For instance when 15 kyu plays against a 10 kyu on a 19x19 board. No handicap, Japanese rules.
At the end of the game, how is the kyu difference used to calculate who won?
no handicap means score normally regardless of the rank
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So if the 10 kyu player wins by one point, the 15 kyu player looses kyu points (rating)?
Hello Samos - I cannot give you the exact numbers as the calculation is much more complex that simple + or - and I would be lying if I said I understand the math behind it.
However if you are looking just for a general explanation I can help:
If a much stronger player wins against a weaker one, the weaker player will lose some points, but a very minimal ammount. If the weaker player won, he/she would get a large boost, while the stronger player would lose a lot of points.
If handicap is present it would be treated as if their ranks were equal.
However the system takes into consideration more data than just that. It also calculates the “confidence” in one’s rank which can affect the points change. Also if someone you have lost to recently rose quickly in rank, some of the points you originally lost can be reinstated, as the person is likely to be stronger than what the system thought at the time and vice versa. These data are recalculated after every game, therefore any change to your rank is NOT a result of only the last match.
Our system is based on Glicko II https://en.wikipedia.org/wiki/Glicko_rating_system if you are interested in math, check it out.
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Well it’s not even that easy to explain, because it’s a stepwise algorithm; it’s not just the application of a single function to the rating of both players.
But that aside, it’s something a little bit like Elo and Elo is explained here:
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Right! So that explains why I have seen my rank change in between games at times.
Very helpful AdamR and smurph.
Thank you.
I think the answers were pretty clear here , i’d just like to react on this point
Go is a game of judgment and balance, ultimately a win by one point will always have the same value as a win by 50 points or resign. Strong players will be able to know when they are ahead, even by a small margin and don’t need to take risks to try to win by 30 more points. On the contrary if you see that you are behind by 5 points it may be better to take some risks and maybe lose by 30 more points than just being happy with a 5 points loss. So yeah if you lose by 0.5 points … you lose 
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That makes perfectly sense Capitaine_Crochet.
Yet, it leaves me with the feeling that the no handicap variation is more in favor of the stronger player then the handicap variation. Especially if there’s a large difference in strength.
Even though more points can be earned by no handicap for the weaker player. Just because it will about never happen that a 20kyu player wins from a 20kyu player without handicap. lol
That might be true, but normally players don’t want to play with much weaker ones with no pressure at all.
If you setup a game, you can limit rank range that can accept the game. I suspect many just don’t bother to change the default settings.
I personally would prefer to play no handicap against stronger players because I can learn a thing or two typically, while handicap game is more like stronger players tricking me. I can still learn something but not fundamentals that I can apply more often to even games. And those tricks are very complex and easy to forget.
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What if two people play two consecutive games?
What of it? I don’t think player’s names matter. Only their ranks. One rank goes down, one rank goes up. I am not sure why the identity of the players should make a difference.
I thought I read somewhere if we play one game after another, the second game is calculated a little differently? I could remember wrong though.
Well the game is calculated a little differently because your ranks changed by the first one. Right now I can’t imagine why there would be any more to it than that, but I can be wrong just as well… As said, I am by no means an expert on these calculations. 
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You could join the Open Study Room: we play long games (at least 30m + 5 x 30s) with no handicap, and the winner (or stronger player) reviews. I will be honest with you though, there are not all that many active 5k+ players.
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Thanks! In my plan, after I finish my readings and maybe up my rank a little.
And what about games won or lost with time-out in correspondence mode? It seems to me that there is hardly any score awarded if at all.
Basically the same, but there is a controversial rule counting only the first timeout against rank, effectively anulling further timeouts.
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Hmmm… no penalty for consecutive lost games in time out could be labelled controversial (easy to beat the system that way).
No positive points for people who win by time out, just because they are lower in the list is just wrong.
It has nothing to do with a position in whatever list. It’s just about consecutive timeouts.
Also ranking points are no reward, they are (an approximation of) a measurement of playing strength. (Well, at least this is a view shared by many, but clearly not be everyone I have to admit.)