Market-Based Paradigm for Designing Multiwinner Voting Systems

by Warren D. Smith February 2010

The "market-based voting" paradigm is an interesting approach to trying to create good multiwinner voting systems. It was invented by me (Warren D. Smith) in February 2010 and is at this point a fairly speculative idea whose performance and value are not yet clear. It is related to certain earlier ideas that had earlier been considered for single-winner voting systems, e.g. Clarke-Tideman-Tullock voting, and to the ideas behind the multiwinner asset voting and reweighted range voting systems. If market-based voting systems actually work as hoped, they should be excellent in terms of delivering utility, proportional representation, and simplicity. But unfortunately, my current understanding of these systems is not really "rigorous and mathematical" at all; instead the approach is to semi-abandon mathematical rigor in favor of "economic reasoning." To some extent this sacrifice is unavoidable, because Von Neumann "game theory" simply cannot be generalized to games with more than 2 players (the voters here are the "game players"); and the known crude attempts to do so (the "Nash equilibrium") are almost entirely unsuited to and irrelevant to voting questions. That obstacle forces both voting theory and economics to be only approximately modelable. The purpose of this page is to explain the MBV idea and begin to explore it, with help from whoever wants to help.

The multiwinner voting system problem is this. There are C candidates. Voters vote, i.e. somehow express information about their preferences among the candidates. From the votes, the system somehow deduces W winners, 0<W<C. The designer of the voting system needs to specify what a "vote" is and how to deduce the winners from the votes. If W=1 this is a single-winner voting system. If W≥2 it is multiwinner. Multiwinner voting systems are a more interesting, difficult and less-understood topic.

We shall be interested in the regime where the number V of voters is very large and also where the number W of winners and C of candidates also are fairly large, but nowhere near as large as V.

Market-Based idea – the crucial ingredients

  1. We need there to be many seats at stake, i.e. W is fairly large.
    Because: an "economy" with only one "commodity" (e.g. gold) is not interesting at all. If it has two (e.g. gold & iron) it still is not enough to exhibit much interest; it won't behave like an "economy" as the term is usually thought of. "Economies" need to have many commodities.
  2. Voting power needs somehow to be approximately turned into a fungible and conserved "money"-like quantity.
  3. Electing A instead of B costs the supporters of A "money."
  4. The rules need to be devised in such a way that "market forces" incentivize voter honesty. What does "honesty" mean? It means basically that the votes for X (i.e. monetary bids) come out proportional to perceived utility benefits from electing X.

Enjoyable presumed consequences of the above design principles:

  1. "Proportional representation" will happen (approximately) automatically because the "market" will assign a "price" to each seat, and then the Red voters will have enough pseudomoney to elect R seats, the "Blues" enough to elect B seats (et cetera) where R and B are proportional to number of red and blue voters.
  2. Subject to that, "utility maximizing" will happen (approximately) automatically because voter honesty implies that the candidates who get elected (because of maximum pseudomoney, i.e. votes, supporting them) must be the ones maximizing utility-sum.

One possible concrete design, based on the idea behind the "Vickrey auction"

  1. Each voter as her vote assigns a pseudodollar value to each candidate between 0 and 1000.
    For example one legal (albeit perhaps stupid) vote might be: Gore=1000, Nader=1000, Buchanan=182, Bush=0.
  2. Each ballot is regarded as having its own "bank account" initially containing 1000 pseudodollars.
  3. The candidate with the most votes wins.
  4. If A won while B came second, then each ballot q which voted $Xq for A and $Yq for B loses an amount of pseudomoney
    Lost moneyq   =   $Xq × ∑all ballots v Yv / ∑all ballots v Xv.
    [Note this lost amount necessarily lies between 0 and Xq.] Thus the total amount of pseudomoney removed from all ballots combined, is ∑all ballots v Yv, i.e. the total amount of money bid for the second-greatest vote-getter, who note did not get elected.
    For example suppose your vote was "A=900, B=800, C=1000, D=0" and then A wins a seat with B being the runner-up. Suppose that was because A got the maximum average vote of $750 while B got the 2nd-maximum average vote of $500. In that case you lose $900×500/750=$600 from your bank account, leaving you with $400 (assuming your account contained $1000 before it was announced that "B wins"). Meanwhile the entire electorate loses an average of $500 per voter, leaving them with $500 (on average, assuming they also began with $1000) in each of their "bank accounts."
  5. All ballots are now rescaled according to the amount of pseudomoney that "remains" inside each of them, and with the candidates who have already won seats eliminated from the picture.
    Thus in the above example, your ballot would now be automatically transmuted to "B=320, C=400, D=0" due to eliminating A and scaling your vote by 40% since your ballot's bank account has now shrunk from $1000 to $400.
  6. Loop back to step 3, until the full number W of winners has been elected.

The point of the magic formula in step 4 was to make this basically be a Vickrey auction. (The entire election thus is a succession of Vickrey auctions, one per seat, with competing voter blocs "bidding" for the right to determine the occupant of that seat.) In a conventional auction, bids on an item are submitted and then the highest bidder wins and purchases the item for the price she bid. William Vickrey, who won the Economics Nobel in 1996, suggested instead that the price she pays should correspond to the second greatest bid. Vickrey's point was that this (provided all the bids are secret until the winner is announced) will incentivize bidders to bid honestly, i.e. the dollar amount the item truly is worth to them.

Here, the votes can be regarded as "money bids" with the candidate winning who has the greatest total amount of "money" bid for him. Then the "price paid" corresponds to the total amount that had been bid for the second-highest candidate. Hopefully, this approximately incentivizes voter-blocs to be "honest" in their "bids."

