## Clarke-Groves-Tideman-Tullock "perfect" scheme for voting-with-money

This is what happens when some economists invent a voting system...

In principle, the ultimate voting scheme would be "honest utility voting" in which each voter states the "utility" (measured in some common, agreed-upon units) of each possible candidate for him, and then the candidate with the greatest utility (summed over all of society) is elected.

Unfortunately, honest utility voting seems unachievable in practice, since (a) there are no common agreed-upon units (and utility is intangible), and (b) just one dishonest-strategic voter, by making some vast exaggeration, could control the election. (However, in certain unusual scenarios, such as where the voters are not dishonest humans, but rather honest robots, and in which utility is easily measured, this voting method would be best possible.)

Therefore (the common thinking before the 1970s was) honest utility voting is a pipe dream – an idealization of no practical interest. However in the 1970s several people realized, some independently, that, at least in a mathematical idealization of voters as rational economic money-maximizing animals, such a "perfect" voting system (that always elects the max-utility candidate) actually is achievable!

The initial idea (1961) was by W.Vickrey in a different context: auctioneering. (Vickrey was awarded the 1996 Nobel Prize in economics, mainly for this idea.) Imagine some moderate number of bidders (e.g. 10-20) want to buy some expensive object.

Vickrey second-price (sealed bid) auction protocol:

1. Each bidder privately estimates the true worth of the object, to him, in dollars.
2. Each bidder submits that secret estimate, in a sealed envelope, as his bid.
3. All the envelopes are opened. The winner is the one who submitted the greatest bid, but now he only pays the amount specified by the second-highest bidder.

Vickrey argued that in this scenario, there is no strategic motivation for bidders to be dishonest in their bids, and plenty of motivation for them to be honest. Bid too high? Risk paying more than it is worth. Bid too low? Risk not acquiring the object at cost below its true worth to you, and does not reduce your payment if you do get the object. Note that in this scheme it is important that the other bids be secret, since a bidder who knew the topmost external bid could then have motivation to be dishonest and bid 1 cent below it to minimize the profit of that enemy bidder. But it also is important that the bids ultimately be revealed since otherwise the auctioneer could cheat by pretending the second-top bid was much higher than it really was.

Clarke-Groves-Tideman-Tullock public choice (i.e. voting) protocol:

1. Each voter as his (secret ballot) vote submits his private estimate of the true worth of each candidate (or election alternative) to him, in dollars.
2. The alternative (or candidate) with the greatest amount of money voted for it/him, wins.
3. We now re-examine all the votes, but now with the name of its voter-author unveiled on each. Suppose that vote "made a difference," i.e. caused the election result to differ from what it would have been with that vote removed. Namely, suppose without some voter's pair (BY, BZ) of vote-bids for candidates Y and Z, alternative Y would have won, but with it alternative Z wins. Suppose the minimum amount of money that vote-bid-difference BZ-BY could have been replaced by that still would have caused Z to win, is MYZ where 0 < MYZ < BZ-BY. Then that voter now must pay a fee (the "Clarke tax") equal to MYZ.

The Clarke-Groves-Tideman-Tullock central claim is that there is no strategic motivation for voters to be dishonest in their votes, and plenty of motivation for them to be honest. Bid too high? Risk paying more than the election Y → Z result-change you caused, actually was worth to you. Bid too low? Risk not acquiring the election result you want, even though you could have done so at cost below its true worth to you and without decreasing your payment if you have to make one. And if the votes indeed are honest, then this voting system therefore is "perfect" and always elects the best choice, as measured in (perceived) dollars, for all of society. Tideman and Tullock felt obliged to point out that nevertheless their method "would not cure cancer."

Voter Y Z
Amy \$5 \$0
Bob \$5 \$0
Cal \$3 \$0
Dee \$0 \$4
Eve \$0 \$7
A Clarke-Tideman-Tullock 2-candidate example-election. Y wins by \$13 to \$11. Amy and Bob then pay Clarke tax \$3 each (because without Amy, Z would have won by \$3), and Cal pays \$1. The other voters Dee and Eve pay nothing.

Unique features. This is the only system claiming to result in "perfection" and "voter honesty." It also is the only one that directly involves money (and therefore sadly cannot be used in abstract scenarios in which the "voters" actually are not money-owning economic entities).

