After Sunday’s eleventh game of the 2014 FIDE World Chess Championship in Sochi, Russia—a repeat victory for title-holder Magnus Carlsen—the Norwegian looked worn, flashing both the mental fatigue of a two-week fight and emotional joy of victory.

“Today was one of the toughest days of all,” said a raspy Carlsen, “but I'm so happy I was able to push through.”

The...