Alternative alternative voting systems, part three: Approval Rate Voting (ARV)

In this series of posts on alternatives to the First Past the Post and Alternative Vote (AV) voting systems, which are the only alternatives the British people are going to be offered for English and UK elections in the proposed referendum in May next year, I’ve discussed AV itself and Approval Voting (which could also confusingly be abbreviated as ‘AV’). So what is ‘Approval Rate Voting’ (or ARV), which I’m going to discuss here?

ARV, which I’ve deliberately named as such in order to position it aggressively as an alternative to AV (and to Approval Voting), is the new-improved version of the single member-constituency system of my own invention that I previously termed ‘PV’ (or the ‘Popular Vote’). I previously described how this works as follows:

  • ARV is a form of range voting. In the same way as a Borda Count (one of the main forms of range voting), it assigns a number of points to voters’ preferences. The maximum number of points that can be obtained by any candidate is determined by the overall number of candidates. For example, if there are five candidates, the highest number of points you can give to any candidate is five (i.e. they would be your first choice).
  • However, unlike a classic Borda Count, voters would not be obliged to rank all of the candidates (e.g. from five to one); nor would they be obliged to award their top candidate five points. They could, for instance, decide to award their favourite candidate any number of points from one to five: a voter’s first-choice candidate would simply be the one to whom they gave the most points. In most cases, voters would give their preferred candidate five points; but it would be entirely up to them how many points they decided each candidate merited, the only restriction being that no more than one candidate could be given the same score. In other words, you couldn’t award the same number of points to more than one candidate, meaning that you’re obliged to demonstrate your preferences in the points you award to each candidate. Voters could also choose to give some candidates no points, by either writing a zero in the box next to those candidates’ names or leaving it blank.
  • When the vote is counted, note is taken of the first preference of each voter, and if a majority of voters selects one of the candidates as their first choice, that candidate is elected. However, if no such majority is secured, the result is then determined by the number of points each candidate has obtained – the winner being the candidate with the most points.

The improvement, as I see it, that I’ve now introduced is to allow equal rankings: voters can give the same number of points to more than one candidate, including their ‘first-preference’ candidates, which would make those candidates the ‘joint favourites’ of that particular voter. This also has the effect that more than one candidate could obtain an overall majority of first-preference votes. The winner in this instance would be the candidate with the largest number of such votes (and in the unlikely event that more than one majority candidate obtained the exact same total of votes, the winner would be the one obtaining the highest number of points).

The effect of this change is to make ARV more like Approval Voting, in that indicating more than one candidate as one’s ‘favourite’ is similar to the way, in Approval Voting, all candidates are effectively counted as if they were voters’ equal-favourite choice. The difference is that my range-voting modification allows voters to exactly grade the extent to which they ‘approve’ of candidates by giving each of them a rating: hence, ‘Approval Rate Voting’.

Allowing equal rankings is an improvement to my first version of ARV (PV) on a number of grounds:

  1. It’s more logically consistent: in the previous version of the system, it was allowed to give more than one candidate zero points; so why not allow voters to give more than one candidate any number of points, so long as those points stay within the permitted range (e.g. nought to five), which is determined by the number of candidates in the election?
  2. Allowing equal rankings is also more consistent with the way voters actually think about candidates and parties: some voters may be genuinely undecided and would wish to rate two or more candidates equally. There isn’t a good enough reason to force such voters to rank the candidates preferentially, as AV does for largely functional reasons to engineer a ‘majority’ choice.
  3. This change also makes the system much more open and liable to untap the latent support for smaller parties that exists in England and the UK at large, in that voters who support, say, UKIP or the Greens can make them their equal-first preference along with whichever mainstream party they have tended to vote for under FPTP (e.g. the Conservatives or the Lib Dems). In this way, the smaller parties would win a large share of the ‘approval points’ that ARV allows voters to award to the different parties; and in some cases, voters’ wish to register strong support for alternatives to the big-three parties in this way could lead to shock wins for the smaller parties, whether on a majority of first preferences or – more likely – on points.

