Monetized Score Voting

Warren D. Smith, June 2013.   warren.wds at

Abstract. In C-candidate 1-winner range voting, each voter assigns each candidate a numerical score from a prespecified fixed range, as her ballot. Greatest average score wins. We now explore monetizing this, i.e. each voter must pay money to vote, the cost being M(UWinner-U2ndPlace) where Uk is the score that voter's ballot gave to candidate k. We investigate several possible formulas for the magic price function M(z). Currently the best seems to be M(z)=(κ/2)z2/(1-z2) where κ is a tunable positive constant and the score range is the real interval [0,1]. Most of our proposed price functions are designed so that, assuming each voter believes certain oversimplified probability models of the behavior of the other voters, then her unique max-believed-expected-profit voting strategy is to honestly state the dollar-value (to her) of each candidate as her score for them on her ballot. If every voter acts that way then the candidate with the greatest summed honest dollar value to all voters, will win. In contrast, with ordinary moneyless range voting, voters usually are motivated to distort their honest scores.

Unfortunately, monetized range voting has serious problems, including:

  1. the payment and expected profit amounts in contemporary USA elections would be tiny (typically between 0.01 and 5 cents) for each voter,
  2. hence the discreteness of contemporary money would destroy the whole justification for monetized score voting even for voters solely motivated by money,
  3. other effects (both economic and non-economic) also swamp these tiny amounts of money by factors>100 and dominate the picture, rendering all the mathematics almost irrelevant,
  4. many of the systems "self-invalidate",
  5. "money" and "utility" differ [albeit it is possible to correct for that to some extent],
  6. this would be unconstitutional for use in governmental elections in the USA under the 14th and 24th amendments.

As a result none of our max-expected-profit-based monetized systems can be recommended for any (or almost any) governmental use. But (a) it remains conceivable there might be a valid use in some kinds of corporate votes, and (b) if the monetization is not based on the max-expected-profit theorem as motivation, but instead merely aimed at the different and weaker goal of discouraging strategic exaggeration by plain range voters, that might work well in practice using the above price formula; (c) one can also consider schemes based on pseudo-money which is only usable for voting and not for other purposes.

Previous work was atrocious. J.K.Goeree & J.Zhang proposed, then E.G.Weyl further examined, the C=2 candidate special case of one of our systems (which Weyl called "quadratic vote buying") but did not see our generalization to permit C>2. Even as such we demonstrate that the papers G, Z, and W published on the internet are seriously flawed. In particular, our proof of the max-believed-expected-profit theorem which underlies this, appears to be the first correct theorem statement and proof, even in their special case, as we demonstrate by providing counterexamples and refutations of G, Z, and W's previous theorem stating+proving attempts. Also there was a stunning absence of real-world numbers in the Weyl paper. The present paper appears to obsolete and/or overturn all that previous work, for both theoretical and practical purposes.

Table of Contents

  1. The voting systems
  2. The purpose of monetization is to encourage/improve voter honesty
  3. Illustrative example where monetized system could produce better societal results
  4. The probabilistic models (in the C=2 candidate case)
  5. C≥2 candidates
  6. Failed numerical sanity checks
  7. The law
  8. Huge motivation to cheat
  9. Comparison with present-day (2013) major USA elections
  10. Utility≠Money; and "Shannonizing" to repair that problem
  11. Conclusion
  12. Tragicomic previous work
  13. Honesty-Inspiring Economic "mechanisms" more generally
  14. Late-breaking news (added Oct. 2014)
  15. References

1. The voting systems

"Monetized score voting" refers to a new family of single-winner C-candidate voting methods (C≥2) in which each voter's "ballot" is an assignment of a numerical score within some fixed permitted range, to each candidate. (We shall consider continuum real numbers as scores; rational numbers would also seem fine.) The candidate with greatest average score wins. So far, what we have described is just plain range voting.

The new "monetization" rule would be that voters have to pay money to vote. Specifically, there is a publicly known formula which, given any ballot, will compute its dollar cost:

Cost = M(U1-U2)

where Uk is the score the ballot gave to candidate k; the winning candidate is 1, the candidate with the second-greatest average score is 2; and M(z) is a smooth nonnegative-valued function (increasing when its input is positive) specified by the voting system (see table below). The voter must submit that amount of money with her ballot. Since actually the price is contingent upon who wins and who places second (which is not known when voting), in practice the voter would need to submit the maximum worst case price – then be partially refunded later if the election outcome turned out not to be the one maximizing her payment. (Call this "pre-payment" and "refund" with the net effect being just "payment.") But to simplify the discussion we mostly shall pretend the voter pays the correct amount just once because both she and the vote-taker magically predict the election outcome, so no refund is needed. In this paper we also mostly shall pretend that "money" is an infinitely divisible real quantity although actually at present in the USA, money is an integer number of "cents." But money-discreteness issues would really matter and will be discussed in §6. After the election is over, the collected monies could (most simply) either be appropriated by the government, or redistributed in nearly equal parts to all voters according to the following scheme: if there are V voters, each voter's payment is split into V-1 equal parts each to be paid to the V-1 other voters. Equivalently, if T is the total money collected, then each voter would receive post-election payment (T-P)/(V-1) where P is whatever amount that voter paid to vote.

SystemScore range M(u) M-1(x) Properties of M(u)
Honest Utility Voting [-∞, +∞] 0 (no payments)
Plain range voting [0, 1] 0 (no payments)
Quadratic money voting [-∞, +∞] (κ/2) u2 (2/κ)1/2 x1/2 Increasing for u>0, concave-∪ for all real u, unbounded above.
Tanh-based money voting* [-∞, +∞] [tanh(κu)+1] u -
κ-1 ln([exp(2κu)+1]/2)
? Increasing for u>0; concave-∪ when |u|<0.77170κ-1;
asymptotes to ln(2)κ-1≈0.69315κ-1 when u→+∞.
Erf-based money voting* [-∞, +∞] 2(κπ)-1[1-exp(-πκ2u2/4)] (2/κ)ln(1-cπx/2)1/2 Increasing for u>0; concave-∪ when |u|<(2/π)1/2κ-1
where (2/π)1/2κ-1≈0.79788κ-1;
asymptotes to (2/π)κ-1≈0.63662κ-1 when u→+∞.
Procrustean money voting [0, K] (κ/2)u2/(K2-u2) [2x/(κ+2x)]1/2K Increasing for 0<u<K, concave-∪ for all u with |u|<K, unbounded above.

Some of the formulas in the table involve tunable positive constants κ, K, which need to be chosen semi-empirically for best performance and whose optimal values might differ in different elections (perhaps rather unpredictably and dramatically), or because of inflation altering the value of a dollar. The asterisks * indicate systems known to be self-invalidating (a concept we'll explain later); quadratic money voting also seems self-invalidating albeit in a different (and perhaps weaker and less-convincing) sense.

Note that all proposed nonzero M(u) functions are asymptotic when u→0, namely M(u)=[u2±O(u4)]κ/2. Also, they obey even symmetry M(u)=M(-u) even though this is not immediately obvious in the tanh case. (In the preceding sentence, and in various places later in the paper, we make claims without providing the details, which is because verifying the claim is at the level of a homework exercise for students in a calculus-1 class.) But they differ at nonzero u. In particular, for tanh- and erf-based money voting, M(u) asymptotes to the constant values κ-1ln2 and 2κ-1/π when u→+∞, which is very different from the unbounded behavior of u2 for quadratic money voting. In other words, in tanh- and erf-based systems, it never costs more than than a constant dollar price to vote, no matter how absurdly extremely large your ballot's |score|-values are; while with the quadratic system the price can be unboundedly great. With the procrustean system, any attempt to make two scores differ by ≥K would require an infinite money pre-payment and hence is forbidden.

