I was involved in an endgame where it was my
King against my opponents
Bishop and I knew he could checkmate me, but he wasn't aware how. I just kept going in circles, dancing around his pieces and he could not figure out how to checkmate me. Into move 140, I was getting frustrated and thought about either resigning or asking for a draw. I eventually asked for a draw and he agreed, so my question is whether you are on the losing end or the winning end, when is it time to mutually agree to a draw?
When to mutually agree a draw depends on the skill level and the position in question. An agreed draw is most logical when neither side believes they have real winning chances (Sometimes people just chicken out in complex positions...). In the case you describe, practically speaking, there are no real winning chances for either side. Alternatively, 2 GMs may rightly agree a draw in many drawn theoretical endgames where mortals such as you and I should probably play on.
For the rulebook perspective... the rules state that "The game is drawn upon agreement between the two players. This immediately ends the game." It is correct to offer the draw after making the move but before pressing the clock. You and your opponent may agree to a draw at any time. In terms of etiquette, it is generally considered poor practice to offer a draw when losing or significantly worse. It is also poor etiquette to repeat draw offers and the TD may penalize you for "annoying the opponent". As a matter of preference, I offer a draw only when I believe that I have no real chance of losing. Rules such as 3 move repetition and the 50 move rule are there to ensure that your opponent cannot meaninglessly torture you in a position such as the one you describe.
In your given situation, it may be useful to claim a draw by the fifty move rule. The fifty move rule states: "The game is drawn when the player on move claims a draw and demonstrates that the last 50 consecutive moves have been made by each side without any capture or pawn move." The case of B and N versus King is a classic example where 50 moves may elapse without a pawn move or capture.Tweet