The US is divided up into political districts. Each voter gets to vote for one member of the House of Representatives and two members of the Senate in this bicameral national assembly. There are 435 voting members of the House so there are 435 congressional districts. The member from your district supposedly represents your interests, but none of the others do. In the Senate the political districts are the states. There are 100 senators, two from each state. The two senators from your state supposedly represent your interests but none of the other 98 do. So 1/435 or 0.22% of the members of the house represent your interests, and 2/100 or 2% of the members of the senate represent your interests.
This is a pathetic situation, and it's even worse if you did not vote for any of the congressman or senators who supposedly represent you. Say you're a Democrat and the congressman elected from your district (whom you didn't vote for) is a Republican. Then arguably you have no representation at all in the House. The same could be said of the Senate if you didn't vote for either of the senators who actually got elected. In other countries where they use other methods for making up the national assembly or congress like, for example, proportional representation, the percentage of the members representing each voter's interests is much higher. For example, if 28% of the electorate (including you) voted for the Green Party, then 28% of the seats in the national assembly would be Green Party members.
In a districtless congress each voter would vote for each representative, and each representaive would represent the interests of all voters. For example, if there are 300 seats in the congress and 500 candidates running for those seats there would be 500!/(300!)(200!) possible congresses or ways that this congress could be made up. In theory each voter could list each possible congress in order of his/her preferences, and then all the voters' specifications could be amalgamated to get the one congress that best represented the electorate as a whole. The problem is that it would be impractical for each voter to study the qualifications of each candidate and then come up with a list of all possible congresses. It would be too much work. However, there are ways to expedite this process as explained in more detail here. If each voter just listed the candidates (instead of the congresses) in order of preference, this list could be translated by software into an ordered list of congresses. Furthermore, the list of candidates could be simplified by using the recommend- ations of the voter's political party or other trusted experts in part or in whole. A customized list could be generated by taking eclectic recommend- ations cafeteria style. Or there could be different lists available to the voter depending on the voter's profile related to his/her political objectives. A simple questionnaire given to the voter could generate a list according to the voter's predilections. There are a lot of different ways any particular voter's list could be generated with the voter having complete control and the final say.
One way of amalgamating the list information is by range voting. Using this method each possible congress would be given a numerical rating, and the ratings for each congress would be added up over all the individual voters to determine the winner - the one with the highest overall social rating. There is no need to rank the possible congresses in order over the entire electorate since only the top rated one would be chosen. Therefore, Kenneth Arrow's model for social choice and his impossibility results as presented in Social Choice and Individual Values are invalid. In fact Arrow's model which calls for a full social ranking doesn't apply to most political as well as most economic situations. The only thing it seems to apply to is combining judges' rankings in Olympic figure skating where it is important to know not only first place but also second and third. In political and economic situations it's necessary only to know the top rated or first place result.