Also note that all proposed M(u) functions obey M(0)=0 and the nonzero ones are strictly increasing functions of |u|.

2. The purpose of monetization is to encourage/improve voter honesty

Honest utility voting would be the perfect voting system if each voter provided her "honest utility" (on some common scale of units) for each candidate's election, as her score for that candidate. Then the candidate with the greatest summed utility for all of society would be elected – the best possible outcome.

But unfortunately, there are no common agreed-upon units for utility – and even if there were then a single dishonest voter could, by providing enormous bogus scores, throw the election. Thus honest utility voting would be ridiculous and useless for most or all applications with human voters, although it might still retain interest for voters who are, e.g, robots.

Plain range voting prevents the use of enormous scores by restricting them to the real interval [0,1]. (Any other real interval of finite nonzero width could be used instead, but mathematically all are equivalent to [0,1].) This stops huge exaggerations by dishonest voters, but also stops the voting system from being "perfect." Indeed, in most realistic situations strategic voters would want to award all-extreme scores (0 and 1 only, not u with 0<u<1) to all candidates, in which case range voting would degenerate to approval voting (in which each voter "approves" or "disapproves" each candidate; most-approved wins). This would reduce range voting's quality. However

  1. There are situations in which approval-style range voting is known to be strategically non-optimal, and
  2. It is fairly clear from numerous range-style polls plus simulated-election exit-poll studies of real human voters, that most humans do not range-vote in approval-style.
  3. Indeed, they often do not even utilize the full score-range, thus intentionally weakening their ballot.

Facts (b) and (c) massively and blatantly contradict and refute what one might call the "naive economist's theory of the world," i.e. that voters aim to maximize their expected utility (or expected profit). Other observed facts about voting (e.g. the behavior of about 85% of Australian IRV voters) even more enormously contradict that theory – most obviously the very fact that voters vote at all. It is massively economically irrational for typical voters to vote in the present-day USA for (say) President, because the expected benefit (chance of altering election times dollar value to that voter of doing so) is tremendously smaller than the cost (in wasted time, transport costs, etc) of doing so. About 108 voters, apparently including a large fraction of professional economists, perform this refutation every election day, but the latter cover their eyes and devote great effort to ignoring this.

Monetized score voting is an attempt to overcome those real and/or perceived defects of plain range and honest utility voting. For example, in procrustean money voting, approval-style voters would have to pay an infinite price. Highly-exaggerating dishonest voters would tend to pay larger prices than honest ones. This would encourage honesty.

Indeed, the point of the particular cost formulas we have proposed is that they, in certain oversimplified approximate models which we shall explain, cause the max-monetary-profit voting strategy to be "honesty." That is, the election of each candidate would cause some benefit or loss to each voter, which she quantifies in dollars. The voter also experiences money-gain or loss due to the imposed costs of voting. The sum of both these is her "profit." This profit is a random variable in the view of a voter who cannot perfectly predict how all the other voters will vote. If the voter adopts certain simple probabilistic models of the other-voter score-totals, and votes in such a way as to maximize her expected profit, then we shall prove that the max-profit voting strategy is to honestly state each candidate's money value (to her) as her score for that candidate. That in turn would cause the "best" candidate – with maximum summed-over-voters money value – to be elected.

3. Illustrative example where monetized system could produce better societal results

A classic (if perhaps over-simplified and over-dramatized) example is the "kill the Jews" vote where the choices are

  1. Kill the Jews and steal their money, using it to reduce taxes for non-Jews.
  2. The Jews (who are a minority) live.
  3. (Possibly other choices too.)

Votes of this nature illustrating the "tyranny of the majority" arise all the time, although fortunately usually in less-dramatic forms.

Almost every A-versus-B voting system ever proposed would kill the Jews with honest voting ("honest" meaning voters' personal economic assessments are honestly reported as votes). This would be bad. The underlying problem is "votes" and "utility" are not the same thing.

Plain range voting (unlike almost every other voting system out there) could permit the Jews to live provided enough non-Jews "honestly" self-weakened their votes by using a small score difference between A and B, while meanwhile the Jews honestly voted with large score difference. (Actual score voting poll data suggests such behaviors do happen to a substantial extent, although it would be better to do experiments intentionally investigating this.)

With monetized voting systems, Jews would be motivated to provide large money-difference values as their votes. If these values were 10× those provided by the non-Jews (costing, under the quadratic system, 100× as much per voter), then the Jews would live (with the quadratic system) even if only a 10% minority. "Utility" and "money1/2" also are unequal, but in this case they approximate each other well enough to improve the situation.

If, further, we throw in a certain amount of self-weakening and/or altruistic "irrational" voting behavior too, then maybe only a 5% minority of Jews still could live. But, note that if the Jews were a small-enough or poor-enough minority then none of the monetized systems could save them from "economically rational" voters, whereas, with plain range voting with actual humans exhibiting actual human voting behaviors, they still might be able to live even in the limit of approaching 0% Jewish minority.

Nasty example: In particular in the quadratic system, consider a single Jew whose total wealth is 100, versus N>1 non-Jews. The single Jew votes his entire wealth 100. The non-Jews see the election as worth 100/N money to each of them, and therefore each vote 100/N, with costs 5000κ/N-2. The vote then is an exact tie. Note for large N, the quadratic system actually here made it less costly to kill the Jew than if the 1/N-votes had simply each cost 1/N.

My point is, it is certainly possible to conceive of situations in which monetized voting systems of the sort we are discussing, could outperform traditional moneyless ones, although that is not guaranteed always to happen. Furthermore, Goeree & Zhang did a pseudo-election experiment we shall discuss later demonstrating that exactly this kind of monetized "rescue from tyranny of majority" really did happen in an artificially created scenario.

4. The probabilistic models (in the C=2 candidate case)

For simplicity in this section we shall assume there are only two candidates "a" and "b"; the voter we are considering prefers "a"; and the amount of her preference is D dollars.

Simplest model: Suppose a voter believes that the probability that her ballot's scores Ua and Ub for candidates a and b will alter the election result (versus if she had not voted) is

Prob(alter) ≈ κ |Ua-Ub|

for some small positive number κ which depends only on how the other voters behave. Further, we shall later assume that every voter has this same model involving the same numerical value of κ. Under this model our voter's expected profit (versus not voting) would be

E(Profit) = κD·(Ua-Ub) - M(Ua-Ub),

which under quadratic money voting would be

E(Profit) = κD·(Ua-Ub) - (κ/2) (Ua-Ub)2.

(If there is post-election redistribution of the collected monies, then our voter's profit will be altered by an additive amount which depends solely on the payments of the other voters, i.e. is unaffected by her own actions. Thus this possibility does not affect her optimal strategic decision making.) Then, our voter's uniquely best strategy (maximizing expected profit) is then to choose Ua-Ub=D. This is because the unique maximum of the function κDΔ-(κ/2)Δ2 of Δ is located at Δ=D. [And the value of the maximum is (κ/2)Δ2, which here happens to exactly equal the price of casting the ballot.]

In other words, our voter's unique expected-profit-maximizing voting strategy is to provide scores Ua and Ub constituting an honest assessment of the money-values of the two candidates, for her. If every voter acts that way the result is that the candidate with the maximum money value (summed over all voters) is elected.

Unfortunately what we really want in an "optimal" voting system is not "maximum money value" but rather "maximum utility," and money and utility are not the same thing (although economists often act as though they are). That will be discussed in §10, but for now we'll just act as though money=utility.

