Its extremely rare that two members of the same party compete for the same seat. If its an important office that is a sign that the party leadership has failed spectacularly in getting consensus, and almost guarantees defeat in a competitive election. Abraham Lincoln famously won the presidency this way when the democratic party split, allowing him to win the presidency with just the West and Northern states (as it happened four different people won electoral votes, being the most fragmented presidential race in history).

For all of its many faults, the primary system that the Republicans are enduring now is designed to ensure that the party eventually unites around one candidate, and in that it has mostly been successful. Thus I think your method will just needlessly complicate the vote in the 90-99% of cases where two members of the same party are not running against each other/

Or you could have a Democratic Government where the majority rules ... Oh Wait

There are some practical reasons why US founders formed a democratic republic as opposed to a pure democracy.  The situation and times they lived in guided them on that.

There were factions desiring peace and reunification with England...there were Colonial independence types who wanted the colonies to go their separate ways once independence was achieved and rule autonomously over themselves.  There were religious and antislavery groups who wanted universal law on their terms.  These and other blocs furiously labored to stir public opinion in their favor and as is true today would use sensational statements to stir public emotion to act.

These lobbying factions could keep a new country in a near state of anarchy--and look at all the new 'democracies' that are in places with traditional tribal and ethnic factions--they were pretty much right.  Even today with two parties, we are bogged down dealing with critical issues and can't resolve them.

The ancient Greeks had a pure democracy but they also had incredibly high expectations of citizens.

The reason in the US for parties was to force groups with agendas into one or two blocs of consensus so practical governance could occur without factional issues always being present.

Changing the manner in which votes are wieghted may have some affect.   But the US system needs much more than that.

The USA electoral system has several basic faults that allow minority parties to take office (W bush, both terms, for example).  One is the electoral college and 'winner take all' (the votes) system thats in place on many levels. Re: votes cast for president: if popular vote, (as opposed to votes ground up and processed thru the electoral college system) were in place, many small mid western states, with very low populations, would find their 'influence' decreased.  Votes are not equal in the USA.  The vote of a citizen in, say, N. Dakota, weighs more than the vote of a citizen in large population states like TX, CA, NY. 

Ditch the electoral college, and make the voting for the US president a truly national vote, not a vote based on how well the politicians do in manipulating 'key' states in an attempt to 'manufacture' a majority in the elctoral college.

Secondly, the gerrymandering that is rampant in the system and is currently so acceptable now must end.  Fifty years ago, politicians were voted out of power because they abused the redrawing of district lines.  Now its so common.  Ever notice how districts often cut up cities, placing a small portion of the city population in with a very large surburban, rural population?  The effect of this is to guarentee that the poor people in the city have NO representive.  They can't pool their votes and elect even one of their own because they are divided into several chunks.  And each chunk is, within 'that' district a small minority.  Both major parties play this game...  (sadly).  Currently, its TX trying to minimize the hispanic-american vote.  And...

The new monster to rise and distort 'democracy' in the USA is the mating of corporate personhood and 'quiet money' financing of election campaigns.  Now Uber-corporations, (many of whom are not 'citizens' of the USA, but more interested in'global' markets,') can speak very loudly in election campaigns, via their immense wealth.   Their voice is louder than mine, and reaches many more possible voters than my voice can.  To make matters worse, the people have no idea who is really voicing an issue, as the uber-corp (and the entrenched plutrocracy that mostly controls corporate agendas) can now spend these mega millions of dollars secretly.  The very idea of privacy, which was intended to protect individuals from the power of organized groups (govt, military, etc) is now being used to shield uber-corps at the very time when technology is virtually eliminating individual privacy fro the masses of people.  (Oh, some individuals can have privacy if they are protected by the walls of a corporation.  "Its good to be the King."  LOL)

just a decade ago, one presidental candidate had issues because 'foreign' interests were contributing to his campaign. The GOP (Greedy 'Ol Plutocrats) were all over this candidate (Clinton, I think) for taking foreign money.  And I agreed with the GOP stance.  I'm not sure how foreign these corporate 'persons' are.  I havn't seen their birth certificates.  But their corporate policies seem to hurt main street americans and favor economic elites in many non-democratic countires.  Why can these 'uber-corps contribute to domestc campaings, and secretly at that? They shouldn't. 

SO thats it.  Popular vote for president, not an electoral college.  Districts with boundries that don't intentionally carve up groups so as to eliminate / dilute their voices.  Remove all private monies from campaigns, severly limit amounts non-individuals (corporate/associations/ etc.) can give, and/or limit to individual giving.  No privacy for corporations, associations, etc., when it comes to election issues.  And apply the RICO - conspiracy/ money laundering laws to corporations who meddle in elections.