For now, committee assignment is a pretty closed process, from what I understand. Here is what we get from the Office of the Clerk of the House:
Before Members are assigned to committees, each committee's size and the proportion of Republicans to Democrats must be decided by the party leaders. The total number of committee slots allotted to each party is approximately the same as the ratio between majority party and minority party members in the full Chamber.
Members are then assigned to committees in a three-step process. Each of the two principle parties in the House is responsible for the assigning its members to committees, and at the first stage, each party uses a committee on committees to make the initial recommendations for assignments. At the beginning of the new Congress, Members express preferences for assignment to the appropriate committee on committees. Most incumbents prefer to remain on the same committees so as not to forfeit expertise and committee seniority. These committees on committees then match preferences with committee slots, following certain guidelines designed in part to distribute assignments fairly. They then prepare and approve an assignment slate for each committee, and submit all slates to the appropriate full party conference for approval. Approval at this second stage often is granted easily, but the conferences have procedures for disapproving recommended Members and nominating others in their stead. Finally, at the third stage, each committee submits its slate to the pertinent full Chamber for approval, which is generally granted.
Has there been any discussion on whether an auctioning process could lead to a "better" outcome?("better" has to be clearly defined though).