@hirashin A little more to ramble on about here... First, to recall your examples:
I know that you say "The game has already begun." (present perfect)
But do you ever say "The game already began"? (past simple)
One issue to consider here is 'already'. Usage is different in BE (british) and AE (american english). In BE, 'already' is generally not used with simple past, while it
is used with present (and past) perfect. OTOH, in AE, using 'already' with simple past is more acceptable (along with using it with present/past perfect).
<digression>
A prescriptivist might say that the 'correct' way to use (and teach) the use of 'already' is that it should not be used with simple past ("That's wrong") and that it is okay to use--can be used--with present and past perfect. Here, the BE model is seen as correct, and the AE usage is seen as sloppy, incorrect, lax, and so on.
A descriptivist would look at English as it is used in many different places, see some differences in how 'already' is being used, and would hold back on judging what is right/wrong, or correct/incorrect. They would teach both, perhaps pointing out (in this case) that geography is a factor in its use.
<end digression>
Another thing I posted about was that 'begin' and 'start' felt different--when used with 'already'--in the examples I posted above: (along with a comment I made)
a) The game already began.
b) The game already started.
Again, I agree with what Lothar said about (a). But then (b) sounds fine to me in a way that (a) does not.
I think one reason 'start' seemed more acceptable to me there is the
form of the verb. As you know, the simple past and participle forms are the same: started/started. However, for 'begin', those are
different: began/begun.
Perhaps as an american, (b) sounds okay to me because it's easy to overlook or ignore one tiny missing piece--a hardly pronounced
's after game: "The game
's already started." In spoken english, it's an easy "who cares" thing. Eliding/dropping the
's is something americans don't miss or worry about. Some part of an american's brain is processing that: "Eh, it sounds like present perfect, so who cares?" So, "The game already started" can kind of
pass as present perfect. (contains two signals of present perfect)
On the other hand, using 'begin' in this kind of example creates flashing lights--or at least a more prominent signal of what's going on. I think the reason is the difference in verb form: began/begun. Because of the different form, 'began' cannot
pass as present perfect.
"The game
's already begun" has three signals that it is past perfect--the 's, the word 'already' (the proper use of it), and the form of the verb.
The example in your initial post, "The game already began," has two signals of simple past: no
's (or 'has'), and a verb form that also strongly signals simple past (began). Those signals clash with 'already', a pretty good signal of present (or past) perfect. ((And I'd offer that the more I look at (a), above, the more I think it is wrong.))
Bottom line: 'already' is a marker of (is used with) present/past perfect. Even slack and slovenly americans*--depending on their education level and demographics--have some awareness of that.
*
I know that TOEIC doesn't test BE/AE differences--they intentionally avoid that (e.g., verb agreement for collective nouns, different to vs different than, etc.), but I'm not sure how they'd view this particular question/issue.
Enough for tonight...!
(And note that 'already' shares some things with the usage of 'yet' and 'just', but let's leave those for some day in the future!)