The math behind the Holy Spirit

By Ask Foldspang Neve

[note: this essay was published just a few hours before the election was over at a few minutes past 6 GMT; as it turns out, Conclaves really are unpredictable].

Papal elections are not like any other ballot out there. Indeed, looked at with modern eyes, it is difficult to tell whether it is best compared with the election of a president or a vote held in Congress. It is steeped in a deeper mystery than even the American Electoral College, although the Catholic college has no Floridian Cardinals who show up two weeks after the ascension of a new Pontiff and pretend that producing their own little puff of white smoke matters. Compared to the US Senate of the 113th Congress, the electorate is also even more male (100% compared to the Senate’s 80%) and aged (median age is 72, compared to the Senate’s 62) though the College of Cardinals is the more ethnically diverse (65% white, compared to the Senate’s 93%) of the two. In theory, what makes the election of the Pope truly unique, though, is the fact that it is not meant to be an election at all. Rather than ‘deciding’ on the new Pope, the two-thirds majority needed to appoint Benedict’s successor is really thought to be only a means of communication of the Holy Spirit that has descended on the crowd. In practice, however, there is another feature of the papal elections that make them special: due to the math behind the Dove’s descent, it is almost impossible to make any accurate estimates of who will end up with its divine favor.

Moreover, we will quite possibly have to wait a while to find out. At any rate, much longer than almost any other election, and that with an electorate that is just about one-millionth the size of the electorate that reappointed Barack Obama for the US presidency a few months earlier. But just how long could it take for 115 well-educated, elderly men to find a leader among them? Well, it can take weeks, and the reason for that lies in the special election system that depends on a two-thirds supermajority. Welcome to an institution that makes even a filibustered election in the US Senate look like a painless decision-making process.

Popes come and popes go, but the Holy See is forever. After Benedict XVI abdicated on the last day of February, the College of Cardinals has gathered in the Sistine Chapel, where they will remain from yesterday until they have chosen the new head of the single largest religious community in the world. Since the election is resolved by a two thirds majority, it has often taken many days, sometimes even months, to find a winner. In return, the voters have them found the absolute ruler of the Church, who is in principle capable of speaking infallibly on matters of Catholic dogma. As such decisions might affect all 1.2 billion Catholics, it is obviously an appointment of some importance (some prominent Catholics, though, beg to differ with the last statement).

Titian - The Descent of the Holy Ghost (c. 1545), Venice.

Titian – The Descent of the Holy Ghost (c. 1545), Venice.

It has not always been a given that the pope should be at the center of the church. Just as Europeans today discuss which political matters ought to be decided in Brussels and which ones are properly resolved in the capitals of the member states, the clerical-political elite of medieval Europe assiduously discussed how authority should be divided between episcopacies, the monasteries, the abbacies, the synods and the Pope. Conduct the simple experiment of exchanging the list of Catholic authorities with names such as Parliament, Supreme Court, the European Council, the Council of the European Union, the European Commission, and you will see the resemblance. And like the discussions in contemporary Europe, clerics and other authorities were not only disagreeing about who had political competence; they also disagreed about the Kompetenz-Kompetenz to solve those issues. The situation of today is, in other words, only just as complicated as it has always been.

The wisdom of these somewhat arcane procedures is more difficult to apprehend unless one knows a little about the causes for their introduction. Over the course of history, the church faced a series of challenges designing a fitting electoral procedure. Different methods were introduced, often with little or no prior knowledge of the consequences of selecting one technique instead of another. Today, the social sciences have developed a number of tools to analyze votes and electoral behavior, and employing these, we can explain why Benedict’s successor has to be found so laboriously. We might even learn from history about the different troubles facing the elected institutions of both Europe and the US today[1].

One of the primary trade-offs when implementing an electoral system is between stability and speed of decision-making. In a qualified majority system like the Catholic, stability has been emphasized over speed. However, this needs qualification: stability here means the lack of a credible alternative after the election, a very real concern in many polities. It does not mean predictability, of which there is very little in the papal elections. The reason for this is the very difficulty of obtaining a two-thirds majority. As voters progressively realize that their preferred candidate, though maybe strong, will be unelectable, they begin speculating about more unusual alternatives whose election will not be an outright defeat in the same way as the election of another major player might. Moreover, as the electorate is (most often) identical with the pool of candidates, it is possible to strategically put divisive candidates in play. Even given that they might not win in the first couple of rounds after they have entered the ballot, their presence might make other voters change their allegiance, as more electors realize that their de facto set of options has expanded. However, contrary to plurality voting, divisive candidates have a psychological effect rather than a purely mathematical one: since a candidate needs a two-thirds majority no matter the number of Ralph Naders in play, the effect is more to show the futility of supporting the original candidate.

