting

Is the method liable to tactical voting? Firstly, as before, if my candidate gets into the candidate set then it pays to rank my candidate first, and the other rankings do not count. Hence the only possibility of tactical voting is in determining the candidate set. As above, I could try to eliminate candidates from the candidate set by exploiting cycles, but I cannot falsely promote candidates into the candidate set, because I would have ranked them highly anyway. In this sense the scope for tactical voting is limited. You vote for the candidate with the best chance of beating your least liked candidate. If this is your priority, then this is arguably an honest vote. It may be that we need to distinguish between types of tactical voting, rather than treat them all as equally disreputable.

## Types of tactics

Some voting systems, such as first-past-the-post, waste votes for genuine first-preferences when they are unpopular. Some hold this to be a good thing, but it means that the votes are cast are not a reliable indication of actual preferences. While it would be ‘a good thing’ for a system not to penalize voters who rank all their preference honestly, a compromise would be a system that never penalizes voters for ranking their true first preference first. Not all tactics are equally harmful. In Condorcet’s system problems arise with ties. If, as above, ties are broken by selecting ballots at random, there is no incentive not to rank one’s first preference first. An alternative method would be to eliminate candidates who were most often worst-ranked. This would encourage voters to take account of how often candidates were likely to be given a low ranking, and so might encourage them to rank a more moderate candidate first. While this is tactical voting, it is not obvious that it would be a ‘bad thing’. Finally, some voting systems (such as first-past-the-post and approval voting) call for tactics that depend on subtle assesments of how others were likely to vote. Under alternative vote, for instance, you want to work out in which round candidates wull be eliminated and who will pick up their next preferences. Tactics which depend more on relatively simple factors, such as who is more moderate, seem preferable. The aim here is not to advocate any particular method, but to cast doubt on simplistic interpretations of ASrrow’s theorem.

Dave Marsay

## AV: Yes or No? A comparison of the Campaigns’ ‘reasons’

At last we have some sensible claims to compare, at the Beeb. Here are some comments:

# YES Campaign

## Its reasons

1. AV makes people work harder
2. AV cuts safe seats
3. AV is a simple upgrade
5. AV is our one chance for a change

## An Assessment

These are essentially taken from the all-party Jenkins Commission. The NO Campaign rejoinders seem to be:

1. Not significantly so.
2. Not significantly so.
3. AV will require computers and £250M to implement (see below).
4. AV Makes votes count twice, or more (se below).
5. Too right!

## A Summary

Worthy, but dull.

• There would be a lot less need for tactical voting
• The results would more reliably indicate people’s actual first preferences
• It would be a lot easier to vote out an unpopular government – no ‘vote splitting’
• It would make it possible for a new party to grow support across elections to challenge the status quo.
• It may lead to greater turnout, especially in seats that are currently safe

# NO Campaign Reasons

## AV is unfair

### Claim

“… some people would get their vote counted more times than others. For generations, elections in the UK have been based on the fundamental principle of ‘one person, one vote’. AV would undermine all that by allowing the supporters of fringe parties to have their second, third or fourth choices counted – while supporters of the mainstream candidates would only get their vote counted once.”

### Notes

According to the Concise OED a vote is ‘a formal expression of will or opinion in regard to election of … signified by ballot …’ Thus the Irish, Scottish, Welsh, Australians, who cast similar ballots to AV, ‘have one vote’. The NO Campaign use of the term ‘counted’ is also confusing. The general meaning is a ‘reckoning’, and in this sense each polling station has one count per election, and this remains true under AV. A peculiarity of AV is that ballots are also counted in the sense of ‘find number of”. (See ‘maths of voting’ for more.)

### Assessment

There is no obvious principle that requires us to stick with FPTP: all ballots are counted according to the same rules.

Should ‘supporters of fringe parties’ have their second preferences counted? The ‘fringe’ includes:

• Local candidates, such as a doctor trying to stop the closure of a hospital
• The Greens
• In some constituencies, Labour, LibDem, Conservative.

AV is blind to everything except how voters rank them. Consider an election in which the top three candidates get 30%, 28%, 26%, with some also-rans. According to the NO campaign the candidate with a narrow margin should be declared the winner. Thus they would disregard the preferences of anyone who votes for their hospital (say). Is this reasonable?

## AV is not widely used

True-ish, but neither is FPTP (in terms of countries – one of them is large), and variants of AV (IRV, STV, …) together are the most widely used.

## AV is expensive

Countries with AV don’t have election machinery. Australian elections may cost more than ours, but it is a much bigger country with a smaller population.

## AV hand more power to politicians

See the Jenkins Commission.