This business of points constitutes a major innovation, which could bring radical changes to the way politicians engage with voters. Election results would include not only the number and percentage share of first-preference votes each party had received, but also an ‘approval rating’, which would be a percentage figure derived from dividing the number of points each candidate and party had received by the total number of points available to that candidate. E.g. in a constituency in which five candidates were standing, the maximum number of points available to each candidate would be the number of people voting multiplied by five. Dividing the actual number of points received by this theoretical maximum and multiplying by 100 gives a percentage ‘approval rating’ for each candidate. This could of course be extended to the UK-wide results, so you’d end up not only with each party’s share of the total vote but their overall approval ratings.

These approval ratings would generally be higher than each party’s share of first-preference votes. For example, assuming that each person indicating the Conservatives as their first preference (effectively, 36% of voters at the last election) awarded them maximum points, there would in addition be some voters that awarded the Tories fewer than maximum points (i.e. for whom the Conservative candidate was not their first preference but who wished to express a degree of approval for the Tories all the same). My sense is that this non-first-preference support for the Conservatives would not have been high enough to give them an approval rating of over 50% at the last election, just as the actual votes they received were not enough to give them an overall parliamentary majority. I estimate their approval rating would have been around 45% at the election, which is almost the same as the share of seats they won.

By contrast, when New Labour won the election in 1997, the level of latent support for Labour among Liberal Democrat and even Conservative voters was so high that I estimate Labour’s approval rating, under my system, would have been around 60% to 65%: again, similar to the share of seats they won. I’m going to consider the question of how proportional the ARV system would be below. But what these figures suggest is that the present FPTP voting system does produce results that are to some extent proportional, not to the share of vote each party receives but to the degree of approval the parties benefit from. The difference is that approval does not always translate to voting in FPTP elections. For example, I reckon that Labour and the Lib Dems would have obtained similar approval ratings at the last election: maybe around 35% to 40%. This reflects the way the parties performed in opinion polls during the election campaign, where the Lib Dems were neck and neck with Labour for much of the time but both were behind the Tories. However, when it came to voting, the latent support for the Lib Dems (what I’m calling the approval for them) was suppressed in a classic two-party squeeze whereby voters were dissuaded from voting Lib Dem in case this resulted in a victory for parties (i.e. Labour or the Conservatives) they opposed.

The ARV system would release this latent support for parties other than the big two (or two and a half) by allowing voters to express different degrees of approval for any number of parties without thereby inadvertently aiding the cause of parties they dislike. Among other effects, this could lead to the smaller parties obtaining some quite high approval ratings. For a start, as I suggested above, voters would be emboldened to award a large number of points, and even (equal-)top points, to parties with which they sympathise but for which they have hitherto refrained from voting so as not to ‘waste’ their vote. So I would expect parties like UKIP, the Greens, the BNP and the English Democrats to record quite respectable approval ratings. It would not be beyond the bounds of possibility, for instance, for UKIP and the Greens to obtain approval scores of between 10% and 20%: consistent with the share of votes and seats UKIP has won in the proportional elections for the European Parliament. Although this would be unlikely to translate into significantly more seats under ARV (as a single member-constituency system), the relatively high approval ratings for UKIP and the Greens would make it more difficult for the mainstream parties to ignore the views of voters on issues such as the EU or the environment, which the present system allows them to do. And in time, voters might become more aware of the power that ARV gives them and would be encouraged to mark smaller parties as their sole first preference in the confidence that voters of opposing political persuasions would also be doing the same (i.e. voting for smaller parties ahead of their tactical choice).

So how does ARV perform in relation to the six criteria by which I have been assessing alternative voting systems? In relation to the first criterion – does every vote count / is every vote counted? – I’d award ARV four points out of five (this compares with three for FPTP and Approval Voting, and two for AV). ARV is clearly better than FPTP in that no vote is wasted: voters can vote both for the candidate(s) they truly support and for the ‘least bad’ candidate with a chance of winning; and they can fine-tune their vote to express the precise degree of support or otherwise they want to give to each candidate. However, ARV doesn’t get a perfect five, as it’s still the case that the shares of the vote and the approval ratings obtained by the parties translate only imperfectly into shares of seats, such that votes for the smaller parties continue to carry less weight than votes for the larger parties.