This has quite often led to the election of popes who never looked like winners beforehand. John Paul II was such a dark horse. The last pope to abdicate before Benedict, Celestine V, is another good example. He was elected after an 822 days-long election. Before he was elected pope, he was a local hermit, with little official interest in church politics. As pope, he promptly reintroduced the harsher rules for papal elections of Gregory X and then stepped down. In total, Celestine wore the pallium for only five months and eight days, that is, less than a fifth of the period that it had taken to elect him. This means that despite what bookmakers say, we know very little about who will become the next bishop of Rome. This is not even due to the famed secrecy of the Conclave and its members, the Cardinals. As the unpredictability arises from the institution itself, even the Cardinals don’t know for sure.

The majority rule employed in the college until 1179 for determining papal successions gave rise to its own, internal problems that in turn left the institution open to outside interference. A special problem arises when there are more than two candidates (and more than two voters) in an election using simple majority rule. Often known as the Condorcet paradox of voting, named after the French enlightenment philosopher and mathematician Jean Marie de Condorcet who first pointed to its existence, the problem consists in the voting system being able to produce non-conclusive results; that is, not producing a unique winner. If we have an electorate of three voters, {1, 2, 3}, and a set of three candidates, {A, B, C}, the following ranking of preferences can appear:


Voter 1

Voter 2

Voter 3

1st priority




2nd priority




3rd priority




From the above table can be seen that with the given preferences, no candidate would stand a pairwise comparison with any other candidate in terms of satisfying voters’ preferences. Thus, if A is chosen as the winner, both voters 2 and 3 will prefer C to A as the winner. If C is then chosen, both voters 1 and 2 will prefer B to C. It goes the same way for B. In more formal terms, it can be said that simple majority rule is like to produce the following electoral cycle:

A > B > C > A > (…), where > denotes an electoral majority.

Normally it is assumed that the individual voter has transitive preferences, that is, if a voter prefers A to B and B to C, she also prefers A to C. The general problem, then, is that even when individual voters (or coherent coalitions of voters) have transitive preferences, it might not be possible to achieve social transitivity. The problem exists when voters act sincerely (as contrasted to tactically) and extends to most of the policy space, that is, for most combinations of voters and preferences, unless voters have essentially equal preferences. Furthermore, only very slight changes from the set of stable preference orderings reintroduce intransitivity to most layers of the field (that is, even to ordering among the top preferences of most voters). In other words, a simple majority system is extremely unlikely to create a system of complete transitivity.

The problems incurred by this paradox are multiple. It points to the fact that most electoral results formed by a majority will, paradoxically, be less satisfactory to a majority than some other option. This can lead to different problematic situations. The first is impossibility in reaching conclusions. If different choices cannot be ranked socially using the election method in question, the electorate cannot easily come to a conclusion. Even if they do, there is a great likelihood that the route to that decision is created through agenda- setter manipulation. The second is the risk that a party in the election will not respect the result altogether, which will lead to competing results or the breakdown of the electoral institution.

This is possible in a surprisingly simple mechanism. When individual preferences are aligned as depicted above, (i.e. prone to producing cycles under majority rule) an individual who can control the pairing of votes can lead the electorate to any position in the policy space of his choosing. In the figure below, the preference set {A, B, C} from before has been continued, with three voters each preferring their marked positions; S is the starting consensus. Now, in a finite number of steps {S, Z…Z’’’}, the agenda-setter is able to move the consensus over the marked positions. The farther away from S, the wider each voter’s indifference circle will become and the bigger steps the agenda setter can take. In the figure below, A and B prefer Z to S. However, then B and C will prefer Z’ to Z. That position, however, is less preferred to C and A than Z’’; finally, A and B prefer Z’’’, the agenda-setters preferred position, to Z’’:

Agenda Manipulation

Agenda manipulation under simple majority rule.

But in the 1100s, the times they were a-changin’. The church was coagulating into a self-owning entity. Hence, its interest in not being shaken by schisms and bloodshed became organizational, and as such became more likely to trump dynastic impulses to disregard church safety. Although contestants for power internally, the newly established College of Cardinals and the Pope had a concurrent interest in keeping the organization a unified whole. Even though the Cardinals were still mostly drawn from the high aristocracy, they increasingly prioritized office interests above dynastic goals upon their ascension, sometimes to the expressed displeasure of their kin. Just as the members of the European Commission are given powerful incentives not to act as representatives of their native countries, the preferences of the cardinals also increasingly came to be shaped by the structure of the church. This in turn strengthened the position of the church, which again made it more attractive to obtain official positions within it and keep these offices as independent as possible.

Until 1163 it was still the prerogative of Roman ecclesiastics to elect the Pope; hereafter arose the custom of including individual external members as well. Not surprisingly, the first external member was a German-speaker. His name was Conrad of Wittelsbach, the archbishop of Mainz. To some southerners, this must have been the equivalence of taking dictates from the ECB today. As the Italian cardinals in the current election still number almost a fourth of the College, it is quite obvious that some members still feel that the election of the Roman bishop should be kept an internal affair.

