user13647: What are the odds that Kirito can build a harem?
I am unclear how asking for a number or set of numbers where I provide most of the input for a calculation is "primarily opinion-based".
For example, if we assume a 75/25 split of male to female players, and compute at the beginning of the anime, then the answer would be (if I recall that semester of combinatorics and probabilities correctly):

 1 in 2501*2500*2499*2498*2497, which is a really small chance.  For reference, the chance of having 1 girlfriend is 1 in 2501, which is still pretty small.

Details:

10,001 players (+1 for Yui, an AI)
7500 male
2501 female
harem size of 5


user32564: Okay, so the first thing I must admit is that I am a mathematical dunce so there are factors that may be beyond my comprehension. That may even be one of the problems with this question in this regard: Some of the necessary evidence is outside of our specific field of expertise.
However, insofar as what I can consider, I suspect that the problem is with additional unknown or otherwise incalculable factors which further complicate the question beyond your expectations.
First of all, this might not be a question that can be answered purely by mathematics. The volition of the females to be in a relationship with a particular somebody, let alone a polyamorous one, may be an incalculable factor. Asuna's willingness to be with Kirito is probably predicated on a presumption of chastity, or in other words his ability to be faithful to a monogamous relationship. Kirito's charisma is considered to be especially high, and in an abstracted question like this it is difficult to ignore that he is the protagonist of the story, which may increase his odds of success beyond that of the average extra. 
Second of all, how do you determine the average cross/gender encounter rate? If we assume that a woman has the same odds of being added to a harem at any given time, then odds of creating a heterosexual harem are zero percent if the player base is completely segregated for the entire duration of the game, and perhaps each time a person meets increases the likelihood that this person can be added to the harem. Now we know for a fact that it is not, but I suspect we still need to know how often people of opposite genders interacted within two years.
How many people constitute a harem? Can it be as few as three or five, or do there have to be at least a dozen women for it to count? You say at least five here, but it is not expressly stated in the question so it is left ambiguous for anybody viewing the question on the main site, and we base a question's open/closed status based upon the content of the question there.
Also, you should take note that your exemplary harem is impossible:

 Please remember that Sachi can not be in Kirito's harem because she dies before any of the other prospective members are introduced. We need to consider how the death rate affects the odds of being able to build a harem over the course of two years.

 Now some of these concerns can be hypothetically addressed if we know enough of the facts of the story, and a question that can be based in facts is usually not Primarily Opinion Based. Moreover, S.A.O. is the sort of story that divulges demographical information, so it is probably one of the better candidates for this type of question because they are more likely to exist in this tale and we are better suited to knowing those facts than a mathematician. In this regard, I am not sure if the question is exactly P.O.B., but I think these are the sorts of factors which may have been considered, and they certainly seems to be a very complicated one that would be difficult to answer.
More importantly, I have difficulty believing that the volition factor can be anything other than pure guesswork, which is probably adequate grounds enough for closure unless a study can be produced which demonstrates otherwise.
One incidental note I would like to add is that, working with assumed rates may solve the P.O.B. issue by giving us objective numbers to work with, but at the same time it would distance it from being an actual question regarding the show so much as the numbers derived from it, and risk putting it outside of our scope.

