Two characters are in a race both have the same base speed. There are no obstacles and both begin running at the same time. In this case chasing rules don't apply I guess.
So what RAW can be used in this case? DEX roll against each other?
Wow,good question. Alternatively you could also use Strength? Or make some kind of endurance check near the end to see who begins to slow down. Another question is whether or not the course is in a straight line? There could be some play with those curves and who is on the inside and outside of them.
For a longer-distance race I would use this rule:
A character can run for a number of rounds equal to his Constitution score on the local scale without needing to rest.
For a race within the Constitution score for running of all participants...not sure.
Since there is no athletics-skill, I would let them roll DexMod+StrMod+d20 against each other. And depending on the length of the race, remember that you can only run for a certain amount of rounds. After this, you have to make a fortitude check, I think. Can't seem to find the rules at the moment though. This page suggests only dexterity rolls.
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Just have them both run, as the full-round action. Both players can run a number of rounds equal to their constitution score, after which you must make a DC10 Constitution check to continue running. You must check again each round in which you continue to run, and the DC of this check increases by 1 for each check you have made. The first character to fail their check loses the race.
Depends on the length of the race.
If the race is shorter than Speed x 4 x Con for the lowest Con score, then by RAW it should go entirely off Initiative. (Optional: Roll initiative each round, but it really doesn't make any difference - it all comes down to one initiative roll, in the end). Other people have defined the rules which cover longer races.