Reading stuff on this and participating in some online conversations on this, I haven't seen the following discussed yet, so I thought I'd get it out there.
The debacle in the House with the Republican's health plan suggests one of two possibilities will happen. Either there will be gridlock for the foreseeable future, because the Republicans themselves can't make legislation that get enough votes, or there will be a change in approach to break the gridlock that brings in enough Democrats as partners to create a majority. For the latter, however, the legislation that goes through must represent real compromise. Is the White House willing to go for that? I don't know, but let's say it is possible.
Now the game theory part a la Prisoner's Dilemma. A good strategy is tit-for-tat, which means this time around do what your rival did the last time around. If the rival cheated the last time, you cheat this time. If the rival cooperated last time, you cooperate this time. This simple approach rewards good behavior and punishes bad behavior.
In politics, tit-for-tat can appear as logrolling. Reciprocation of favors is how things get done. When I was first taught about logrolling in social studies during high school, it was presented as sleazy behavior. That sleaziness is relative to an ideal where each issue gets debated on the merits and the better ideas that emerge from the debate then prevail. In that ideal world issues aren't linked for strategic reasons. But that simple world can be gamed for individual or small group advantage. And it has been gamed - quite a lot, actually. From the point of view that such gaming is inevitable, logrolling can look better than the alternative. The alternative is not the ideal (which is not feasible). The alternative is gridlock.
If the White House actually wants to bring in Democrats as partners to get something done, how does one get there from here? The above suggests there is a need for some up front show of cooperation on the part of the Republicans. Yet in the Senate, as distinct from the White House, it seems evident that Gorsuch will now get through but with fewer than 60 votes, perhaps even requiring Vice President Pence to break the tie.
It appears too late at this point to have a different candidate, one who is more moderate and hence who could garner 60 votes. But, the President proposed Gorsuch before the health care debacle in the House, under the then thinking that he could have it all his way. Surely that thinking no longer holds sway. Indeed, the President (and/or his close advisers) may now regret having nominated Gorsuch.
If there is such regret it would be evidence that the White House recognizes there is a trade-off here. The choice is either gridlock for some time to come or pull Gorsuch as a first step towards cooperation. In saying this, one should recognize a different possibility, bringing the Freedom Caucus on board. But that bridge was apparently burned by having a rush job on health care. This other bridge hasn't been burned just yet.
If I were betting I'd put my money on gridlock from here on out. But until the nuclear option gets exercised and Gorsuch gets approved, one can still hope that the improbable will happen.