A new poll of likely voters in the Third Congressional District suggests the Democratic primary winner in the race to succeed US Rep. Niki Tsongas will be elected without a clear mandate from voters.

It’s become a familiar story in Massachusetts, where incumbents dominate and the rare open seat attracts so many candidates that the winner typically fails to garner support from a majority of the voters. The problem is so pervasive that Paul Schimek, in an analysis written for CommonWealth, concluded that democracy isn’t working in Massachusetts.

The poll of voters in the Third Congressional District illustrates the problem. Tsongas in August announced her decision to leave office at the end of this term, setting off a scramble among would-be replacements. The open seat has attracted 13 candidates in the Democratic primary, an unwieldy number that makes it difficult for voters to keep track of the challengers, let alone discern differences among them. The end result is typically voter apathy and low turnout.

Professor Joshua J. Dyck of the UMass Lowell Center for Public Opinion, which conducted the voter survey with the Boston Globe, highlighted the problem in his report on the poll.

“A campaign that has thus far been waged as a series of fundraising disclosures shows an electorate that still is relatively unfamiliar with many of the candidates,” he said. “More than half (59 percent) of Democratic primary likely voters surveyed said that they have not yet made up their mind about who to vote for in this race. Thus far, the electorate reports modest interest in the race with just over one third of likely voters (36 percent) saying they haven’t really been following the race much at all.”

The goal in such crowded fields is not to win over a majority of the voters in the district but to beat out the other candidates. Since there are so many candidates, it doesn’t take that many votes to win. Candidates in such situations often tailor their message to narrow groups of voters. Making matters worse, the winner of the Democratic primary is heavily favored to win the general election.

In the UMass Lowell poll, Rufus Gifford, a recent transplant to the district and a former ambassador to Denmark, led the pack with 11 percent support. He was followed by state Sen. Barbara L’Italien at 7 percent, Lori Trahan at 5 percent, and Rep. Juana Matias and Dan Koh at 4 percent. The list of candidates in the Democratic primary is so long that the Globe, in its story on the poll, didn’t even list all of them.

A number of states are experimenting with ways to overcome the problem. One approach that is gaining some currency in Massachusetts is ranked choice voting. Instead of voting for just one candidate, voters cast ballots that rank all of the candidates from top to bottom. If no candidate wins majority support in the initial count, the candidate with the fewest votes is tossed from the race and the votes of his or her supporters shift to their second choice. The process is repeated until a majority winner emerges.