Who shall we kill?

I've seen several versions of the story in which someone overhears two people discussing the best way to kill someone. The eavesdropper is always highly alarmed, until it's revealed that the conversation is actually about the plot of a book or a play.

Crime and mystery writers find themselves in this situation every time they begin a new book. Who shall we kill and how shall we do it?

First, is it necessary to kill anyone? Dorothy L. Sayers, one of the greats of the Golden Age of British crime and detective fiction, wrote one book in which nobody died and others in which there was no murder. 

In an Agatha Christie type mystery, the murder happens near the start of the book. The character who is murdered was created solely for that purpose. He or she has no other function in the story. The reader will not have come to know the victim well. He or she is often an unsympathetic character whose death is not greatly regretted by those around him or her. 

What about other types of crime or mystery novels? Is it essential to have a murder? 

A murder, or at least a suspicious death, increases the tension. It raises the stakes; the investigation must succeed, or there will be a murderer going free, with the possibility of more deaths. A murder increases the danger for the character who is investigating the mystery, especially if he or she is an amateur sleuth. 

If the death happens some way into the book, the victim's character will have been developed to some extent. His or her relationship to the central characters will have been established. 

If the victim was an unpleasant person, will anyone, whether the reader or a character within the novel, care very much about his or her death? Will it have any dramatic or emotional impact?

If the victim is someone who was close to the investigator, that will increase his or her determination to solve the mystery and catch the criminal. On the other hand, if the victim was close to other characters in the story, they have to be allowed time to grieve, and also to deal with the practicalities surrounding a death. There is a danger that the plot is neglected while this is happening, and the pace of the story is slowed.

And more importantly, if a writer kills a character whom readers have come to know and like, there is a risk that they will be alienated. They might not finish the book. They might not want to buy any more books by that author.

It might be the case that, in the author's  opinion, killing that character is absolutely the right thing to do in the context of the plot. Should the author go with what feels right for the book, or do what is most likely to please readers?

So, who shall we kill?