Proportional Influence voting schemes

(Note: The original voting method examined in this article, although it was interesting to me at the time it was written, was a terribly implemented idea. I have removed most of the nonsense cruft in this update.)

Under democracy one party always devotes its chief energies to trying to prove that the other party is unfit to rule – and both commonly succeed, and are right.
― H. L. Mencken is one of the most interesting political websites around.  Its contributors advocate and research various ways of dramatically improving the voting methods in use in democracies, including especially the titular Range Voting method.  Range voting performed in voting districts could be just as useful for the public as nationwide proportional representation (P.R.) votes, and potentially even better due to its tendency to elect centrist candidates.   Both range voting and proportional representation would be wonderful reforms here in the U.S.


But recently it’s been made clear to me that all such voting methods are missing out on one feature that I find quite desirable.  Political representation does not imply political influence.  Here in my town there have been weeks of hoopla about our local politics, with thousands of working mothers out protesting nightly the perceived invasions by their legislature.  Oddly and wonderfully enough, they seem fully cognizant of the fact that their protests will have zero actual effect on governance, and they are maintaining morale even so.  Reflecting on the futility of the passionate minority’s efforts, it occurred to me belatedly that despite the most perfectly chosen Condorcet-winning slate of candidates or the most ideally proportionally representative legislative body, because they use ordinary plurality voting in the legislative body’s votes, the majority faction is granted total control.  This effect is strikingly undesirable (to the minority!) and feels to me deeply antagonistic to the ideal of collective self-government.  So I set about finding a voting scheme for legislative bodies that supports an ideal that I am provisionally calling “Proportional Influence” (P.I.), which applies when voting on uncompromisable questions and means each faction will tend to win a proportion of the votes close to the proportion of their members in the legislature.  Also of interest is an ideal I’ll provisionally call “Proportional Satisfaction” (P.S.), which applies when voting on compromisable questions and means the winning compromise position of a vote will tend to give a level of satisfaction to each faction close to the proportion of their members in the legislature.  An assumption of mine behind the search for a P.I. voting scheme is that minorities often have a few non-negotiable issues which, if they can just secure their way regarding those, they will rankle less at the rule of the majority.  Thus a P.I. system might encourage a greater sense among voters that their union is intended for and reflective of the common good.

There’s one very simple P.I. scheme, which is to select a random ballot out of all those cast and use it to decide the question.  Then each question will be settled with probability equal to the proportion of votes for each position.  But I suspect people have good reason to find nondeterministic votes distasteful, so I’m going to ignore that scheme and restrict my attention to deterministic systems.  Readers of may immediately think of using Reweighted Range Voting (RRV) for each vote sequentially.  I like the idea, but the math of implementing it is exceedingly ugly, so I propose here a simplification of sequential RRV that may be satisfactory.  I’m no expert on these matters, and I don’t know if something like this has already been discussed or named, or if there’s a clearly superior system that I missed.  In any case, here are the rules of what I’ll provisionally call CasellaVickrey Voting (CVV):

1. Each member of the voting body starts with one whole vote, which is divisible into some specified fraction such as 10 decivotes, one million microvotes, or any other according to the needs of the voting body. (I’ll call each individual fraction of a whole vote, whatever its size, a “token”.)
2. When a yea-or-nay question is put to a vote, each member secretly pledges from zero to all of his tokens to either one side or the other.
3. When the vote is complete, all the bids are revealed, and the option with the most tokens pledged to it is the winner. The price that will be paid by each winner, however, is the second-highest pledge total for an option divided by the number of winners. For example, if the options are just “Yea” and “Nay”, and 60 Yea voters together pledge 8000 tokens and 40 Nay voters together pledge 6000 tokens, then each Yea voter will pay 100 (6000/60) tokens even if they bid more or less than that, and the Nay voters will pay 0 tokens.
4. All token payments are deducted from the members who pledged them and go into the pot.  (The pot need not be a literal physical pot.)  After some set amount of time, the pot is redistributed evenly to all members regardless of how or whether they voted.

Overpledging [optional but extremely useful]:
5. When a question is put to a vote that will select one winning option from more than two candidate options (such as electing one of many candidates to an office, or selecting the most-preferred version of a bill out of a set of different versions), each member may pledge from zero to all of her tokens for each option, since only the pledge to the single winning option will be deducted from her total.
6. When a question is put to a vote that will select some specific greater-than-1 number of winning options from more than two candidate options (such as electing members of a committee), each member may pledge from zero to all of his tokens for each option. The first winner is the option with the most pledged tokens, and it is locked in as a winner and the pledges for it are paid in the usual way.  For selecting each subsequent winner, any member’s pledges are only counted up to the total of her remaining tokens. The second winner is the remaining candidate option with the most pledged tokens, and the pledges for it are collected from the voting members based on the third-highest pledge total (counting only remaining tokens).  Repeat until all winning options are selected: the third price based on the fourth-highest pledge total, the fourth price based on the fifth-highest pledge total, and so on.

