One interpretation of the play is that all three characters were at one time real living people. Deeley met Anna first and slept with her, then later met Kate at the movies. Kate may or may not have been the friend Anna spoke with at the party. Deeley began dating Kate, and Kate found out that Anna was trying to steal him from her, so she killed Anna. Anna's death upset Deeley (he stared longingly into her empty bed), and Kate then killed him, too. Once he was dead, Kate's mind took over, imagining him hopelessly in love with her. She has lived the past 20 years in a fictional world where Anna and Deeley love her instead of each other.
Another interpretation is that Kate and Anna are different personalities of the same person, Kate being the prominent one. Deeley met "Anna" first, and the friend at the party was one of the many friends Anna had that Kate mentions in the first scene. Deeley then met Kate at the movies. Deeley cried in the chair when he discovered Kate's mental issue, and stared sadly at the empty bed before hugging Kate. Kate "killed" Anna for Deeley's sake. 20 years later, she tells him that Anna is returning, and he does all he can to keep Kate from allowing Anna back into her life, ultimately succeeding by the end of the play, when Kate kills Anna again by recalling the first time she killed her.