I'm not an arbitration company, nor do I specialize in criminal justice. I cannot say what means would be available to arbitration companies to determine guilt. However I can say with reasonable confidence that it would be very much in the interest of competing justice companies to make very sure they deliver swift and sure justice, and so they would certainly develop extremely advanced means of doing so.
If DNA proved to be unavailable for example, perhaps the victim would volunteer to have his/her memory reconstructed in the interests of identifying his/her attacker.
As to whom they are proving it too, it would be to themselves, their clients, and every hawkish competitor just waiting for a chance to discredit them. Most people in a society where we defend ourselves, would have the means to capture evidence of an attacker. There would be a wide variety of methods for self defense easily accessible to most everyone. One of the perks of a defense contractor's services might have to do with this.
Its impossible to say what innovations might occur. What is clear and provable is that competition breeds excellence whereas government monopoly breeds incompetence.
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