Overpledging is what enables support of proportional influence, basically by mimicking Range Voting. The main two differences from range voting is that in CVV members may have different top scores that they can give, and vote totals will be used rather than vote means. In addition, organized factions can use overpledging to enable proportional representation in multiwinner votes. A simultaneous combination of proportional influence and proportional representation can occur, too. For example, suppose the legislature is voting on ten committee members and my party knows it can win three seats on a committee if the party members together pledge at least the Droop quota for all our committee candidates. We have seven party members who have put themselves forward as candidates for the committee. The party tells me my allotment of the Droop quota is 4 decivotes, so being a loyal party member I overpledge 4 decivotes to all seven of our candidates (28 total decivotes pledged, of which only 12 will be used unless the other parties act stupidly). I currently happen to have 15 decivotes, so assuming we will win three committee seats and only 12 decivotes from my overpledge will be required, I can distribute 3 more decivotes Range-style to my favorites among our candidates without endangering the Droop quota.)

An auction where the highest bidder wins but pays the price bid by the second-highest bidder is called a Vickrey Auction. If the bids are secret and there is no collusion among the bidders, it has the nice property that the dominant strategy is to bid what winning is really worth to you. If you view CVV from the perspective of the political party, and treat the party as a single bidder, then the same is true here. The parties do best to pledge the number of tokens that winning is really worth to them. But of course the parties aren’t unified. They are made up of members who may be tempted to skimp on their pledge in order to keep more tokens for themselves, in the hopes that their fellow party members will make up the slack. The pledge payment method described in step 3 partially helps to fix that. An honest bid is no longer strategically dominant, but it remains the optimal bid according to a minimax decision theory, which is to say that it’s the option that is guaranteed to be least bad. Imagine a representative about to cast her secret vote thinking through her strategic options:

It’s too late for me to change the other representatives votes, so my strategy now only depends on mine. I can’t vote for the other side, since my party would stop supporting me. So I’ve just got to live with the consequences of voting for my side, however many tokens it costs me. If I pledge less than winning is worth to me, and I make us lose, then clearly I made a mistake. If we lose but my pledge wouldn’t have the diffence, then it makes no difference to me. But if we win, skimping on my pledge doesn’t save me any tokens. On the other hand, if I pledge more than winning is worth to me, and we lose, it doesn’t cost my anything. It will drive up the price paid by the opposing side, though, which is a nice benefit. If we win and my pledge doesn’t make the difference, it doesn’t change how much I pay. If we win my pledge is what made the difference, but I bid more than it was worth, then I’ll definitely pay more than it was worth to me. If I pledge what it’s really worth to me, then I don’t risk the mistakes of either side.

An obvious problem is that the order in which votes are taken may be critical. The strategizing in selecting the order in which to bring bills to a vote would be intense, since for small majorities CVV can introduce rapid swings in the balance of power from vote to vote. There could even be a “nuclear option” whereby the party could save up its votes until the end of the session and then overturn everything that their opposition previously passed. This chaotic behavior would be conclusively enough for me to reject the practicality of CVV if it were the only behavior available. But in fact the scheme has a range of behaviors between two extremes: At one extreme, a majority which is unwilling to cede the balance of power may retain it indefinitely by finding compromise bills. At the other extreme, a majority which is unwilling to compromise will occasionally lose the balance of power, and for these circumstances, if CVV was implemented, it could be buttressed by a rule forbidding changing the same law twice in one session. (Heck, that kind of rule would be nice even in our present system, since it would force the House GOP to move on from its pretenses of repealing the ACA that it has spent so much of its time on, and instead perhaps actually accomplish some of their goals.)

Granted: No party in power would implement such a system, unless perhaps they foresaw themselves permanently falling short of majority power in the future and could overcome the eagerly ascendant minority. But I still think systems with characteristics like this one are worth considering. An organization wanting to find an agreeable mode of governance for polarized factions may wish to use one. It might be possible to implement CVV physically with real tokens if the available technology is limited. Finally, this scheme could be neatly combined with Asset Voting or directly democratic Vote Delegation, so that an elected/delegated official would start out with tokens equal to his assets or delegated votes.

Democracy is not limited to rule of the majority.

If liberty and equality, as is thought by some, are chiefly to be found in democracy, they will be best attained when all persons alike share in the government to the utmost.
― Aristotle


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