Eight – The New Belfry

The Reverend Charles Lutwidge Dodgson was the most conservative of men, in politics and nearly everything else. When he went on a rail trip, he was careful to map the route and compute exactly how much he would spend on tickets, meals, newspapers, and cab fare. He would put the exact change for each expense in pockets of a multicompartment purse, where it would be ready at hand.

At the theater, the reverend would mentally compile pointers for improving the production. In London, he saw a play in which a character was thrown off a bridge. It bothered Dodgson that there was no splash. “A little bit of realism … would be very welcome,” he wrote the leading man, appending a sketch of a device for producing a suitable sound effect on cue.

Dodgson was thin because he ate only one meal a day (carefully recording its contents in his diary). One eye, and one shoulder, was slightly higher than the other. A speech impediment caused him to freeze in mid-sentence, unable to get out the next word. This was a [p.150] serious handicap for a man who was not only a preacher but a mathematics lecturer at Oxford’s Christ Church College. His lectures were said to be “dull as ditchwater,” and his sermons were undistinguished except for the fact that he sometimes worked himself into tears.

The windows of Dodgson’s rooms at Oxford overlooked the gardens of the deanery, the residence of the worldly dean Henry George Liddell, his wife, and his huge family. Liddell, a classics scholar who coauthored the first lexicon of classical Greek, was the polar opposite of Dodgson in many ways, including politics. Liddell supported the Liberal Party.

The new dean, recruited with great fanfare from Westminster School, had scarcely unpacked his bags when he began an ambitious architectural program for Christ Church College. This commenced with a MeMansion-style renovation of his own living quarters. (Mrs. Liddell did not move in until it was completed.) The dean then turned his daring eye to the rest of campus. Both the expense and the aesthetics of his new construction became sensitive issues.

Dodgson adored architecture—the older the better. He thoroughly despised the dean’s changes. What particularly ticked off Dodgson was the dean’s new belfry. The cathedral’s bells had long hung in its crumbling spire. The dean moved the bells to a new, temporary wooden structure above the grand staircase of the dining hall. The new structure was a box. Dodgson was appalled.

The Victorian equivalent of blogging was pamphleteering. When one of Oxford’s professors got worked up on an issue, he could write a pamphlet, have it printed handsomely on Oxford’s university presses, and distribute copies to that ever-so-small circle who might care. Dodgson responded to the dean with a satiric pamphlet titled “The New Belfry,” in which he tartly labeled the belfry style “‘Early Debased’: very early, and remarkably debased.” The pamphlet sold for six pence in local bookstores and was a modest success.

Oxford held an architectural competition for a new and more [p.151] permanent belfry. This entailed committee meetings, parliamentary rules of order, and, above all, voting. Liddell was a charismatic politician, as Dodgson was not. Unable to compete with the dean on those terms, Dodgson turned his rigorously logical mind to something else: What is the fairest and most logical way of voting?

Electoral reform was in the air. The 1872 Reform Act gave British voters a secret ballot for the first time. Dodgson is known to have attended some of the House of Commons debates on this very matter.

“The following paper has been written and printed in great haste,” Dodgson began the first of several pamphlets he would write on voting, “as it was only on the night of Friday the 12th that it occurred to me to investigate the subject, which proved to be much more complicated than I had expected.”

This pamphlet, dated December 18,1873, starts by remarking on the “extraordinary injustice” of a plurality vote. Dodgson then proposed a system (“whether new or not I cannot say”) that is in fact identical to Jean-Charles de Borda’s.

This raises the question of whether Dodgson knew of his eighteenth-century French predecessor. Evidently not. The work of Borda (and Condorcet) had by then been utterly forgotten. In the twentieth century, Kenneth Arrow’s rival, Duncan Black, went so far as to examine the 1781 volume of the French Academy proceedings containing Borda’s article in the Christ Church Library. The pages were still uncut in the 1950s.

The very day that Dodgson’s first pamphlet was published, his (Borda’s) system was adopted in the voting for a readership in physics. There were three candidates, and in the tally used, a high score was good. A candidate named Becker got forty-eight points. That was just one point more than the second-place candidate, Baynes. Becker should have won, but because it was so close, they held a second, runoff vote with just the two front runners. This time Baynes beat Becker, 11 to 9. Baynes was declared the winner. [p.152]

The runoff wasn’t part of Dodgson’s plan, and in his diary he reported, with a trace of disappointment, that “we partly used my method.”

