E.Glenn Weyl: Quadratic Vote Buying
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2003531
It begins by warning "many of the results are conjectural" and
hopefully proofs will be added later. (Strange.)
The paper is poorly written (e.g.
theorems involving undefined terms), intricate with
a lot of buzzwords, journalese, and random-looking math, and very long (57 pages)
making it hard to provide any rapid verdict. And I guess if and when he does add the
missing proofs it'd then be 100 pages long?
(Nevertheless I will find a flaw that invalidates
his entire paper as currently written. But I do not pretend to have completely
digested the paper.)
Weyl's method is only intended for "binary" decisions, not 3-or-more choice elections.
That's a major handicap.
Weyl's method:
Each voter pays N^2 dollars for N votes. (N varies with the voter.)
Side with most votes wins. After election over, all monies
are divided evenly among all voters so in net none is lost, although there is redistribution.
The number N of votes can be any real>=0, it does not need to be an integer.
Weyl near start says he does not intend to convince us his method is "optimal" but rather
robust and simple and practical.
The notion of optimality hinted in the abstract is this. If you have bought N votes,
but want to buy N+1 instead, you need to pay 2*N+1 extra dollars.
You will consider this worth it if this 2*N+1 cost is less than the money benefit for you if
your side wins election. In which case you do it. Hence eventually we reach an "equilibrium" state
where everybody has paid nearly that much, and say it then is nearly balanced election --
then you ask "what if I just bought 1 more vote and that tipped it?"
And you do so if 2*N+1<benefit, and hence this final move involves each voter putting
in a number of dollars proportional to her benefit (given that we ignore
the "+1" as comparatively tiny), and hence the most-benefitted side
wins, which is optimal outcome, QED.
Well, that reasoning seems a crock... but a funny thing happened when I wrote down
the explanation of why it was a crock:
1. I've paid $1000000 to buy 1000 votes and am willing to now pay an extra $2001 to
buy 1 extra vote because the election win is worth $2003 to me.
Bullshit. In reality, I'd never pay the $1000000 in the first place.
2. Linear, not quadratic, vote buying under his normal vote-total assumption
means my chance of altering election is proportional to my #votes is proportional to my $ spent.
That not quadratic seems to cause "optimality"... except say my chance of altering
election is c*N if I spend N dollars, where c=0.00000001. Under those circumstances, is
it worth me spending ANY amount of dollars? If I spend X dollars and
the election is worth $1 to me (if win) then I get c*X expected benefit at cost X,
so no, spending anything at all was not worth it.
3. Aha, but now we see Weyl's point!! Say my chance of altering election with
his quadratic system is c*squareroot(N) if I spend N dollars. This IS worth me
spending money on, because of the vertical asymptote of the squareroot function at zero.
My expected benefit (if an election-win is worth $U to me) is
U*c*sqrt(N) versus cost=N, and the former is larger if N is small enough
(for any c>0 no matter how tiny). Solving c*U*sqrt(N)=N for the point where cost=benefit
we find that N=(c*U)^2 is the max amount of money I should spend voting.
It sure would be nice if money were real numbers, not discrete, because in reality it is discrete
(1 cent is minimum spendable amount) and that in view of the tinyness of c and the fact U
for many people is small, probably invalidates the whole idea right there since most people
are being asked to spend, say 0.00123 cents. (I did not notice Weyl mentioning this
whole discreteness issue.) But hey, let's improve the world to allow
continuum money to solve that, and continue on under that assumption. [Or solve this
by demanding K*N^2 dollars for N votes, where K is some LARGE preselected constant...
except then you may run into other problem almost nobody wants to vote.]
If everybody does that, then the number of votes on each side will be proportional
to the sum of the U-values on that side (i.e. utility sum) and hello -- optimality: the
max utility side wins the election.
[Course, there also is some redistribution of money, but it tends to be from richer to poorer
which is utilty-sum-increasing if utility is a concave-down increasing function of money.]
So that is the deeper explanation of why Weyl's system is "optimal" in a balanced
election situation with a normality-assumption.
Now what if we had an UNbalanced election where it was clear which side was going to win.
The probability of an unexpected election result is exponentially tiny like 10^(-100).
In that case neither side finds it "worth it" to vote more than an exponentially tiny amount of dollars.
