I have this situation:

There was an easy question posted (why $19 \mid 5^{2n+1}+3^{n+2} \cdot 2^{n-1}$ for every natural $n$), and I knew the answer [which is quite obvious]. I posted my answer, and I saw that it was submitted. after 14 minutes, someone posted another answer, almost the same solution. but only because of his/her reputation, the answer got 27 up-votes, and I got 3. Although I answered earlier. Why would this happen?

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    $\begingroup$ Welcome to math.se! This is, indeed, somewhat unfair. But the reasons why this happens are so deep rooted in the human psyche that worrying about it is not going to make you (or anyone else) happier. Things will improve quite a bit when you gain a reputation as a dependable answerer (which correlates with, but does not equal to a high rep score). Admittedly this is a bit extreme given that you were a few minutes earlier than the other poster. $\endgroup$ – Jyrki Lahtonen Jan 22 '17 at 12:53
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    $\begingroup$ @JyrkiLahtonen You mean people in math.se would judge an answer by reputation of the one who answered? So if I'm not a high-rep user, my answers are not worth reading! This is what it seems like! $\endgroup$ – 1Emax Jan 22 '17 at 12:56
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    $\begingroup$ Not quite. The mechanisms at play are (my guess): 1) voters have learned to trust certain posters. If they know that this user can most certainly do this question, they may vote without stopping to check for alternatives. 2) if one of the answers has more votes than its rivals many voters, again a bit lazily, just upvote that answer without glossing over the others. So what probably happened is that a few voters operating under 1) came by, and the rival answer gained a bit of headway, then the rest came from mechanism 2). $\endgroup$ – Jyrki Lahtonen Jan 22 '17 at 13:02
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    $\begingroup$ The situation there seems quite special. The 27 is voted out of proportion, not so much because of its quality or that of its owner, but rather to take a stance 'against' the accepted answer (further augmented by the question getting a lot of exposure via a Hot Question list). And those felt they did not make the point twice. What @Jyrki says also plays a role, and yes it is unfair in a way, but the main effect is something else in that case. $\endgroup$ – quid Mod Jan 22 '17 at 13:05
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    $\begingroup$ The reputation probably has a role in this, but there are different factors. One is that once that answer's score was significantly greater than your answer's, that alone caused upvoters to concentrate on the higher scoring one. Another factor is presentation. The other answer is presented more compellingly than yours. It's easier to follow what's going on there. $\endgroup$ – Daniel Fischer Mod Jan 22 '17 at 13:06
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    $\begingroup$ @JyrkiLahtonen that starts to be off-topic here, but after all the detailed one got accepted. Presumably because it was clearer to the person that asked it. There are also poor choices of accepting an answer, but I am not sure this one qualifies. (Of course for you and I the short ones are quicker to read, but we would not have asked the question to begin with.) $\endgroup$ – quid Mod Jan 22 '17 at 13:17
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    $\begingroup$ Trying to make sense of the arbitrariness and capriciousness of voting is a sure road to ruin. Just keep on writing good answers to be recognized as a reliable sort, and before long somebody will post a meta question asking why SSepehr has twice the number of votes even when that other person's answer came first. ;) $\endgroup$ – J. M. ain't a mathematician Jan 22 '17 at 15:50
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    $\begingroup$ I saw that question evolve in real time. When it entered the Hot List the high-voted answer had the highest number of votes (but only a few more than others). Hot List votes tend to go mostly to the highest voted answer, probably because most readers sort by votes, and many drive-by readers lose interest after reading the first answer. Later answers (e.g. mine) typically get very few votes (even though they may yield greater insight due to the author taking more time to think about the essence of the matter). Blame goes to SE design $\endgroup$ – Bill Dubuque Jan 22 '17 at 23:32
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    $\begingroup$ Its worse when someone with higher rep edits the original question and your answer gets down-voted because it no longer answers it. $\endgroup$ – user400188 Jan 23 '17 at 2:20
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    $\begingroup$ @user400188. that seems to be worth discussing in a different question (perhaps started by you), as that's a different matter from the one here. $\endgroup$ – J. M. ain't a mathematician Jan 23 '17 at 5:06
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    $\begingroup$ You are a talented high school student and you have a long road ahead. Just try to enjoy math and don't care about these minor issues. The world is generally not a fair place at all! We just learn to cope with unfairness, gradually... $\endgroup$ – polfosol Jan 25 '17 at 8:29
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    $\begingroup$ Since upvotes are free, I always try to upvote all the answers that satisfy me, which usually include those coming from seasoned users. $\endgroup$ – Matemáticos Chibchas Jan 26 '17 at 3:23
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    $\begingroup$ This has happened to me once. I answered a question perfectly well, and later along came a more famous user and answered the question with a mistake. When I pointed this out, he said 'so?' and edited his answer. He got far more upvotes. $\endgroup$ – user50229 Jan 26 '17 at 23:13
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    $\begingroup$ Formatting and presentation makes a big difference. Your answer is much, much harder to read. The other answer is easily understandable almost at a single glance. $\endgroup$ – Wildcard Jan 27 '17 at 0:51
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    $\begingroup$ @RutgerMoody: Please flag any such post. The moderators have busted many such voting rings in the past. Deleting so called sock puppets (together with their votes) is something we do weekly. $\endgroup$ – Jyrki Lahtonen Jan 31 '17 at 7:44

