Hybrid of Score & Asset voting (possible way to elect the parliament of Canada)

Warren D. Smith, October 2015

We describe a voting system that is a "hybrid" of score voting within ridings, with a new "top-up" system based on Simmons' version of asset voting. If T=0 our proposal simplifies to become pure score voting to elect MPs in ridings. If T=W (which I would not recommend) it becomes pure asset voting countrywide. This system is designed to resemble some current widely known ideas in Canada, but which however failed to enjoy nearly as many good properties as our scheme.


  1. Design goals / Properties
  2. Rules of proposed voting system (on a single page)
  3. Criticisms (and responses, some of them good)
  4. Possible modifications

Design goals / Properties

The goal is to elect a W-member parliament, from W-T ridings (0≤T≤W), in such a way that:
  1. The system has two stages. In the first stage, exactly one MP is elected from each riding (hopefully the one that riding considers the best available!); call these the "local" MPs.
  2. In the second stage, a further T "top up" MPs are elected countrywide (call these the "top up" or "nonlocal" MPs)...
  3. In such a way as to obtain or approach "proportionality."*
  4. The rules of the voting system in no way even mention or require the existence of "parties," nor assume anything about the existence or nature of "leaders" or "pecking orders" within those parties, and thus hopefully in no way inherently hurt independent candidates or nonconformist parties. (That is important because PR countries – which future-Canada would become under this system – usually have a lot of small parties, with independents as "parties of one" being the extreme case of that. E.g. the Netherlands, with population about half Canada's, as of year 2015 has 17 parties holding seats.)
  5. System is simple.
  6. The ballots are "ratings style" which allows voters to provide the most possible information, but also (if that voter desires) less.
  7. The system is immune to "candidate cloning."*
  8. It is never suboptimal strategy, for a voter to give her honestly favorite candidate, the max rating.*
  9. It is never suboptimal strategy, for a voter to give her honestly most-detested candidate, the min rating.*
  10. Not voting, is never a superior voting strategy than voting honestly ("participation property")*; and raising a candidate's score on your ballot cannot hurt him*, while lowering it cannot help him* ("monotonicity properties").
  11. The system is "precinct countable," i.e. each precinct could publish a compact table of "subtotals," in such a way that the whole election process could be performed using only these precinct subtotals tables as input.
  12. The system obeys a version of NESD property, but unfortunately it is a weaker version than I would like. Specifically: even if every voter naively exaggerates her vote by making the two most-major parties have "max" and "min" scores, then it still remains entirely possible for a third-party to win seats. (This is good, because if that voter behavior had made this impossible, then conjecturally this system would be highly likely to fall into permanent 2-party domination.)
        Those seats can come in two ways. The first is, if the two major parties get, say, 55% and 45% top ratings, but meanwhile some third party gets 66% of voters to give them nearly-max ratings, then that third party will win local MP seats. The second way is: the third party could share top-ratings and thus get a nonzero amount of "juice" which in the second stage is converted into top-up seats. My criticism of that is: voters may feel impelled (by the same strategic considerations which made them exaggerate the two leading parties to "max" and "min") to give all third parties only nonextreme scores. In that case, these third parties will only be able to win local, and will not be able to win top-up, seats. That would inherently disadvantage third parties, which still might be severe enough to drive a country into eventual 2-party domination.
        In contrast, the simpler psi-voting-based top-up scheme [II(C)1 here] obeys full-strength NESD: For it, third parties can elect both local and top-up MPs, and plenty of them, even if 100% of voters top-score only the two major parties.

* More precise claims:

Proportionality theorem: under a "party loyalty assumption" that candidates always prefer own-party colleagues over foreign-party candidates, and if voters always give their own party's candidate the maximum rating and all other-party candidates the minimum rating, then the final parliament will have the same party composition – up to small errors forced by rounding to integers, and up to limitations which could sometimes be forced by T being too small so that the top-up simply does not have enough seats to achieve exact proportionality in cases where the first stage provided a too-disproportional winner-set – as the electorate.

