[asphaltgambler]

From experience. Since real street racing is illegal, there are no rules except what the actual racers agree to. To answer your specific question, some will run what's called 'Chase is the race'. Which means either car can leave early with no penalty. Depending on what everyone agrees to, if the other racer 'sits' and does not move the car that left first is automatic loser. A different twist is the other car must chase after and hopefully get to the finish line 1st.
But in this case a sucker bet for sure. Basically by agreeing to that you're giving away any shot at winning, my opinion.

But a lot of guys have big egos, fast cars and money to gamble away. They'll agree to chase is race with no penalty and are certain they can still run the opponent down.

To reference the OP scenario, the guy that gave chase was stupid. If he didn't move, he would've won as that was what the racers agreed to in that race. But as a driver it's hard to just sit when the other guy leaves, it's instinct.