First of all, I used to be often frustrated by this phenomenon myself. I would post answers similar to (well, better than, IMHO) those of other users with much higher reputation, only to find my answers receive half of the upvotes, and virtually none of the acceptances of these others. There was a time during which I hustled for upvotes, only to find that I hated the hustle. For quite a while now I only occasionally answer questions when I find one I feel is interesting and to which I think I have something to add. I'm not looking for upvotes or green check-marks, but really making posting itself the goal to providing answers. Granted, it's a bit easier to do this when you don't have any more privileges to earn.

On Stack Exchange a user's reputation is meant to be a rough measure of how much the community "trusts" that user. Most reputation is gained through the posting of clear and useful questions, as well as clear, helpful (and correct) answers. At a certain point (and I don't known when this happens), a user will have their posts trusted (and upvoted) because their previous posts have been trusted, and this will in turn lead to a further increased trust of their future posts: it becomes a positive feedback loop. Part of this is also that particular users become known as experts in certain areas. If you're wading through multiple answers to a single question, it is easier to focus on those by users you are already pretty sure will have posted a good answer. (I'd be curious to see what would happen if robjohn were to finally post an answer in the tag.)

In connection to the last point above, I tend to think that voting is fairly depressed on math.se. I think many users are miserly with their upvotes, and only bestow them on certain favoured users. I should stress that I feel this is only a guess, and I have no real data to back any of this up.

Is any of this fair? Surely not! At the same time I'm not certain what can be done about this. We certainly cannot mandate that users upvote "similar" answers to ones they have upvoted (how would the system judge this?). While there are nudges, such as the Sportsmanship and Electorate badges, to encourage users to pass votes around more, not everyone cares about them. It would be curious to see what would happen if all posts were anonymised on the site. Experiments of this sort have been suggested before, but declined.

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    $\begingroup$ I see that most answers support the idea of having anonymity still the suggestion declined, $$\Huge \ddot \frown$$. $\endgroup$ – A---B Jan 22 '17 at 17:45
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    $\begingroup$ @A---B I'm doubtful about how much it would change things as there is really no such thing as anonymity. Many (high rep) users here have such a particular style in their choice of words, presentation and use of TeX that it's easy after a while to know who has written an answer before reaching the end of the post. $\endgroup$ – Winther Jan 22 '17 at 18:08
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    $\begingroup$ @Winther Don't know about you but I can't predict any one except Bill from the style of writing answer. $\endgroup$ – A---B Jan 22 '17 at 20:29
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    $\begingroup$ @A--B I'm pretty confident that I could predict Asaf Karagila and Noah Schweber, at least. $\endgroup$ – Patrick Stevens Jan 28 '17 at 13:03
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    $\begingroup$ @Patrick: I'll take that as a compliment. $\endgroup$ – Asaf Karagila Mod Feb 3 '17 at 7:34
  • $\begingroup$ @AsafKaragila As it was intended :) $\endgroup$ – Patrick Stevens Feb 5 '17 at 22:27

As you suggest, one reason why robjohn's answer got more upvotes than yours may be that he has a lot more rep, and is a familiar face to many people on math.SE. Both of these facts may cause some voters to trust his answers more, without necessarily taking the time to check them in detail, than they would trust a similar answer from an unfamiliar low-rep user.

