In context of the exact question you refer to, I would personally vote for neither. That said, your best course of action is to edit the original question, and let others decide whether it should be reopened.
Ultimately, if you wish to make your case for reopening the question, you must do better than 'there is good information available that I think would be helpful'. You specifically need to address the core close reasons, and more specifically, address how you could change the question to limit these reasons.
It is important to note that in opening or closing a question, we generally do not address any current or potential answers. That is; arguments based off answers are invalid for closing or opening a question. I personally find that answers can give us insight into how others interpret the question, but if you comment that "this question should be opened, because the answer is really helpful", it's just not going to happen. Sorry.
First and foremost, you should never ask a new question if you already identify it as a duplicate of a previous question.
But what if the original question was closed?
If it was closed, your identical question will probably be closed for the same reasons. In this case I think there is more to it than that, but I will talk more about that, below.
If you instead edit the original question to better suit site policy, you will not necessarily see the question reopened. The edit will flag the question for the "Vote to Reopen" queue, and other members will have an opportunity to vote towards reopening the question. Keep in mind that once opened, further users may still vote to close the question, again. If you truly believe the question is on topic for current site policy, make an argument about it in the comments.
But what if I think the current answers are wrong?
If you think the current answers are wrong, down-vote them. If you really wish to be helpful, add a helpful comment explaining what you find to be wrong. You can add your own answer, and if your answer truly shines above the rest, it will eventually be up voted above the rest. If you do not have an answer, but still believe the other answers to be incorrect or of low quality, post a bounty. These options are all available if and when the question is reopened.
But what if the question is not reopened?
If the question is not reopened, that infers that the community in general does not see this question as being on topic. You can ask about it more specifically, here in meta, and perhaps other users will provide more helpful dialogue on why the question is not a good fit for here. Ultimately, if it comes to this, any identical question you post will meet with quick closure and potential down voting.
But what if it really IS a good question?
Just because a question is good and potentially helpful, does not mean it is a good fit for StackExchange. Good examples can be seen in primarily opinion-based questions. Other elements to the question might make it a poor fit for this sort of site, and ultimately make it far too difficult to uphold the general quality we strive for. I will talk more about why I think this is one of those questions, below.
In regards to the question in context, I would personally still vote to close. As well as being "Not Constructive", I can see this question being "Too Broad", "Primarily Opinion Based" and even "Unclear".
The question "What are the basic game activities" is currently closed as "Not Constructive". I do not believe we close questions for this reason, any more. That said, if we do, that is the penultimate argument. This question has already been closed, and will surely be closed again. One of the users who voted on this question is now a mod1, and if they agree the question is still off-topic, they do not require any more votes to close the question.
Ignoring the current close status of the question, the major issue I see is broadness. What do you mean by "game activities", exactly?
I interpret it as "what can we tell the player to do". If we stick to the basics, this question is not very broad, at all. At least, not from my opinion. One constant in my studies has been that, when it comes down to it, we can only really give the user 8 different core objectives. That said, I am sure others would disagree, entirely. That is their opinion. If an answer relies to heavily on a users opinion, that might also make the question primarily opinion-based, which is another sort of question we don't accept.
The first answer interprets the question as specifically asking about the basic activity for each genre in existence. With 4 up-votes and no down-votes, I think it is clear that others have shared this interpretation. This particular answer screams too broad. A complete answer would have to include every game genre - well outside the scope of the format of Stack Exchange.
The second answer appears to interpret the question as specifically asking about aesthetics. I am personally unsure how the user could have made that interpretation, aesthetics and activities are greatly different from each other. All aside, the answer has received almost as much up-votes as the first, despite tackling a completely different question all together.
It is clear to see that this question received mixed interpretation. I can personally understand why this question is considered not constructive, and would vote to close this question, myself.
1 Forgive me if I miss-inferred; As far as I know, users are listed in order of when they placed their votes, so I infer that Byte56 was not a mod when they cast this particular vote, as the question would have been automatically closed before another user could cast their vote.