To ensure that everybody recognized their authoritativeness, Alexander III decided to proclaim his new election rules at the Third Lateran Council in 1179, speculated by the scholars Josep Colomer and Ian McLean to be inspired by the Venetian voting system in place at the time. No matter whether you regarded the highest authority on the rules of the church to be its general assemblies or its presidency, you would have been accommodated. It was made clear that cardinals of all three sacramental orders were entitled to cast their vote and at the same time the two-thirds rule was established. It was intended to make it less likely that the elected pope would face opponents who themselves held credible claims to the honor. However, the price paid for this was immense difficulty of reaching a conclusion. To counteract this, the Conclave (literally, ’with key’, that is, under lock) was developed by the Roman authorities. Gregory X even decided that the Cardinals had to live in increasing discomfort as the days went by with no result, so that in the end, they would be served only bread and water. This ought to make them to tune in to His frequency. More important than the temporary loss of comforts was probably the simultaneous loss of profitable tithes, which members of the conclave were not entitled to while in session. While not especially effective, instituting such a rule for today’s political classes would probably prove popular among their jaded electorates.

Ask Foldspang Neve

Ask Foldspang Neve is currently pursuing his MA in political science at the University of Aarhus,  Denmark. He is writing on religion and politics from sociological and political science perspectives.


1] The technical properties of the papal voting systems were first analyzed in a 1998 article by Josep M. Colomer and Ian McLean in the Journal of Interdisciplinary History, which this essay draws on.


2 Thoughts.

  1. Hej søde Ask.
    Meget interessant arbejde, du har lavet.
    Lige et spørgsmål til dig, den unge mand med den store viden 😉
    Som du måske husker, er religion generelt et emne som har min interesse, i forhold til dig nok mest det “at tro” – den for mange manglende susbstans der er heri.
    Nå men tilbage på tråden. Langt hen ad vejen forstår jeg dit skriveri; men kom til at tænke på om din grund for at gå i gang med lige dette emne.
    Pavevalget tiltrækker jo en kæmpe gruppe mennesker ikke bare katolikker.
    Er det selve måden valget foregår på, altså traditionen som dog er undergået forandringer igennem de mange århundreder?
    Eller er det grundlaget for valget, at mennesket vælger et menneske som “regerer” lige under Gud?
    Sandsynligvis kommer du med en helt anden forklaring 😉
    Jeg håber, at du, din familie og din kæreste har det godt.
    Din studietid er nok snart slut, det bliver spændende, hvad fremtiden bringer.
    Knus og hav det godt.

    • Kære Tove,

      Tak for din kommentar. Jeg tror, at pavevalget tiltrækker så megen opmærksomhed af to-tre årsager:

      For det første er den katolske kristendom verdens største organiserede, religiøse bevægelse. Katolsk lærdom om fx abort og prævention har stadig stor betydning i store dele af verden – herunder både i Afrika, Nord- og Sydamerika og til dels i Sydeuropa. Og selvom især mange yngre katolikker er uenige med den officielle lærdom fra kirken omkring eksempelvis prævention, har valget stadig stor symbolsk betydning – som det også havde, da kirken i 60’erne officielt fordømte antisemitisme og diskrimination mod jøder dermed blev ugleset også i de fleste konservative kredse.

      For det andet tror jeg, at den tradition, der ligger i valget – som du også nævner i din kommentar – og som er helt afgørende for den katolske kirkes syn på sig selv, har en vis fascinationskraft, ikke mindst i disse årtier (de rige, men krigeriske 00’ere og de fattige 10’ere), hvor mange i Vesten til dels i usikkerhed søger forskellige traditionelle rødder: amerikanerne længes tilbage til de gode årtier mellem 2. Verdenskrig og Vietnam; og de europæiske befolkninger har hver deres guldalder, der i det nationale narrativ står som den tid, hvor alting var mindre kompliceret, politikerne ikke løj og der var sammenhæng mellem indsats og udbytte.

      I forlængelse af det sidstnævnte er det nok også især i protestantiske lande en sport at undres over, hvordan katolikkerne dog kan tro, at paven har et særligt forhold til det guddommelige. Jeg tror ikke, at der kan være nogen tvivl om, at selv “kultur-lutheranere” – og endda protestantiske ateister som jeg selv – til en vis grad ubevidst definerer vores kultur som modsat den katolske. Men vi tilhører jo også både langt mindre trossamfund og traditionelt langt fattigere lande (omend dén balance er skiftet i de sidste 100 år). Katolikkerne har aldrig defineret sig selv i forhold til nogen anden trosretning (katolsk betyder ‘almen’), mens det ligger i protestantismens navn, at den definerer sig som en modstand mod (visse dele af) det katolske – og ikke mindst idéen om paven som overhovede for kirken.

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