Another reason why his answer got more upvotes than yours is that, at least to my eye, it's a lot better typeset and more readable. I can look at his answer, understand the form of the argument he's making, and verify that all the steps are correct in just a few seconds. It's nothing but one simple (and nicely laid out) chain of basic algebraic manipulations and modular equivalences, leading from the given expression to zero (modulo 19).

Your answer, meanwhile, is a dense and compact block of inline math, with two separate chains of equivalences. I've been looking at it for over a minute now, while writing this answer, and (if it weren't for the fact that a lot of other math.SE users have surely vetted it a lot more thoroughly by now) I still wouldn't be able to tell for sure whether it's correct or not. In your second chain of equivalences, especially, each step involves a bunch of non-trivial algebraic manipulation that really would benefit from being written out. Correct or not, posted early or not, I can easily see how many readers may have taken a look at your answer, seen a terse answer with a bunch of messy formulas that they can't immediately parse, and decided to ignore it since there are other, more readable and interesting answers too.

Finally, as others have noted, most readers will tend to sort answers by score, and pay most attention to the top-scored answers. Some readers may also interpret a high answer score as an indication of community trust (just as people interpret rep the same way), and thus be more inclined to upvote already high-scoring answers. For better or worse, these effects tend to create a positive feedback loop that amplifies scoring differences: the more votes an answer already has, the more votes it's likely to get.

In particular, as quid noted in the comments above, in this particular case robjohn's answer may have acquired some extra upvotes from readers who didn't like the (IMO excessively long-winded) accepted answer, and therefore chose to upvote the "leading competitor" instead. Not all of these readers necessarily took the extra time to also go over all the other answers to see if they'd deserve upvotes too — they just saw an answer they liked sorted underneath an answer they didn't like so much, and decided to upvote it.

I suspect the combination of all these effects is what led to robjohn's answer acquiring so many more upvotes than yours: the few initial voters saw a clear and well-written answer by a familiar and trusted user, and decided to upvote it, while ignoring the terse and kind of hard-to-read competing answer from a low-rep user they didn't recognize. And once robjohn's answer got enough upvotes to raise it up above yours in the default sorting order, the "snowball effect" also started working for him.

Of course, by now, the "meta effect" has also caused a lot more people to take a closer look at your answer and upvote it, since it is, after all, a correct answer to the question. Thus, your answer is now at score 16 (+13 from before you posted this meta question), while robjohn's is only at 28 (+1 from before). The other (also, presumably, equally correct) lower-voted answers still languish at 2 to 3 votes, though — because, honestly, even though all of them are valid and correct, they still don't add much to the other, more popular answers.

  • $\begingroup$ Your last paragraph is nonsense (at least as far as it concerns my answer, since it explains the genesis of the congruence, unlike other mechanical verifications). The voting is completely skewed even more than normal towards elementary (and quicker) answers only because the question was on the Hot List. There is no hope for deeper answers once a question reaches the Hot List (and, alas, often even before so). Blame goes mostly (but not completely) to very poor platform design $\endgroup$ – Bill Dubuque Feb 2 '17 at 14:37
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    $\begingroup$ @BillDubuque The question was long off the HNQ when this meta question was posted, and the high number of upvotes that SSepehr's answer received since is not because of the HNQ, but very very likely due to the meta exposure. That your answer didn't receive many upvotes due to the exposure may be because viewers concentrated on SSepehr's answer, or it may indicate that viewers agree with Ilmari that it doesn't add much to the other answers. But the HNQ has no part in that. $\endgroup$ – Daniel Fischer Mod Feb 2 '17 at 14:42
  • $\begingroup$ I saw the question evolve in real time on the Hot List, so I am well aware of the effect it had (both here and on many similar questions). The subsequent Meta effect played little role in the issue that I addressed. The damage was already done by that point. Not to mention sockpuppets running rampant (do you plan to address that or should I contact the SE team?) $\endgroup$ – Bill Dubuque Feb 2 '17 at 15:09
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    $\begingroup$ I agree wholeheartedly with this answer, particularly the second section. People often greatly underestimate the importance of presentation. $\endgroup$ – user14972 Feb 2 '17 at 15:25

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