These composers have common approaches. Cage is known for prepared piano and aleatoric music, he is more piano and vocal, often beautiful. Kagel is more theatrical/absurdist/collage/distortion. Often you will need to know a particular piece for correct answer. (Cage, Kagel)

Being both contemporary composers, Cage is more of an artist, while Lucier is a scientist. Cage is known for prepared piano and aleatoric music, he is more piano and vocal. It can be interesting to read about how Lucier made his different recordings using different physical experiments. (Cage, Lucier)

Compare Cage aleatoric music (often slow and beautiful) and Bolcom neoromantic eclectism (often using popular songs). (Bolcom, Cage)