It would be completely up to you - But it's killing your ducks - so....
More lives are being lost.
Many or one? --- I wouldn't want to, but it might just be a must.
I'm extremely sorry for your losses. I would call Animal Control or even the Game Wardens to see if they can do something, and also ask $30 - $40 per duck.
If you still can't any results - If the birds were fenced in on YOUR property than you have the right to do away with the dog. This would be...