The King and Queen of all the Worlds
The live chess… Has there ever been a better metaphor for how the Doctor lives his life, and how he defeats his opponents? Of course that game is at its most basic an analogy for how the Doctor wins throughout the episode: By conceding the game, he gets what he wants.
But here’s what I missed initially - River is the Queen. As everyone has noticed (and complained about at great length), River has throughout her life been treated like a pawn, been used (and mis-used and hurt and abused and exploited) for other people’s ends and moved across the board over and over. But River isn’t anybody. The result of all the maneuvering is that by now she (metaphorically) has 40,000 volts running through her, and the next person who touches her is going to burn. And that is the River at Lake Silencio. Someone who has been moved one time too many and will rather let universes collapse than be forced against her will again.
For most of the episode the Doctor tries to reason with her, tries to move her the way he’s moving and manipulating everyone else around him, but all he gets in return are deadly sparks - even with all of creation hanging in the balance.The key turns out to be just the same as in the real match - concede the game. Stop trying to force her, stop using her, and instead acknowledge her for what she is - his equal, his partner, his Queen, his wife. And so they play the game together.