The Economist December 18th 2021 Holiday specials 17
and soft Brexit—each could beat one of the others but
not both. If asked, Britain would have preferred a soft
Brexit to remaining a member of the eu, because even
hardened Eurosceptics thought some kind of exit was
better than none. Voters would have preferred Remain
to a hard Brexit, because of the economic chaos en
tailed by such a disruptive departure. But they would
have preferred a hard Brexit to a soft one, because if the
country was going to go to all the trouble of leaving the
euit might as well regain control of immigration.
Carroll proposed a number of alternative voting
methods to bring races to a more satisfactory conclu
sion. One was a version of the “method of marks”. Each
voter is given a number of points or “marks” to spread
across candidates as they see fit. They would give the
most marks to their favourite candidate. But they
might also give a few to acceptable alternatives. This
would allow voters to express both the fact of their
support and its intensity.
Polarising figures like Mr Trump would attract lots
of marks from some voters, but none from others.
They might therefore lose to choices that earned a few
marks from everyone. “This Method would, I think, be
absolutely perfect,” Carroll wrote in 1873, if only voters
were “sufficiently unselfish and publicspirited” to re
serve some marks for their second or third choices. But
in practice, he thought, voters would “lump” all of
their marks on their favourite candidate—if only be
cause they worried that others would do the same.
across the quad
Carroll never really overcame this problem. The intel
lectual baton instead passed to another eccentric enig
matologist, who provided the ingredients for a sol
ution shortly after the second world war. Like Carroll,
Lionel Penrose was educated as a mathematician (at
Cambridge not Oxford) before becoming a medical
doctor and geneticist at University College London.
Like Carroll he was fond of riddles, paradoxes and in
conclusiveness. He and his son documented some of
the “impossible objects” (such as the neverending
staircase that leads up to its own foot) made famous by
M.C. Escher. In his 16thcentury country home, he was
amused by a tree with roots that grew upwards.
According to one obituarist, he also knew how to be
“difficult” in committee if things went the wrong way.
Indeed, his ingenious contribution to the theory of
voting was to write down the mathematics of getting
one’s way. Being difficult, he proved, is often easier
than you might think. The power of a single commit
tee member was surprisingly hard to dilute.
To see why, imagine a committee of three people—
call them Tom, Dick and Harry. Their votes can in prin
ciple fall eight different ways. (A proposal can gain
support from all three or none. It could win backing
from Tom and Dick alone; Dick and Harry alone or Tom
and Harry alone. Alternatively, these same three pair
ings could oppose it.) In four of these combinations,
Tom can change the outcome, turning a knifeedge
victory or defeat into its opposite. A voter’s chances of
finding themselves this pivotal position serves as a
good index of their “power”. So in a small threeperson
committee, their power is four out of eight or 50%.
What if the committee were three times bigger? You
might think that would divide a voter’s power by three.
You would be wrong. In a committee of nine, there are
512 different combinations of votes. An individual vot
er can decide the result in as many as 140 of these, be
cause the other votes are evenly split. By the same in
dex, the voter’s power is 140 out of 512 or 27.3%—down
by less than half.
Why is that? The voter is now only one of nine, di
luting their influence. But this dilution is partly offset
by another consideration, as Richard Baldwin of the
Centre for Economic Policy Research has pointed out:
with nine members, many more potential knifeedge
combinations of votes exist. A voter’s power diminish
es as a committee grows in size, but it does not shrink
proportionately. It shrinks at a more gentle pace, pro
portionate not to the size of the committee but to the
square root of its size. The voter’s power, Penrose
pointed out, diminishes as the roots grow upwards.
Writing just after the second world war, Penrose
had bigger concerns in mind than university politics.
He was intrigued by the best way to allocate votes to
countries in a “world assembly” like the United Na
tions, established only months earlier. Giving every
country a single vote, regardless of its size, was un
democratic. The obvious solution was to award votes
in proportion to population. But that would give big
countries too much clout, Penrose argued.
He instead proposed a middle way between the
two. Each country should be allocated a number of
votes corresponding not to the size of its population,
but to the square root of its population. The popula
tion required for one thousand votes might be one
million. For two thousand, four million. For three
thousand, nine million. To put it another way, the pop
ulation required for a given number of votes should be
that number multiplied by itself (or squared).
Penrose’s idea reappeared when the eumulled re
forms to its voting rules in 2007. Poland worried that
big countries like Germany would have too much
sway. So it adopted an odd slogan for the summit: “the
square root or death”. The phrase “neatly combines ob
scurity, absurdity and vehemence,” as Gideon Rach
man of the Financial Times(and formerly of The Econo-
mist) put it, “capturing the spirit of the modern eu”.
Despite this vehemence, the proposal got nowhere.
But something similar in form has been proposed by
Glen Weyl, a political economist now at Microsoft Re
search. He calls it quadratic voting or qv. In its sim
plest version, each voter would be given a budget of
“marks” as Carroll might call them or “voice credits” as
Mr Weyl calls them. Voters could use these credits to
“buy” votes for a candidate or proposal. The first vote
for a candidate costs one credit. But casting two votes
for a single candidate costs four credits (ie, two
squared); casting three costs nine (three squared), and
so on. Under this scheme, people buy votes with their
credits just as countries “earn” votes with their popu
lations in Penrose’s imagined assembly. In both cases,
the aim is to give voters as much sway as their popula
tion or passion warrants. But no more so.
Compared with the method of marks, qvmakes it
harder to “lump”’ votes. That is because each addition
al vote for a single candidate costs more than the last
one did. (A second vote costs an additional three cred
its; a third vote costs an additional five.) Thus instead
of buying increasingly expensive votes for their num
berone choice, voters are nudged to cast some rela
tively cheap votes for second or thirdchoice options.
In this way, the method encourages compromise.
In the book “Radical Markets”, Mr Weyl and his coThe rules
of the race
can change
the course
of nations