Who, you ask? Names are not important, and both of them have many anyway. One is the Doctor, Theta Sigma, the Oncoming Storm, Ka Faraq Gatri, Destroyer of Worlds; the other is the Master, Koschei, the second survivor of the Time War, the Merciless One- names. Just names.
Once, they were Theta Sigma and Koschei, playing a game to pass the time at the Time Lord Academy away, and the game was simple. There would be one winner, one loser, when the game was over. Each of them knew the rules, and neither would break them; they played fairly.
And then they gazed into the Untempered Schism, and everything changed. Koschei had the drums, the omnipresent one-two-three-four in the back of his head, and Theta wanted to run more than ever. Now when they played, Koschei played to win, even if it meant cheating and lying, and Theta played absentmindedly, head in his hands as he watched stars die and be born, galaxies spin and twist, and daydreamed about what he could find there, a faraway look on his face.
The Master (for he was the Master, by then, and Theta Sigma the Doctor), mowed down the Doctor’s pawns in droves, laughing at the looks on their tiny little faces, and the Doctor retaliated in broad strokes that wiped out the Master’s bishops, knights, rooks, when they fell. The Master had bright, elaborate plans, intricate and dramatic. Sometimes they worked, partly, but the Doctor usually pulled something out of his sleeve in the last moments, and won battles. Other times the Doctor’s plans were the grandiose ones, desperate and quick, and a few of those times the Master won the battle and other times the Doctor did. It doesn’t really matter, at that point. The Master may be playing to win, but he wants the Doctor by his side just as much as he wants the victory, so he doesn’t attack the Doctor’s king. It puzzles the Doctor, of course. He never was very good with relationships.
And then there was the Time War, and the sweep of other Time Lords on the game board made them lose sight of the game, even as they had lost track of the original reason for it so long ago. They both fought, and then it got worse and worse, and the Master ran, and hid, taking only his king and leaving the others behind to fall.
The Doctor won the war (of course he did, with his only true opponent not fighting), killing billions with one brilliant, huge, stroke, and he had only his king left as well, alone on the board. And then one of the fallen turned out to have lived, and the Master came back, and the game went on. The Master found himself some other pieces, and the Doctor had his companions, his pawns, as always. They fought, still, but sometimes neither of them was quite sure which side they were fighting on. Rassilon came back, and Rassilon and Gallifrey fell again, but they played on, until it was just them, Gallifrey with the other Time Lords back but far away where they couldn’t interfere.
Just them. King against king, one square step to a turn, no bishops, knights, rooks, pawns, no last-minute plans. It was a dance, perhaps, or a courtship, the kings inching forwards and backwards, cornered and twisting away, slow and stately. Eventually, they stood across from each other, the pieces so old they were no longer white and black, but scarred, burned, chunks of wood missing. Across from each other, neither willing to make the first move. It was stalemate, in its way; as soon as one of them moved towards the other, the game would end.
The Mater hates endings, just as much as the Doctor does, even if only the Doctor would ever say it aloud. So they wait, in their way, until one of them makes the decisive move.
Neither of them ever will.
(Eventually, entropy catches up with them. But it takes a long, long time. All you have to do to make sure summer never ends is steal a time machine, after all.)