The more I think about it, the more I realize that Leary can be perceived in different ways.
What I see is Leary taunting Horrigan over and over again, making him relive the most painful moment of his life, taunting him about his wife leaving him, his drinking, and that he has nothing worth living for. Leary went after what was left of Horrigan's soul, trying to hurt him over and over again. He had expected that he'd make him fail to save another president.
there is enough evidence to be found to sustain that point of view. The only reason I post again is to try to explain why I see Leary differently. This will ba a long post, because there are some very valuable and good points I have to reply to, so all those not so interested...

I've always assumed that Leary did choose Horrigan. But how can we be sure it was not a hazard? On the "murder-collage" there are clips of several incidents involving Secret-service agents. John, F. Kennedy, Robert F. Kennedy, Ronald Reagan. So, how can we be sure Leary wanted Horrigan, and noone else? I don't know, it just always seemed obvious to me that he wanted Horrigan as an adversary. Why? There were certainly, among the other attacks, some secret service agents who would have been intelligent, capable adversaries in the game. So why Horrigan? Because by accident he was the one to inspect the apartement? I just can't believe this to be an hazard. And, if Leary wanted Horrigan, why? To taunt him, hurt him? To have better chances for succeeding, since Horrigan had failed once? Again, I've just assumed that there were other reasons. I felt that Leary did want to give someone a chance he himself did not have. Leary was hurt too much, he knows he has to die. But Horrigan, as I pointed out before, could maybe succeed in redemption. And Leary gets him back in the game several times. I know that this scene can be interpreted in many different ways. But this is how I saw it. When Horrigan finally gets into that ballroom, Leary is smiling. The smile then fades, as he concentrates and aims at the president. I've understood this as if Leary was glad to see Horrigan join in at the last moment. This way, the game is more exiting, but Horrigan also still has the chance to proove that he will take the bullet. Then, later, in the final dialogue in the elevator, this is what Leary tells Horrigan:
I am waiting for you to show me some go... gratitude. Without me, you'd still be another sad-eyed, piano-playing drunk. I brought you into this game, I let you keep up with me, I made you a god.... hero today (...) I redeemed your pathetic $#!tty life
Horrigan then betrays Leary by talking to Raines and Leary at the same time.
These are the words left on the answering machine:
Hello Frank. By the time you hear this, it'll be over. The president is most likely dead, and so am I. I wonder Frank, did you kill me? Who won our game? Not that it matters, for among friends like you and me, it's not wether you win or loose, but how you play the game. And now the game's done, and it's time to get on with your life. But I worry that you have no life to get on with, Frank. You're a good man, and good men like you and me are destined to travel a lonely road. Goodbye and good luck
I am perfectly aware of the fact that you can understand this as rubbing salt into wounds. But I never did. I understood this speach, spoken in a soft, tender voice, as a real good bye and good luck. He realizes that Frank's life will not be easy, but he only speaks of lonelyness, not of demons! I added emphasis on the words that made me think that this was a sincere good bye.
He had fun with him and he enjoyed having a worthy adversary but he thought all along that he was going to be successful in the end.
I also interpreted that Leary is not that sure of succeeding. To me, Leary behaves like a wounded animal. He's dangerous because hurt, but in reality he needs help. Since it's too late to help him, Leary tries to help someone else...