The most obvious flaw in the simple model: The model would be correct if the difference between the two candidates' score-totals (before our voter acted) were a real random variable uniformly distributed within an interval of width 1/κ. That is clearly not going to be exactly correct in real life... and even if it were, it would become incorrect once we hit the ends of the interval. That is, it could be correct that our voter by saying Ua-Ub=Δ on her ballot, alters the election with probability=κ, and by instead using 2Δ obtains probability=2κ, by using 3Δ obtains probability=3κ, etc, but eventually κ+κ+...+κ will exceed 1, at which point clearly this model must break down. The model will be approximately valid in the sense that every smooth probability density locally, near any generic support point, is approximately uniform – but this approximation clearly is only an approximation, and for "richer" voters (who are willing to pay more money to to get a higher-impact vote) the approximation is worse than for "poorer" voters.

The tanh model: A less-bad approximate model is as follows. Each voter believes that the probability that her ballot's scores Ua and Ub for candidates a and b will alter the election result (versus not voting) is

Prob(alter) ≈ tanh( κ |Ua-Ub| )

for some small positive number κ which depends only on how the other voters behave. Note that tanh(z)=z-z3/3+O(z5) when |z|→0, but the tanh(z) function asymptotes when z→±∞ to tanh(z)→±1 with the approaches to ±1 being exponential. This behavior is at least qualitatively correct, whereas the original simple model was obviously incorrect in this respect.

Under the tanh model our voter's believed expected profit (versus not voting) would be

E(Profit) = D tanh( κ·[Ua-Ub] ) - M(Ua-Ub),

using the tanh-based money voting formula for M(Δ). Then, our voter's uniquely best strategy (maximizing expected profit) is then to choose Ua-Ub=D. This is because I have cleverly designed the M(Δ) function so that the unique maximum of the function

D tanh(κΔ) - M(Δ)

is located at Δ = D. (This corresponds to the unique zero of its derivative. The derivative also approaches 0 when Δ→∞ but that corresponds to a local minimum.) Again, the profit motive inspires voters to provide honest dollar-value assessments as their votes, which if every voter behaves that way causes the maximum-dollar-value candidate to be elected.

The erf model: An alternative less-bad approximate model is this. Each voter assumes/believes that the probability that her ballot's scores Ua and Ub for candidates a and b will alter the election result (versus not voting) is

Prob(alter) ≈ erf( 2-1π1/2κ |Ua-Ub| )

for some small positive number κ which depends only on how the other voters behave. Note that erf(2-1π1/2z)=z-πz3/12+O(z5) when z→0, but the erf(z) function asymptotes when z→±∞ to erf(z)→±1 with the approaches to ±1 ultimately being faster than exponential.

Under the erf model our voter's expected profit (versus not voting) would be

E(Profit) = D erf( 2-1π1/2κ·[Ua-Ub] ) - M(Ua-Ub),

using the erf-based money voting formula for M(Δ). Then, our voter's uniquely best strategy (maximizing expected profit) is then to choose Ua-Ub=D. This is because I have cleverly designed the M(Δ) function so that the unique maximum of the function

D erf(2-1π1/2κΔ) - M(Δ)

is located at Δ = D. (This corresponds to the unique zero of its derivative. The derivative also approaches 0 when Δ→∞ but that corresponds to a local minimum.)

Again, the profit motive inspires voters to provide honest dollar-value assessments as their votes, which if every voter behaves that way causes the maximum-dollar-value candidate to be elected.

Self-invalidation: However, both the tanh and erf models, even though "obviously better" than the simple model underlying quadratic voting, are "self-invalidating." That is, due to the "bounded price" properties of the M(u) functions that these models yield, rich-enough voters are happy to submit enormous "dollar values" for the candidates. Under the model they then believe they are essentially certain of altering the election, but know this will only cost them a bounded amount of money. If enough voters behave that way, then the election becomes an absurd "who can name the biggest number?" game, plus the whole underlying probability-model is invalidated.

In contrast, the quadratic M(u) from the simple model seems better behaved and might avoid self-invalidation. There are, however, arguments that it may also self-invalidate:

  1. Suppose the election is for the position of "dictator." Doctor Evil arranges for some voters to vote enormous "dollar values" for Doctor Evil, who is duly elected. Once elected, Doctor Evil can repay the enormous debts his supporters racked up, by redefining/printing "money," then repaying them from government coffers.
  2. Suppose that (continuing the last few decades' historical trend in the USA), wealth disparity continues to increase, eventually approaching the point where a country contains one ultra-rich person (the "lord") while everybody else (the "serfs") has zero wealth. Historically, situations of that ilk have happened. In that case, the only person who can vote is the lord, who then "democratically" elects himself, achieving economically-reckoned "optimality."
  3. The preceding two pathologies actually happen with any monetized system at all in which a voter with enough wealth achieves "dictator" status (i.e. ability to control the election result by himself).
  4. The very fact (which we already pointed out) that the underlying probabilistic-belief that voters must have, is self contradictory, seems to constitute a self-invalidation of the quadratic system. Indeed, it was clear the tanh and/or erf systems were superior, but they self-invalidated, which constitutes by "transitivity" a self-invalidation of the quadratic system too.

Given that, it might be of interest to go in the opposite direction, which is the idea of procrustean money voting. In this system, a rich man cannot achieve unboundedly great voting power via unboundedly great wealth. Hence the procrustean system largely loses the alleged advantage of the quadratic system of obtaining the "economically-best" result for all of society (which entails giving the rich much more voting power), but still retains the advantage of discouraging exaggerated approval-style range voting and encouraging more "honest" such ballots. That alone could be a very substantial advantage because honest range voting is known to be greatly superior to approval voting.

The quadratic system is a limiting special case of the procrustean system and by adjusting the tunable parameters κ,K we effectively can make that approximation better or worse.

However, before getting too thrilled with the procrustean system, the reader would be wise to consider the following apartheid scenario:

  1. Suppose there are two kinds of voters: "blacks" and "whites" (for simplicity assume these two types are equipopulous and all whites are identical and all blacks identical). Due to racist policies the blacks are an economic underclass, the whites are richer. Voting on the question of "shall we keep the blacks getting ≤20% of the pie?" both blacks and whites honestly have the same financial amount at stake, but the blacks do not have that amount. So the whites can always win the vote with honest quadratic vote buying. (The payments due are then the same from all and there is essentially no redistribution – or in the government-appropriates-monies version of course the whites get most of the benefits the government then spends that money on.)

This is realistic: The US state of Georgia initiated a poll tax in 1871, and made it cumulative in 1877 (requiring citizens to pay all back taxes before being permitted to vote). Every former confederate state followed its lead by 1904. Although these taxes of $1-$2 per year may seem small, they were beyond the reach of many poor black (and white) sharecroppers. Kousser 1974 (pp. 67-68) estimated that Georgia's poll tax probably reduced overall turnout by 16-28%, and cut black turnout in half. That poll tax is rather like a simplified version of monetized voting in which only two fixed bid levels are permitted, one of which is "zero."

This problem would seem to arise for every nontrivial monetized voting system, which is probably why the USA enacted the 24th amendment.

5. C≥2 candidates

Really, all dollar values need only be specified by each voter up to an overall additive shift, because only the dollar-value differences among the candidates matter in all of these systems. The voter in particular must consider the dollar value of replacing the winning candidate 1 by each possible rival r=2,3,...,C. Now assume that every voter believes in the "3-way ties unlikely" model: "The probability that a voter will be able to alter the relative ranking of candidate 1 versus candidate r, is if r≥3 negligibly tiny compared to the probability she will be able to alter 1-versus-2." (Where candidates 1 and 2 are whoever will finish first and second.) We have now proven:

MAX-BELIEVED-EXPECTED-PROFIT THEOREM: Assuming the voters all believe in the 3-way ties unlikely model (plus whichever is appropriate among our previous models), and assuming that the voters can never predict with 100% certainty in advance who will (or will not) win and come second, then the quadratic, tanh, and erf systems we have proposed each are "optimal" in the sense that the unique max-believed-expected-profit voting strategy in a C-candidate election (C≥2) is to honestly state your dollar values for all candidates. And if all voters adopt that behavior, the max-summed-dollar-value candidate will win.