With respect to the second criterion – is the system proportional? – ARV scores only three out of five (compared with two for AV). ARV is not intrinsically a proportional system in that it is a single member-constituency system; and these always carry a bias whereby parties obtaining a plurality of votes can win a majority of the seats. However, ARV would change the basis on which proportionality itself would be measured, as I suggested above: the degree to which the shares of seats were proportional would be assessed in relation not only to vote share but approval ratings. As I pointed out above, New Labour’s landslide victory in 1997 was highly unsatisfactory when measured against vote share, as Labour secured only 43% of the votes. But measured against approval ratings, Labour’s 63.4% share of the seats was probably quite proportional.

This is in part a mathematical coincidence in that, for the leading parties, the share of seats won in single member-constituency elections and the parties’ latent approval ratings, as I define them, are both higher than their vote share. However, under ARV, in seats where no majority of first-preference votes is obtained, the results would be decided on the basis of approval points, where the parties would be more closely matched. In addition, the number of constituencies where no party secured an overall majority would be higher than under FPTP as a result of the change in voter behaviour described above, whereby voters would increasingly favour minor parties at the expense of the larger parties for which they have previously voted tactically. The overall election results would therefore be increasingly proportionate to the relative standings of the parties in terms of their approval ratings.

In this way, ARV would be much more effective than AV at transforming UK politics into a pluralist, multi-party environment in which results would almost always be decided on approval ratings (because no party would win a majority based on first preferences) and would therefore more accurately reflect the true degree of support enjoyed by each party (as measured by the approval rating). There would still be a bias that would tend to transform a strong approval rating into the potential for an overall majority UK-wide, even if a majority of first-preference votes had not been obtained by that party. But those results would still be relatively proportional to the approval ratings themselves and therefore would produce an outcome that voters as a whole would be by definition more content with, given that the approval rating provides an index of the real level of popularity of each party. Such would have been the case in Labour’s 1997 landslide: seats that were in fact won by Labour on a plurality would still have been won on the ARV points system, as Labour enjoyed genuine cross-electorate approval; so that Labour’s big majority would have been relatively proportional to its approval standing.

Criterion No. 3: Does the system foster accountability? Here, I’d award ARV four points out of five (compared with three for both AV and FPTP). In common with those other single-member systems, MPs are directly accountable to their electorate under ARV. But ARV represents an improvement over AV and FPTP in that it is much easier for voters to punish MPs and parties they are disenchanted with by, say, awarding approval points to other candidates while still assigning points (even maximum points) to the incumbent MP. This aspect of ARV would be likely to provoke the greatest opposition from the likes of the Conservatives, who would fear that left-of-centre voters would gang up against Conservative MPs by awarding equal-top points to the Labour, Lib Dem and even Green candidates. The same criticism tends to be made of pure Approval Voting, as I discussed in my previous post in this series.

However, ARV differs from Approval Voting in that it allows voters to fine-tune their votes to express the precise degree to which they are prepared to lend their support to each candidate; and it’s far from obvious that enough left-of-centre voters could be mobilised to give maximum points to all left-of-centre candidates in order to aggressively oust incumbent Tories. For a start, English people are generally quite fair-minded and could well be hostile towards any attempt to persuade them to vote negatively in this way. In addition, it’s likely that many Lib Dem and Labour voters would be naturally disinclined to award equal-top marks to each other’s candidates, especially as the system exempts them from having to do this, unlike Approval Voting. Besides, many Lib Dems and even some Labour voters might even award a certain number of points to the Tory MP if they thought (s)he was a good constituency MP. So, in essence, the ability to punish the parties cuts both ways, and ARV places an extra premium on MPs being both good servants of their constituents and courting the support of voters whose main preference is for other parties. It’s not, however, possible to give ARV a full five points against this criterion, as ARV still contains a bias towards creating unrepresentative parliamentary majorities or pluralities, which are one of the main causes of unaccountable government.

