Just like in the Road Rage mode, you will see passengers with a green circle around them. Now, you can actually bump the other player off of this person and take them - but beware! They can do the same thing to you as well. You can actually do this throughout the fare's ride, and it does not matter who gets the passenger there, but the one who actually delivers the passenger will get the $1,000 not necessarily the one who picked them up in the first place.
Make sure that you use your knowledge you gained from the Sunday Drive mode to where the alley ways, side streets, and short cuts are so that you can lose your opponent and get the passenger to their destination quickly and without worrying about the other person stealing them from you.
Once you drop this passenger off, you will see a new passenger appear somewhere close by and you will want to make sure that you get there first and get away before the other player can bump you. Remember, one bump is all it takes and the passenger is flipped to the other car.
Whoever gets to the set amount first is the winner of the head to head competition in the Simpsons Road Rage and you will then have to start over.