Proportionally elected congress
The most significant advantage to a proportionally elected congress (right now at least) is that it prevents the stagnation of a two party system. When even a small percentage of the population can elect people who directly represent them in government, they are less likely to throw their support behind candidates that they merely “
disagree with less than the other guy”.
One way to impliment such a system is to calculate the number of votes a party gets nationwide for a set of positions (eg House seats), then starting with the party that has the least votes required to win a single seat (more than 1/435th of the popular votes) the district with the highest percentage of votes for that party is assigned that seat. Repeat until all districts have been assigned a representative.
Alternative voting methods
The election of a president should be decided by some more optimal voting method. I'd suggest the Schulze method, since it has a lot of nice features and is simple for the end user (just rank the candidates in order of preference). the most obvious benefit is that satisfies the independence of clones criterion (which is to say, it doesn't suffer from “
spoiler” issues). It also satisfies the Schwartz criterion (which is to say that the winning candidate will always be from the set of candidates who is unbeatable by candidates outside the set... to put it another way, the order in which candidates are eliminated won't result in an unpopular candidate being elected, as can happen with instant runoff voting).
The electoral college should be eliminated. It serves no beneficial purpose in modern voting.
Requiring a district to be continuous and convex would eliminate a great deal of gerrymandering. To state it simply, a valid district would guarantee that a straight line drawn from any point in the district to any other point in that district would not cross any point outside the district.
There should be hysteresis, i.e. redistricting should only occur when population exceeds the nominal value (the intended average size of each district) by a specific threshold (eg 50% larger or smaller). This results in districts being more stable and prevents rapid redistricting. Also, redistricting should affect the minimum number of districts and the non-triggering districts should be chosen to ensure that the resulting districts are as close to the nominal population as possible.
National districts should not be dependant on state boundaries. This is particularly true given that states already have representation based on their own boundaries: the Senate. Even without the Senate, it seems more appropriate for national decisions to be based on the entire populace of the nation than on artificial subdivisions.
Redistricting could be done on a strictly mathmatical basis, such as the shortest splitline algorithm (rangevoting.org).
Another mathmatical alternate would be some algorithm that minimizes the total length of all equally populous district borders, similar to how soap bubbles equalize size based on pressure. In this case, the pressure inside the soap bubbles would be inversely proportional to the population it contains: high population means low pressure, which induces the bubble to shrink, until all bubbles contain the same population.
The Senate should be elected by the state legislatures, as it was done originally. This ensures that there is at least one party in the federal government interested in maintaining states' rights.