This is the first correctly stated and proven theorem of this kind, because (as we shall later demonstrate), the preceding papers which tried to produce such theorems, failed. However, they did have the right idea.

6. Failed numerical sanity checks

As simple sanity checks, let's perform some arithmetic using some fairly realistic numbers.

Contemporary USA presidential election: Suppose there are V=108 voters and the typical voter is affected about D=$1000 worth by who wins. Suppose the chance said voter's ballot will be able to swing the election is 1/50000.

Then the max-expected profit voting strategy (we shall use the formula for quadratic money voting, which as we said is asymptotically the same as all our other money-voting systems in the small-|u| limit) is to honestly state that the top two presidential candidates differ in value by $1000. The chance of this altering the election being 1/50000 means that the parameter κ ought to be κ≈1/50000000=0.00000002 with money measured in units of dollars, or κ=0.000002 if we instead measure money in cents.

In that case using the quadratic system, the voter would expect to profit (κ/2)D2 thanks to her $1000 vote, which with our numbers is 1 cent. She also would need to pay 1 cent to be allowed to cast this vote. On the bright side, these are enjoyably tiny, meaning the voting process itself would not distort the economy much, etc. But unfortunately:

One cent is not enough profit to motivate an economically-rational voter to vote at all, considering wasted time, transit costs, money-motion paperwork costs, etc, which exceed this profit by at least a factor 100. In view of this, any arguments that our voters will act "economically rationally" are massively refuted before she even reaches the voting booth. These amounts also are tiny compared to other distortionary effects such as "altruism" (which are, in fact, the biggest reason the vast majority of voters presently vote; e.g. 71% of Americans say they "feel guilty when they don't vote," Bittle & Rochkind 2009). In short, the whole idea that this kind of economics is going to inspire voter honesty is pretty absurd because the economic motivations seem tiny and "lost in the noise" compared to other economic and non-economic (but very real) influences.

For example, suppose typical voters decided to live large by paying a full dollar (50× larger). This would enormously distort the election.

But it gets worse! That 1 cent payment obviously is coming up against the discreteness of money in a serious manner. This also invalidates the whole system until/unless money becomes granulated in units around 1000× smaller than today's. And suppose voters were interested in paying 0.49 cents hoping that would be rounded down to zero. Again, that would be an enormous distortion of the election.

Similar arithmetic, reaching similar conclusions, can be done (and I did it) for Clarke-Tideman-Tullock voting. The conclusion is that these kinds of monetized voting systems are best used in smaller elections and are inherently stupid for something like USA presidential election.

Statehouse race: In my state (New York), a state assemblyman race typically involves 40000 voters, with incumbents usually winning by large margins (often running unopposed) due to massive gerrymandering, which makes the outcomes about 98% predictable a year in advance. This in turn means the chance your vote can swing the election is about 1/10000. There are 150 assemblymen. A typical voter in one such race might consider the race's outcome worth $10. From that we compute that κ≈1/100000 with money measured in dollars. Therefore typical expected profits and payments (with honest dollar evaluations used as votes) to a typical voter would be of order 0.0005 cents. Monetized voting here is even stupider.

Town highway supervisor race: My town has a historically high level of corruption, and the highway supervisor controls a large budget, about $70 million. The highway supervisor race typically involves 23000 voters, which is about 8% turnout (300000 registered voters). Each might honestly assess the race result as worth about $23 (i.e. $7 million swing, which assuming 300000 registered voters is $23 per voter) although usually the assessment is highly noisy since most voters do not know much about highway supervisor candidates. This race is actually competitive fairly often, so let's optimistically say the chance one vote could alter the election is 1/230 (even though actually a tied or near-tied race, such that one vote could alter it, has never happened in the history of the town, just like no any individual vote in any US house, senate, or president race has ever affected the outcome in all history up to 2013). This would suggest κ≈0.00019 with money measured in dollars. In that case using the quadratic system, the voter would expect to profit (κ/2)D2 thanks to her $23 vote, which with our numbers is 5 cents. She also would need to pay 5 cents to be allowed to cast this vote. In this election, monetized voting makes more sense than the preceding two examples, but it still is pretty absurd.

A corrupt mafia contractor with $1 million at stake in this election would (by honestly voting that amount) have the same power as 43478 of our $23-each voters (but remember there are actually only 23000 of them) even if 100% of them were voting against him! However, he would be charged $95000000, i.e. 95 times $1000000, to cast this vote (which remember, was for him an honest vote) so the plan of honestly voting would not make any sense for him. Also, the whole point of the method – encouraging honest dollar assessments as votes – here has failed. How can this be? The reason is that the "uniform distribution" approximate probability model underlying quadratic voting has broken down. This failure of the underlying mathematical justification, however, has the good effect in the present case of stopping the mafia don from singlehandedly swinging the election.

But being mafia, plus also not an idiot, he instead probably would arrange for, say, 50 henchmen to vote D=$1000 each instead, thus buying the same effect as a unified bloc of 2174 typical $23 voters. That would almost certainly be enough to throw the election in a competitive race, which would seem a big indictment of quadratic monetized voting – it made the election more corrupt, more complicated, and worse than a plain moneyless simple majority vote. Each of these henchmen would be charged (κ/2)D2=$95 to vote, costing the mafia don a mere $4750 in all. (And more realistically our mafia don probably could get 500 voters [not merely 50] to each vote $100 with the same election swinging effect as the 50 at 1/10 the cost to him, or he could get a bit more certainty by having them each vote $200...)

Most of the judgeship races I am offerred the opportunity to vote in might behave similarly to the Highway Supervisor example.

So in summary, quadratic money voting would be worse than simple majority vote in essentially all the kinds of elections I currently vote in. Hence it is not a good idea for governmental elections in general.

Comparatively small elections (by USA government standards), with unusually large amounts of money at stake, would seem the most hopeful ones for both the present and Clarke-Tideman-Tullock monetized systems, but even then they are highly dubious. This in turn suggests that corporate, not governmental, votes might be the best place for either kind of monetized system, plus that would have the advantage of being constitutional (and similar conclusions were reached for Clarke-Tideman-Tullock voting).

7. The law

Monetized voting systems appear to be unconstitutional in the USA under the 14th and 24th amendments (about equal protection and poll tax).

8. Huge motivation to cheat

A "fat cat" rich voter (with κ=1 and the quadratic system for simplicity) would be foolish to spend $1000000 to buy a mere 1000 votes, when he could instead purchase 500× more votes for the same money by paying 1000000 voters $2 each for them each to make a $1 vote (they keep the other $1).

In theory this would be prevented by the "secret ballot" and/or prosecution risks, but in reality those have been circumvented many times in massive manners in the past (contrary to Weyl's statement in his §4.4 that "de-merger into two individuals... is probably all that is feasible in most cases"), and with modern technology such as cell phones that send real time video, might be easily defeated today in new ways. The motivation might increase with monetized voting systems, and if so the corruption might increase, which actually might make things worse.

9. Comparison with present-day (2013) major USA elections

It can be argued that USA elections today already work like Goeree/Zhang and Weyl's "quadratic vote buying" (which is the C=2 special case of our "quadratic money voting"). First their QVB systems were only for binary elections – USA elections right now almost always effectively involve no third choices... check.

People right now attempt to "buy" votes using political dollar contributions. If you donate N dollars, that effectively buys some function f(N) votes, and the plot of f(N) looks qualitatively similar to the plot of √N. I.e. if a typical USA voter hears some putrid political commercial 10 times, I doubt that influences her 10 times as much as hearing it once. I think the factor is considerably below 10. So the plot of f(N) has a concave-down character resembling squareroot. And just hearing one initial ad can have a lot of effect, especially if it contains new information, i.e. initial money seems to have a much larger, even huge, effect, just like the squareroot function has a vertical asymptote at 0.