## AV supporters are sceptical

Opposition to FPTP is split between variants of AV, with single-member constituencies and forms of PR. The Jenkins Commission recommended AV+, seeking to provide the best of both. The referendum is FPTP and hence can only cope with two alternatives: YES or NO.

I don’t know that AV supporters are sceptical against a move away from FPTP – just differ on what would be ideal.

• The NO campaign is playing down the ‘strong and stable government’ argument. The flip side is that an unpopular government can survive.
• A traditional argument for FPTP was that it encourages tactical voting and hence politicking, and hence develops tough leaders, good at dealing with foreigners. We haven’t heard this, this time. Maybe the times are different?

## AV: the worst example

According to the NO campaign the Torfaen election shows AV in the worst light. Labour won with 44.8%, followed by Conservative (20%), LibDem (16.6%)  and 6 more (5.3% or less each). The No campaign claim that under AV the 8th placed, an Independent, could have won. But to do so Labour would have had to have picked up less than 5.3% from the other candidates, including LibDem, and the Independent would have had to be ranked higher than the others by a majority. In particular, the Independent could not have won without support from Conservative voters.

Is it reasonable for Conservatives to complain?:

• Conservative votes contributed to the victory.
• Don’t the Conservatives prefer this to Labour?

It is also worth noting that the Independent would have to have picked up most second-rank votes from the Greens and UKIP, and so on, which also seems unlikely.

AV pros and cons

Dave Marsay

## The choice: FPTP or AV?

The UK has this choice on May 5th. (AV is also known as Instant Runoff Voting.) The debate so far hasn’t been particularly enlightening. Here I consider the advice from wikipedia and the UK Jenkins Commission, with a short note on tactical voting.

## Wikipedia

Wikipedia gives a comparison of AV to other voting systems. We are interested in FPTP, a variant of plurality voting. Wikipedia has a table  showing that both methods are about equally common. But which is best?

Wikipedia shows the following advantages for FPTP:

• Preservation of “one person, one vote” principle.
• Moderation
• Fewer minority parties.

It notes that IRV (AV) is also generally regarded as satisfying ‘one  person one vote’, so this can be discounted. The other two are partially true, but one needs to consider the whole truth.

The UK ‘No’ campaign makes a number of other claims, but they don’t seem to have any validity.

### Moderation

Wikipedia notes:

Under a first-past-the-post system, voters are often afraid of “wasting” their vote on a candidate unlikely to win, so they cast their vote towards their most preferable choice possible of victory. Advocates of plurality voting suggest that this results in most candidates having to field a fairly moderate or centrist position.

This suggests that a moderate or centrist result is desirable. Where FPTP relies on tactical voting (which seems not to be so very common) to achieve this, AV tends to achieve it by design. It also achieves what wikipedia calls the mutual majority criterion. This is quite technical, but links to the notion of majority rule.

### Majority rule

Majority rule is the binary decision criterion that if most people prefer A to B then A will be selected. For three or more choices all deterministic methods are technically vulnerable to tactical voting, so one needs to decide which desiderata are essential, and which can be compromised.

Suppose that one has a tribal society with the biggest tribe commanding 26% of the vote using its majority to repress the other tribes. Suppose that all other tribes would prefer a representative from anything other than the biggest tribe. Then majority rules demands that they get one. But if all tribes put up a candidate then the biggest tribe may win, due to vote splitting.

AV is not liable to vote-splitting and respects the mutual majority criterion, and hence is ‘democratic’ in a different (fuller?) sense than FPTP where it seems to be considered a virtue that the opposition parties must form a pre-election agreement on a common candidate.

### Fewer minority parties

FPTP encourages both tactical voting and strategic agreement, coalitions or unions of parties, to avoid gross vote splitting. This leads to fewer minority parties, either because tactical voters don’t vote for them or because they merge with other parties. Under AV voters can vote for a minority party without wasting their vote. Voting will also generally be less tactical, and thus, unlike FPTP, a minority is not disadvantaged in building to a majority over a series of elections.

### Summary from FPTP perspective

FPTP rewards politicking and ‘strong parties’ by rewarding tactical voting and pre-election agreements between candidates or parties. It disadvantages minority parties, such as the UK Greens and BNP. Are these good things?

Under FPTP a minority party which was most voter’s last choice could gain or retain power by ‘divide and rule’. Is this a problem?

Wikipedia notes many advantages of AV, including the mutual majority criterion. It has this special case:

Instant-runoff voting [aka AV] also passes the Condorcet loser criterion, which requires that if a candidate would lose a head-to-head competition against every other candidate, they must not win the overall election. First-past-the-post does not meet this criterion, indeed it is usually violated in elections with more than two popular candidates.

