When you are in a situation where you cannot "win", one potential solution is to remove yourself from the situation. Quit. This is sometimes the best solution, but not always. Bad situations often arise and fester because nobody tries to improve them. Leaving usually is easier than making positive changes, but does not provide the rewards of a job well done.

Some people stay in bad situations (bad jobs, bad relationships, etc.) out of a sense of loyalty to others in the situation. Loyalty is an important part of one's character, but it can go too far. You have a responsibility to yourself. If TheyCanFireMe, then there is nothing wrong with me firing them.

Related to loyalty is the desire for others' affection or respect. If you leave, they might hate you or blame you. You may not be able to LetItBe when they turn on you.

You may have other places to place your loyalty, for example to your family, to your dreams, to your principles and values. Conflicting loyalties is all-too-common.

One reason people stay in a situation they cannot win is that they have a belief that there is "justice". An unfounded belief that some power above will correct the situation and bring a just end. An extremely valuable skill to be able to recognize when you cannot win AND get out before it destroys you. It is said that it doesn't take strength to hang on but it does take strength to let go. 

Another reason people stay in bad situations is a belief that "Winners never quit, and quitters never win." Stick-to-it-iveness is another valuable trait that can go too far. No-win situations do exist, and failing to recognize them as such leads to misery. Hanging on to the bitter end may make you feel like a hero, but don't be surprised if others fail to recognize your achievement.

Finally, many people stay in bad situations because they want to avoid change or risk. The bad situation may be bad, but at least it is a well-known situation and it might be bearable. If you change your situation, you might make it worse. It takes courage and insight to make changes for the better.
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A really innovative way of leaving a situation which is bad is simply this - take a vacation! Someone I knew did this once and midway through the vacation called and informed his boss that he would not be returning, he relayed that while on vacation he discovered an opportunity he just couldn't refuse. He followed up with a letter of resignation, and still has a standing offer to return to his job at any time. He cleverly crossed the bridge to the other side without destroying his ability to return. 
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A vacation can also be a false sense of freedom for short term. You may leave and feel good for a week if you like that sort of thing, but you always have to come back... in the end. (unless some rare magical situation occurred, like you found a new job while on vacation). You are just simply running away for short term. It could bring about a good thing... but I think that's rare and possible wishful thinking. But then again, I don't even like holidays that much.
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See also: WinWinOrNoDeal, ChangeYourOrganization, LifeVectors