The runoff must have shaken Dodgson’s confidence. The candidate who won by the Dodgson (Borda) system was revealed to be less popular than another candidate in a two-way vote. This possibility had been the nub of Borda and Condorcet’s dispute, and it happened the first time Dodgson’s voting scheme was used.

Dodgson went back to the drawing board. In 1874 he issued a second voting pamphlet “in the immediate prospect of a meeting of the Governing Body, where matters may be debated of very great importance, on which various and conflicting opinions are known to be held.” The matters of great importance included another vote on the belfry.

Nothing more was said of the Borda system. This time Dodgson suggested that they conduct two-way votes of every pair of candidates (belfry designs, actually). The candidate who beats every other candidate in two-way votes deserves to win. This is exactly what Condorcet had said, and again Dodgson apparently came up with it on his own. (The closest copy of Condorcet’s ghastly Essay was in another Oxford library, the Bodelian. Duncan Black found that that copy had an uncut page in the section on elections.)

This plan was followed at the June 18, 1874, meeting. There were then four proposals for the belfry. In successive two-way votes, George Frederick Bodley’s design beat those of Thomas Graham Jackson, Thomas Deane, and George Gilbert Scott. Bodley’s design, the Condorcet winner, was chosen.

At the time that Henry Liddell assumed the post of dean, few could have imagined that Charles Dodgson, a minor and peevish mathematician, would one day be among the best-known men in Britain, a man [p.153] whose renown would far eclipse that of Oxford’s famous dean. This was the case, and Dodgson’s dislike of Liddell was rooted in something deeper than a difference in architectural taste.

It had to do with Liddell’s daughter Alice. Dodgson first met Alice in April 1856 while photographing Oxford’s cathedral, the one whose belfry would become the point of contention. Just shy of four, Alice became Dodgson’s favorite photographic subject. By late 1856, Mrs. Liddell sensed something unwholesome in Dodgson’s interest in Alice and her sisters. She told him to stop taking photographs of her children.

Soon afterward, the dean and his wife went on vacation, leaving the children in the care of the governess, a Miss Prickett, who let Dodgson resume seeing the Liddell children. These meetings continued even after the parents returned.

On July 4, 1862, Dodgson took the three Liddell sisters on a boat trip, accompanied by a Reverend Duckworth of Trinity College. Dodgson entertained the children with an improvised tale of an Alice who fell down a rabbit hole. The real Alice begged him to write the story down. He did so, greatly expanding it in the process.

While Dodgson was in the course of writing Alice’s Adventures Underground, as he called it, something happened that again caused the Liddells to ban Dodgson from seeing their children. By the time he completed his handwritten and illustrated copy of Alice’s Adventures Underground, Dodgson was unwelcome in the Liddell home. He sent the manuscript to Alice as a Christmas present in 1864. There is no indication in Dodgson’s generally obsessive diary that the Liddells ever acknowledged the gift.

In 1865, Dodgson published the manuscript as Alice’s Adventures in Wonderland. He used the pen name Lewis Carroll to preserve his privacy. The book became a sensation and was followed in 1871 by a sequel, Through the Looking Glass. Most biographers believe that Dodgson was in love with the real Alice, and Morton Cohen speculated that the author even brought up the possibility of marriage. Alice was eleven, [p.154] and Dodgson was thirty-one. The Liddells would have been outraged, not only by the age difference but by their none-too-high opinion of Dodgson.

The scandal overshadowed Dodgson’s later dealings with the family. In Duncan Black’s colorful surmise,

we will simply state what appears to us to be the almost inescapable conclusion. It is that by about 1872, when Alice Liddell. a girl of outstanding beauty and charm, reached twenty, surrounded by some of the most eligible young bachelors in England, and Dodgson reached forty without much reputation in Christ Church, he realized that Alice Liddell, who had meant so much to him, was slipping out of his life… Dodgson felt frustrated and humiliated. His reaction amounted to the attempt to triumph over Liddell and lower the stature of the Dean. By his pen, the instrument of his genius, he would alter the direction of architectural policy. and college policy, the objects at the centre of the life which he still shared with the Liddells.

In Black’s half-plausible reconstruction, Dodgson’s interest in voting was motivated by attempts to reconcile his own conflicting feelings about Alice and other child-friends. Not everyone has bought Black’s psychoanalysis. One Carroll biographer, Roger Lancelyn Green, wrote Black, “Frankly I think you are barking up a Tum Tum tree.” (It’s a tree in “Jabberwocky.”)

