Wow. Why does it not surprise me that OP lives in scandinavia? There's shitloads of flaws in your system, I'll try to list some the others didn't yet. A pirated game is not a lost sale. Thus your "compensation" for the company is not compensation, it's additional money that would not have necessarily been paid to them were it not for piracy. Not every household with internet access plays games. There's no reason for them to pay an industry they don't care about. The big companies that you would give the money to are not the people who made the game. Most developers have to sell the license of their games to publishers to finance it, so the money you propose to give the "big companies" would probably go right past anyone who actually worked on the game in question.

Crime, especially theft, or piracy, whatever you define it as, is not solvable like health care or unemployment money is. This is not a social service issue, you can't make everyone pay for other's crimes. You can't transform the public into an insurance company for game publishers. The publishers don't deserve the money and the public has no reason to pay it.

Edited:

You gotta remember, the "copyright holders" are not the people who made the content.