Supporters of direct representation cite the following benefits:5
Suppose there is a congressional district with 400,000 residents eligible to vote and three candidates running to represent it, A, B, and C. Further suppose that in the election, A receives 90,000 votes (45% of total), B receives 70,000 (35%), and C receives 40,000 (20%), with the remaining eligible voters declining to vote. Under the plurality voting system, such as that used to elect members of the United States House of Representatives, A would "win" the election and hence only A would be allowed to vote in the legislature, with the weight of a single vote. Under a direct representative system, A, B, and C would all be allowed to participate in and vote in the legislature, with each candidate being able to cast the number of votes equal to the number they received themselves in the election. Hence if B and C happen to agree on an issue that A disagrees with them on, with direct representation they can out-vote A on that issue (110,000 to 90,000) since they represent the choice of more voters in their district, whereas with representation determined by a plurality of voters, only A can cast a vote on the issue, while B and C, who together represent a majority of the electorate, are completely shut out from the legislature.
A direct representation scheme was proposed in Robert Heinlein's The Moon is a Harsh Mistress:
Suppose instead of election a man were qualified for office by petition signed by four thousand citizens. He would then represent those four thousand affirmatively, with no disgruntled minority, for what would have been a minority in a territorial constituency would all be free to start other petitions or join in them. All would then be represented by men of their choice. Or a man with eight thousand supporters might have two votes in this body. Difficulties, objections, practical points to be worked out— many of them! But you could work them out. . . and thereby avoid the chronic sickness of representative government, the disgruntled minority which feels— correctly!— that it has been disenfranchised.
Dave Robinson. "Direct Representation". http://directrep.org/article.html ↩
Anatole Beck (2001). "A More Proportional Representation". http://www.math.wisc.edu/~beck/proxy.html ↩