As for my fourth criterion – Does the voting system allow voters to express the full range of their political and personal preferences, and send a message to politicians? – ARV scores a strong four out of five, compared with only two for AV and one for FPTP. ARV allows voters to express the precise degree of their support for all the parties, and it permits them to indicate more than one party as their first preference, which ironically gives more weight to many voters’ actual favourite candidate and party than AV. That’s because, even though voters would tend to indicate their actual favourite candidate as their sole first preference, under AV, if that vote is for a smaller party, it’s inevitably going to be transferred to a lower-preference candidate representing one of the larger parties. In other words, despite being a first preference, that vote is effectively of nil effect and is not retained in the final result.

By contrast, a first-preference vote for a smaller party continues to be recorded and treated as such in the ARV count. Indeed, it is recorded in two ways: as part of the total of first-preference votes for each candidate, and as part of that party’s points score and approval rating. This means that voters can demonstrate support for key policies of minor parties in ways that politicians are forced to take note of: the percentage of first-preference votes for the minor parties will be considerably higher than under FPTP and AV, because there is a real purpose in specifying candidates from those parties as your first choice (unlike in AV) and you will not be damaging the chances of your tactical vote by so doing (as in FPTP). And the higher the approval rating for those parties, the more politicians will not be able to ignore voters’ support for those parties’ policies.

The reason why I have not given ARV a score of five out of five against this criterion is that, despite the fact that it allows voters to express the full range of their views and send a strong message to politicians, that communication is still relatively disempowered: ultimately, the full range of voters’ opinions will not be represented in parliament, as only one candidate and set of party policies can be ratified by the electorate in each constituency, unlike in multi-member-constituency systems.

Fifth criterion: Does the system mitigate the need to vote tactically? Here, I’d award ARV three points, compared with two for AV. It’s impossible to eliminate tactical voting completely in single-member systems, as there will always be a motivation to vote tactically to spoil the chances of strong candidates that supporters of opposing candidates do not like, and to improve the chances of ‘second-best’ or least bad candidates for voters whose first-choice candidates do not stand a chance. That said, ARV does enable people to vote for both their real preference and their tactical choice, in contrast to AV where often only the last (tactical) preference is the ‘real’ vote and first preferences are of nil effect, as discussed above.

Above, I also touched on the potential of ARV to enable voters to pool their vote behind all of the left-of-centre candidates in order to defeat Conservative candidates that might actually obtain the largest number of first-preference votes. In response to this concern, I’d stress the fact that if a majority, or the largest majority, of voters indicates the Conservative as their first-choice candidate, then that candidate is automatically elected under ARV. Second, as I said before, I’m sceptical as to whether enough left-of-centre voters could be mobilised to ensure that one left-wing candidate was able to win a majority of first preferences. They might, however, win on points. And if those candidates do win either a majority or a points victory, then that would tend to indicate that the Conservative candidate – as an individual as well as a party representative – was sufficiently unpopular with enough voters not to merit being elected anyway.

Finally, how easy is the system for voters to understand, trust and use effectively? Here, I’d give ARV three points: the same as FPTP and one more than AV. On the face of it, ARV is more complicated and fastidious than FPTP, where you just have to mark a single cross against your favourite candidate. But ARV is in fact quite simple and employs rules that voters are quite familiar with from situations where they have to assign a numerical value to indicate the degree to which they like certain things or agree with particular statements, such as in marketing surveys and opinion polls.

When it came to polling day, it would be easy to explain the rules of ARV on the ballot paper in no more than six sentences of plain English, as follows: ‘Indicate the degree to which you approve of each candidate by writing a number from 0 to 5 [in the case of five candidates] in the box next to each candidate’s name. 5 indicates the maximum possible support for any candidate; 0 equals the absence of any support for that candidate. You may give the same number of points to more than one candidate. The candidate(s) to whom you award most points will be considered as your first-choice candidate(s) for the purpose of determining whether the election has been won by any candidate on the basis of an outright majority. You are not obliged to give a maximum of five points to any candidate. Leaving a box blank is equivalent to filling it with a zero.’