In QVB rich voters are motivated to cheat by voting through paid proxies. Similarly in the USA today, the Koch brothers and similar enormous political donors continually try to hide their monetary donations through a multiplicity of front groups, with disguised sources, "astroturfed" fake-grassroots organizations, and money-laundering groups like "Crossroads GPS."

So the two systems – QVB and present-day USA elections – actually are very similar. Enjoy "optimality."

10. Utility≠Money; and "Shannonizing" to repair that problem

Another issue is the (experimentally massively supported) fact that "utility" and "money" are not the same. Claude Shannon once proposed that a better approximation was utility=log(wealth), but this clearly still is not quite right – a still better approximation might be that utility=τ1log(debts)+τ2log(assets)+τ3log(|assets|+|debts|), where τ1, τ2, τ3 are suitable positive constants. But that still is not right, e.g. omits knowledge of cliff-edge effects when your wealth falls below survival threshold. And further, utility also depends on things other than money.

Anyhow, if we just use Shannon's simple approximation, now with the correct goal of making the election-winner maximize summed-utility instead of summed money, then we would (to good approximation) obtain the monetized voting systems above except that

  1. "The election winner" would no longer be "the candidate with the greatest summed score" but rather "the one with the greatest weighted-sum of scores" where voter v's ballot is weighted by (Wv)-1 where Wv is voter v's wealth.
  2. Thus the "constant" κ would become wealth-dependent (proportional to 1/W for voter with wealth=W).

Unfortunately, if this were done, then voters would need to state their wealth on their ballots, causing "voting" to become "as complicated as doing your taxes," ruining ballot privacy, plus subjecting us to new kinds of voting fraud... requiring detection and prosecution... which all would be a nightmare. That all would be totally unrealistic and unacceptable in most practical applications.

Fortunately, we can achieve something similar without requiring voters to state their wealth, as follows. Assume/approximate a voter's "wealth" as proportional to the max-min candidate dollar difference that voter states on her ballot. If so, then the ballot would (via the 1/W weighting) effectively automatically become rescaled in such a way that every ballot had the same max-min score difference.

Furthermore, given that those modifications were made, it then would become strategically foolish for any voter to "admit" they were wealthy by scoring candidates with huge dollar amounts – they'd get the same voting power by "pretending to be poor," but pay less. Farewell to inherently incentivized voter honesty.

Thus, "Shannonizing" our voting systems would cause them in yet another sense to "self-invalidate." Given that every voter would be motivated to hide their wealth in a Shannonized monetized system designed to incentivize the election of the max-summed-Shannon-utility candidate, we might as well force all ballots to live within a restricted score range and use a price function M(z) not depending on voter-wealth.

Thus the net effect of "Shannonizing" to repair the utility≠money flaw, and then repairing the resulting self-invalidation effect of wealth-hiding voters, would effectively bring us back to something like plain (no money) range voting or the procrustean system (even if the original system had unbounded score range)!!

We now have come full circle: the improvement of plain range voting by monetization has, after still further improvements, led us back to plain range voting (or perhaps the procrustean money-voting system)! But the exercise has been educational. For example, those economists who wanted monetized voting schemes and disparaged plain range voting, now might want to reconsider.

11. Conclusion

All the monetized schemes in this paper based on the max-believed-expected-profit theorem, would work badly in USA governmental elections (even ignoring unconstitutionality). There might be a valid niche for some corporate votes.

We recommend plain range voting, or if we insist on a monetized system then use the procrustean one with κ and K chosen not to motivate honest dollar assessments as ballot-scores via the max-believed-expected-profit theorem, but rather with the goal of discouraging strategically-exaggerated/distorted range voting (e.g. approval-style voting).

The quadratic, tanh, and erf-based monetized systems all invalidate themselves and/or do so when we attempt to use Shannon's utility≈log(wealth) approximation to repair the money≠utility flaw in the original logic. As a result, the procrustean monetization method is the only one (from our original table) which remains standing.

12. Tragicomic previous work

This all started for me when the popular press article Posner 2013 was brought to my attention. Although ostensibly about a public bicycle rental scheme in New York City, it contained the following remarkable quote

Recently, however, an economist at the University of Chicago named Glen Weyl has developed an ingenious new mechanism that is simpler and more robust, and could help a city decide whether to introduce bike sharing. He calls it Quadratic Vote Buying...

and then the QVB scheme is described and discussed. The scheme Weyl proposes is equivalent to our "quadratic money voting" if we restrict attention to the special case C=2. As Weyl described it, his system was unusable if C>2.

However, it appears the Posner quote was false, invalidating the main claim in his article. Specifically, the inventor was not Weyl 2013, but rather Goeree & Zhang 2012. I am saying this since Weyl cites G&Z, but G&Z do not cite Weyl; also G&Z was dated earlier by about 1 year. (See also below about Hylland & Zeckhauser 1980.)

Goeree & Zhang (also) defined the QVB system (and also demanded C=2, and also had no proposal for handling utility≠money), and allegedly proved two theorems ("propositions") about it, and organized an pseudo-election experiment with 250 human voters to validate the concept.

Unfortunately, both of Goeree and Zhang's theorems are wrong, as we shall demonstrate by counterexample to their Proposition 1, and by finding at least 2 errors in their "proof" of Proposition 2. Further, we'll explain why their experiment is much less meaningful (or different-meaning) than the naive reader might imagine.

I should point out that the Goeree & Zhang (as well as Weyl) used undefined terms galore in their theorem statements, as well as probably many unstated assumptions – both of which are absolutely unacceptable behavior. If you do that, then you do not have a "theorem." But as a result, perhaps some G+Z defender could argue that really their theorems are rescue-able. This is true in the sense that we have demonstrated such things already here, and now in a rigorous fashion! But here is my contention. If I produce a counterexample to your "theorem" or "proof," then you say "with extra assumptions etc, the counterexample will no longer work," then I produce another counterexample to the repaired theorem, then you add still more assumptions and wording changes, etc, etc (this is in fact exactly what was happening in my emails with Weyl), then that is not "science," but rather "a waste of time." I am quite angry about all this behavior by G&Z and by Weyl. "Science" is falsifiable. A "scientist" makes well-defined and falsifiable statements (except that "theorems" ought never to be falsifiable!). They didn't. Given that all theorems/proofs in their article are wrong, I regard G&Z's paper as an embarrassment. However, nevertheless it has value since they did invent an interesting and at least somewhat meritorious main idea (albeit largely obsoleted by the present discussion). So they had a good idea – the problem was their execution was atrocious.

Weyl's 2013 paper may be even worse. It was 57 pages long. Near the start was the rather remarkable quote "many of the results are conjectural" accompanied by the claim that hopefully proofs of everything will be added later within 6 months or so (presumably increasing the length to about 100 pages?). I've never seen that behavior in any paper before. This is not to say that Weyl's paper lacks mathematics; on the contrary it is packed full of it. Unfortunately as we shall explain there is a foundational error on page 7 which invalidates essentially the entire paper and renders most of those mathematical manipulations and claims irrelevant. When I pointed this out to Weyl by email, he then made various (all false and sometimes self-contradictory) claims about why the error was not really an error, could be repaired, was inconsequential, etc. As far as I can tell (mid-June 2013) he still refuses to admit it is anything more than a minor typo. But I still claim he was wrong. This is a major foundational error and would force virtually the entire paper to be rewritten and redone quite differently to have any notable claim for relevance and correctness.

Another remarkable quote early in Weyl's paper was:

While I show that QVB is in some sense uniquely limit-efficient, my goal is not, as in many mechanism design papers, to argue that QVB is exactly optimal or perfectly robust under a well-defined but relatively narrow set of assumptions or criteria. Instead it is to persuade you that, unlike other proposals economists have made (Subsection 6.6), QVB has the robustness and simplicity necessary to be a practical alternative to voting that is superior in many settings.