A voter-bloc that dishonestly bids too high for some candidate risks paying more than his election truly is worth, if he wins. (And that loss of pseudomoney matters because it reduces that bloc's voting power for the purpose of deciding further seats. Observe we do not need to employ real money to cause money to "matter" provided the election is multiwinner. Forcing voters to employ real money would have been unconstitutional in the USA.)

On the other hand, a voter-bloc that dishonestly bids too low for some candidate risks failing to elect him, even in situations where they could have elected him while "paying" a price below his true "value."

Assuming that a "market" develops that determines a "price" per seat, one might infer that a voter-bloc with N times more voters, will have N times more pseudomoney and will thus be able to determine the occupants of N times more seats – "proportional representation."

Observe that in the single winner (W=1) special case, the above system becomes range voting, known to be one of the best single-winner voting systems.

Criticisms

  1. There are voter "blocs" casting strategic votes, and there are individual "voters." These two idealizations do not necessarily agree. In particular, a voter "bloc" lowering its collective bid for winning candidate A below true value, will not pay any less for A's victory (due to the Vickrey "second price" rule) hence has no collective incentive to dishonestly shrink their vote for A. That's good. However any individual A-voter will realize savings by lowering her vote for A, thus "free riding" on the A-supporting bloc vote (provided A still wins). That seems bad. Also, economists generally regard "economies" as being about self-interested individual actors, in contrast to "voting" which involves a collective attempt by a large bloc of voters to achieve a common goal. Those two idealizations differ and also (of course) both are incorrect.
  2. Vickrey's requirement for "bid secrecy" corresponds to "ballot secrecy" – except for the fact that pre-election public opinion polls might largely defeat both. (Of course, people could strategically lie to the pollsters...)
  3. In the pseudo-economy, the different bids interact in mysterious ways. The idealization of modeling this as an "economy" with "prices per seat" is only a vague approximation, while the reality is very complicated. (Indeed, the whole notion that economies obey simple laws is only approximately true and the truth is very complicated!)
  4. Criticism: If you honestly vote Bush=1000, Gore=500, Nader=0, then later (after Bush wins) you still value Gore above Nader by $500, but due to your payment when Bush won a seat, your rescaled vote dishonestly values Gore above Nader by only $300 (if there was 60% scaling).
    Response: That might have been true if voters were using real money, but we are speaking of pseudomoney. And pseudodollars are worth 66.7% more after Bush wins a seat than before (since 40% of the pseudomoney supply vanished). Hence the rescaling actually does not diminish your honesty.
    Critic counters: But some voters lose more money than others, e.g. somebody who voted Bush=0, Gore=500, Nader=0 pays nothing when Bush wins. That voter now supposedly values Gore over Nader more than the Bush=1000, Gore=500, Nader=0 voter!?!
    Response: True, but that voter deserves to have more power because she has no representation (yet). That payment was for getting representation by Bush. Such payments seem essential to achieve anything like proportional representation, and also seem essential to make pseudomoney have real value.
  5. The "Vickrey based" rules I just suggested are certainly not the only possible rules. There are many other possibilities, some of which might be better.
  6. Critic: Consider an election involving the Red Party versus the Blue Party. The voters are 75% Red and 25% Blue and there are 4 seats at stake. I believe the correct result, therefore, obviously ought to be 3 Red winners and 1 Blue winner. However, if the Red-voters (75%) vote 1000 for each Red and 0 for each Blue, while the Blue-voters vote 1000 for each Blue and 0 for each Red candidate... what happens? A massive screw-up. One Red candidate wins, then all the Red voters lose all their pseudomoney. Then a Blue candidate wins, whereupon all the Blue voters lose all their pseudomoney. Then weird fringe voters determine the two remaining seats. (Or maybe if the Blues voted only 999 then they have some pseudomoney left over so they get the 3 winners, not the Reds, in spite of voting lower and being in the minority!) This is not right!
    Response: The critic is trapped in voting OldThink, i.e. "if we Reds want to elect more Reds, we should award them more 1000s! More more more!" Well, no.
    Actually, those votes were very dishonest and (hence, since this voting system incentivizes voter honesty) stupid. The critic's postulated Red voters are saying with each of their "1000" votes "electing just this one Red is so important to me, I'm willing to give up 100% of my voting power to do it." Then, they get exactly what they said they wanted – electing one Red – and pay exactly the price they bid for that privilege – all their pseudomoney – and then they complain "I didn't really mean it! I just made a super-unwise bidding/purchasing decision! Oops!"
    Really, the Reds would have been both more honest, and smarter, to bid about 400 for each Red candidate, trying to save some money so they would have enough left to elect more Reds.
    So the critic's example actually is not a bad example for our voting system, it is a good example demonstrating that it works as hoped to incentivize voter honesty! (The critic was so used to bad voting systems that incentivize voter dishonest-exaggeration rather than quantitative honesty, that he actually confused himself into thinking that was a good thing!) Of course, after a few lessons of this kind, the voters would quickly get wise and learn that "dishonest exaggerated" and "stupid" voting were the same thing. Indeed the Red-party and Blue-party strategists would already know that on day one, and would advise people how to vote smart. (On the other hand, if a voter really did honestly feel that just electing some one Red was worth all her voting power, then she'd have nothing to complain about – she reached nirvana, and her honest vote was pretty clearly the strategically optimum vote.)
    Of course, some people would remain dumb. They would refuse to (or be unable to) honestly quantititively assess how much each candidate was worth to them. They would be unable or unwilling to submit honest bids. Those people would pay for their stupidity and dishonesty with reduced voting power. But personally, I do not see why it is a bad thing if dishonest, stupid, and quantitatively-handicapped people have less power in some country. It is probably a good thing, and would give that country an advantage over countries in which dishonest, stupid, and unquantitative people had greater power.

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