Strictly speaking, there is no luck in chess. And yet it has often been said that to win a major tournament, a player must be both lucky and good — meaning that his opponents must make enough errors at critical moments, and the player must be good enough to exploit them. If Fabiano Caruana should win the Candidates tournament underway in Berlin, which he now leads, he will be able to point to Round 4 on Wednesday as a turning point, one in which he was fortunate and skilful. Caruana, an American, beat Vladimir Kramnik, a Russian ex-world champion, in a game in which both players held significant edges at different points. In the end, a terrible blunder by Kramnik ceded the point to Caruana. It had to be a bitter pill for Kramnik, the leader after Round 3, as it came one round after he played one of the best games of his career. Caruana now has three points (each win is worth one point and each draw is worth a half point), while Kramnik is tied for second with Shakhriyar Mamedyarov of Azerbaijan, each with 2.5 points. The Candidates is being organized by World Chess, the commercial partner of the World Chess Federation (FIDE), the game’s governing body. The prize fund is 420,000 euros. The winner will receive 95,000 euros, but, more importantly, he will earn the right to play Magnus Carlsen, the world champion, for the title in the Championship Match in London this November. In addition to Caruana, Kramnik and Mamedyarov, the other competitors are Alexander Grischuk and Sergey Karjakin, both of Russia, Ding Liren of China, Levon Aronian of Armenia, and Wesley So of the United States. The tournament is a double round-robin, with each player facing all the other competitors twice, once with each color. The venue for the tournament is Kühlhaus (or “cool house” in English), an industrial building in central Berlin that was built in the early 20th century as a cold-storage facility for fresh produce. Among the principal sponsors of the tournament are PhosAgro, a giant Russian fertilizer company; Kaspersky Lab, a global cybersecurity firm; E.G. Capital Advisors, an investment management company; S.T. Dupont, a global luxury goods maker; Prytek, a venture capital firm; and Isklar, a Norwegian mineral water company. Matthias Deutschmann, a popular German kabarettist (a form of satirical review), made the opening move for Round 4. In the game between Kramnik and Caruana, Kramnik had White. He opened with 1 e4 and Caruana chose the Petroff (also known as the Russian) Defense. That is thought to be one of the most soporific openings in chess. Indeed, it often leads to draws. The game between Kramnik and Caruana seemed to be heading in that direction, but both players kept pressing and probing, hoping for an error from the other. It finally came on Move 23, when Kramnik played 23 c5 instead of 23 gh3. He evidently miscalculated, or just overlooked Caruana’s reply of 23 … f5. Kramnik was soon in trouble and had to give up a piece for Caruana’s advanced g pawn. Kramnik was not without compensation, however, as he had a far advanced d pawn. And Caruana was down to only seconds on his clock, with 15 moves to make before the first time control. Initially, Caruana was able to steady his nerves and find the best moves, but he finally went astray and Kramnik was able to get his queenside pawns rolling, while Caruana’s were stuck on the kingside. After some additional inaccuracies by Caruana, the players reached the time control and the position was far from clear. It even seemed as if Kramnik might have an edge. After another mistake by Caruana (41 … h5, instead of 41 … f3), Kramnik did have a clear edge. But now it was Kramnik’s turn to miss the best move (44 c4 instead of 44 b4). After some exchanges, the players seemed to be heading for a draw, but Kramnik then made a bad move (58 Kb1, instead of 58 Kc1) followed by a terrible blunder (59 Kd1, instead of 59 Rf6). Caruana did not miss his chance and after 60 … Rd1, Kramnik realized that he could not take the rook by 61 Kd1 because after 61 … h2, he could not stop the h pawn from queening. (62 Rh6 Bh4.) Kramnik resigned a few moves later. Aronian had lost badly to Kramnik in Round 3, so psychologically, as well as from the standpoint of the tournament standings, he wanted a win. Karjakin, the winner of the 2016 Candidates tournament, was in the same situation vis-à-vis the tournament standings. Karjakin, who had White, opened with 1 d4 and Aronian chose the Ragozin Variation of the Queen’s Gambit Declined – a popular and double-edged system. Aronian is at home in complicated positions, so the opening suited his style, whereas Karjakin is more comfortable in positions with clear strategic themes. Karjakin decided to not back down from a fight, sacrificing a pawn for the initiative. He soon found himself in an unfamiliar position, however, and gave up a second pawn in a desperate effort to maintain his initiative. Aronian was able to give back one pawn to force some exchanges, after which Karjakin was in trouble. Karjakin is noted for his defensive abilities, but this time he was not able to overcome his disadvantages and succumbed after 68 moves. The win puts Aronian at 50 percent and leaves him in the hunt. Though there are still 10 rounds remaining, Karjakin’s situation in the standings (he has one point) is already dire. Grischuk may have been a little upset with himself after his game with Ding, as he seemed to miss a golden opportunity. The opening was the Anti-Moscow Gambit of the Semi-Slav Defense. (Who doesn’t love the names of chess openings?) The system is one of the most heavily analyzed and sharpest in the game. On Move 12, Grischuk sacrificed a piece – a known idea – that left Ding’s king permanently exposed. Initially, both players handled the complications fairly well, but after 21 f4, Ding made what could have been a fatal error by playing 21 … gf4. If Grischuk had replied 22 Bh4, he would have had a terrific attack after 22 … Bf6 23 Qg4, though it was necessary to see that follow-up. Instead, Grischuk continued 22 Rf4. The missed opportunity put Ding in the driver’s seat, but he missed the best follow-up as well. From there, the game see-sawed back and forth, with some hair-raising tactics. In the end, though, after many exchanges, it petered out to a draw. Grischuk and Ding each have two points and are tied with Aronian, one point behind the leader. The quietest game of the day was between Mamedyarov and So. Though the opening was a sharp line of the Queen’s Gambit Declined, it has been heavily analyzed and both players had clearly studied it beforehand. They played rapidly and arrived at a position after 20 moves in which neither player had an edge, despite a slight imbalance in material and pawn structures. Mamedyarov, who was White, forced a draw by repetition after 31 moves. So, like Karjakin, has one point and really has his work cut out if he is to try to get in contention for first (which, he has said himself, is the only place that matters). Round 5 is at 3 PM, local Berlin time. The tournament can be watched live at www.worldchess.com, the official site of the World Championship.