Protagonist (main character in the battle or conflict) and antagonist (person opposing the protagonist) are terms originating from and applicable to Greek tragic drama, and much serious drama since Sophocles. But Pirandello is using the conventions of modern drama to make a dramatic statement about a philosophical notion – the family of characters in the play cannot define themselves or act “in character” because they have been disengaged from their putative creator, the inventor and life-giver called the author. As a metaphor for the existential situation of existing without essence or purpose, the play moves into the philosophy by reversing the “normative” expectations not only of an audience, but of society itself. “Where is our direction, our climax, our reason for being?” This is an early work compared to the French Existential or Absurdist movements of Sartre (his seminal treatise, Being and Nothingness, is dated 1943) and Camus (his play, The Flies, was performed that same year and was how the two met). Pirandello’s dramatic inquiry, while early, postdates such works as Buchner’s Woyzeck (1836) and even Calderon’s Life is a Dream (1635).
(The internal plot, it could be argued, has the father as the protagonist and the director of the rehearsal as the antagonist, in that he obstructs the father in his pursuit of getting his story told. But this approach to the play's "plot" is unrewarding.)