What's the criterion for the sequence of the Four Gospels?

It seems likely that they are ordered by the year in which they were written, with the exception that Matthew and Mark are switched. It seems natural that Matthew is placed first because it is written to a Jewish audience and has such a high incidence of Old Testament quotation. It is a natural bridge from the Old Testament to the New Testament.

The early manuscripts present the Gospels in a variety of orders:

1. Matthew, Mark, Luke, John (most manuscripts)
2. Matthew, John, Luke, Mark (Codex Bezae, Codex Washingtonianus, the Old Latin)
3. Matthew, Mark, John, Luke (Curetonian Syriac)

(source)

The sequence Matthew, Mark, Luke, John is strongly associated with Augustine, but can be traced even earlier, to Origen of Alexandria (3rd century), who recorded:

Among the four Gospels, which are the only indisputable ones in the Church of God under heaven, I have learned by tradition that the first was written by Matthew, who was once a publican, but afterwards an apostle of Jesus Christ, and it was prepared for the converts from Judaism, and published in the Hebrew language.

The second is by Mark, who composed it according to the instructions of Peter, who in his Catholic epistle acknowledges him as a son, saying, 'The church that is at Babylon elected together with you, salutes you, and so does Marcus, my son.'

And the third by Luke, the Gospel commended by Paul, and composed for Gentile converts. Last of all that by John. (as preserved in Eusebius historia ecclesiastica 6.25.4-6)

The Patristic evidence (the writings of early Christian historians) favor two possibilities:

1. Matthew, Mark, Luke, John (Origen, Augustine, implied by the Muratorian Fragment, sometimes Irenaeus)
2. Matthew, Luke, Mark, John (implied by Clement, sometimes Irenaeus)

It is noteworthy that Matthew is always first. In the early sources there's no dispute, debate, or competing theory. Matthew is always first. The "Western order" which puts John 2nd is often thought of as an order of prominence--it puts the apostles first and the the associates of apostles (Luke & Mark) after the apostles.

It is also noteworthy that Origen of Alexandria disagrees with his own teacher, Clement of Alexandria. Origen puts Mark 2nd; Clement puts Mark 3rd.

The evidence is abundant that the early church believed that Matthew's Gospel was written first and John's Gospel was written last, and thus it comes as no surprise that Matthew's Gospel is always listed first and John is usually last. However, the early church appears less certain on the sequence between Mark & Luke; it is not difficult to hypothesize why this may be.

The Gospels were written in different parts of the Mediterranean (most sources put Matthew in Judea or Syria, Mark in Rome, Luke in Achaia, John in Ephesus). If Mark and Luke were written within a few years of each other, some churches would have obtained Mark's Gospel before Luke's, and some churches would have obtained Luke's Gospel before Mark's. Notably, the highly influential church in Rome is where Mark's Gospel is said to have originated. I explore this topic further in this video on my channel, where I argue that Mark's Gospel was written third, but that the Christians in Rome did not have Luke's Gospel until a few years after Mark was written.

If this argument is correct, the reason for the different ordering would be as follows:

• Matthew, Luke, Mark, John: the chronological order in which they were written
• Matthew, Mark, Luke, John: the chronological order in which the highly influential church in Rome received the Gospels

I recognize that "Markan Priority" (the view that Mark was the first of the Synoptic Gospels written) is highly popular in academia today. In this video on my channel I review the major arguments for Markan Priority, and show that each is circular, reversible, or both: Why I Do Not Believe in Markan Priority.