A new pope will be chosen in the papal conclave starting Wednesday 7 May 2025. Omitting some (I believe irrelevant) details, the procedure is roughly as follows: All cardinals under 80, who are well enough to do so, go in conclave at the Sistine Chapel at the Vatican and proceed to each individually and anonymously nominate (vote for) a person to be pope in a series of scrutinies (rounds) until one “candidate” has more than two-thirds of all votes. This person is then the new pope.

Between reading Robert Harris’ “Conclave” and hearing (Nobel Laureate) Eric Maskin’s recent Schumpeter lectures on the “Theory of Voting,” I consider myself in a good position, if not to analyze fully, but to at least provide some reasonably informed discussion of the papal voting procedure. The exact voting procedure used for electing popes has not been studied much yet. For a good starting point, you might look at the recent paper by Andrew Mackenzie and some of the references therein, but my “treatment” here is a bit different. I will put more focus on the role of communication throughout the multiple rounds of voting.

While I do want to get to analyzing the particular voting procedure used to elect the pope as quickly as possible, I think it is helpful to establish a bit more general voting theory context. First, Eric Maskin’s lectures (based on his recent paper) have convinced me (to see why see my earlier blog post on how a book club should choose which book to read next) that there is a best voting rule if voters can be assumed to vote sincerely. To vote sincerely means to provide your true preferences: You don’t try to strategically manipulate the election outcome in your favor by misrepresenting your preferences. This best rule is the (more than 200-year-old) Borda rule (or Borda count). It works as follows. Suppose there are n candidates to choose from (in practice these are all the cardinals, but, in theory, they could be all catholic men – the last pope that was not a cardinal was Pope Urban VI in the 14th century). Then every cardinal at the conclave provides a full ranking of all the possible candidates. This ranking translates into points: the top-ranked candidate gets n points, the next best-ranked candidate gets n-1 points, and so on down the ranking, until the least-ranked candidate gets 1 point. These points are then summed up and the candidate with the highest number of points is the pope. If there is more than one candidate with this highest point total, then the holy ghost decides by means of a throw of a coin or dice or something like that.  

While reading this, you immediately have one counterargument to this procedure in that it seems impossible to provide a full ranking over all candidates (especially when we consider all catholic men), but one could probably allow a cutoff under which the cardinals don’t have to rank candidates any further. If there is sufficient agreement among the electing cardinals as to a small set of possible suitable candidates, then this should be possible and sufficient.

A bigger problem with the Borda rule is that it is strategically manipulable. Let me first explain why and then discuss whether we think this is a problem – after all, cardinals swear this oath: “I call as my witness Christ the Lord who will be my judge, that my vote is given to the one who before God I think should be elected.’’ I have this from page 719 of the paper by Andrew Mackenzie, who has this from reliable sources and something along these lines is also repeatedly stated in Robert Harris’ novel.

How is the Borda rule strategically manipulable? Suppose, for instance, that there are only three serious candidates, A, B, and C. Suppose that just more than one half of all cardinals prefer A over B over C (the “A supporters”) and the remaining (just under a half of all) cardinals prefer B over A over C (the “B supporters”). If they all provide their true ranking, candidate A will be pope by the Borda count. Now, suppose that the B supporters suspect that this is the situation. Then, if (at least a sufficiently large fraction of) the B supporters get together and strategically declare their preference as B over C over A (instead of B over A over C), then their favorite candidate B would win by the Borda count. Thus, by strategically misrepresenting their preferences, they would get their favorite candidate, B, to win.

Would the cardinals do that? It all depends on how the B supporters interpret “that my vote is given to the one who before God I think should be elected.” I think it is not hard for them to rationalize to themselves that when they think about it, they actually do prefer C over A even if this is not really so. Or they could be pragmatic and read this line as to mean that they should do everything in their power to have that candidate elected that they truly think is best. We will come back to this point.

In any case, given that the Borda rule is strategically manipulable, maybe it should not be used in such a context. And, of course, it is not used to elect the pope. But the problem is that no voting rule is non-manipulable (unless it is dictatorial – meaning that one person decides who is pope). This is the essence of the Gibbard-Satterthwaite theorem.  But is strategic manipulation always bad? In fact, I have the feeling that the papal voting procedure does a pretty good job of utilizing strategic “manipulation” in such a way that the pope is always a pretty good (compromise) choice.

