If there is to be a successful resolution of the debt ceiling and government shutdown standoff, it will be because House Republicans come to grips with an important concept that they have, to date, showed little appreciation for. It is called the sunk cost fallacy.
A sunk cost is something you're not going to get back. If you watch five episodes of a television show you decide you don't like very much, you're never getting that time back. If you date somebody for six months before figuring out you're incompatible with them, same deal.
We should only make decisions based on the future costs and benefits of those decisions. The past cannot be undone. The sunk cost fallacy is the mistake people make when they allow sunk costs to shape their future actions: continuing to watch that TV show, or date that person. In extremes, the sunk cost fallacy can be disastrous. World War I and the Vietnam War were examples of nations falling for the error, with combatants doubling down on failed strategies, in part because they had already invested so much, making decisions based on what had happened in the past rather than solely on the costs and benefits for the future.
In their resistance to making a deal to reopen the government and raise the debt ceiling, House Republicans risk making the same mistake (one that would be a great deal less costly than World War I, but quite a bit more costly than watching all of a lousy TV series).
House Republicans pushed a hard line in the runup to the government shutdown, demanding a repeal of Obamacare in exchange for agreeing to fund the government. There was never any way that the White House or Senate Democrats would go along with that, but that was their strategy, and it led to the shutdown of the government.
Two weeks later, Republicans have started to accept that they will not get a full repeal of the health reform legislation, and are trying to work on more attainable goals. But there is a strong current within their caucus that sees the fact that they have shut down the government and attendant decline in popularity as a reason that they must continue to fight.
That's the real message of a much-cited quote from Rep. Marlin Stutzman (R-Ind.): "We're not going to be disrespected. We have to get something out of this. And I don't know what that even is.” Put another way, as Rep. Scott Garrett (R-N.J.) told National Review, "They may try to throw the kitchen sink at the debt limit, but I don't think our conference will be amenable for settling for a collection of things after we’ve fought so hard.”
The fact that House Republicans have "fought so hard" is irrelevant to the future costs and benefits of any deal. The more the caucus is making decisions based on what happened in the past, the less likely they are to make strategy decisions that are best for both the country's and their own future prospects.