Additional stronger claim: if voters give other-party candidates not necessarily minimal, but merely submaximal, ratings, then we still guarantee exact proportionality (based on top choices for each voter only) provided the "zeroing negative juice" substep inside step 5, does not occur, and provided enough top-up seats are available to allow proportionality. The zeroing problem arises for local MPs who won because they are the favorite of few, but the 2nd-favorite of many. It can cause a party containing many such "compromise" MPs to be somewhat over-represented. All proportionality theorems ever proven for voting systems, necessarily make some extreme partisan-behavior assumption of our ilk. (Specifically, ours was that all voters score their party MAX and all others MIN; we had proved our PR theorem under this assumption.) Proportionality advocates then have to hope that with more realistic voter behavior, their voting systems will actually elect better parliaments for their country than arise from the simpler-to-model extreme voter-behaviors. These "better" parliaments might have greater proportionality-errors, but will enjoy other benefits hopefully more than compensating for that. The point of a proportionality theorem is as a safety backstop. It tends to prevent parliaments from becoming too-poorly reflective of the electorate's composition. No proportionality theorem can always assure a perfect mirroring; and there usually is no way even to define what that would even mean, or measure whether we have it. (E.g, if you scored the Liberal 8 and the Tory 6, does than mean you are 4/7 liberal and 3/7 Tory for purpose of assessing your "composition"?) The best that PR theorems can do is to assure that in certain crudely simplified circumstances where it is possible to clearly define what a "perfect mirroring" would mean, it is achieved.

Cloneproofness, Participation, & Monotonicity theorems: Always valid for first stage. These all also valid about the top-up stage provided we assume both party loyalty and a "party sameness model" that all candidates from any one party are equivalent.

Comments: A scheme suggested to me by one Liberal MP was similar to ours except (i) involved the first stage instead done via Instant Runoff Voting (IRV), and (ii) the second top-up stage via a party-list system. That's worse because:

(i) IRV would cause the first stage to disobey monotonicity, participation, and precinct countability, and the favorite-safe property H and its mirrored property I; and would force voters who wish not to express an opinion about a candidate, to provide one (which, if they were ignorant, would just be forcing them to guess); and would make everything more complicated.

(ii) Party-list based top-up would inherently kill independents, by law. One could counter-argue that independent MPs have been neither very common nor very effective in Canada – but in future-PR-Canada they would be more common. Independent thinking is very important for making good decisions! Should we kill it all, as a matter of law?! Should we mandate the existence of political parties? And also, countries with such top-up systems have found it necessary (to prevent "gaming the system") to, by law, restrict the ability of MPs to change parties! Should we, as a matter of law, force MPs to stay in a party they dislike? (That'd take "whipping" to a whole new level. Remember, the ultimate cure for the disease of too-whipped parties, is for unhappy MPs to be able to switch parties.) I think not.

So in my view, the scheme suggested here is similar in spirit to the one that MP had in mind, but repairs many of its most severe disadvantages. Also, given that Canada's Liberal Party seems likely often to be victimized by IRV's center squeeze pathology, it may be against the Liberal party's own interests to use IRV (i.e. that Liberal MP would, by enacting his system, have risked hurting his own party as a nasty unintended side effect!).

Simplicity for voters: In the view of any voter, our proposal is simple: just give the candidates ratings on an 0-9 numerical scale (leaving some unrated if that is too difficult). Done.

I don't mind if the candidates – they are pros usually with helpers – have to work hard during election, but Amy The Average Voter must have it easy and simple.

Rules of proposed voting system (on a single page)

I will regard voters as "female" and candidates as "male" just to make wording clearer. Various numbers (2, 9, 1000, 3) in our description have been selected pretty arbitrarily and could perfectly well be replaced by other numbers. And we could demand T=⌊W/5⌋, where again "5" is a pretty arbitrary value.

1. Each voter rates each candidate in her riding, on an 0-to-9 numerical scale. (Voters also allowed to leave some candidates unrated, if that voter wishes not to express any opinion about that candidate.) Refinement suggested by J.Quinn: "Any voter who did not explicitly rate any candidates at 0 is considered to have given 0 to any candidates they left blank. Any voter who did rate at least one candidate explicitly at 0 is considered to have expressed 'no opinion' about any candidates they left blank. (I think this rule is the safest way to interpret 'intent of the voter'.)"