While I agree with the goal (and non-goal) expressed in that quote, the fact is that vast bulk of the immense amount of argumentation in Weyl's paper was about trying "to argue that QVB is exactly optimal or perfectly robust under a well-defined but relatively narrow set of assumptions or criteria."

But it gets worse. I also pointed out the money-discreteness and "tiny utilities dwarfed by the noise" (and constitutional/legal issues) to Weyl – and the fact that real voters are nowhere near to being "rational economic animals" – he claimed those were nonsense and some secret analysis he had that he'd left out of his 57-page paper (which never mentions any of those issues) shows that – plus human voters merely "overestimate their pivotality density" and aside from that are rational economic animals (which is garbage refuted by overwhelming experimental evidence from range and Australian instant runoff voters).

In summary, every scientific paper and popular press description I found about QVB was massively wrong and misleading. So one reason I wrote the present page was so that I could actually say something correct. Another was that I have extended their ideas considerably and reached different conclusions.

The gory details of the massacre: We begin with Posner. A key ingredient of journalism ethics is "full disclosure" of conflicts of interest, defined as

The disclosure of any connection between a reporter (or publisher) and the subject of an article that may bias the article.

As the Society of Professional Journalists Code of Ethics (2013) puts it, journalists should "avoid conflicts of interest, real or perceived" and "disclose unavoidable conflicts." The naive reader of Posner's Slate piece would have thought Posner was just a reporter describing a great new politico-economic scientific invention by (the totally unconnected person) Weyl. But actually, Posner was not "just a reporter." He was a law professor at University of Chicago, the same institution Weyl worked for, i.e. was a faculty colleague of his. And not only that, he was a very close colleague. Posner had in fact coauthored at least 7 prior papers, op-eds, etc with Weyl, plus was working with him on at least one paper about this very quadratic vote buying topic. This is approximately the most extreme conflict of interest imaginable.

(I, in fact, was such a naively-trusting reader for about a week before I caught on, and despite having email exchanges with Weyl during this period, in which he never mentioned his relationship with Posner.) As two random examples of how to behave unethically:

  1. Glenn Beck repeatedly used his one-hour weekday Fox News forum to encourage viewers to buy gold (even noting that, during one of his broadcasts, the price of gold rose $50/oz). If Beck's had been independent thought and research, then Beck would have been acting as an honest ethical journalist. But Beck happened to be a paid spokesman for the gold vendor Goldline International, a fact he rarely if ever disclosed while doing this.
  2. Economist John R. Lott Jr. (also once employed by Univ. of Chicago, same department as Weyl), author of the interesting albeit controversial book More guns less crime, was often praised on the internet by "Mary Rosh," a former student of Lott's. E.g. Rosh wrote one review of Lott's book. She not only praised Lott's academic writing, but also called him "the best professor I ever had." The only problem was that Mary Rosh did not exist, and was a fake persona created by Lott himself.

As two random examples of how to behave ethically:

  1. Joan Connell, writing an article "Alterman's arrest" for The Nation (18 June 2007) about the arrest of fellow journalist Eric Alterman at a political debate in New Hampshire, stated early in the piece "(full disclosure: I was [Alterman]'s editor at from 1997 to 2003)" and in its first line described him as "Nation columnist Eric Alterman."
  2. Economist and NY Times columnist Paul Krugman, writing in NYT (24 Jan. 2001) about Energy Deregulation, wrote in paragraph #3 "Full disclosure: Before this newspaper's conflict-of-interest rules required me to resign, I served on an Enron advisory board that turns out to have been a hatchery for future Bush administration officials. (What was I doing there? Beats me.) I can't say that I got to know Mr. Lay well, but I presume that he is an honorable man..." But Andrew Sullivan (also a prominent journalist) attacked Krugman's disclosure as ethically inadequate because it did not reveal how much Enron had paid him, which (Sullivan claimed) was $50000. Krugman later defended himself by noting that since his standard fee for giving a lunch lecture was $20000 plus travel and boarding expenses, his payments from Enron were small by comparison. (For comparison, the median income among full-time-employed US women during 2001 was $29215.)

You can decide for yourself where on this continuum Posner's behavior fell. My personal judgment is he should have stated all those facts about his relationship with Weyl in paragraph #1, and anything less was unethical; and the icing on the cake was that Posner's article mentioned neither "Goeree" nor "Zhang," who appear to be the real inventors of the QVB concept, contrary to the quote I repeated above. I have no objection to Posner writing puff pieces praising the economic work of himself and his friends; but say so up front. I object to such pieces disguised as independent reporting about bicycle rental schemes.

We now move on to Goeree & Zhang. The good news is, G&Z's paper contains two theorems. The bad news is, each theorem uses undefined terms, I have a counterexample to theorem 1, and I have found at least two reasons the proof of theorem 2 is wrong.

G&Z's Proposition 1: "For any symmetric value distribution, the fraction of the total surplus realized by majority voting falls from 1 when the size of the electorate is one to limn→∞ Wvoting/Woptimal = E(|v|)/√E(v2) which is less than 1 (for non-degenerate distributions) by Jensen's inequality. The total surplus loss Wvoting-Woptimal diverges in the limit."

Problems: What is "symmetric"? What is "value distribution"? What is "total surplus"? What is "Woptimal"? What is "non-degenerate"? They never define any of these. There is a reason theorems only should contain defined words and explicitly stated assumptions. What if I produce a counterexample to their theorem? Will they then say "wasn't really a counterexample since (moving definitions and unstated assumptions)"? That would not be science. "Science" is falsifiable.

But anyhow, one counterexample to theorem 1 is: N voters. Throw N dice and flip 1 coin (all independent, identically distributed, and fair). If coin=heads, then every voter has value in {1,2,3,4,5,6} where the Mth voter's value arises from dice-roll M. (Here positive values mean "pro" vote.) If coin=tails, then instead every voter has value in {-1,-2,...,-6} similarly. This is a symmetric and nondegenerate value distribution. In the limit, the majority vote will 100% of time elect an outcome favored by all. Hence the "less than 1" conclusion of theorem is false. Q.E.D.

Next, we move on to their only other

"Proposition 2. The bidding mechanism is budget balanced and individually rational. For large electorates, truthful bidding constitutes a Bayes-Nash equilibrium and limn→∞ Wbidding/Woptimal=1. Furthermore, voters of (almost) all types are better off under bidding compared to voting."

Problems: what are "individually rational," "Bayes-Nash equilibrium," and "almost" and "types"? Again these are not defined in the paper. Again: Using undefined terms in theorem statements is simply unacceptable behavior. But this proposition actually has a proof provided, unlike prop. 1! Hooray. Unfortunately the proof is bogus:

QUOTE FROM PROOF: Since G(0)=1/2 [probability of election outcome is 50-50 if voter buys zero votes] voter i's payoff when she bids zero is equal to the rebate, which is non-negative, so the bidding mechanism is individually rational. END QUOTE.

No. First of all, G(0)=1/2 is not stated as an assumption plus seems false, i.e. is contradicted by 96% of major USA elections. Second, even if we assume it, then I do not understand why the fact that a voter's "payoff is non-negative if voter bids zero (i.e. does not vote)" logically forces "the bidding mechanism is individually rational. "

Huh?? I mean, essentially every bidding mechanism ever conceived of, no matter how rational/not, satisfies that. So what?

Their next sentence is "Moreover, for a large electorate, the central limit theorem implies that..." which is also wrong in the sense the central limit theorem is not applicable at all unless further (never-stated) assumptions are true. I have no idea whether they knew that, or what – but I will say this: it is unacceptable to leave major assumptions out of the theorem statement.

So: both theorems in this Goeree-Zhang paper are (1) refuted, and/or (2) unproven and in both cases the theorem statement itself is simply unacceptable.

