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Sharon - why do you like seeing games thru to the end? If you are hopelessly lost, what are you hoping for? That the opponent will time out? give up in disgust? die?
tonyh Since this is a game server that encourages community along with the games, there is no harm in opponents playing out their games or even having friendly chats in the face of definite defeat.
Since BK offers no doubling cube for backgammon, it's really hard to take the game too seriously here. Backgammon with a doubling cube and backgammon without a doubling cube are two entirely different games. Add betting and money to the mix, and you've changed the games even more.
If I find myself in a hopeless position in a game, I usually congratulate my opponent and tell him/her that I'm going to resign on my next move. This avoids a lot of misunderstandings and keeps the tone friendly. If I have mathematically won a game, I never ask an opponent to resign. To me it's really rude to ask that of someone. They may want to have some fun by learning from you.
Being a member of a gamesite such as BK is more than BKN ratings and amassing numbers in one's win column; it's social games and community.
If you're obsessed with wins and ratings, a BG site such as FIBS is probably better suited for you than BK.
Peacepickle; I took the liberty of looking at your profile and see that you play only Backgammon. This is a game where you can have remarkable turn-rounds, even when everything looks hopeless. And you have a super rating, which means that you lose rarely!
A game like chess, though, can be totally won; getting the win quickly can take ages!
When I play games for chat and fun and I was in a lost position, I would suggest that I resign and start another with the same opponent.
But a game like chess
Actually In backgammon it is frequent that a position is reached when one player cannot lose, even if (s)he rolls 2-1 and the opponent 6-6 for the rest of the game. Obviously this is only applicable if there are no overlaps and therefore no chance of capturing. In these cases you should resign unless both players are happy to keep chatting. It is really only tournaments that I would apply this to since a lot of other players may be waiting for the outcome before proceeding.