Of course the alternative has always been Griesbach, with the theory that Mark wrote, then Matthew expanded on Mark, and Luke on Matthew. It's cleaner, but it always rubbed me wrong. I actually don't know why. There were also alternative version which accepted not Markan, but Matthean priority and thought of Mark as being a shorter version of Matthew. This never made any sense to me, not because I thought that Mark was less complex (which it isn't!) and therefore had to "evolve" into Matthew, but because if you're going to write a Gospel and make your unique contribution, why give less information rather than more?
Scaer with his catechesis theory has really changed my mind on this in a big way. So, his argument as I read it, seems to be a kind of variation on Griesbach, that accepts Matthean priority. Arthur Just from what I've read by him seems to buy into this as well.
So, Matthew was the first Gospel and it was probably written some time in the 40s to serve as a Jewish Christian catechism. It possesses five great discourses and therefore exemplifies a new Torah or instruction. The named "Matthew" is even very close the word for "catechumen," which explains to me why he is called Levi elsewhere- i.e. he never was named Matthew, he just wishes to be understood as the ideal catechumen of Jesus, like John wants to be understood as the ideal witness. Anyways, Luke read both Matthew and Mark (probably in the later 50s or early 60s) and constructed his Gospel as a Gentile-version of Matthew's Jewish catechism.
Now, Scaer's a little bit more vague about Mark. But he likes the idea of mystagogue and its presence in the very early Church. This would make sense because mystagogue grows out of the Jewish rabbinical practice (present in the ministry of Jesus) of distinguishing an inner circle of those initiated and are therefore allowed to know certain teachings, and those who are not.
In the early Church, the unbaptized were not allowed know a lot of things. One thing they weren't allowed to hear were the words of institution. They were sent away after the sermon and only the baptized could hear them. This explains why the Didache has no words of institution when discussing the Eucharist. Mark, as you might recalled, has a shorter version of them- so perhaps he thought that hearing part of them was alright, but not the whole thing.
Hence, we can infer that Mark intended to a Gospel for those not initiated into the mystagogue through baptism. He knew Matthew, which was for those who had been baptized and therefore wrote a catechism for those who had not been baptized. Luke then wrote for Gentiles who had been baptized and initiated into the Church's mystagogue. This makes sense that he writes to Theophilus ("one who loves God"- I take this to mean just any Catechumen), who has already been instructed (as the prologue says- probably by reading Mark who Luke utilizes) and now is moving onto more advanced teaching.
This would explain the order, the cultural form and the literary relationships between the Gospels. A version of Griesbach makes remarkable sense, if you assume that the Gospel writers were attempting to fill a literary need for catechesis among particular populations in the early Church.