2. In each riding, the candidate with the highest average score wins that riding's MP seat. Slight optional refinement: each candidate could receive 1000 artificial scores of 2 before voting begins. Then the candidate with the highest average among both his genuine and artificial scores, wins.

[At this point we have elected W-T of the W seats in Parliament, each via the excellent "score voting" single-winner system. The "top up" now commences, whose mission now is to fill the remaining T seats.]

4. Each voter is viewed as having given 1 cup of "juice" to her top-rated candidate. (If a voter rates more than one candidate co-equal top, then her juice is divided equally among them.) Let J denote the total amount of juice. (Which, note, equals the total number of voters.)

5. Remove J/(W+1) cups of juice from each candidate who has already won a seat. (Except: If this would cause that candidate now to have a negative amount of juice, instead just give him zero.)

6. Candidates now negotiate. Any candidate may give any other some or all of their juice.

7. But candidates must pre-specify an ordering listing all the other candidates, meaning "if I am obliterated, then award all my juice to the first still-unobliterated candidate on my list." These orderings are made publicly available before the election. The election authority, when the deadline comes, will dispose of all juice that the candidates had not already given away before the deadline, as follows: find the candidate with the least juice, obliterate him, and give his juice to the first non-obliterated candidate on his list; continue doing that until only T non-obliterated and as-yet-unelected candidates remain. They fill the remaining T seats. (In other words, to oversimplify slightly, "the T 'juiciest' candidates win the T seats.")

Criticisms (and responses, some of them good)

Proportionality criticism: The critic does not like the possibly-proportionality-damaging effect of the "zeroing negative juice" exception-substep of step 5.

The critic realizes that: The whole "juice" mechanism was designed to force proportionality in a simple manner while at the same time avoiding any explicit mention of "parties" in the rules, and also granting all candidates some freedom of action to try to mould the parliament in ways that will improve it in their view. There might well be negotiations about juice transfers between candidates in the same (or different) parties, which might allow even candidates or parties who fail to win seats, still to influence policy in return for their juice. This can

  1. yield a finer degree of "proportionality" than any PR method not involving such candidate-involvement,
  2. tend to yield a more-harmonious parliament with less risk of "gridlock" because the winners will become enriched with proven negotiators & compromisers, and those the other MPs appreciate. (If Canada adopts a PR system such as ours, then majority-control by a single party will become less common, and it will become more common that "coalition governments" are needed, so gridlock will be more of a risk, so this feature will be very useful.) This beneficial side-effect also can tend to protect MPs their party-cohorts consider valuable (e.g. respected leaders) from losing their seat, which otherwise would have denied Canada an MP widely considered useful.

Response to that critic: The zeroing problem arises for local MPs who won because they are the favorite of few, but the 2nd-favorite of many. It can cause a party containing many such "compromise" MPs to be somewhat over-represented. However: in a riding in which Compromise Candidate Claude wins, he is the "favorite" candidate of that riding, and therefore, in some reasonable approximate sense, of its voters. Therefore, if we were to steal juice from all of Claude's rival (losing) candidates in that riding (say stealing F fraction of each's juice) to give it to Claude to bring his negative juice up to zero (F would have to be chosen just right in each riding to accomplish that) then the net effect of such thefts would not constitute a horrible distortion. They are just removing juice from individually-favorite candidates to give it to a collective-favorite candidate, the collective effect of which is not hugely distortionary. And since this only happens when Claude is a compromise candidate, e.g. a Centrist who here would be stealing juice from Leftists and Rightists, this theft is inherently less distortionary than if, say, a Leftist was stealing juice from a Rightist. So if, after we did all those thefts, the juice-mechanism proceeded normally from then onward, we still would expect a not-horribly-distorted parliament.

So we've just argued that a certain modification of our proposal, involving "juice thefts," should, while imperfect, still work well. But now, we argue that doing nothing (i.e. no such juice-thefts) should be simpler and actually better! Because: the whole problem was Claude's Compromise Party getting overrepresented in parliament. The juice-theft plan would actually amplify that problem by not only giving Claude extra juice, but also taking it away from the rival parties. So without juice-theft, just the negative-zeroing substep of step 5 alone, we should approximately halve the problem.