Additional work. For some reason, interest in this scheme apparently died after 1984 and it was almost entirely forgotten. (Later note: actually it was not forgotten; a remarkable 2001 book by Bailey uses it as a key ingredient.) One of the last works (besides Bailey) on the matter was by Tideman who conducted experiments on actual use of the scheme. Specifically, Tideman paid several college fraternities to do their decisions by his process instead of plurality voting for about 1 year. Tideman's experiments failed to reach a convincing conclusion. The fraternity members (polled afterward) preferred plain plurality voting to Tideman's method, 108-to-56, because of increased administrative work! But they preferred Tideman (112-to-42) if the issue-proposer was willing to pay a administration-fee to get his issue considered by the Tideman method – but that in fact never happened during the subsequent year.

Tideman's process yielded different decisions than plain voting in about 10% of the fraternity decisions. This led to about 2.25% extra perceived dollar benefit. However the "Clarke taxes" paid were larger than that (namely 3.04%) so in some sense it wasn't worth it. However, frat houses are small – and in larger elections the Clarke taxes would be comparatively negligible. Finally, there may have been collusions or other effects (see criticism list below) which really caused CTT voting to perform even worse, and if so Tideman had no way to know it.

Criticisms. Unfortunately there are several reasons that Clarke-Tideman-Tullock, in practical use in large elections, would in fact fall considerably short of perfection.

1. In the large elections typical in governments, the probability is extremely small that any one vote will "make a difference." (In all the 50,000 or so US presidential, congressional, and senatorial popular elections so far, there has never been a case where any single vote has affected the outcome.) This contrasts mightily with the situation in most auctions (typically there are 10-20 bidders so each has a reasonable chance – 1/10 or 1/20 – of winning) and in most small fraternity-house elections. And that contrast, as we shall see next paragraph, does matter.

2. In auctioneering and fraternity houses, the problem of bidders not paying, is a minor problem. But in nationwide elections in the very rare instances when payment was required, there might be a severe nonpayment problem. Even if all voters were honest, some would die during the election. And if only one election in a million required the typical voter to pay, that might encourage a culture of dishonest exaggerated votes, followed on the rare occasions every 3000 years (!?) where payment actually was required, by a culture of tax-evasion. How could voters be forced to pay? Throwing the election the other way in response to nonpayment would not work because the other side also might not pay! This all could lead to a nightmare scenario giving a new meaning to the term "election fraud."

The problem here is the combination of 1 and 2. To defeat it we could require each voter to submit his payment in "escrow" (to be refunded in most cases). That would avoid the nonpayment problem. However, it would cause a new problem: the cost in time, hassle, and interrupted investments to place one's money in an escrow account – a cost greatly and unfairly varying from person to person -- would vastly exceed the actual expected value of the Clarke tax. (If the Clarke tax must be paid 1 time in 40,000 then even missing out on one day of interest even at only 1% annually, would exceed the expected value of the Clarke tax.) Furthermore, the escrow account set-ups and necessary financial motion might defeat the goal of secret voting. (And, as with Vickrey auctions, secrecy is essential to prove the CTT central claim on which everything rests.) Actually, the fact that all voter-names are revealed to somebody to cause payment, seems to mean that CTT voting is incompatible with ballot secrecy. Therefore, in practice the economic rationale of this sort of voting would be dominated by these distortionary effects and the CTT goal of a voting system in which undistorted true utility dominates would not be achieved at all.

To minimize these effects, the government could give each voter a "free donation" of a certain small amount of money into his escrow account, provided he voted, and could pay interest on the escrow accounts. These moves would attempt to null-out transaction costs. However, there would unavoidably be remaining inequities and mismatches in that nulling-out attempt, which would usually still dominate the true expected utility estimates made by, and true "costs of voting" for, each voter.

The point of this has been that certain "negligibly small" effects such as the cost of moving money and cost of voting, while they are negligibly small in the case of auction of an expensive object, are in fact the most important thing in the voting case.

The two problems of auctioneering and voting might seem almost equivalent except that some numbers are changed, but the problem is that these numbers change so vastly that the approximations of certain effects as negligible, become completely invalid.

Margolis 1982 made similar points. He observed that in a typical election situation the expected Clarke tax on a typical voter would be 0.00001 penny, and hence expressed doubt that voters would bother to work out and use their optimum-utility votes. The transport, money-motion, and other transaction costs experienced by a typical voter would dwarf this and hence, rather than being "negligible" would in fact "dominate" the thinking of rational voters.

