user17136: Fide has a rule saying that if one player is rated more than 400 points higher than his opponent, their difference should be set to 400 when calculating rating gain/loss. That means no matter how weak the opposition is, a GM (or anyone) will receive minimum 0.8 rating points for winning. What is the reason behind this?
To me it just seems like a giant weakness in the rating system, making any strong GM able to get as high rating as he wants just by playing weak opponents. Wouldn't it be better if the gain approached 0 when the difference increased (like the basic formula for calculating rating actually does)?
EDIT: As most of the answers don't really seem to get the point, I will add an example which points out the problem:
One of the top 10 players in the world (say Magnus Carlsen), wants to increase his rating to 3000. He has a good friend who has 1200 in rating, and who is willing to play a match of 200 games (2 each day for 100 days) against him. His friend will of course play his best, so this is by no means considered cheating.
According to https://wismuth.com/elo/calculator.html, Magnus will have way more than 99 % chance of winning a single game, but for simplicity, let's say he has 99 % chance of winning a game (which is an understatement). If we then forget about draws (for simplicity), Magnus would be expected a score of 198/200. That means a rating gain of 0.8*198 - 9.2*2 = 140 rating points. That is if he scores as expected. And he used less than 1/3 of a year, and is extremely close to 3000 already. What if he does it again? (And again?) Even if he plays really poorly and loses 5 and draws another 5 out of the 200 games, he will gain 85 rating points. That is by scoring FAR below expected.
If there was no rating difference cap, playing an expected result (say 198/200), would by definition give a rating gain/loss of 0. So who came up with the idea of making a loop-hole for any top GM to become extremely high rated?
Of course it would be looked at as bad sportmanship to do something like this, and I doubt any of the top players of the world would consider doing so, but that doesn't justify the possibility being there. In a serious sport with a serious rating system there shouldn't be these kind of loop-holes.
Can anyone find any arguments justifying this rule?

user18809: Previous answers to this question were correct but are now outdated. As of 2023, the 400-point-gap rule is only applying to one game per month per player rather than to every single game.

user13147: It all evens out when an underrated opponent defeats the GM. 
Assume the GM only plays weak opponents, so he gets a considerably large rating.
Then when he gets inevitably famous (he's a GM) people are going to look up his playing history and notice a discrepancy. This guy is only good because he plays patzers!
So when an underdog defeats him (this underdog being several hundred points stronger than he is on paper) the GM's rating will shift towards a truer description.
EDIT As a response to your edit, you do realize everyone around the world can see Carlsen's playing history, right?
Don't you think this would affect his image somewhat...I don't know...NEGATIVELY?
You're just asking a theoretical question that doesn't have a basis in real life. That's why you're not getting the answer you want: there is no solution.

user224: For starters, eight-tenths of a point really isn't that much of a ratings gain, especially for someone rated 2600 or more. At that rate, it would take 100 wins (without an upset loss) against these lower rated players just to gain 80 points. Given that a GM would mostly have to limit him/herself to the 3 to 6 round weekend tournaments that most of us lower rated players primarily participate in, it would take a while to gain those 80 points. Also, some tournaments might not allow a GM into the lower rated sections. Additionally, the prize money would be almost insignificant for a player at that level. As noted in Jossie Calderon's answer, such a point grab would invite challenges from other GM's looking for a big point gain. All in all, it just doesn't seem like a smart way to gain points.
As I see it, this isn't about guaranteeing a minimal increase for a GM. It's more about capping the gain for a lower rated player and minimizing the loss for the higher rated player if somehow some patzer manages to upset a GM, or a seriously underrated player beats the GM.

user9684: The higher rated player should get something for winning a game, and 0.8 seems to be the least to award the winner and still get recognition.
https://en.wikipedia.org/wiki/Claude_Bloodgood exploited the this system to achieve a rating of 2759.

user825: The rule makes total sense to me as statistics for rating is not so accurate when the difference in playing strength is too much. Capping to 400 minimises the statistical damage.
No GM would risk a hefty rating penalty by playing only weak tournaments. Too much risk for very little gains. A single tactical oversight (GM is human too!!) could be deadly.

user4418: 
Fide has a rule saying that if one player is rated more than 400
  points higher than his opponent, their difference should be set to 400
  when calculating rating gain/loss. That means no matter how weak the
  opposition is, a GM (or anyone) will receive minimum 0.8 rating points
  for winning. What is the reason behind this?

