This is a topic where there’s a lot of smoke but fire is illusive. Probably the most ardent and vocal critique concerning this was Bobby Fischer with the Soviets his target. He wrote a 1962 article for Sports Illustrated accusing three of the top Soviet players of pre-arranging draws (all 12 games against each other drawn, all in less than 18 moves) in the 1962 Candidates Tournament. After his article, the rules for the tournament to determine the challenger for World Champion were changed. But this wasn’t the first instance where allegations were made against the Soviets. 10 years earlier in the 1952 Interzonals, the top 5 Soviets drew all 10 games with each other. Again no proof but certainly the results were noted and commented upon at that time. A web search on “Botvinnik collusion” will bring up many articles that have second hand statements where Botvinnik is alleged to have stated that there was “state sponsored” pressure on players to not perform at their best levels against certain players but I couldn’t find any specific interviews by him where he said this. Ditto for statements allegedly by Botvinnik that Stalin directed players to throw games to him until he (Botvinnik) found out and had it stopped. So for state sponsored collusion, accusations but no confirmation.
As to individuals, one really has to look no further than the grandmaster’s draw to see that final tournament scores aren’t always what they could be if players always played their utmost in the game. Does playing to an artificial or pre-arranaged draw qualify as tournament fixing? In some cases probably not but in others, it’s less clear. An example which didn’t affect the tournament winner but could have changed the last position is in the 2015 Capablanca Memorial. In the second game between Yu and Nepomniachtchi, a draw by repetition was agree to after 9 moves. This game was played in the last round and a loss by Nepomniachtchi would have dropped him into a tie for last (Yu won by 1.5 points so other than ELO points, the game didn’t affect him). So the impact of the game was an undisputed 5th for Nepomniachtchi (out of 6) versus a possible tie for 5th-6th. Not world shaking but possibly annoying to you if you were the 6th place finisher. I only call out this game because you have to work to come up with a shorter draw than this in a tournament. A quick draw with more impact was a last round, 13 move draw by repetition between Alexei Fedorov and Sanan Sjugirov in the Cappelle la Grande Open. Both players ended up in a tie with six others for first with Sjugirov taking the title on tiebreakers. In fact, five players entered the last round tied for first and four played each other to draws. The fifth played a draw with a player .5 points behind. None of these games lasted longer than 21 moves. At tournament end, these five players ended up tied for first with 3 others who won games in the final round. Given that 16 players were a half point behind these players, if either of the three last round draws involving the players tied for first had ended in a win, there would not have just been a clear tournament winner, the loser would have been severely shoved down the totem pole into a tie for 8–25th. There are also quick draws earlier in a tournament whose overall effects are much harder to gauge. There’s no real way to assess the impact on later games if a player essentially get a bye via playing a 5 minute draw versus fighting it out for a couple of hours but clearly he or she will be more rested than other players who are engaged in mental combat. There’s certainly nothing new in taking a quick draw as shown by this sentence in “Max Euwe: The Biography” concerning the London 1922 tournament, “Capablanca won the way he wanted to, with thoughtful play and here and there a tactical gentleman’s draw (with Alekhine and Rubinstein).” Looking at these drawn games, Alekhine’s game was 17 moves long and in round 9 (of 15) while Rubinstein’s 13 move game was in round 14. Checking the standings and the games subsequent to these, both could have significantly affected the standings, at least as far as prize money went.
Getting back to the initial question concerning “fixing”, I don’t think anything I’ve said really proves tournaments have been fixed, at least at an egregious level. Certainly if Fischer was correct, the actions of the involved Soviet players could be called fixing the tournament. But as far as I know, it’s never been admitted or proved. There’s a statistical analysis paper, “Did the Soviets Collude? A Statistical Analysis of Championship Chess 1940-64”, by two university professors but statistics don’t “prove” a result, just provide confidence that you know what’s going on so it falls short of demonstrating the fix was in. As for grandmaster draws, it’s in the eye of the beholder whether that constitutes fixing a tournament. Clearly a 9 move draw by repetition is suspect and in this particular case might have changed the last place position but is that really fixing the tournament? In the London tournament, the quick draws could easily have resulted in significantly different overall results if they had been fought with effort from all parties. But how does one call that fixing when Capablanca played what he thought was the approach that give him the best chance to win: drawing with dangerous foes while preserving strength for the next opponent? In the case of his game with Rubinstein, a loss plus a loss in the final round would have dropped him down to a tie for first while the quick draw was a guaranteed tournament win. How does one knock that strategy? And looking at it from Rubinstein’s perspective, he was offered the opportunity to walk away with a draw against someone who hadn’t been defeated since 1916. Hmmm, play for the first win against Capa by anyone in 6 years or take a draw? Tough choice there. Defending Capablanca’s quick draw with Alekhine is a bit more difficult if one believes that chess should always embody a spirited fight until it’s obvious there will be no winner. This draw certainly didn’t guarantee him the tournament but he walked away from a dangerous foe without losing ground and with a clear head for his next game with Reti which he won. His gentleman’s draw may not have been in the absolute best spirit of chess and may ultimately have affected the positions below him but I don’t think anyone would consider him guilty of “fixing” the tournament or trying to accomplish anything other than getting himself on the top of the podium. And so we finally get to the Cappelle la Grande Open. Of all of the tournaments, this is the one I’d toss to the ethics teachers and have them weigh in. The top five players entering the last round all played to draws and ended up tied for first prior to the tiebreaker. If any of them ended up on the losing side of a game, the loser would have dropped like an anchor, tying a herd of players at 6.5 points. So how are these drawn games characterized and were there any sort of agreements made such their outcomes were decided other than across the board? It can’t be proven so in the American judicial system you’d have to say the players are innocent.

I've been following the chess world championships closely. It's interesting that betting companies are now prominent sponsors of these tournaments. Anyway, on completely unrelated note, I found a really well written response to a question asked on another site about possible match fixing in chess. It would be a shame if top level games tournaments really were marred by this terrible trend that affects so many other sports. Perhaps with the amount of analysis we have available to us in chess, however, we stand a better chance at identifying it and punishing players who may be playing for a specific result in return for greater rewards.
https://www.quora.com/Is-there-match-fixing-in-chess-matches-and-tournaments