I think voters would soon wrap their heads round this system and begin to understand how they could use it to their best advantage in the ways described in this article: to punish parties and candidates they disapprove of; to vote for their actually preferred candidate as well as tactically; and to express the full range of their political opinions. So despite being more complicated and difficult to understand than FPTP, ARV earns the same score for this criterion than FPTP by means of enabling voters to use their vote to the best advantage.

Finally, here is a summary of how ARV performs by comparison with the other voting systems I’ve discussed in this series so far:

Criterion FPTP AV AppV ARV
Does every vote count?





Is the system proportional?





Does the system foster accountability?





Does the system let voters express their views?





Does the system mitigate tactical voting?





How user-friendly is the system?





Total scored out of a maximum of 30





In conclusion, ARV is as good as or better than the other single-member voting systems discussed so far according to every criterion against which I have measured it, apart from how easy it is to understand and use to best advantage. I would argue it compares very favourably with the only properly proportional single-member systems, known as Mixed Member Proportional (MMP), which combine single-member constituency ballots with the election of non-constituency MPs on a proportional basis from national or regional party lists.

I’m going to consider one such system – AV+ – in my next post.


16 Responses

  1. Borda voting IS NOT a form of range voting.

    Although it can be made arbitrarily similar to range voting by adding an arbitrarily large number of irrelevant candidates.

  2. Also, these criteria you’re using to evaluate voting methods are completely subjective. You are choosing which criteria to consider, and how much weight to give each one.

    That would be like assessing race car performance based on weight, drag, engine power, tire quality, etc. Does dropping 10 kg improve average speed as much as adding 5 addition HP to the engine? You don’t know – you’re just giving them subjective weights.

    The right way to measure performance is to put the cars in timed trials and get a statistically significant number of races in. Thus you will include EVERY criterion, even ones you are completely unaware of — and you will give them all the correct weight, because it will be empirical.

    There is a way to do this for voting methods. It is called Bayesian regret, and it makes your criteria-based analysis obsolete.

    • I take your point, but I disagree that there is a fail-safe way to determine the validity of different voting methods. The criteria I have selected are those that seem to me to be the most important, and are also generally recognised issues: proportionality, tactical voting, accountability, etc. I think if you were going to purchase a car, you might also run a similar analysis based on the criteria that seemed most important to you, rather than some ‘objective’ measure for the merits of the car that might give different weight than the purchaser to some of the criteria, and add criteria that were not relevant to the purchaser.

      The Bayesian Regret system you link to suffers from the limitation that you inevitably have to make massive assumptions about what voters’ real priorities and motivations are for voting in the way they do. By contrast, ARV (which I couldn’t have tested according to Bayesian Regret anyway, as it’s a completely untested electoral system that hasn’t been used anywhere and I don’t have access to the full Bayesian testing apparata) is quite simple with respect to how the result relates to voters’ wishes and expectations: 1) it respects the majority principle: the candidate with the largest majority of first-preference votes wins; and 2) the approval-rating method guarantees that, in the absence of a majority, the winner is the most strongly supported candidate. I’d have thought that’s a sufficient case in favour of its merits – and its likelihood to eliminate excessive levels of ‘regret’ on the part of voters.

  3. David,

    First off, I’m only talking about single-winner (thus, non-proportional) methods. We do not know of a robust way to calculate Bayesian regret for multi-winner methods. And I don’t think there’s an appreciable difference in “accountability” for single-winner voting methods either.

    As for other criteria, like “tactical voting”, the point is that there is no meaningful way to measure the significance of tactical voting aside from calculating Bayesian regret. For example, Score Voting gives voters better expected satisfaction with 100% tactical voters than Instant Runoff Voting does with 100% sincere voters. So even if IRV had some magical property that caused 100% of voters to be sincere (it doesn’t), and even if Score Voting had some magical property that caused 100% of voters to be tactical (it doesn’t), Score Voting would still be better than IRV from the point of view of a random voter.

