Question

Hello Rabbi,
I wanted to ask a common question. When damage occurs—say, bodywork damage in a car accident to both cars.
The injured party claims with certainty, while the damager says he is unsure. On the other hand, the injured party also committed a traffic violation during the incident, but it is not certain that this is what caused the event.
The two sides exchanged details. 
Is the damager obligated to compensate the injured party in the heavenly court? Or does the whole obligation begin only if they go to court and the judge rules whatever he rules?
Because the damager genuinely is not sure how much he has to pay, what the value of the damage is, whether he really caused it, and it is not clear that he can rely on the injured party’s appraiser, etc., etc.
On the other hand, the injured party has no interest at all in going to court over this kind of damage. There is even a concern (for the sake of making the point, admittedly remote) that he might lose and even end up paying the damager.
So in short: when the damager would be happy to go to court—in particular since they would presumably reach a compromise—and the injured party has no time, is that what is called waiver? Or is there an obligation in the heavenly court to pay the injured party an unknown sum of money? (And what exactly does that mean?)
At first I thought simply that this is called waiver, and that this is exactly why there is a religious court—not only for dishonest people, but also to clarify the law.
But when I spoke with an avrekh, he claimed not so, and that this is the meaning of an obligation in the heavenly court.
I thought that the meaning of the heavenly court comes only after the human court, but not before.