Generally, for major concerts, set lists are supposed to be filed. I have had no first hand dealings with this, but I would think that, if, say, Peace, knows that his songs have been played in concerts by a major artist, he can ask ASCAP to look into the documentation that the venues has provided, and, if there is a discrepancy, have ASCAP straightened it out.

You do have to be a member of ASCAP, and you do need to register your songs, in order to collect, but ASCAP acts on behalf of the songwriter in dealing with performers and the venues.