The pages in Dodgson’s diary that appear to bear on the break with the Liddells are missing, removed by razor. Elsewhere, the diaries are full of pleas for God’s forgiveness of a sin that is never specified. “God help me to lead a new and better life,” Dodgson wrote just eighteen days after the river trip that prompted the Alice story (July 22, 1862). Later the same month, he postpones a preaching engagement: “Till I can rule myself better, preaching is but a solemn mockery—‘thou that teachest another, teachest thou not thyself?’ God grant that may be the last such entry I may have to make!” [p.155]

That prayer went unanswered. Over a hundred similar admonitions were to follow it in Dodgson’s diaries. One of the most anguished is dated June 5, 1866: “Gracious Lord, send Thy Holy Spirit to dwell in this cold love for Thee, to strengthen this failing faith, to lead back from the wilderness this thy wandering sheep, to make real my repentance, my resolution to amend my struggles against the temptations of the devil, and the inclinations of my own sinful heart.”

The most important of Dodgson’s voting pamphlets was printed in March 1876, days before the publication (on April 1) of “The Hunting of the Snark.” It was again occasioned by the ongoing grudge against Dean Liddell. Oxford philologist Max Müller had announced his intention to retire from teaching in order to translate the sacred texts of the Orient. The University of Vienna offered him an appointment that would give him the freedom to do so. Dean Liddell presented a “decree” to keep Müller at Oxford. This proposal would pay Müller’s salary and hire a successor at half the usual salary.

Dodgson objected to hiring a new teacher at half price. The Oxford community sided with its dean, Most saw the decree as a referendum on whether to keep Müller at Oxford. “The advocates of the Decree persisted so much in praising Max Müller,” Dodgson told his diary, “and ignoring the half-pay of the Deputy that I rose to ask them to keep more to the point.”

Liddell’s decree was put to a simple vote and won by 94 to 35. A jubilant crowd pressed congratulations on Liddell as he himself walked to Müller’s residence to tell him the good news.

Dodgson occupied the next week thinking more about voting. Exactly eight days later, he “spent the afternoon in writing out a Method for Taking Votes which I sent to the Press to be set up in slip.”

This, Dodgson’s third and most original publication on voting, was titled “A Method of Taking Votes on More Than Two Issues.” The title page contains the disclaimer “Not yet published” and the author’s note: [p.156]

(As I hope to investigate this subject further, and to publish a more complete pamphlet on this subject, I shall feel greatly obliged to you if you will enter in this copy any remarks that occur to you, and return it to me…)

The pamphlet was interleaved with blank pages for comments. “What responses he got, if any, we do not know,” Duncan Black wrote, “but it is likely to have been negligible.”

Dodgson’s pamphlet treats the paradox of voting, going well beyond Condorcet’s discussion. His interest in paradoxes is of course evident throughout the Alice books. The story’s young heroine is forever encountering authority figures oblivious to the illogic of what they say and do. One episode in Alice’s Adventures in Wonderland, the “Caucus-Race,” may bear on voting. The Dodo

marked out a race-course, in a sort of circle, (“the exact shape doesn’t matter,” it said) and then all the party were placed along the course, here and there … they began running when they liked, and left off when they liked, so that it was not easy to know when the race was over. However, when they had been running a half hour or so … the Dodo suddenly called out “The race is over!” and they all crowded round it, panting, and asking, “But who has won?”
&nbsp ;    This question the Dodo could not answer without a great deal of thought … At last the Dodo said, “Everybody has won, and all must have prizes.”

American readers are apt to ask what Alice does—”What is a Caucus-race?” Caucus was and is a political word on both sides of the Atlantic. In England, it is a derogatory term for factions that vote as a bloc. It is less clear what Dodgson meant by a caucus-race. Critics have assumed that he was lampooning Oxford committees that ran around in circles while getting nowhere. [p.157]

In his third voting pamphlet, Dodgson presents majority rule as an absurd race that everyone can win. “The majorities may be ‘cyclical,’” he wrote, “e.g. there may be a majority for A over B, B over C, and C over A.” It was Dodgson who coined the term cycle for this situation.

Dodgson described and critiqued several schemes for resolving cycles. He concluded, however, that none of these schemes can be entirely satisfactory. He ended with the dispiriting suggestion that the result of a “persistent” cycle should be “no election.” The vote should be tossed out.

This may not be practical. It does show what a provocation a voting cycle is to any reasonable theory of voting.