Pre-deposit idea to combat Margolis' 1982 criticism: The government would contribute and pre-deposit into voter escrow accounts, a moderate amount of money – enough usually to pay for all transport and time-costs – but these moneys would only be available to people who actually voted. Further, the government would, by law, require bank transfers into and out of your voter-escrow account to be free of cost (or automatically compensated by the government). In that case, once a voter has decided to vote, he shuts these transaction costs out of his mind. They are no longer relevant; the voter already has the money that more-than-covered those costs. The voter can now purely concentrate on: what is my best voting decision?

Problem is, this does not quite work, because many voters will have moneys tied up in other forms than bank accounts.

3. Most people would not agree utility is the same thing as money -- even though (economist) Tullock may think it is! If alternative A leads to a rich man dying while alternative B kills five poor men, then CTT voting would choose alternative A. More generally this system might do whatever the rich and fanatical want, and might exhibit a systematic bias against poor people. Indeed, for this reason CTT voting is unconstitutional in the USA under the 24th amendment.

But recently Professor Marcus Pivato has proposed a way to make CTT voting be truly based on utility not money! Pivato's idea is that the bids not be in terms of money, but rather in terms of probability. Suppose we can identify two magic events which we can agree have about the same utility-difference for everybody. When you "pay" the Clarke tax, in Pivato's non-monetary scheme, that now means you get subjected to a lottery in which the worse-event happens to you with probability P (where P is your tax) and the better-event with probability 1-P. To make this work it is essential that the two events be enough-different in utility that they (for all or almost all voters) exceed the utility difference arising from who gets elected. (Otherwise we get "truncation effects" distorting the picture.)

4. The Clarke taxes would in fact be paid into government funds which would then be used to reduce (ordinary) taxes. Hence in reality the true amount voters would effectively pay, would differ from the Clarke tax. That in practice would distort the system away from the ideal of employing "perfect utilities." (One nightmare: consider a candidate whose platform was "if elected, I will refund all Clarke taxes.")

One suggestion to combat(?) that criticism is to award all the Clarke taxes to one randomly selected voter, the "Clarke-tax lottery winner."

5. Collusions: Suppose "Nixon" bids some enormous amount of money and "Agnew" also does, to make the Nixon-Agnew ticket win. Now, each of {Nixon, Agnew} alone did not change the election result because their bids were each vastly huger than everybody else in the country combined, so Nixon could argue "I would have been elected anyway thanks to Agnew's bid" therefore Nixon pays zero Clarke tax. Similarly Agnew also pays zero and both get elected for free. This pathology could reduce everything to 2-man teams of colluders each trying to "name the largest number they can" – which would be a ridiculous dysfunctional state of affairs.

However, this Nixon-Agnew criticism largely falls to the ground if escrowed vote moneys are demanded. That makes ridiculously large bids impossible. But that, as we said, leads to other difficulties. (Another related to #3 is: what if my utility exceeds the amount of money I am able to raise in cash form? Consider abortion rights. Jill in losing her abortion rights, is losing a lot of expected dollars. The ratio of abortions to live births was approximately 32:100 in the 1990s USA. Suppose it costs \$100,000 to birth and raise a child but only \$300 to get an abortion (1990s dollars). Suppose Jill is going to have 3 children and hence (on average) one abortion. That means \$100,000 of expected utility is at stake for her if there is a vote on a law that would make it impossible for her to abort. (Actually she could still try to get an abortion in a foreign country, or try to abandon her child for adoption, so it is less than \$100,000, but we shall ignore that.) But more utility is at stake the younger Jill is. However, the younger Jill is, the less likely she has \$100,000 to vote with! Now on the other side are the moralists who believe abortion is murder. From their view, it is worth all the money some fetus would ever be able to earn, to avoid being murdered. However... fetuses don't have any money and can't vote. So it appears both sides in this issue are going to regard the CTT voting system as unfairly biased against them and invalid. How do CTT-proponents answer them? )

New randomization idea to reduce the collusion problem: Suppose every vote-bid X is automatically replaced, before doing the election, with a random number in the interval [0, 2X] with mean X. (The problem of making everybody confident the randomizations were genuine, independent, and unbiased, while keeping everything secret until the end, is a solvable cryptographic problem.) One possible probability distribution would be the two-mass one with either 0 or 2X chosen by a coin flip.

