What is the fewest number of cardinal electors are needed to validly elect a pope?

Question

What is the fewest number of cardinal electors are needed to validly elect a pope?

Pope St. John Paul II, in his Apostolic Constitution Universi Dominici Gregis on the vacancy of the Apostolic See and the election of the Roman Pontiff has this to say on the electors of a papal election, but does not provide us with the minimum number of cardinal electors needed to validly choose a pope:

33. The right to elect the Roman Pontiff belongs exclusively to the Cardinals of Holy Roman Church, with the exception of those who have reached their eightieth birthday before the day of the Roman Pontiff's death or the day when the Apostolic See becomes vacant. The maximum number of Cardinal electors must not exceed one hundred and twenty. The right of active election by any other ecclesiastical dignitary or the intervention of any lay power of whatsoever grade or order is absolutely excluded.

What happens if the conclave is to be held during a time of war or some other event and the possible number of cardinal elector at Rome dwindles to 1, 2, 3 or 4. Is there a minimum number of cardinals needed to produce a sovereign pontiff?

We know that conclave of 1277 had the fewest number of cardinals in a papal election:

The papal election of 1277 (May 30 – November 25), convened in Viterbo after the death of Pope John XXI, was the smallest papal election since the expansion of suffrage to cardinal-priests and cardinal-deacons, with only seven cardinal electors (following the deaths of three popes who had not created cardinals).3 Because John XXI had revoked Ubi periculum, the papal bull of Pope Gregory X establishing the papal conclave, with his own bull Licet felicis recordationis, the cardinal electors were able to take their time. After six months of deliberation, the cardinals eventually elected their most senior member Giovanni Gaetano Orsini as Pope Nicholas III. From the end of the election until Nicholas III's first consistory on March 12, 1278, the number of living cardinals—seven—was the lowest in the history of the Roman Catholic Church. - Papal election, 1277

I have heard about the possibility that 4 cardinal electors would be allowed, but I can not find any documents to support such a number.

Answer 1

There is no minimum number of cardinals set in the apostolic constitution Universi Dominici Gregis, written in 1996 by Pope St. John Paul II. This document is the current norm for papal elections. It prescribes such things as the length of the dead pope's funeral, the procedure for managing Vatican City and the Diocese of Rome while the See is vacant, and the procedure and timetable for setting up the conclave to elect the next pope.

There are groups, or at least one rotating group, of cardinals, called the "Particular Congregations", which oversee day-to-day operations of the Vatican:

The Particular Congregation is made up of the Cardinal Camerlengo of [the] Holy Roman Church and three Cardinals, one from each order [that is, a Cardinal Deacon, Cardinal Priest, and Cardinal Bishop]. ... The office of these Cardinals, called Assistance, ceases at the conclusion of the third full day, and their place is taken by others.

The Particular Congregations are to deal only with questions of lesser importance which arise on a daily basis or from time to time.

(Universi Dominici Gregis, sections 7–8)

Perhaps this is where your "four cardinals" idea comes from.

The apostolic constitution specifies that all the cardinals who are not "legitimately impeded" (section 7) must come to the conclave as soon as they are informed of the pontiff's death. No further mention is made specifically of any number required in order to cast a valid vote for the pope. Presumably this is to allow an election regardless of how many cardinals are able to make it to Rome.

Answer 2

It seems true that there are no explicit restrictions on the minimum number of cardinals with voting rights that can participate in the Conclave (notice other cardinals also participate but do not vote). Yet, logically, because of the voting rules, there is a minimum.

For a start, there is a rule, first brought forward by Pope Gregory XV (Aeterni Patris, 1621 and Decet Romanum Pontificem, 1622), that voting cardinals cannot vote for themselves. Furthermore, there is the rule that the potential winner should not be counted among those who elected him. Thus, logically, N=1 is ruled out.

What about N=2? Well, by definition, each of the has to vote for the other or null. But given that another very important rule regarding the design of the voting system is that of anonymity (i.e. who votes for whom cannot be known), even if one of them votes null (the only way the deadlock could be broken), anonymity will not hold.

N=3 is safe however, because there is more than one way in which a winner can arise. First, there is the "consensus", whereby one candidate receives two votes and another one (that of the winner, provided he respected the no self-nomination rule). Second, when two of them vote null and the other votes for one of those who voted null. It is impossible to tell which occurred, beyond that at least one of the non-winners voted for the winner.

In conclusion, based on the voting system rules and principles, 3 is the minimum of cardinals with voting rights that can participate in a papal conclave.

PS: Some background notes and analysis of the voting rules can be seen in this document.

Answer 3

There is nothing in Canon Law about a number. It does provide general guidelines:

Canon 176 Unless it is otherwise provided in the law or the statutes, the person who has received the requisite number of votes in accordance with Canon 119, n. 1, is deemed elected and is to be proclaimed by the person who presides over the college or group.

And 119:

Canon 119 In regard to collegial acts, unless the law or the statutes provide otherwise: 1° in regard to elections, provided a majority of those who must be summoned are present, what is decided by an absolute majority of those present has the force of law. If there have been two inconclusive scrutinizes, a vote is to be taken between the two candidates with the greatest number of votes or, if there are more than two, between the two senior by age. After a third inconclusive scrutiny, that person is deemed elected who is senior by age; 2° in regard to other matters, provided a majority of those who must be summoned are present, what is decided by an absolute majority of those present has the force of law. If the votes are equal after two scrutinizes, the person presiding can break the tie with a casting vote; 3° that which affects all as individuals must be approved by all.