First, New York City Police Reports don't normally apportion fault for the accident on the police accident report (MV-104), unless the police officer actually witnesses the accident. Even then they usually just provide each drivers or pedestrians version of events, even when the officer issues a traffic violation to one of the parties.
As for your ability to sue "in small claims for compensatory damage, treble punitive damages..."; cases in NYC Small Claims Court are limited to a maximum claim of $5,000.
If it is the adverse party's insurance company that covered the loss, they would not have paid you unless you first signed a general release in favor of the adverse party. In that case you would be barred/estopped/precluded from suing them.
If your insurance company covered your loss, the insurance company would have the right of subrogation, that is to "stand in your shoes" and sue the adverse party to recover $ they paid you. In that case, your our own policy may bar/prohibit you from bringing the suit. Note that if your carrier covered the loss and you subsequently sued the adverse party and recovered damages, you would be required to repay the $ your carrier paid you.
PLEASE NOTE: this response is not legal advise and does not create an attorney client relationship. I implore you to consult with a licensed attorney so that they may ultimately guide you accordingly. Please also note that there is a time limitation on the length of time you have to file suit in NY and if suit if not filed with a certain period, you may be forever precluded from filing an action.
Forget the treble punitive damages; there is nothing in this scenario that would justify that; you can sue for your deductible. You will have to decide whether that is worth your time and effort. Good luck.
The author of this post is licensed to practice law in the District of Columbia, Maryland, and Virginia. This post is intended as general information only, and is not provided as legal advice in connection with any specific case, and does not create an attorney-client relationship.
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