## Tactical voting

All methods are vulnerable to tactical voting,  so some compromise is required. Under FPTP vote-splitting can encourage you to vote tactically to avoid your worst-case choice winning. Wikipedia notes that, while not perfect, “alternative vote is quite resistant to strategy” (i.e., tactical voting). Under AV you are encouraged to vote tactically when:

• There are preferences that form a cycle, as in AB, BC, CA.
• You can be sure that the other candidates’ supporters aren’t voting tactically, or if they are, what proportion is voting tactically, and how.

The first condition is normally considered rare, and is where all methods have a problem. The second makes tactical voting much more risky: whereas under FPTP tactical voting is usually straightforward, under AV it is for from this.

## Jenkins Commission

The UK’s Jenkins Commission took a broad of the political implications of FPTP versus AV and others. The defects of FPTP were:

• A tendency to result in landslides.
• Disadvantages third parties, even strong ones.
• Disadvantages parties with even support across constituencies.
• The essential contest is fought over a few ‘marginal’ seats.
• It leads to ‘perverse’ results.
• It advantages the ruling party.

The report noted the above advantages. Further:

Fairness to voters is the first essential. A primary duty of an electoral system is to represent the wishes of the electorate as effectively as possible. The major ‘fairness’ count against First Past the Post is that it distorts the desires of the voters. That the voters do not get the representation they want is more important than that the parties do not get the seats to which they think they are entitled.

And:

It [AV] would also virtually ensure that each MP commanded at least majority acquiescence within his constituency, which is far from being the case under FPTP, where as we have seen nearly a half of members have more opponents than supporters, and, exceptionally, a member can be elected (as in Inverness in 1992) with as little as 26% of the vote.

There were no criticisms of AV at the constituency level. The most significant criticism was that it isn’t proportional. AV is not always any more proportionate than FPTP, but this is not at issue in the UK referendum. In any case, some people prefer methods that tend to lead to enhanced majorities.

## Consequences

Wikepedia seems to provide a relatively independent summary of voting systems, including the unavoidable problems and the pros and cons of FPTP and AV (aka IRV) considered across many countries and years. A comparison with the UK’s current YES and NO campaigns seems instructive: not everything they say should be taken at face value.

One way to decide would be to rate yourself on the following scales:

1. We need a system that gives the sitting candidate / party an advantage … or not.
2. We need a system that gives the two leading candidates / parties an advantage … or not.
3. We need a system that discriminates against minority parties such as the BNP … or we don’t want to discriminate against parties like the Green party.
4. We want a system that tends to leave us with the same old two main parties … or we want a system that allows new parties to grow and potentially overtake the old parties, as long as they have support.
5. We want a system that rewards politicking  and tactical voting … or not.
6. We want every vote to be counted once (in a technical sense) … or we want to make sure that votes are not split.
7. We want a system that tends to stable government, by giving the ruling party an advantage … versus we want a system that will enable us to oust the ruling party when the majority wish to do so.
8. We want the winner to have the most first preferences … or we want to reject every candidate that belongs to a group such that some majority prefers all of those outside the group to all of those inside (as in the criterion above).

Of these alternatives, the first would indicate FPTP, the second AV. Another way to choose is to think about the current situation. If the government does well the Conservatives are likely to get a working majority next time. If the government does badly then labour should do well. The voting system should only affect things if the government performance is debatable, so if you support one of the major parties then the voting system will not matter unless your party’s policy turns out to be wrong. What might favour the liberals? If the coalition does okay and many floating voters think that it could have gone much worse without the liberals’ moderating influence. Thus, if the liberals’ manifesto was essentially correct they might do better than last time, more so under AV. This could be seen to be reasonable. I can’t think of a credible situation in which AV favours a party whose manifesto ‘line’ was clearly wrong. The longer term impacts, such as suppressing the BNP and the Greens, seem more significant. For example, labour might split into ‘new’ and ‘old’, allowing the electorate to choose. similarly, if the government does badly the Conservatives might split into ‘wet’ and ‘dry’, giving voters a choice. Under FPTP the two main parties are effectively coalitions with only party members choosing the ‘flavour’. AV would give others a say, which would presumably be a moderating influence. Which is more reasonable?

Which of all these considerations is more important? My current view is that:

•  AV better respects the wishes of the majority
• FPTP encourages tactical voting
• AV gives candidates which have a lot of first preferences some advantage over ‘wishy washy’ candidates that are ranked reasonably high by most but which are the first preference of few. This is not as much of factor as for FPTP, but seems reasonable.

David Marsay

## 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.

### Synthesis

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.

### Barriers

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.)

Conclusion

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.

#### February

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.

#### March

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.