The game already began

I'd like to ask American people.
I hear that in American English the past tense is used in the case the present perfect should be used. For example, some Americans say "Did you eat lunch?" instead of "Have you eaten lunch?" Would they be used interchangeably in the United States?

How about in this case? Which would someone say when you don't arrive at the stadium on time?
(a) The game has already begun.
(b) The game has begun already.
(c) The game already began.
(d) The game began already.

Thanks in advance.
Hirashin

I know people would use the last two in spoken English but they sound odd to me. Also a small correction, if you're talking about the event as a past event then you would need the past tense of the verb. "(a) The game had already begun. (b) The game had begun already." If your friend is at the game and you want to make sure you are not late you may ask, "Has the game begun yet?" to which your friend might use (a) or (b) as a response.

On the other hand. If you're talking to someone at the same game as you then you can use them like you did in (a) and (b). "Oh man. The game has already begun." or "I see the game has begun already." this is because it's a present observation of an event that has already passed.

Thanks for the help, OoTmaster.