So I guess in that case, Weyl would say (or hope) the
result was inevitable, it happens, and hardly any money was lost, so that's "optimal" too.
By the way, note what I just said was one hell of a lot shorter than 57 pages.
So why the 57 pages?
Apparently because Weyl has an overpowering desire to suck up to
everybody in every economics department in the world,
in vast preference to being simple and clear.
In sec 4.4 Weyl discusses the "de-merger" problem where it is far more cost effective under
his system to buy votes thru lots of proxies. There is a huge incentive to do that, which
might undermine whatever claims his system supposedly has going for it.
I had anticipated he was going to say "secret ballot"
here but he does not. I also thought he would argue the more people you use, the bigger
your risk of being caught, and some kind of prosecution risk would worry you.
But he does not say that either.
He says "de-merger into two individuals... is probably all that is feasible in most cases"
for some unknown reason. I can tell you that the Kansas City machine that elected Harry Truman
maintained registration lists of huge numbers of fake voters, comparable to or
outnumbering the real voters in the city. And they controlled the judges (Truman was one)
hence not worried much re prosecution.
Weyl never mentions the 24th amendment, which is what
says his scheme is unconstitutional in USA.
(It would appear that minor little things like US history and US constitution are of no
importance to Weyl.)
Near the bottom of p.7 Weyl makes a flat out false statement about central limit theorem. Since
his entire paper rests on this false statement that is not good.
The false statement is:
"the sum of values of all but one individual converges [to a normally distributed random variable] by the central limit theorem...when the number of individuals, and hence the variance... grows large."
This is false. False. False. Period. Full stop. Complete and utter garbage. It invalidates
Weyl's entire paper right there.
It is NOT the case, that a sum of random variables converges to a normally distributed random variable
(after rescaling).
Period.
But it would be true if, say, the summands all were identically independently distributed random variables
each with identical bounded variance and mean=0. But Weyl explicitly assumes that his
values are NOT identically distributed. And they sure seem unlikely to be independent either.
In particular, suppose person number M (M=1,2,3,...) has value +-1/M with the +- sign
got by a random coin flip. This way we get total independence because I am extremely
generously handing that to Weyl for free out of the kindness of my heart.
Now I will further, also extremely generously, let M--> infinity (infinite population)
going right to the limit immediately.
Then: Is it the case that SUM(for M=1,2,3...)OF (+-1)/M
is (after some rescaling) a normally distributed random variable? NO. Is it the case
that its variance goes to infinity? NO.
And might a power-law distributed society like that be a reasonable model of a society?
Quite possibly. It is observed that many ecosystems involve such power law distributions.
It also is observed that, e.g. "80% of the wealth is owned by 20% of the people"
and empirical economic laws of that nature, which note ARE power law distributions.
Sorry Weyl.
Is this flaw fatal? Not necessarily. I do not think Weyl actually ever really
needed normality. All he needs is there be SOME limit distribution
which in an unbiased election situation
looks uniform near the balance point. The uniformity would be generic behavior for any
probability density with a smooth CDF. The existence of a limit distribution is not
obvious.
Still, it is not a promising start to have him shoveling hafalutin and false baloney at you,
when actually all he needed was to talk about generic behavior... if he can get existence
(which I'm not sure he can).
Now above, I spoke of two cases "balanced" and "very unbalanced" elections.
But what is probably most interesting is a third intermediate case where say
the 2nd option is probably going to win, say with estimated chance 70%, but not
enormously near 100%. Then what? In that case, your chance of altering election
might behave noticeably nonlinearly as a function of your number of votes.
In that case the argument quadratic ==> optimality is not exactly true anymore.
But the whole quadratic voting system should not behave too badly.
Other incidental notes: Clarke-Tideman-Tullock voting is discussed here including some ideas Weyl did not
mention:
A quadratic-voting idea for forcing honest utility-revelation was invented and explained here by me
sometime before 2009 but for that purpose seems supplanted by this:
So in conclusion, while I've been kind of nasty in this review, I think Weyl is correct
that his proposed system is fairly simple, and would look fairly reasonable and practical
at least if we had continuum money, no US constitution, and if "paying for votes"
and "reliable uncorrupted secret ballots" were compatible notions. And I also think his optimality
argument at its core after you strip the bullshit away, contains a goodly amount of
approximate truth.