Having dispatched the theory in this paper, we now enquire about their experiment. In this experiment, they set up artificial binary election scenarios involving pre-determined and pre-designed money payoffs to 250 real voters (depending on the election result; 20 preliminary followed by 20 further elections). The scenario was set up so that the voters, in net, would lose money if they employed naive simple majority vote. However by using quadratic vote buying the voters with larger amounts to gain could express that fact in their ballots and thus could hope to overcome the "tyranny of the majority." By means of preliminary elections the voters got used to using the bidding mechanism and apparently learned, to at least some extent, to exploit it. This worked: G&Z claim that with 99% statistical confidence, QVB delivered more money to the voters than simple majority voting.

I.e. that experimentally proved the system works.

The problem with that: The most obvious presumed or suspected flaw in QVB involves what happens when the voters are highly financially unequal. I.e. about 1.4 million USA citizens went bankrupt in the year 2012, and media estimates of the number of homeless ones ranged from about 300,000 to 3 million. Meanwhile other USA citizens are multibillionaires, e.g. Bill Gates, believed as of 2013 to be the richest person in the world with $73 billion. In contrast, G&Z's 250 experimental subjects all were near equal in life status (students at University of Zürich and ETH) and I doubt included even a single homeless person or multimillionaire (anyhow if they did, their paper did not say so). G&Z artificially restricted all vote-scores to lie within a set of 10 specified dollar values, with all others forbidden.

In the event all voters are thus made near-equal, then instead what the experiment addresses is not really the QVB system, but rather something more resembling our "procrustean system." While I'm happy they've experimentally validated my (probably improved) idea, the problem was the lack of verification of their idea.

Finally, we consider the 57-page tome by Weyl. On page 7 it says

QUOTE: the normal distribution (to which the sum of the values of all but one individual converges by the central limit theorem) is well-approximated about its peak by a uniform distribution when the number of individuals, and thus the variance of the distribution, grows large. If such a linear-in-value equilibrium exists for one such quadratic payment scheme, it should exist for any, as the multiplier applied to value to generate votes may be "undone" by individuals rescaling the number of votes they purchase linearly. Thus one should expect an asymptotically linear and thus efficient equilibrium to exist. END QUOTE.

Weyl's cavalier 1-sentence claim about "the central limit theorem" (CLT) is (1) utterly wrong and unjustified, and (2) foundational for his paper (a huge amount, certainly over half, of the paper rests on this 1 sentence).

A counterexample is if the "value distribution" were that the Jth voter has value ±1/J with the ± signs arising from independent fair coin tosses. The random variable ∑1≤J≤N±J-1 is not normally distributed including in the limit where the number N of individuals (summands) tends to infinity. The usual central limit theorem requires summands to be identically and independently distributed. Of course in the real world of voters, it is absolutely absurd to pretend the voters are identical (in fact Weyl's entire point is that they are not) and also it seems rather absurd to claim independence, for example the votes/opinions of husbands and wives seem probably dependently distributed, and if any voter sees polls about the other voters, she is likely to react (another kind of dependence) probably in a manner which destroys normality. Of course, as Weyl pointed out to me in email, there are fancier kinds of central limit theorems, but that is irrelevant because the counterexample I just gave involves a limit distribution that is not normal (which, incidentally, was studied by Rice 1973); hence it applies to every possible kind of CLT, that ever existed or ever can exist. (Also, while I used the power "-1" for simplicity in the counterexample, any power <–½ would work.)

Is this kind of counterexample a mere "mathematician's monster" or is it plausibly economically/politically realistic? The latter. "Power law" like wealth distributions in which poor citizens are much more common that rich ones, and where, e.g. the "top 20% richest own 80% of the total wealth in a country" are often seen. (Other numbers such as 90-10, and 95-5 have been claimed in different countries or different times; in the 2011 USA the top 20% were estimated to own 84%. The 2001 Survey of Consumer Finances found the top 1%, next 6.5%, and next 42.5% of USA households each owned about an equal share – about 32% – of all the wealth; these equalities would be consistent with the hypothesis that the Jth wealthiest household's wealth is proportional to J-0.63 albeit it is inconsistent with the bottom 50% owning 4% of the wealth.) And indeed a real live economic model predicting this kind of power law behavior was published by Bouchaud & Mezard 2000. In addition to situations like my counterexample being inside published economic models, we now shall show it also resembles actual reality.

The "Forbes 400" list of the 400 richest Americans, year 2012. (The top 400 super-richest continued to grow in wealth and pull away from the rest of America and the merely rich, with only 2 members of the 400 losing wealth versus in 2011. Much of their gains were driven by the top three – Gates, Buffett, and L.Ellison – alone.) Names with asterisks (*) are rich primarily because they inherited it (not that this matters for the present purposes). The formula in the final column fits the rank to within a factor of 1.64, or equivalently the Jth-richest person in 2012 USA had wealth W≈$9.5×1010J-0.73 accurate to within a factor of 1.44 for 1≤J≤400. If this were extrapolated to J=3×108 it would predict W≈$61578 which is certainly an overestimate (32.5% of USA families had net worth ≤zero in the year 2010 according to Federal Reserve survey data, and one-third of USA citizens do not pay their bills on time according to a 2012 National Foundation for Credit Counseling survey) suggesting the truest power is not -0.73, but closer to -1 exactly as in my counterexample.
RankNameNet Worth $W (9.5×1010/W)1.37
1Bill Gates66 billion1.64
2Warren Buffett46 billion2.70
4Charles Koch*31 billion4.64
8Alice Walton*26.3 billion5.81
16Forest Mars Jr.*17 billion10.6
32Anne Cox Chambers*10.7 billion19.9
64"Richard LeFrak & family"*5.2 billion53.5
128Ron Burkle3.1 billion109
256Daniel Gilbert1.9 billion213
512?about 700 million?835
8192?about 100 million?12008

What we are really interested in for our present purposes is more like "disposable wealth" or "politically-affected wealth" than "wealth" – e.g. "how much would you be willing to spend to buy votes?" and "how much will political election results affect your wealth?" – which are not quite what was modeled nor have they ever been measured. I am highly certain that a goodly fraction of registered voters (if polled on this question) would say they were not willing to pay any money to buy votes (the majority of USA citizens do not vote now even for free), in which case the contention (which Weyl made in email to me) that these amounts were bounded below by a positive constant, is refuted. Weyl also contended the Bouchard-Mezard model "had no variance" which meant it was absurd (I think they'd disagree...) and also contended he was an expert on wealth distribution and that the real wealth distribution is nowhere near posing any problem for CLT (refuted by above data) etc. Indeed Weyl then had a long list of other bogus arguments he sent me one by one in emails, which I refuted by return mail each time only to receive a new one (some contradicted previous ones, too) – but I'll spare the reader. I told Weyl he needed to discard dependence on the CLT entirely and outlined how to do so, but he vehemently denied this, which is why I did it here. Normality perhaps could be justified experimentally after long experience but seems beyond hope of mathematical proof under realistic assumptions, or anyhow if there can be such a proof, Weyl's 57-page paper clearly had no idea whatever how to produce it. I also have no idea how to produce it (although I do know how to produce more counterexamples to various attempts).

13. Honesty-Inspiring Economic "mechanisms" more generally

More generally, economists have been busily awarding each other Nobel prizes for decades for inventing Honesty-Inspiring Economic/Financial "Mechanisms." One of the earliest and most famous was the Vickrey Second-Price Auction, published in 1961.