This idea has the disadvantage that it adds extra "noise" to the election and that it can stick unlucky voters with a higher than-expected Clarke tax bill (although they could compensate by underbidding by a constant factor). But its advantage – and point – is that it causes the Nixon-Agnew collusion we just described to become very risky. Randomization causes your collusion-teammates to become effectively "untrustworthy" even if they actually were totally loyal.

Counterattack: With 100 colluders, it becomes exceedingly unlikely (probability 100·2-99) that any colluder will be forced to pay. So the colluders by organizing moderately large teams can defeat this idea.

Incidentally, Margolis in 1983 tried to argue using differential equations that the CTT scheme was the essentially unique one with its properties. Margolis unfortunately did not actually state an explicit theorem with proof, but suffice it to say that Margolis was wrong in the sense that we have just exhibited this randomization generalization constituting an infinite number of ways to generalize CTT. So no, CTT is not unique at all, but yes, it is unique if we restrict to deterministic voting schemes in which voter payments are made as small as possible.

Other ideas to reduce the collusion problem:

1. Require "reasonable" bids: e.g. the top 5% highest bids could be collapsed to the same amount.
2. Randomly split the voters into 3 groups. All 3 groups are likely to agree on a winner. However, if only 2 agree, then both those 2 groups have to pay. This is a different randomization technique but with the same goal of making collusions become risky. (Of course, this might work even better with more than three groups. Having more groups makes collusions more risky, but also makes the election more noisy and less ideal.)
These ideas may slightly destroy the mathematical "perfection" of the CTT scheme, but it may be worth the sacrifice.

6. Howard Margolis offered another criticism: which is that voter "ancillary utilities" (Margolis called it "altruism," but Professor Marcus Pivato convinced me that was a bad name and "ancillary utilities" is a better name) would invalidate CTT. That is, in many cases the amount of money at stake for each voter is small, so that voter is willing to be altruistic and lose money to further what he considers to be a good cause. Or that voter simply gets a certain amount of joy simply from pulling the voting lever, which is not the expected amount of joy she gets from the chance that vote will alter the election winner. The problem is this "ancillary utility" is likely to far-exceed the utility CTT is trying to measure.

That seems to me to be an excellent and completely valid criticism: the only reason almost anybody votes at all, is "altruism"/"ancillary utilities" since it is totally economically irrational to vote considering the cost in time and transport to vote versus the low probability of that vote having an election-swinging effect. So ancillary utilities (aka economic irrationality) are not "negligible"; they are the "dominant" effect in voting! Given any such huge distortionary effect on utilities, CTT's claims of "perfection" fall to the ground.

The net effect of all these criticisms is very serious and probably render the Clarke-Groves-Tideman-Tullock voting system practically useless (or at least uncompetitive enough to remove it from consideration) for large governmental elections. CTT also is useless for (or at least dubious for) small elections such as in Tideman's frat house study (since for them the Clarke taxes exceed the benefit of switching to the system). However, for elections of intermediate size (500-50000 voters?) and in which utility does closely correspond with money, and assuming the collusion problem is overcome, this system may make excellent sense. (Stockholder elections in corporations, with Clarke taxes donated to charity?) It might also make sense if future techno/political developments alter the nature of "money" so that large anonymous and undetectable monetary transfers become easy and cheap.

### References

Martin J. Bailey (with editing by Nicolaus Tideman): Constitution for a future country, Palgrave 2001.

Edward H. Clarke: Demand-revelation and the provision of public goods, Ballinger Cambridge MA 1980. (Second ed. 2000.)

T. Groves & J. Ledyard: Optimal allocation of public goods, a solution to the free rider problem, Econometrica 45 (1977) 783-809.

H. Margolis: A thought experiment on demand-revealing mechanisms, Public Choice 38 (1982) 87-91.

H. Margolis: A note on demand-revealing, Public Choice 40 (1983) 217-225.

H. Margolis: Selfishness, altruism, and rationality, Cambridge Univ. Press 1982.

T.N. Tideman: An experiment in the demand-revealing process, Public Choice 41 (1983) 387-401.

T.N.Tideman & G. Tullock: A new and superior process for making public choices, J. of Political Economy 84 (1976) 1145-1159.

W.Vickrey: Counterspeculation, auctions, and competitive sealed tenders, J.Finance 16 (1961) 1-17.