I've been pondering this one a while, I'm curious as to what the thought processes are for the people who make each choice.
According to the statistics, 81% of players killed Duck themselves.
I am not one of those players. I read "Of Mice and Men" at a very young age, and I believe that story has always stuck with me. Spoilers for that book. If you haven't read it, I strongly advise you do so. In it, there is an old man (Candy) who has a dog, who is also very old. Old enough that he needs to be put down. Candy cannot bear to kill his good friend himself, so sits inside while his dog is shot by one of the other ranch-workers.
Later on, he says to George (the main character) that "I ought to of shot that dog myself, George I shouldn't ought to of let no stranger shoot my dog." He expresses that he will forever feel regret.
The book ends with Lennie (the deuteragonist) having accidentally killed a woman. A lynch mob is after him. George learns from Candy's mistake and shoots Lennie himself, rather than let a bunch of strangers do it for him.
That's why I help Kenny shoot Duck. It's his son, and for that reason I think he would regret not doing it himself. He later confirms that in Episode 5, that he was glad to be the one that did it.
So why do you pick your choice?