That argument has now "proven" that our whole top-up proposal should perform decently about proportionality, in all circumstances, despite the so-called "zero-floor flaw" inside step 5. The word "proven" is in quotes to indicate that this is not really a proof – it's admittedly a somewhat flaky argument, which indeed my critic did not like – but there nevertheless is a lot of truth in it.

Whipping criticism: The fact the candidates donate their juice is "horrible" because they will be "whipped" by evil party bosses to do whatever those party bosses want! Yikes!

Response: This criticism is about as wrong as it could possibly be, for the following simple reason: the vast majority of the donated juice is going to come from losing candidates who are donating their juice as their last official act before vanishing! It is hard for bosses to pressure anybody who's leaving government employ. Furthermore, this pressure can work in both directions: MPs can influence who their bosses will be, and who their colleagues will be, by using their juice.

Logistical criticism: Steps 6-7 could be a logistical nightmare. And what if the internet breaks, possibly due to a cyberattack? And what if it doesn't break, but some candidate pretends that it did?

Response: First, we expect future-Canada will have better internet than now. Second, television could be used as emergency backup if the internet breaks. Third, we've also proposed (in step 7) an emergency backstop measure involving juice-transfer priority lists, that could be used even if all communication broke down (and obviously any election scheme is going have trouble if all communication breaks). Fourth, the main logistical problem we actually expect is related to the X-ing ordering in step 6. There will come a time when we ask "who do we X-off next: Joe or Tom?" And then it will be a near-tie. And then the entire country will stall until every last vote is precision-counted in a major lawsuit nightmare, just to get past that point. I.e, the problem is, the system is vulnerable to near-ties of this kind causing a stall, and there are a huge number of opportunities for them to arise.

However, there is a simple and effective workaround: Allow the election authority (EA), whenever Joe and Tom are within 0.5% of each other, to just order them wrong in any way it likes. Well, more precisely, it has to order them according to whatever it thinks the vote counts are, but Joe is not allowed to complain unless they were more than 0.5% wrong. (That may allow the EA to be slightly evil, but live with that.)

Note the whole top-up stage will be unable to start until everybody is nailed down to better than 0.5% accuracy. Also, even if future Canada will somehow be able to attain the 0.5% accuracy instantly, then still there will be a long delay while all the very-low-juice fringe candidates (who will have very little effect on the final result) perform their donations. My point is, these initial delays will give all the candidates plenty of time to scheme and plan their juice-transfer strategies well ahead of time based on preliminary counts; but thanks to the "0.5% fuzz" workaround, there should be no stalls thereafter.

"Fake MP" corruption criticism: To understand this criticism, you have to know that a form of corruption which has occurred in Canada is the "fake shadow MP." That is, the official MP from some riding has little power because he is in an opposition party. That can allow a fake unelected MP from the governing party to, because he has the right friends, effectively have more power than the real MP. So, if he wants, he could run around telling people he is "the shadow MP" for his riding, and that he, not the official MP, is the guy to go to for funding and problem solving. Then later, hope to really get elected. (In the USA, this happens too, except in this different flavor: they do not call it a "shadow MP," they call it "retiring from congress to become a big-money lobbyist.")

Now (the critic continues) with our proposed system some ridings will actually have two MPs (if one was a top-up) and if those two are from the opposition and governing parties, then the latter can proclaim he is the "real" MP with all the power, while his colleague is useless, so don't bother going to that joker for funding and problem solving. This is just like the fake MP problem, except likely worse because he's real, not fake! Also, the critic dislikes the fact some ridings will have two not one MP, which could have a geographically-unfair effect.

Response: These problems actually will happen not only with our system, but indeed with any riding-based system trying to achieve PR via "top up" schemes. I.e, sorry, they are unavoidable. We can proclaim that the top-up MPs are "nonlocal" i.e. do not represent any riding – they were not elected by their riding, but really by slight effects from the whole rest of Canada – but if those MPs disobey that proclamation, then the critic's problem arises!