This is an historical anomaly. Currently, apart from your first ever rating, every game is rated individually. This means that you can never lose rating points by winning a game (LRPBWAG). It wasn't always so.
I don't remember exactly when the change happened, 2008 I think, but before this ratings were calculated per tournament. What happened was that your rating was calculated using your score for the tournament, the average of your opponents' ratings and your expected score against them. This opened up the possibility of winning the tournament but losing rating points because some low rated players in the tournament, who you crushed, lowered the average rating for the tournament so much that even though you scored well against similarly rated players as yourself your rating went down.
The original solution to this was to limit the difference to 350 points. Later this was changed to 400. Now that games are rated individually it could be scrapped.
There was an article on Chessbase 6 years ago discussing this point between Jeff Sonas, Ken Thompson and John Nunn.
Part of John Nunn's contribution is worth quoting -

The original rating rules had no 350-point rule. It must be remembered
  that in 1970 the rating list was very small and it was never envisaged
  that it would be extended substantially. Only top tournaments were
  rated (and these were invariably all-play-all events) and all
  calculations were done by hand. The 'Losing Rating Points For Winning
  A Game' problem didn't really arise. The top players always skipped
  the first round or two of an Olympiad, and in any case any weak
  opponents would be unrated, so the games simply didn't count.
But times changed, the rating list expanded and started including more
  and more weaker players. Open tournaments appeared, and the LRPFWAG
  problem started to become significant. I don't remember exactly when
  the 350-point rule was introduced to deal with this, but I believe it
  was around 1980. Later, in 1988 or 1989, Karpov was unhappy about
  losing points when he had won a tournament, arguing that the aim of
  taking part in a tournament was to win it, and if you have achieved
  that then you shouldn't lose points through not having won by a large
  margin. For a time the 'Karpov rule' was in effect, that you couldn't
  lose points for winning an event. This was later scrapped.
Rating games individually is a fairly recent innovation, and for the
  vast bulk of the lifetime of the Elo system it was done on a
  tournament basis (because it made the calculations easier, especially
  for all-play-all events). The current rule is in a way a hangover from
  the earlier times, but I think there is some logic in it, although one
  can argue over the exact details. Rapidly improving young players, for
  example, can be massively underrated.


17136: <p>Fide has a rule saying that if one player is rated more than 400 points higher than his opponent, their difference should be set to 400 when calculating rating gain/loss. That means no matter how weak the opposition is, a GM (or anyone) will receive minimum 0.8 rating points for winning. What is the reason behind this?</p>

<p>To me it just seems like a giant weakness in the rating system, making any strong GM able to get as high rating as he wants just by playing weak opponents. Wouldn't it be better if the gain approached 0 when the difference increased (like the basic formula for calculating rating actually does)?</p>

<p>EDIT: As most of the answers don't really seem to get the point, I will add an example which points out the problem:</p>

<p>One of the top 10 players in the world (say Magnus Carlsen), wants to increase his rating to 3000. He has a good friend who has 1200 in rating, and who is willing to play a match of 200 games (2 each day for 100 days) against him. His friend will of course play his best, so this is by no means considered cheating.</p>

<p>According to <a href="https://wismuth.com/elo/calculator.html" rel="noreferrer">https://wismuth.com/elo/calculator.html</a>, Magnus will have way more than 99 % chance of winning a single game, but for simplicity, let's say he has 99 % chance of winning a game (which is an understatement). If we then forget about draws (for simplicity), Magnus would be expected a score of 198/200. That means a rating gain of 0.8*198 - 9.2*2 = 140 rating points. That is if he scores as expected. And he used less than 1/3 of a year, and is extremely close to 3000 already. What if he does it again? (And again?) Even if he plays really poorly and loses 5 and draws another 5 out of the 200 games, he will gain 85 rating points. That is by scoring FAR below expected.</p>

<p>If there was no rating difference cap, playing an expected result (say 198/200), would by definition give a rating gain/loss of 0. So who came up with the idea of making a loop-hole for any top GM to become extremely high rated?</p>

<p>Of course it would be looked at as bad sportmanship to do something like this, and I doubt any of the top players of the world would consider doing so, but that doesn't justify the possibility being there. In a serious sport with a serious rating system there shouldn't be these kind of loop-holes.</p>

<p>Can anyone find any arguments justifying this rule?</p>
 18809: <p>Previous answers to this question were correct but are now outdated. As of 2023, the 400-point-gap rule is only applying to one game per month per player rather than to every single game.</p>
 13147: <p>It all evens out when an underrated opponent defeats the GM. </p>

<p>Assume the GM only plays weak opponents, so he gets a considerably large rating.</p>

<p>Then when he gets inevitably famous (he's a GM) people are going to look up his playing history and notice a discrepancy. This guy is only good because he plays patzers!</p>

<p>So when an underdog defeats him (this underdog being several hundred points stronger than he is on paper) the GM's rating will shift towards a truer description.</p>

<p><strong>EDIT</strong> As a response to your edit, you do realize everyone around the world can see Carlsen's playing history, right?</p>