user63: The novel and anime mentions rough numbers about how more than thousand (2,000) died in the first month before the first boss raid (see episode 2 of the anime). Your question make many assumptions on what people should know or should regard as fact without providing any specific citations beyond stating certain numbers. Questions of the legitimacy of tertiary sources aside. This whole assumption is a bit "What IF" scenario with little to no bearing on the series or franchise.
The author does not give any indications of gender distribution amongst the 10k users, we can make a generalized guess based on the number of characters that have appear, but outright generalizing is biased. Furthermore, Yui is not a player and can't be classified as one. Kayaba Akihiko is not one of the 10k players either as he just inserted himself into the game by digitizing himself (by remodeled FullDive machine to to transfer his consciousness into the virtual world, permanently, mentioned in epilogue/aftermath of the Ainclad Arc, with Rinko [Vol 4. Cp. 9]).
A harem builds itself by the whims of the author. You also forget that a harem does not have to be made up of a gender. Not all females will be interested in Kirito, despite his MC status. There are guys (Klein, for one) in Kirito's harem as well. Not all males won't be interested, for varying reasons. Without an acceptance criteria it's anyone's guess. As it stands as a question for this site, it's not a good fit. Maybe it would be for Code Golf or Puzzles. Claiming that is is based purely based in mathematics does not make it less opinionated. The math might not be but the context sure is.
This question is opinionated not unlike the if A and B fought who would win... The late Stan Lee answers it well "The person who'd win in a fight is the person that the scriptwriter (i.e. author) wants to win." Making biased assumption to formulate the basis some sort of question isn't going to get us anywhere, except nitpick more are potential holes left by the author or writer. If we remove mentions of the series from the question it becomes a hypothetical statistical problem.
Adding math to an assumption does not may it any less of an assumption and just serves to skew public opinion of one's favor. Certain news outlets to it all the time to make ppl seem more favorable, while the information is technically true, the way it is presented is questionable.
tl;dr Because there as no acceptance criteria made and the assumptions were made with no regard towards actual canon ("every one know that..." is a very big generalization, and introduces a bit of cognitive bias [the question is leading]), with minimal use of citations from tertiary source. I am skeptical of the basis of the question, and regarded that it should be put on-hold is "generally opinionated". 
OP should take some time to review the basis of the question by make necessary citations and connecting the questions to be better align with the series or franchise itself instead of posing one that can be regard as a statistical math puzzle with an SAO skinned setting. 
Here is a question asking how Misaka's Railgun can be so destructive? The sort answer is, from a pure real-life (RL) physics standpoint it's not. But there maybe in-series canon cite that there is some underlining at play in their in-series universe. The question is generally acceptable because the context asked is regarded to be in series. RL examples can be provided to prove of disprove the series, mathematical models can help but is not generally required to answer. 
The mentioned question falls short of that as it is posed in the derived context of the OP's generalization (only females can be in be in a harem, all females are interested in becoming a port of the harem, the amount of female players [without subtext], who are considered players, the fact that Kirito would initiate such a scenario, etc).

13647: <p><a href="https://anime.stackexchange.com/questions/49612/what-are-the-odds-that-kirito-can-build-a-harem">What are the odds that Kirito can build a harem?</a></p>

<p>I am unclear how asking for a number or set of numbers where I provide most of the input for a calculation is "primarily opinion-based".</p>

<p>For example, if we assume a 75/25 split of male to female players, and compute at the beginning of the anime, then the answer would be <em>(if I recall that semester of combinatorics and probabilities correctly)</em>:</p>

<blockquote class="spoiler">
  <p> 1 in <code>2501*2500*2499*2498*2497</code>, which is a really small chance.  <em>For reference, the chance of having 1 girlfriend is 1 in 2501, which is still pretty small.</em></p>
</blockquote>

<p>Details:</p>

<ul>
<li>10,001 players (+1 for Yui, an AI)</li>
<li>7500 male</li>
<li>2501 female</li>
<li>harem size of 5</li>
</ul>
 32564: <p>Okay, so the first thing I must admit is that I am a mathematical dunce so there are factors that may be beyond my comprehension. That may even be one of the problems with this question in this regard: Some of the necessary evidence is outside of our specific field of expertise.</p>

<p>However, insofar as what I can consider, I suspect that the problem is with additional unknown or otherwise incalculable factors which further complicate the question beyond your expectations.</p>

<p>First of all, this might not be a question that can be answered purely by mathematics. The volition of the females to be in a relationship with a particular somebody, let alone a polyamorous one, may be an incalculable factor. Asuna's willingness to be with Kirito is probably predicated on a presumption of chastity, or in other words his ability to be faithful to a monogamous relationship. Kirito's charisma is considered to be especially high, and in an abstracted question like this it is difficult to ignore that he is the protagonist of the story, which may increase his odds of success beyond that of the average extra. </p>

<p>Second of all, how do you determine the average cross/gender encounter rate? If we assume that a woman has the same odds of being added to a harem at any given time, then odds of creating a heterosexual harem are zero percent if the player base is completely segregated for the entire duration of the game, and perhaps each time a person meets increases the likelihood that this person can be added to the harem. Now we know for a fact that it is not, but I suspect we still need to know how often people of opposite genders interacted within two years.</p>

<p>How many people constitute a harem? Can it be as few as three or five, or do there have to be at least a dozen women for it to count? You say at least five here, but it is not expressly stated in the question so it is left ambiguous for anybody viewing the question on the main site, and we base a question's open/closed status based upon the content of the question there.</p>

<p>Also, you should take note that your exemplary harem is impossible:</p>

<blockquote class="spoiler">
  <p> Please remember that Sachi can not be in Kirito's harem because she dies before any of the other prospective members are introduced. We need to consider how the death rate affects the odds of being able to build a harem over the course of two years.</p>
</blockquote>

