The Old Man starts out being kind of anonymous, hence the fact that he doesn't get an actual name. When we first meet him, he's just chilling in his rocking chair on a little platform jutting out from the stage.
From the stage directions, we know that only May and Eddie can hear/talk to him, but we don't know too much more until we really dig into the play… and find out that he's not only May's father, but Eddie's father as well.
We could probably leave our description there, but there are a few other key details we need to drop about the Old Man.
Yup, that's how he ended up with two kids that met and canoodled without realizing they were half siblings. We get all the dirty deets while Eddie is filling Martin in:
EDDIE: He had two separate lives. That's how come. Two completely separate lives. He'd live with me and my mother for a while and then he'd disappear and go live with her and her mother for a while.
THE OLD MAN: Now don't be too hard on me, boy. It can happen to the best of us. (496-497)
Um, yeah, we're not sure that bigamy is the kind of thing that can just kind of "happen to the best of us," Old Man, but if that's what you need to tell yourself, then go ahead…
Remember when the Old Man called Eddie a fantasist? Well, it seems that Eddie learned from the best, since the Old Man seems very interested in fantasy—in fact, he thinks that his fantasies should be recognized as reality, even if that reality only exists in his own mind:
THE OLD MAN: Ya' know who that is?
EDDIE: I'm not sure.
THE OLD MAN: Barbara Mandrell. That's who that is. Barbara Mandrell. You heard a' her?
THE OLD MAN: Well, would you believe me if I told ya' I was married to her?
EDDIE: (pause) No.
THE OLD MAN: Well, see, now that's the difference right there. That's realism. I am actually married to Barbara Mandrell in my mind. Can you understand that?
THE OLD MAN: Good. I'm glad we have an understanding. (118-126)
Although it's not immediately clear what "understanding" the Old Man thinks he's forged with Eddie, we figure it out toward the very end, when May is threatening the "reality" the Old Man has created for himself with the truth of what happened to Eddie's mother. The Old Man is furious when May says that Eddie's mom had killed herself after the Old Man abandoned her:
THE OLD MAN: Now tell her. Tell her the way it happened. We've got a pact. Don't forget that.
EDDIE: (calmly to THE OLD MAN) It was your shotgun. Same one we used to duck-hunt with. Browning. She never fired a gun before in her life. That was her first time.
THE OLD MAN: Nobody told me any a' that. I was left completely in the dark.
EDDIE: You were gone. (557-560)
You can see that the Old Man is pretty delusional—he's complaining that nobody had told him anything, but he fails to acknowledge that he just wasn't around to hear anything in the first place.
And he seems to want to remain in his little alternate reality bubble even right at the end of the play, after all the truth bombs, as he sits in his chair ranting about how the woman in the picture on the wall (Barbara Mandrell) is the woman of his dreams—and his property:
"Ya' see that picture over there? Ya' see that? Ya' know who that is? That's the woman of my dreams. That's who that is. And she's mine. She's all mine. Forever." (584)
Well, after the mess he made of his flesh and blood relationships, perhaps it would be safer for him to stick to fantasy women from now on? Fine by us.
Oh, and we should also mention that when the Old Man mentions "a pact," it echoes something Eddie said earlier to May—a nice clue Shepard drops to remind us of how similar those guys are.