When two players agree to play a timed game, they have agreed to take a loss if their timer ends before they can defeat their opponent. As such, time is a resource that affects every such match - time management should not be considered secondary to advantage on the board. Chess skill is not simply about being able to find the right moves, it's about finding the right moves fast. Long classical time formats place an emphasis on extreme precision, depth, and accuracy by affording the players more calculation time to find deeper ideas faster. Shorter rapid time controls promote speed and quicker recognition. In any timed situation, an opponent who thinks the same way you do but faster will be stronger than you are.
Part of the strategy of a no-increment game is playing fast enough to put your opponent in serious time trouble, and since time can never be regained, often this means winning the game by expiring your opponent's clock. While it may be sometimes referred to by the unflattering term "dirty flagging", it is no less legitimate of a win than a checkmate or a resignation.
I recently found myself in a drawn position (king and rook for each side) in a zero increment online game. I had about 10 seconds on my clock, my opponent had about 20 seconds. He ignored draw offers, and I eventually lost on time (despite trying to premove as much as possible) as we shuffled our pieces back and forth.
How could I have better handled this situation?
The fact that you are in a drawn position with only ten seconds on your clock is a huge disadvantage. In fact, while the position might be a draw with best play, your opponent has managed to pull you into this endgame while severely exhausting your time - the phrase "drawn endgame" here is a misnomer, as your odds of holding on for a win are very slim! The ideal way to deal with this situation is to not get into it the first place.
This might mean practicing playing moves in the opening much faster, or training oneself to spend less time calculating earlier on in the game, or simply not playing games with no increment in the first place. In fact, you may have even made a mistake by going into this positionally drawn endgame while down time - perhaps you could have noticed that you would lose on time in an endgame and tried to keep your pieces on the board? Other options may not have been theoretically sound, but if you're going to be down to only ten seconds on the clock, it may be better to start a quick unsound mating attack than to play out the draw and guarantee yourself a loss!
Inevitably, though, all players will find themselves in the situation described in the question: down on time and scrambling to hold onto a position, especially with no increment. There are some tricks to try and salvage games, though:
- Look for stalemating options. If you already have few pieces, you may wish to simply throw away material in order to reduce the number of legal moves you can make and raise your opponent's odds of causing a stalemate. You may even intentionally pin your pieces to your king to try to get a stalemate out of nowhere, or run your king towards dangerous squares on purpose in the hopes that your opponent will accidentally leave you with no moves.
- Move as fast as possible. Obviously, every millisecond you spend increases your chance of losing - try to think on your opponent's time where possible. Premoves are your friend, try to string together as many as you can.
- Take the initiative. Look for any attacks, checks, and forcing moves. Your opponent will likely have to spend more time than normal when placed into check or when their powerful pieces are attacked - make them react to your moves if you can. This will give you more time to think, and may put your opponent into time trouble as well. By contrast, if you have to react to your opponent's checks, you will quickly lose time.
- Anticipate your opponent's moves. If you are defending, look for your opponents checks. A smart opponent may prioritize playing checks and forcing moves quickly to flag you, so be ready to step out of check or block nearly-instantly.
- Insufficient material. If you can trade down to a situation where the opponent doesn't have enough material to mate you (King vs. King + Bishop, etc), the game will instantly be drawn! If ahead or in a drawn position, try to trade an opponent down to just a lone king, so that if your clock runs out, it will be a draw rather than a loss! Be aware that pawns for either side can prevent the insufficient material draw from triggering, since with a pawn or two on the board, there may still be legally-possible checkmate positions that could use a pawn to limit its king's movement enough for a checkmate. For example, while checkmate can't be forced in a KNN vs K endgame, a mate could still happen with errors by the defense. Thus, draw by insufficient material will not occur.
- Forget about accuracy. If you don't see any better objective to go for, find any legal move and have it ready - force your opponent to spend time and effort refuting your inaccurate play rather than guaranteeing yourself a loss by taking time to play accurately. You might hang several pieces, or allow a forced mate, and your accuracy score for the game is undoubtedly going to suffer. But your odds of winning will still be increased, since your only hope of survival is to keep your clock running for as long as possible. And you just might play some of the best moves by chance, too, so be on the lookout for your opponent's replies to your moves. Give your opponent a chance to make a big mistake!
- Go for draws by repetition. Try to repeat moves as often as possible, look for perpetual checks and shuffle around two or three squares and hope for a draw.
- Offer a draw. No, you don't deserve one - your opponent would be a fool to take it -, but you may just find someone who will. However, one or two offers is enough. Any more than that, and you're simply wasting your own precious time - if they want one after that, they can send you one themselves if they missed the first few. Plus, spamming draw offers to an opponent who has already rejected your offer could be seen as unsportsmanlike - I imagine the argument would be that it's an intentional distraction when overdone?
- Be unpredictable. You may get some leverage out of playing nonsensical or counterintuitive moves, especially if your timer woes are also affecting your opponent. Your opponent might not see that your rook was unguarded and bite on your tricky move. Or your opponent might have premoved something that will stalemate you if you do something unexpected.
Once in this situation, you'll probably lose, so the stakes are already at their lowest. You'll definitely lose if you don't go for something! Try to avoid getting into this position, and spend an appropriate amount of time thinking for the time control being played.