Dodgson insisted that “The Hunting of the Snark” contain no picture of the title’s mythic beast. He did not want to frighten his child-friend readers or dispel the mystery he intended. The point of this book is to unmystify Dodgson’s (Condorcet’s, Arrow’s) beast. It is helpful to have a “picture” of a voting cycle, if only to decide how terrified we should be.

You can think of the picture on the next page as a precinct map. The eighteen black dots mark the locations of voters’ homes, which cluster in three small towns. The region is voting on the location of a new nuclear power plant. The ballots label the three proposed sites A, B, and C.

In this case, politics is simple. Every voter wants the nuclear plant RIMBY (“right in my back yard”). Everyone hopes to get a job at the plant, and no one likes a long commute.

Start with a vote on whether to put the plant at site A or B. Twelve of the voter dots (in the communities of Springfield and Shelbyville) are closer to A than they are to B. Those twelve would vote for A. The other six voters (in Capital City) would favor B. Site A wins.

Now try B versus C. Twelve of the dots (in Springfield and Capital City) are closer to B. Site B beats C, 12 to 6. [p.158]

Finally, let’s have a vote between C and A. Shelbyville and Capital City are closer to C (slightly), so C beats A, and again it’s by a twelve-to-six margin.

This is a voting cycle. Of course, in the more usual kind of election, the landscape is one of ideology. Each candidate stakes out a position on the issues of the day. Voters tend to favor the candidate whose views are closest to their own. A good voting method minimizes political “distance” between what voters want and what they get. [p.159]

The diagram is a “trick picture.” I had to place the dots carefully. A haphazard scattering almost certainly would not be a cycle. That probably means that voting cycles are rare in the real world. The diagram indicates some of the conditions that have to be met for a cycle to exist. Voters must cluster in three (or more) groups that are not in a straight (ideological) line. Some or all of the candidates must be off-center from the clusters. The center is relatively empty of voters and candidates.

The overall arrangement creates shifting coalitions. In one vote, Springfield allies with Shelbyville; in another, it joins with Capital City against Shelbyville; and in still another, Springfield stands alone.

Many follow Dodgson in saying that a symmetrical cycle like this ought to be declared a tie. (“Circular tie” is another name for a cycle.) This is reasonable provided that two-way votes are the only information we have or can have. This example is different: we know that everyone actually wants to minimize commuting time. Degrees of preference can be measured in miles or minutes saved. So why not figure the total commuting distance (from all eighteen voters’ homes) to each site? The site with the smallest total would be “closest” to the voters overall. It’s unlikely that this would result in an exact tie.

You don’t even have to measure. A glance at the diagram shows that site A is the best choice for commuters. It is adjacent to Springfield and reasonably close to Shelbyville. The downside is that A is a long drive from Capital City. The other two sites aren’t much better. Capital City commuters aren’t going to be delighted with any of the three choices.

The worst location is B, way out in the middle of nowhere. You may wonder how it managed to beat C in a two-way vote. The explanation is that B is the lesser of two evils. Being barely closer to Springfield and Capital City than C is, B picks up these communities’ half-hearted votes—even though it’s not a convenient choice for anybody.

This is the fundamental problem with two-way comparisons. There is no accounting for degrees of preference. Springfield commuters [p.160] must greatly prefer site A to B. It will save them ten miles, say, each way. In a two-way vote, that counts no more than the fact that choosing site B over C will shave two miles off the Springfield commute. Cycles result from giving equal weight to unequal preferences. In this case, the cycle is all smoke and mirrors. The paradox obscures the fact that the voters really do prefer one option.

Charles Dodgson intended to publish a book on his voting theory. He never did. He was frustrated by his inability to devise a truly satisfactory system. “A really scientific method for arriving at the result which is, on the whole, most satisfactory to a body of electors, seems to be still a desideratum,” he conceded in December 1877.

Dodgson died on January 14, 1898. Dean Liddell followed him four days later. Alice Liddell had thirty-six years left. In that time she would lose her husband, two sons, her happiness, and her wealth. Only the Sotheby’s auction of the original manuscript of Alice’s Adventures Underground, the year before the stock market crash, saved her from genteel poverty. “But oh my dear I am tired of being Alice in Wonderland,” she wrote toward the end of her long life. “Does it sound ungrateful? It is. Only I do get tired.” Dodgson’s voting pamphlets were preserved by Lewis Carroll completists, who scarcely knew what to make of them. Consequently, for the second time, the would-be science of voting was almost completely forgotten. Dodgson’s most enduring public legacy was to political cartoonists, who gained an inexhaustible metaphor for the mad tea party of politics. [p.161]