<p>Don't you think this would affect his image somewhat...I don't know...NEGATIVELY?</p>

<p>You're just asking a theoretical question that doesn't have a basis in real life. That's why you're not getting the answer you want: <strong>there is no solution.</strong></p>
 224: <p>For starters, eight-tenths of a point really isn't that much of a ratings gain, especially for someone rated 2600 or more. At that rate, it would take 100 wins (without an upset loss) against these lower rated players just to gain 80 points. Given that a GM would mostly have to limit him/herself to the 3 to 6 round weekend tournaments that most of us lower rated players primarily participate in, it would take a while to gain those 80 points. Also, some tournaments might not allow a GM into the lower rated sections. Additionally, the prize money would be almost insignificant for a player at that level. As noted in <a href="https://chess.stackexchange.com/a/22329/224">Jossie Calderon's answer</a>, such a point grab would invite challenges from other GM's looking for a big point gain. All in all, it just doesn't seem like a smart way to gain points.</p>

<p>As I see it, this isn't about guaranteeing a minimal increase for a GM. It's more about capping the gain for a lower rated player and minimizing the loss for the higher rated player if somehow some patzer manages to upset a GM, or a seriously underrated player beats the GM.</p>
 9684: <p>The higher rated player should get something for winning a game, and 0.8 seems to be the least to award the winner and still get recognition.</p>

<p><a href="https://en.wikipedia.org/wiki/Claude_Bloodgood" rel="noreferrer">https://en.wikipedia.org/wiki/Claude_Bloodgood</a> exploited the this system to achieve a rating of 2759.</p>
 825: <p>The rule makes total sense to me as statistics for rating is not so accurate when the difference in playing strength is too much. Capping to 400 minimises the statistical damage.</p>

<p>No GM would risk a hefty rating penalty by playing only weak tournaments. Too much risk for very little gains. A single tactical oversight (GM is human too!!) could be deadly.</p>
 4418: <blockquote>
  <p>Fide has a rule saying that if one player is rated more than 400
  points higher than his opponent, their difference should be set to 400
  when calculating rating gain/loss. That means no matter how weak the
  opposition is, a GM (or anyone) will receive minimum 0.8 rating points
  for winning. What is the reason behind this?</p>
</blockquote>

<p>This is an historical anomaly. Currently, apart from your first ever rating, every game is rated individually. This means that you can never lose rating points by winning a game (LRPBWAG). It wasn't always so.</p>

<p>I don't remember exactly when the change happened, 2008 I think, but before this ratings were calculated per tournament. What happened was that your rating was calculated using your score for the tournament, the average of your opponents' ratings and your expected score against them. This opened up the possibility of winning the tournament but losing rating points because some low rated players in the tournament, who you crushed, lowered the average rating for the tournament so much that even though you scored well against similarly rated players as yourself your rating went down.</p>

<p>The original solution to this was to limit the difference to 350 points. Later this was changed to 400. Now that games are rated individually it could be scrapped.</p>

<p>There was an <a href="https://en.chessbase.com/post/elo-oddities-the-tortoise-and-the-hare" rel="noreferrer">article on Chessbase</a> 6 years ago discussing this point between Jeff Sonas, Ken Thompson and John Nunn.</p>

<p>Part of John Nunn's contribution is worth quoting -</p>

<blockquote>
  <p>The original rating rules had no 350-point rule. It must be remembered
  that in 1970 the rating list was very small and it was never envisaged
  that it would be extended substantially. Only top tournaments were
  rated (and these were invariably all-play-all events) and all
  calculations were done by hand. The 'Losing Rating Points For Winning
  A Game' problem didn't really arise. The top players always skipped
  the first round or two of an Olympiad, and in any case any weak
  opponents would be unrated, so the games simply didn't count.</p>
  
  <p>But times changed, the rating list expanded and started including more
  and more weaker players. Open tournaments appeared, and the LRPFWAG
  problem started to become significant. I don't remember exactly when
  the 350-point rule was introduced to deal with this, but I believe it
  was around 1980. Later, in 1988 or 1989, Karpov was unhappy about
  losing points when he had won a tournament, arguing that the aim of
  taking part in a tournament was to win it, and if you have achieved
  that then you shouldn't lose points through not having won by a large
  margin. For a time the 'Karpov rule' was in effect, that you couldn't
  lose points for winning an event. This was later scrapped.</p>
  
  <p>Rating games individually is a fairly recent innovation, and for the
  vast bulk of the lifetime of the Elo system it was done on a
  tournament basis (because it made the calculations easier, especially
  for all-play-all events). The current rule is in a way a hangover from
  the earlier times, but I think there is some logic in it, although one
  can argue over the exact details. Rapidly improving young players, for
  example, can be massively underrated.</p>
</blockquote>
