Who decides the result of a game?

Last year during a meeting of the FIDE Rules Commission one discussion turned to the topic of who actually decides the result of  a chess game. I had simply assumed the game was decided by the players by actually playing the game (with a few minor exceptions) and if I remember correctly made this point. I was a little surprised that there was a strong argument that in fact it is the arbiter who decides the result, which was put forward by some of my colleagues. There position was that as the arbiter is required to ensure that the Laws of Chess are followed, they have the final say, as there are situations where the Laws may not be followed (eg outside assistance, intimidating the opponent) or simple situations like losing on time or claiming repetitions.
While I realise that we may be splitting hairs on this topic (it would be a shortly unemployed arbiter who just changed results based on a whim), it still may be a distinction worth making.
Round 9 of the current FIDE Grand Prix event saw an incident which is related to this topic. There was a repetition claim in the game between Karjakin and Caruana, and after checking the arbiter declared it a draw. Then the arbiter realised a mistake had been made (due to misreading the scoresheet) and the claim was incorrect. Although both players signed the scoresheets (with a draw recorded as the result) they were asked to recommence the game, which they did (It was still drawn somewhat later). In this case the arbiter was correct in fixing up the mistake, which supports the argument that the arbiter decides the result. Of course if the players had been happy with a draw at that point they could have agreed to one, which then supports my argument!

(As an aside, I am no longer a member of any FIDE Commissions. Short reason: Elections have consequences. Long reason: The subject of a future blog post)