<p><br> Now some of these concerns can be hypothetically addressed if we know enough of the facts of the story, and a question that can be based in facts is usually not Primarily Opinion Based. Moreover, S.A.O. is the sort of story that divulges demographical information, so it is probably one of the better candidates for this type of question because they are more likely to exist in this tale and we are better suited to knowing those facts than a mathematician. In this regard, I am not sure if the question is exactly P.O.B., but I think these are the sorts of factors which may have been considered, and they certainly seems to be a very complicated one that would be difficult to answer.</p>

<p>More importantly, I have difficulty believing that the volition factor can be anything other than pure guesswork, which is probably adequate grounds enough for closure unless a study can be produced which demonstrates otherwise.</p>

<p>One incidental note I would like to add is that, working with assumed rates may solve the P.O.B. issue by giving us objective numbers to work with, but at the same time it would distance it from being an actual question regarding the show so much as the numbers derived from it, and risk putting it outside of our scope.</p>
 63: <p>The novel and anime mentions rough numbers about how <em>more than</em> thousand (2,000) died in the first month <em>before</em> the first boss raid (see episode 2 of the anime). Your question make many assumptions on what people should know or should regard as fact without providing any specific citations beyond stating certain numbers. Questions of the legitimacy of tertiary sources aside. This whole assumption is a bit "What IF" scenario with little to no bearing on the series or franchise.</p>

<p>The author does not give any indications of gender distribution amongst the 10k users, we can make a generalized guess based on the number of characters that have appear, but outright generalizing is biased. Furthermore, Yui is not a player and can't be classified as one. Kayaba Akihiko is not one of the 10k players either as he just inserted himself into the game by digitizing himself (by remodeled FullDive machine to to transfer his consciousness into the virtual world, permanently, mentioned in epilogue/aftermath of the Ainclad Arc, with Rinko [Vol 4. Cp. 9]).</p>

<p>A harem builds itself by the whims of the author. You also forget that a harem does not have to be made up of a gender. Not all females will be interested in Kirito, despite his MC status. There are guys (Klein, for one) in Kirito's harem as well. Not all males won't be interested, for varying reasons. Without an acceptance criteria it's anyone's guess. As it stands as a question for this site, it's not a good fit. Maybe it would be for Code Golf or Puzzles. Claiming that is is based purely based in mathematics does not make it less opinionated. The math might not be but the context sure is.</p>

<p>This question is opinionated not unlike the if A and B fought who would win... The late Stan Lee <a href="https://www.youtube.com/watch?v=L4_zFYnnn2Y" rel="nofollow noreferrer">answers it</a> well "The person who'd win in a fight is the person that the scriptwriter (i.e. author) wants to win." Making biased assumption to formulate the basis some sort of question isn't going to get us anywhere, except nitpick more are potential holes left by the author or writer. If we remove mentions of the series from the question it becomes a hypothetical statistical problem.</p>

<p>Adding math to an assumption does not may it any less of an assumption and just serves to skew public opinion of one's favor. Certain news outlets to it all the time to make ppl seem more favorable, while the information is technically true, the way it is presented is questionable.</p>

<p>tl;dr Because there as no acceptance criteria made and the assumptions were made with no regard towards actual canon ("every one know that..." is a very big generalization, and introduces a bit of cognitive bias [the question is leading]), with minimal use of citations from tertiary source. I am skeptical of the basis of the question, and regarded that it should be put on-hold is "generally opinionated". </p>

<p>OP should take some time to review the basis of the question by make necessary citations and connecting the questions to be better align with the series or franchise itself instead of posing one that can be regard as a statistical math puzzle with an SAO skinned setting. </p>

<p>Here is <a href="https://anime.stackexchange.com/questions/2235/how-is-misakas-railgun-so-destructive">a question</a> asking how Misaka's Railgun can be so destructive? The sort answer is, from a pure real-life (RL) physics standpoint it's not. But there maybe in-series canon cite that there is some underlining at play in their in-series universe. The question is generally acceptable because the context asked is regarded to be in series. RL examples can be provided to prove of disprove the series, mathematical models can help but is not generally required to answer. </p>

<p>The mentioned question falls short of that as it is posed in the derived context of the OP's generalization (only females can be in be in a harem, all females are interested in becoming a port of the harem, the amount of female players [without subtext], who are considered players, the fact that Kirito would initiate such a scenario, etc).</p>