At the start of the conclave there seem to me to be three kinds of uncertainties that the voting cardinals have. First, they probably don’t all know each other very well: some cardinals might not fully know another cardinal’s “qualities” and also not know their points of view on the many policy issues they care about. Second, even if a cardinal knows every other cardinal’s views, they don’t necessarily know every other cardinal’s preferences over which cardinal should be elected pope. Third, and finally, even if a cardinal knows every other cardinal’s preferences, this does not immediately mean they know which pope every other cardinal would (strategically) vote for.  

It seems clear to me that having time and communication helps with all three types of uncertainties. If there would be just one round of (even Borda rule) voting, then whoever wins may be the best only according to the partial information the voting cardinals have. This is nicely and very plausibly demonstrated in Robert Harris’ novel when new information appears (and spreads) at the conclave about two of the front runners (Adeyemi and Tremblay). Each of them was almost elected, but that would have been a very bad choice in hindsight – as each had made some (at least according to the Church) bad mistakes in the past that would not be acceptable in a pope.

So having a bit more time helps to have this information spread around and having some early rounds of voting (with no real expectation of electing a pope just yet) helps identify the front runners and, thus, focuses the communication on the relevant individuals. Having multiple rounds of voting also seems essential to me to solve the third problem of strategic uncertainty.

To see this, consider the following situation. Suppose that there are only three candidates (A, B, and C) in the running and the voting cardinals either have preferences A over B over C, B over A over C, and C over A and B (with A and B more or less equal) with say roughly 30% of the cardinals with the first preference and another 30% with the second, and about 40% with the third. If you have read Harris’ novel you can think of A as Bellini, B as Lomeli, and C as Tedesco (A and B are liberals, and C is a traditionalist). At one point in the novel, Tedesco tells Lomeli that he thinks it’s good that Lomeli is getting votes, thereby splitting the liberal vote. One can think of the first round of voting more of an opinion poll rather than a vote as it is almost impossible that it will result in a two-thirds majority for any one candidate. Suppose that cardinals in the first round simply vote according to their preferences. Then we would see roughly 30% voting for A, roughly another 30% for B, and roughly 40% for C. If we had plurality rule (meaning the candidate with the highest number of votes wins) C would be pope. But now, one can imagine that the A and B supporters (who all dislike C) would rethink their choices. Suppose that B had slightly more votes than A. Then it seems plausible that the A voters (the least chosen in the first round) now start voting for B. If eventually all of them do, then B will get roughly 60%, while C remains at 40%. By the way, this is roughly what seems to happen in Harris’ novel. At least the main character, cardinal Lomeli, whose thoughts we know best, occasionally switches his votes from candidates that seem to have no more chance to his next favorite. He starts with voting for Bellini, and when Bellini no longer seems to have a chance, then to Tremblay, then eventually to Benitez. There is an interesting dialogue between cardinals Lomeli and Bellini, in which Lomeli says that he has to keep voting for Bellini as Bellini is his true most preferred pope and in which Lomeli refers to the above oath as justification. To this Bellini says that this is nonsense and not how this oath should be interpreted, otherwise no pope would ever be elected (or some such words).

In any case, one can show that the behavior I have just described is a (certain kind of) empirically plausible Nash equilibrium in this simple voting game with just the three candidates as Costel Andonie and I have shown in a paper not too long ago. There is also some empirical evidence from lab experiments by Forsythe, Myerson, Rietz, and Weber in their 1993 paper that people behave in this way and an (admittedly but necessarily somewhat complicated) theoretical analysis of why they do so in a paper I wrote with Daniel Rodenburger. These papers only deal with this special setting with only three candidates and I am not aware of any papers that have tried to extend this approach. Nevertheless, I do think that they provide a bit of a justification for the papal voting procedure, in which multiple rounds of voting induce strategic manipulation that, however, ultimately provides a pretty good (compromise) outcome. Note that in the case with three candidates described above the (equilibrium) behavior also described above leads to the Borda rule choice!

Finally, why the conclave? In my three-candidate example above, the vote will soon be at roughly 60% for B and roughly 40% for C. We could now be stuck here forever. But two thoughts will now go through the voting cardinals’ minds (both mentioned in Robert Harris’ novel): for their own personal comfort they don’t want to be in conclave forever and also for the sake of the reputation of the church and the new pope they want the election over after not too many days of conclave. This gives the cardinals an incentive to switch from C to B (even if this seems to violate their oath yet again), so that finally a new pope is eventually elected.