The challenge facing a hostage negotiator is knowing if the hostage taker is willing to kill the hostages or not. If the hostage taker has killed before or is suicidal the risk to the hostages is very real and giving in to the hostage taker within reason is appropriate. Often this involves publicizing some grievance the hostage taker has.
The current situation with the Republicans holding the financial health of the United States hostage in the debt limit debate is an example where the Democrats have to determine how seriously the Republicans are in allowing the government to default. The Republicans have become an anti-government party except for defense and veterans programs. For them destroying the American government is not the same risk as it is with the Democrats who value government involvement in the lives of its citizens. The Republicans right now resemble a hostage taker who is willing to kill their hostages than surrender. They see backing down as a worse outcome for them instead of a default. They are the most dangerous type of hostage taker. The most likely outcome is that the Democrats will compromise in ways that they feel minimize the impact on the government to prevent the default. Look for the Republicans to play "chicken" with default right up to the last moment. Any gain they get out of hostage-taking will only embolden them to use the same tactic every time they can because damaging the government is one of their aims.