He spends most of the movie inebriated, and when people take his booze away from him, he gets downright violent. Yet he’s Peeta and Effie’s advisor, being the only other person from District 12 who ever won the Hunger Games. As much as they don’t like him, they have to trust him.
Why? In addition to being a survivor, he’s a celebrity to the people in the Capitol. He can lobby the rich and powerful to send goodies to Peeta and Katniss. That can be the difference between life and death in the arena, and having been through it himself, he knows how important those goodies can be.
Initially, Haymitch looks like an alcoholic nitwit who cares only for the next bottle he’s about to guzzle down:
No mention of, you know, helping the kids who are about to go to slaughter? Not a great look, Haymitch. Having said that, we’re inclined to cut old Haymitch a little slack. After all, he’s been doing this ever since he won the Hunger Games himself, and every year he gets to meet two nice young people who subsequently die horribly within a few days.
Every. Single. Year.
That can wear on a guy, but because he’s obligated to do this, he has to deal with it over and over again. So yeah, he drinks. We can’t condone it, but we can at least see where he’s coming from.
This time, though, Haymitch sees a glimmer of a chance in these two. (Or maybe he does it with every District 12 Tribute, we may never know.) He pushes them harder to earn sponsors, to behave in ways that draw their attention, and to otherwise do everything they can to make it through the Games.
He knows how to do this because he’s played the game for years. Appearance is everything in this world, and being memorable is often more important than being tough. Look, for example, at the advice he gives Katniss on impressing the judges:
HAYMITCH: I don’t know how else to put this: Make sure they remember you.
Unfortunately, Haymitch (Woody Harrelson) won’t be playing in the new upcoming Hunger Games launching in 2018.