Commission on Presidential Debates
The Commission on Presidential Debates (CPD) is apparently a private, nonprofit corporation which represents the interests of the Republican and Democratic parties, but Issuepedia does not yet have a definitive source for that information. It has had a de-facto monopoly on all major presidential debates since 1987.
According to CPD's web site:
The Commission on Presidential Debates (CPD) was established in 1987 to ensure that debates, as a permanent part of every general election, provide the best possible information to viewers and listeners. Its primary purpose is to sponsor and produce debates for the United States presidential and vice presidential candidates and to undertake research and educational activities relating to the debates. The organization, which is a nonprofit, nonpartisan, 501(c)(3) corporation, sponsored all the presidential debates in 1988, 1992, 1996, 2000, 2004, 2008, and 2012.
Political advocacy organization Level the Playing Field filed a complaint against CPD sometime prior to September 2014, and then filed a petition with the FEC to change the debate rules on 2014-09-10.