This critic suggested a partial fix for his own problem. Regard the top-up MPs as representing multi-riding regions. Then they will have official local roles, but the region involved will be larger than just one riding. (We indeed could design some kind of hierarchical division of Canada into ridings, coarser-grained super-ridings, etc, and my critic wondered if a system that did that could be better...)

But all that is at best only a partial fix because any such MP still will have motivation to focus on his home riding to try to prepare for a future in which he might be elected from it later as a local MP – and also just because he lives there.

So this criticism seems nasty and largely uncurable. The only solution I can see is to entirely abandon "top-up" approaches, at considerable cost in simplicity. But even if that were done, I believe the problem still would arise in an altered form. For example, if each riding elected 5 MPs (emulating Ireland's PR system) then the MPs who were in the governing party could announce they were "real" while their colleagues in the same riding from opposition parties were "useless." And even if we devised a PR scheme with exactly one MP per riding (which can be done, although only by forcing some ridings to elect an MP whom they very much do not want) even then ridings still would get effectively highly unfair representation, since the MPs from the majority party would have more power while the ridings with opposition MPs would suffer.

There is, however, one saving grace (which is so good it may pretty much nullify this whole criticism): in future-PR-Canada, majority control of parliament by one party would become rarer, and coalition governments necessarily more common. That in turn will diminish all of those problems. It is even possible, perhaps, that the net effect would be that our new system actually is an improvement in this respect! That question is hard to answer since it is hard to predict the future, but this guess certainly seems possible.

Criticisms of the whole juice-transfer PR approach: Will MPs try to be nice to those from other parties so that the "juice transfers" at the next election will be favourable? I doubt it – I think the parties will just do deals (like: we'll transfer votes to you in one riding, you transfer to us in another, then we will form a coalition government and give patronage appointments to all the defeated candidates from our parties). Voters will feel like they are being left out if that happens, and they may feel the elected parliament is less legitimate.

Candidates also will be loathe to say, during the election campaign, who they would transfer their juice to. You are just weakening your campaign message if you talk like that. But in a juice-transfer ("asset voting") system, that is exactly what voters will want to know before they vote!

Response: The whole "patronage appointment" thing is a form of corruption, or abuse of power, which can happen regardless of voting system, but perhaps asset voting would worsen it. On the other hand it could be argued that whenever such inter-party deals are made, that is good, in the sense that some party which (in terms of the votes for it) deserves power but would under prior voting systems lose it, now would effectively regain power via these deals. In other words, the deals may sometimes have a bad smell, but they necessarily will tend to act in the right direction as far as restoring power balances is concerned.

I had wanted to respond that inter-party dealmaking would be hindered because most of the juice-donors are losing candidates, whom, therefore, the party bosses cannot boss very well. But the patronage-appointment criticism would hurt that response!

Even so, though, keep in mind that most candidates probably will behave like "party droids" i.e. only give their juice to same-party. Then only at the end of the within-party juice-transfers, will that party realize "we have 42% of a seat worth of juice left over. What should we do with it?" and then consider making some deal with a foreign party. If there are 10 parties, that means at most 10, and oprobaby about 5, seats will be involved in such deals. My point is: 5-10 seats is not a lot. There simply is not huge scope here for corruption.

And to respond to the "loathe to say" critique, not we are demanding all candidates pre-specify juice-transfer priority lists (recall them from step 7; these were planned to be used as an emergency backup measure in case that candidate's internet connection failed or negotiations deadlocked).

"I like PR less or more than you do" criticism: Proportional representation versus single winners – which is the better way to run the world? This is a debate that has continued for at least 100 years with no conclusion. The critic may not like our proposal above since he regards it as too far to one side or the other of that debate.

Response: The ideal of PR is that if the parliament were to exactly mirror the demographics of the population in every way, then every decision the parliament made, would be the same as the whole population would have made in a (usually impractical) countrywide referendum. So everything will be as good as it could hope to be. That's an excellent ideal. But the reality is that every PR country that has so far existed, has elected parliaments that usually massively fail to mirror the population with respect to, e.g. gender, economic situation, and life-experience, education & expertise in varied areas. The best PR so far has ever managed to do is to mirror "party identification" counts, which is a very crude kind of mirror.