    To take that example to another level, consider a voting method that actually is strategy-proof: voters rank the candidates, and then two candidates are selected at random and the winner is the one preferred to the other on the most ballots. While the system is completely immune to tactical voting, it is so highly affected by randomness and other problems that it is still worse than lots of voting methods which aren’t immune to strategy.

    Your analogy about purchasing a car seems sensible, but doesn’t quite work here. Different people might value different things in a vehicle, just as different voters might value different things in candidates. With Bayesian regret, the voters start with preferences, or “values”, for the different candidates. We then see which system gave people the best expected value.

    Bayesian regret does not have to make any assumptions about voters’ priorities and motivations. All we’re measuring is the ability of a voting method to give people what they want. We don’t have to care why they want what they want.

    You can calculate Bayesian regret for your system by downloading our source code in C. You would just need to add your voting method, which may not be possible if you don’t know C. But you may find it interesting if you have a friend who does and could help you.

    Respecting the majority principle is a flaw, not a benefit. Mathematical proof of that here:

    Your method could be good, but there’s no way to really know without running Bayesian regret calculations.

    • Thanks for the reply, brokenladder. And thanks for explaining how I could test my system. I don’t think I’ll do so, though, as I don’t know C or know anyone who’d do it for free, so I’d probably just spend a lot of time getting it all wrong. But don’t let me stop you if you want to test it yourself.

      I agree that my criteria are to some extent arbitrary and subjective: they are what seem to me to be the most important issues, but admittedly that’s not the case for everyone. I prefer more ordinary-language and pragmatic ways of looking at the issue of the satisfactory or unsatisfactory nature of electoral systems – which can arrive at conclusions not dissimilar to your mathematical method. For example, it’s not hard to see how AV could lead to considerable voter dissatisfaction. Assuming no candidate has won an outright majority in the first round, then the eventual winner is the first choice of a minority plus a subsequent choice of an even smaller minority, meaning that a majority will still be to some extent dissatisfied with the result: all those who didn’t indicate the winner as their first choice. This is similar to FPTP, except that voters whose votes have been transferred to the eventual winner could feel more dissatisfied than under FPTP, because they could have been led to believe there was some point in voting sincerely for their first choice, only to find that that choice was to no avail, just as under FPTP. Meanwhile, if AV has the effect that fewer people vote for the eventual winner in the first round, and more people vote sincerely but unsuccessfully in the first round, then the ratio of dissatisfied people will be higher than under FPTP.

      By contrast, my ARV system would appear on the face of it to optimise the chances of satisfaction: if there isn’t a majority of first preferences, then the winner is the candidate that is most strongly supported by the electorate as a whole. As I put it in a previous post, the winner is then not the candidate whom most prefer (i.e. the plurality) but who is most preferred, and would probably also be the Condorcet winner in most instances. Calculating approval ratings, or popularity indices, for all the candidates would be a way to make the election result more transparent and acceptable to voters. In most instances, the approval rating of the winner would be over 50% or close – so people would think, ‘well, (s)he wasn’t my favourite, but at least (s)he is more popular with the voters overall than not’.

      In response to your point about majorities, it is the case that the winner of a majority of first preferences would not necessarily be the candidate with the highest approval rating, as voters could award fewer than maximum points to their favourite, allowing another candidate with fewer first preferences to gain a higher points total. This would, however, be exceptional and would probably result from a concerted campaign to punish the candidate in question, such as in a Labour stronghold where voters wanted to express their dissatisfaction with the party while remaining loyal to it. But then, in this instance, you could say that the allocation of points by such voters was insincere, as the winning candidate was still supported strongly enough to command a majority of first preferences.

      • You don’t want to optimize the chances of “satisfaction”. You want to optimize the expected value, which is probability of all outcomes times their value.

        E.g. which option do you think most people would take?

        A) 95% chance of receiving 1$.
        B) 1% chance of receiving 1 billion $.

        The latter gives you a much greater chance of getting the outcome you less preferred (getting nothing instead of the billion bucks), but it has a better expected value, so a sane person would choose option B.