But, oddly enough, in spite of the praise economists have heaped upon each other for these inventions, the rest of the world has been remarkably uninterested in actually using them. For example, the Vickrey auction has rarely been used in spite of seeming one of the simplest and most-practical of them all, and in spite of the fact that auctions happen every day, and in spite of it being over 50 years old and awarded a Nobel prize. (Indeed the main users, to the limited extent Vickrey auctions have been used up to 2013, seem to be in the internet/tech community, not the financial community, despite the latter having a much greater number of economics degrees.) In many cases, these mechanisms suffer severe problems in the real world which the economists awarding the Nobels mainly studiously ignored [Rothkopf 2007]. (Incidentally, the computer scientists who figured out how to satisfy the Vickrey auction's anti-fraud, verification and secrecy demands via advanced cryptography in my opinion did far more difficult work than Vickrey, indeed I doubt Vickrey even could have understood it, but got no Nobel since the economists did not regard them as "one of us.")

The present voting-with-money example (as well as the older Clarke-Tideman-Tullock voting which had many of the same problems) is yet another episode (worse than usual) of that recurring bad soap opera. It might be interesting to review the mechanisms area to see which ones really do have practical interest, why the failures failed, and what lessons we should draw from this.

One possible idea would be to monetize, not highest-average-wins range voting but rather the "greatest median score wins" system. As a concrete proposal to consider (I am not saying I advocate this):

  1. Each voter as her ballot assigns a non-negative dollar amount to each candidate X ("how much it is worth to me for X to win election").
  2. The candidate W with the greatest median value wins.
  3. All voters who rated W above median, now pay some amount. One payment scheme is: the amount is W's median value. An alternative payment scheme which presumably would obtain more revenue is: each such voter pays the amount bid by the voter immediately below her's in sorted-by-W-bid increasing order.

This scheme has the disadvantage that it elects the "wrong" person (economists presumably would contend the candidate with the greatest summed money value, not the greatest median, should win). Also, it still would seem to succumb to the apartheid example.

However, on the plus side, it would appear to incentivize honest dollar-assessments as votes, which might overall outperform a "better" voting system contaminated by many dishonest votes (or we could at least hope this). Also, the payment amounts involved would be far larger than the ridiculously small ones arising in our quadratic-money-voting examples, now causing an actual relation to reality and real incentives for voter honesty. The reason this is possible is this scheme does not fall into the straightjacket of the systems we described in the Abstract, i.e. the price you pay does not depend solely on your ballot and who wins, but also depends on the behaviors of the other voters.

But unfortunately, the whole appearance that this method "incentivizes honest voting" is false in this sense: a voter could argue "my chance of affecting the election outcome is small, but my expected payment (with honest voting) is large, therefore, a better strategy for me is not to vote at all." This failed design experience suggests the following (admittedly vaguely worded)

IMPOSSIBILITY CONJECTURE FOR MONETIZATION: There is no reasonable way to monetize any reasonable voting method for large elections, in such a way as to incentivize honest voting more than every other kind, and/or to ensure election of the optimal winner if all voters behave economically rationally.

PROOF SKETCH: In any reasonable election method with a large number of voters, any single voter's chance of affecting the outcome, is tiny. That implies that (for almost all voters) her expected profit from voting optimally (versus not voting at all, or voting sub-optimally) is tiny. That implies that either

  1. The monetization must impose profit/loss |amount|s far larger than the expected utilities for voting,
  2. Or they must be comparable or smaller.

If (b) then it is economically irrational for typical voters to vote at all, the profit motive will be swamped by other economic and non-economic effects, etc. If (a) with large possible losses then voting becomes financially risky and poor voters will be motivated to avoid that risk by not voting. Also if the whole idea of monetization is to encourage voter honesty, then there always is a trade-off faced by voters of being strategic to increase election-altering chances and thus get more expected profit, versus costs imposed by the monetized voting system which counteract that. It appears inherently impossible to make optimizing this trade-off always incentivize honest voting, unless the imposed costs are of the same order of magnitude as the voter's expected profit without monetization.

That was, of course, nowhere near being a genuine proof and the words "reasonable" and "honest" were not defined (and for a fully general voting system, seem hard to define). Nevertheless this obstacle seems difficult to overcome.

A possible way to solidify this impossibility conjecture would be to argue that for any voting system based on a ratings-style ballot and involving a price function of that ballot alone, then (within reasonable probability models) systems like the ones discussed in the present paper are mathematically forced; and then finally prove they then must always encounter at least one of the problems we've discussed.

Another idea was A.Casella et al's storable votes. This would probably be a logistical nightmare if used over long time periods involving many elections, plus seems to be fixing something that is not broken. Also both this and the median-based proposal might fall victim to initiatives worded basically to try to bribe a minority to vote some special interest some advantage. (I thank Jameson Quinn, Tracy Goodwin, and A.u-R. Lomax for these points.)

But anyhow, in multiwinner elections (which are common in many countries, although not the USA) one could imagine "storable votes" of some ilk as pseudo-money just usable in that one (multi)election. Every voter could be given the same amount of pseudo-money at the start of the election (since it would not be real money, rich voters would have no advantage) and then honesty-encouraging "mechanisms" could be used. In this way, objections about paying real money to vote would vanish, and so would objections about the "logistical nightmare" of storing votes between elections. And indeed, an extremely practical multiwinner voting system of the resulting kind was already proposed (by me, among others): "asset voting."

14. Late-breaking news about earlier works (this section added Oct. 2014)

It turns out the quadratic vote buying idea was due neither to Weyl 2013 nor to Goeree & Zhang 2012... in the sense that it had been described in an unpublished, but nevertheless fairly well known, paper by Hylland & Zeckhauser 1980! (Weyl was born in 1985.) I thank Nicolaus Tideman for pointing this out to me; these were not cited by Goeree/Zhang.

Also, I note that in a 2010 post to the Center for Election Science, I already had put forward the possible idea of vote-buying using "pseudo-money" whose "value" would arise from its usefulness for future voting. That is, in a multiwinner election scheme, your vote would cost you some amount of pseudo-money, and the more it cost, the less you would have left to affect further winners. This idea has the potential to overcome several of the objections to the vote-with-money schemes we've discussed here vis-a-vis single-winner elections only.

15. References

Scott Bittle & Jonathan Rochkind: Voters' Experiences in 2008 and the Future of Engagement, Public Agenda Report, 15 January 2009.

Jean-Philippe Bouchaud & Marc Mezard: Wealth condensation in a simple model of economy, Physica A 282 (2000) 536-545.

Alessandra Casella: Storable Votes: Protecting the Minority Voice, Oxford University Press 2011.

Jacob K. Goeree & Jingjing Zhang: Electoral Engineering, one man, one bid, there are at least three versions of this paper on the internet, dated 9 July 2012, 27 August 2012, and 18 March 2013. (I'm told this paper was presented at some conference, but if so I do not know the details.)

Aanund Hylland & Richard Zeckhauser: A mechanism for selecting public goods when preferences must be elicited, unpublished paper, Harvard, 1980. A 1979 version of this paper was discussed in §8.3 of Dennis C. Mueller: Public choice III, Cambridge University Press 2003, as well as at the web page dated "1996-2013" written by Robert Loring.

J. Morgan Kousser: The shaping of Southern politics: suffrage restriction and the establishment of the one-party South, 1880-1910, Yale University Press, New Haven 1974.

Eric A. Posner: The Good Way to Buy Votes, Did New Yorkers really want bike sharing? The city should have tried Quadratic Vote Buying to figure that out. Slate, 5 June 2013.

Stephen O. Rice: Distribution of ∑an/n, an randomly equal to ±1, Bell System Tech. J. 52,7 (Sept. 1973) 1097-1103.

Michael H. Rothkopf: Thirteen Reasons Why the Vickrey-Clarke-Groves Process Is Not Practical, Operations Research 55,2 (March-April 2007) 191-197.

E. Glen Weyl: Quadratic Vote Buying, Social Science Research Network paper 2003531, dated 1 April 2013.

More bilge from Weyl: he publishes false claims honeybees use quadratic voting

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