Pope Benedict XVI — né Joseph Ratzinger — has announced that he will step down at the end of this month. In doing so, he becomes the first pope to resign in 598 years. The last resignation, in 1415, occurred when Gregory XII stepped down to end the Western Schism in the Catholic Church, in which rival popes and antipopes, each recognized by a different set of secular governments in Europe, claimed sovereignty over the church.

Which is to say that this is a pretty strange occurrence. But, as with normal papal successions, it will prompt the vote of the College of Cardinals, a group of up to 120 church leaders (current estimates put the number around 118) below the age limit of 80 who convene to elect new popes. Exactly how that process works, however, changes frequently, and indeed has changed since the election that elevated Benedict in 2005.

NYU political scientist Joshua Tucker and PM at Duck of Minerva have compiled a good set of political science research into papal elections. It's a sensitive subject because, as GWU professors Forrest Maltzman, Melissa Schwartzberg and the late Lee Sigelman put it in their paper on Benedict's election, "Officially, Ratzinger’s selection was attributed to the will of God, a force not amenable to any empirical test that is in our power to conduct."

But unofficially, Benedict was selected in accordance with the wishes of his predecessor, John Paul II. For most of John Paul's tenure, papal elections were subject to a supermajority requirement, with a two-thirds majority required to finalize a selection. As Maltzman et al show, by the middle of 1990 John Paul had already appointed two-thirds of voting cardinals. Assuming his appointees all agreed on a candidate, they could have outvoted any previous appointees from 1990 until John Paul's death in 2005 and installed a candidate along John Paul's preferred lines:

But as the above chart shows, a funny thing happened in 1996. John Paul II issued Universi Dominici Gregis, a document revising the two-thirds requirement. In filibuster parlance, he went nuclear. As the authors note, the timing here is funny. He already had a supermajority of appointees in the college. This seems to refute the notion that the change was intended to help secure a future pope who would continue John Paul-like policies.

Instead, they argue, what drove revision was a desire to prevent gridlock. There were three likely candidates for pope in 2005 (according to these authors; others disagree). There was Benedict (then Ratzinger), a Vatican insider with a reputation as a doctrinaire conservative. There was Carlo Maria Martini, a quite liberal Italian cardinal and former archbishop of Milan who died last year and supported same-sex civil unions, a right of the dying to refuse medical treatment and the distribution of condoms as a "lesser evil" to AIDS transmission. And there was Jorge Mario Bergoglio, the archbishop of Buenos Aires, who earned support from cardinals in the developing world and holds fairly mainline Catholic views. Not only did no block have a clear majority, but a "voting paradox" was at work:

To see what's going wrong here, suppose you're in the Martini bloc. Getting either the Ratzinger bloc or the Bergoglio bloc on your side would put you over a two-thirds majority. You prefer Bergoglio to Ratzinger, so you go to Bergoglio first. But he prefers Ratzinger to you, so he turns you down. You could go to Ratzinger, but you really don't want to make concessions to that faction. And Ratzinger and Bergoglio can't put together a two-thirds majority between them, and even if they could, Ratzinger doesn't like Bergoglio as much as he likes Martini. Everything's deadlocked. A majority election wouldn't eliminate the possibility of deadlock, but it would make it much less likely.

Maltzman et al hypothesized that even if John Paul II hadn't noticed this potential problem, he likely was talking to someone who had. In 1994, they note, John Paul appointed the Pontifical Academy of Social Science, meant to provide the church with advice from political scientists, sociologists and economists of note. One of the original appointees was Kenneth Arrow, the Nobel-winning economist whose most famous work concerns voting paradoxes. His Arrow impossibility theorem proved that it is impossible to take the ranked preferences of a group of voters and turn them into a societal ranking that conforms with certain basic rational and fairness requirements (for example, one requirement is that one person ranking an option higher shouldn't hurt its societal ranking). The relevance of that work to the Pope's dilemma should be clear enough.

Regardless of whether the political scientists' history is right and Arrow's views really did push the cardinal election process in this direction, their point is important for this next papal election because Benedict has reversed John Paul II's repeal of the two-thirds requirement, the very repeal that enabled Benedict to be elected in the first place. In the past, this has led to compromise selections like, well, John Paul II, but if a voting paradox arises, the church could be in for considerable gridlock.