Short answers, pulled from the story.
F. S. Boas published Shakespeare and His Predecessors in 1896 to introduce this label for three specific plays by William Shakespeare. The book appeared through the publisher John Murray during its third impression that same year.
Boas selected All's Well That Ends Well, Measure for Measure, and Troilus and Cressida as his primary examples of this new category. These three works sat uneasily between traditional comedy and tragedy when viewed through earlier classification systems.
E. M. W. Tillyard published Shakespeare's Problem Plays with Chatto and Windus in London during 1949. This publication occurred long after Boas first defined the category in the late nineteenth century.
Neil Rhodes argues that the defining characteristic is its controversial plot rather than any fixed set of rules. He suggests the subgenre becomes less distinct as scholars continue to debate controversies in straightforward tragedies and comedies.
A.G. Harmon argues that in four of the plays he categorizes as problem-plays, the social order is restored when faulty contracts are properly amended. Chaos is vanquished by the institution of a new order following this pattern.