He should be required to pay whatever costs the court believes were reasonably incurred. Why should anyone else but Lehrmann be required to pay a cent when the court has found him to be a rapist? If people who bring cases like this aren't liable when they lose, they're really taking a "Heads I win, tails you lose" approach to litigation. He was trying to impose a heavy imposition by way of damages on those he brought to court. Having lost, he should now have to bear the full weight of it himself.