      • To me, that sounds a bit like a different way of saying the same thing.

  4. To me, that sounds a bit like a different way of saying the same thing.

    And yet I just showed you proof that it isn’t.

    • I know it isn’t mathematically the same thing; but a result that maximises the chance of achieving expected value is also effectively one that maximises voter satisfaction. Anyway, I can’t argue with you on your chosen ground of reasoning – but I think we’re actually in agreement, and for similar reasons, that AV is a rubbish system, and Approval Voting and range voting is better.

      • a result that maximises the chance of achieving expected value is also effectively one that maximises voter satisfaction.

        I just very simply proved that is wrong. In example (a), there is a 95% chance that you will achieve (actually, exceed) the expected value of 95 cents. In example (b), there is a 1% chance you will achieve (actually, exceed) your expected value of 10 million dollars.

        The former gives you 95 times the likelihood of achieving your expected value. But the latter has a higher expected value, hence any sane person would take the latter.

        I was responding to your point that “the ratio of dissatisfied people would be higher than with FPTP”. Well, that may be true, but it is really a flaw in FPTP.

        The reason that you have a very tiny chance of voting in a way you’ll later regret, with FPTP, is that whenever there are two clear frontrunners strategizing voters will only make them even stronger, by following the tactic of voting for their favorite frontrunner. FPTP strategy has a “convergent” effect, like with essentially every voting method which violates the Favorite Betrayal Criterion.

        But consider Approval Voting, on the other hand. Say you have two main camps of voters, with preferences like this:

        49% X=10, Y=7, Z=0
        51% Z=10, Y=6, X=0

        Now if polls show that X and Z are the frontrunners, then the first group will vote for X but not Z, and vice versa for the second group.

        But, because the election appears to be close, many voters on each side might also vote for Y. They both figure that if Y wins instead of their favorite then they lose only 4 or 5 points of happiness, whereas if their least favorite wins instead of Y, they lose either 6 or 7 points of happiness, so it’s a better gamble.

        But then Y wins. After seeing the final results, the Z voters realize they could have had Z if only they had “bullet voted” for Z, so the ratio of “dissatisfied” people (people who regret the way they voted) will be MUCH higher than with FPTP.

        But there is nothing wrong with that. They knew the risks going in. If they had bullet voted for Z and then X would have won, then they would have been kicking themselves for bullet voting instead of also supporting Y.

        So it turns out that the tendency of a voting method to make voters regretful in hindsight is actually counter-intuitively a benefit.

        But the larger point is that whether or not you regret how you voted is a horrible metric of a voting method’s quality. If you had taken the 95% chance of 1 dollar, there would be a 94% chance you’d be better off. And only a 1% chance you’d be worse off than if you chose the other lottery. But you’d still be better off with option (b), because of the magnitude of the better-off-ness if you did win the billion dollars.

        I prefer more ordinary-language and pragmatic ways of looking at the issue of the satisfactory or unsatisfactory nature of electoral systems – which can arrive at conclusions not dissimilar to your mathematical method.

        Well, despite what you prefer, Bayesian regret is the right metric, and regardless of how similar alternative analyses might be, they are not the same. Since we have Bayesian regret values, it’s better to just use them, instead of approximations which, despite being similar to understand, have an unknown amount of inaccuracy.

  5. […] proportional run-off’. This is another method that I’ve invented, in the wake of ARV, which I discussed in the last post in this […]

  6. […] FPTP results are distorted by tactical voting. These problems do not exist in either of the ARV or TMPR voting systems discussed in previous posts: ARV always awards the win to the most popular […]

  7. Hello “David”
    I see other people in the range voting community (such as it is) have discovered your posts, although I am only just doing so now. I’d be interested to possibly
    correspond with you (with one aim being, I’d like to include some of your material on the web site, and I’d like to credit you, which would involve me knowing who you are…)

    My email is warren.wds AT .

    The “range voting community” tends to hang out at the range voting yahoo group and the election science foundation google group (electronic bulletin boards you can join for free)

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