Making a hit movie on a budget is as hard as Hollywood makes it look. "We're in a business where the solution is almost always to write a check," said Joe Drake, the departing co-chief operating officer of Lionsgate, the studio behind "The Hunger Games." "The Hunger Games" opens Friday, tracking toward an opening weekend ticket take of perhaps more than $100 million. The movie, about a future dystopia that pits teens in televised fights to the death, cost around $80 million to make. That amount...