As best I can tell, there are two different games being played. One is the attempt by each side to make sure that either it gets the credit for solving the problem or the other side gets blamed for not solving it. That game is basically about rhetoric and PR.
The other and more interesting game, now that the administration has dropped its demand for tax increases, is about whether or not to raise the limit by enough to get past the next election. From Obama's standpoint, the answer is, I think, obvious. Having the option of deficit spending is almost always a benefit for those currently in power, since it lets them buy votes without obvious cost. Concern with the size of the national debt may have changed that, at least for a while, but I think more likely not. Hence Obama would like to be able to spend as much money as he wants through the election while satisfying demands for fiscal responsibility via cuts, possibly imaginary, in future expenditures.
The Republicans, on the other hand, would like to be in a position to force real reductions in spending, both because many of them think reductions are a good thing and, I suspect, because many of them think that reductions in spending by Obama will cost him the votes that the spending would have bought him. One way of doing so is to arrange things so that a second increase in the debt limit will be needed before the election, and make their support for such an increase conditional on serious reductions in expenditure—which are not, so far as I can tell, happening on either of the current plans.
There is one other feature of the situation, one which may explain Obama's failure to hold out for tax increases even while orating in favor of them. An election is coming up, and Democrats to Obama's left have no realistic alternative.