In my grandmother’s kitchen, when you slammed one cupboard door, another one somewhere would pop open; if you slammed that one, a third would open.

If we try to find a completely satisfactory way to handle three-candidate elections, we run into a “cupboard door” problem. This is called “Arrow’s impossibility theorem,” proved in 1951 by Kenneth Arrow.

At the moment, we elect the governor using the “plurality system.” Each voter casts a vote for one candidate, and the candidate with most votes wins. In a three-candidate race, a candidate needs only a few more than a third of the votes to win; if there are four candidates, the winner needs just about a quarter. The biggest tribe is not necessarily a majority.

Can we find an election system that does not elect minority governors? Yes, in about a dozen ways.

A citizen initiative has placed “Hare voting” on Maine’s November ballot. It is one of several species of “instant runoff” voting. In Hare voting, each voter lists the candidates in order of his preference. If nobody has a majority of first-place votes, the candidate with the fewest first-place votes is dropped. Each ballot with that candidate as first choice is then reassigned to the ballot’s second choice and the ballots are recounted. If, still, nobody has a majority, the candidate with fewest top-choice votes is eliminated and the ballots are reassigned and recounted. The process continues until some candidate has a majority.

Here in Maine, we use a third system to elect school boards and other similar groups: “approval voting.” Suppose there are three seats to be filled and five candidates. Each voter chooses any candidates he would be satisfied to see on the board. All votes for each candidate are counted, and the three candidates with the most votes are elected.

This system also would work to elect the governor.

Let’s look at the 2014 Maine gubernatorial election to see how these electoral systems work. Because it was conducted on a plurality basis, we do not know who would have been the second choice of voters, but we can make some guesses.

The actual result was Gov. Paul LePage, 295,000 votes; Mike Michaud, 265,000; and Eliot Cutler 52,000. Let’s guess that nearly all Michaud voters would have rated Cutler their second choice, nearly all Cutler voters would have rated Michaud their second choice, and a quarter of LePage voters would have rated Cutler a second choice.

LePage won a plurality election with 48 percent of the vote. If the Hare system were used, Cutler would be eliminated and, in the runoff, Michaud would receive Cutler’s 52,000 second-place votes, winning the election over LePage 317,000 to 295,000.

Using approval voting, LePage once again would have 295,000 votes. Michaud would have the votes of his 265,000 supporters plus the second-place votes of Cutler’s 52,000, giving him 317,000. But Cutler would win the election with 390,000 votes: He would have his own 52,000 votes, plus 265,000 second-place votes from Michaud’s supporters and 74,000 second-place votes from LePage supporters.

Surprising? After all, Cutler received the fewest votes in the plurality election. But he was “everybody’s second choice” — the candidate nearly everybody could live with.

What are some of the problems with the Hare system that will be voted on under the citizen initiative?

First, and probably most important, it requires special, new voting machines. Voting machines contain software to throw out “spoiled” ballots. The criteria for spoiled ballots are new and complex for Hare ballots.

Second, most electoral systems — for instance, “plurality” and “approval” — allow for counting the ballots where they are cast and phoning the results to Augusta. To make Hare voting work, all ballots must be counted in a single batch, so the ballots themselves must be packed up and carried to Augusta to be counted there. This delays election results and is a considerable expense. (Hare voting has been most successful in local, municipal elections, where the votes are already counted at a single, central place.)

Third, Hare voting is not “monotonic”: A candidate can lose because he is too popular. How can that be? Which candidates advance to the second round of Hare voting depends on who is eliminated in the first round and where those votes are transferred. Popular Candidate A can knock out an opponent who then transfers strength to A’s most important rival. It’s like what happens in a basketball tournament, in which receiving a higher seed may place you in a tougher bracket.

Choosing an electoral system is not a simple affair. Although the Hare system does solve one important problem, it does not solve every problem: No system does or can.