I'm not going to vote, it's not something comparable. Despite seemingly coming to the exact same setting independently (I believe Susan Collins when she says how the idea came to her) they are doing very different things.
In Battle Royale, it mostly a Lord of the Flies affair, about what humans become when civilisation is taken away and the restraints are put on them. The focus is on how the characters change and the alliances form.
And it's entertainment, the deaths are played as funny or exciting or dramatic, etc. There's a clear hero, a heroine and a villain.
The Hunger Games on the other hand, has the conceit that humans are no different in or out of the arena, in the situation every person does what their nature is naturally inclined to anyway. Instead the focus isn't on the change of behaviour of the people in the arena, but on the nature of humanity that we allow horrific things to happen as part of the status quo.
In short, it's attempt is to convey the brutal reality of the setting and as such, the deaths aren't exciting, but traumatic and the action is obscured because it's not what you're meant to enjoy. In fact in doesn't try to make you enjoy anything, instead it's much more about exploring than making the audience happy. It's protagonist is broken and wrong, sharp defensive, aggressive and morally incorrect. There is no clear villain and the clearest villain is only the human dicator twisting things to serve his power and removing him isn't as easy as expected. It's acknowledged that life is more complicated, that things aren't suddenly over when Katniss wins. But it still has optimism, there are people like Rue who are good and for the rest of us imperfect people trapped in a harsh uncompromising reality, we can take hope in at least sacrificing ourselves to protect the innocent.
I can't compare them. Battle Royale is more fun, but the Hunger Games isn't aiming for fun. The Hunger Games has a much firmer and interesting grasp on reality but Battle Royale isn't trying to be realistic. Both have interesting social commentaries, but on different attitudes, if I had to say which one was more correct, I'd tend towards the Hunger Game. Yes society keeps us normal, but the person who you bring into the arena is yourself and society at large commits atrocities every day by not caring enough about the people who are dying right now.
I can't even say which deserves to stand the test of time. Which would I prefer to watch? Depends on how I'm feeling.
I think I'd just rather have both
EDIT: The Hunger Games is far more introspective than Battle Royale and because it doesn't use story tropes, or at least tries to follow them, even if they appear, although the film is good, there is a lot missing from the book and I'm probably clouded by reading the books. For example every practical complaint in this thread is integral in the book and so woven in that it doesn't feel like an explanation, just how things work. Specifically food packages are the focus from the very start and it's established that they're based on popularity and get more expensive as time goes on to make the games interesting. Whats more Hamish's relationship with Katniss is expressed through the packages and he's guiding her by using them. Peeta doesn't have the star power, they don't have the money, he's signalling to her that she needs to play up the romance etc... Finally Peeta from the very start had planned to use his romance idea and ability to convey an audience to create the social pressure that would make the results of the Hunger Games feel like a bad story to the viewers if both of them died and so as was his intent from the start, with the help of Hamish (who is more connected than he seems) to force the gamemakers into a position where they would at least make it seem like Peeta and Katniss don't have to compete. Then Katniss outmanouevered them at the end.