So in view of this, I do not advocate "too much" PR. Do not overdo it trying to chase the will-o'-the-wisp of PR "perfection" – that would be pointless. Instead I advocate using PR proportionality theorems merely as "safety backstops" to prevent parliament from becoming too distorted a reflection of the population, such as Canada 2008 entirely giving the Greens zero seats.

Meanwhile, the (unfortunately conflicting) ideal of non-PR "single winner" elections is maximizing individual accountability and quality. Each MP is elected by 1 district, maximizing simplicity, voter↔candidate mutual knowledge, and clarity, and hopefully assuring that MP the best choice available (at least if a good-quality single-winner voting system were used – Canada's plurality system is a horribly bad way to try to determine who that best option is, score voting or approval would clearly be superior).

That also is an excellent ideal, but in reality it can (and many times in Canadian history has) yielded highly disproportional parliaments and shut important views out of power entirely.

The nice thing about our proposal is that it is tuneable to seek the optimum tradeoff between these two ideals. The designer can vary the fraction T/W of top-up seats; making T larger causes better PR but at the cost of having fewer directly elected seats. I suspect, based partly on data and partly on extrapolation, that the optimum tradeoff will be reached when T is about 15-30% of W, i.e. 15-30% of the parliament is top-up seats.

Possible modifications

1. If one wanted the top-up MPs to be "regional" rather than "Canada-wide" then use the same proposal, just within regions only. For example if each region were 1/25 of Canada, then use our system to elect 1/25 of the parliament, doing that independently within each region; then finally combine all 25 subparliaments to obtain the whole parliament. This would sacrifice some proportionality-accuracy, but hopefully the advantages of greater MP regional identification would outweigh those disadvantages, at least if it is "tuned" optimally (i.e. by adjusting the number "25" to get the optimal tradeoff).

Whether or not you buy the whole idea that "regional" MPs are inherently more motivated or better than "nationwide 'at large' MPs" the regional scheme still enjoys the following important advantage. Suppose that party composition is highly correlated with geography, e.g. "the Liberals all are located in the East." Now suppose it rains in the East on election day, depressing turnout. With nationwide PR schemes, Canada would then artificially suffer a disproportionately low-liberal parliament, purely because it happened to rain on that day. (Or if it rained in the West instead, then the opposite problem.) With regional top-ups, and each region's number of MPs based on its population (not its voting population), that problem is avoided.

2. Suggested (in essence) by professor S.J. Brams: replace score voting by approval voting, using an "approval plus one favorite" style ballots where each voter approves(score 1) or disapproves(score 0) every candidate (intermediate scores forbidden) and also designates one favorite among her approved candidates. All that voter's juice goes to her favorite.

Approval is simpler than score, but surveys show people nevertheless prefer score, and score is better both in principle and in tests.

3. Elect a "weighted parliament" in which the MPs are not all equal. They each have a "voting weight" equal to their amount of "juice" when they were elected.

Pros for weighting:

Cons for weighting: Weighted parliaments obviously are more complicated logistically – both for conducting votes, and also other issues arise (e.g. should the MPs be paid unequally and have unequal staff?). But if the parliament is equipped with an instant computerized voting system (which many or all do, nowadays) that might be no obstacle. Also the whole "rains in the East" problem pointed out above, is another good objection to weighting seats.

4. Jameson Quinn suggests versions in which voters are allowed to delegate some or all of their "juice" to one candidate of their choice anywhere in Canada (quite possibly not from their home riding; the plan above had forced that voter to give all her juice to the candidate she'd scored topmost in her home riding). She could do this by writing a code number (read from a separate booklet) on her ballot.

5. Quinn also suggests in step 7 that the eliminations be ordered, not be incresing juice-ownership, but rather by disposable juice-ownership. In other words, if you own more one than one "quota," i.e. >J/(W+1) cups of juice, then your disposable juice is the amount you have above that quota.

Overview of modern voting systems options for Canada (the present page is just one).

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