In the view of the ancients, agency and representation was the same as being there. If the elders and friends were there representing the centurion, so then was the centurion. This was a reality of ancient culture, and the evidence shows that this was accepted to the point that actions directed by another could be directly attributed to that person.

For example, in the Babylonian creation story Enuma Elish (Heidel, Babylonian Genesis, 188), Tablet 6 clearly says that Marduk gave a plan to create man to the god Ea, who then went and executed the plan.

Here is another example, but from the Bible:

> John 19:1 Then Pilate therefore took Jesus, and scourged him.

Pilate didn't leave the scourging to his soldiers, did he? The Latins had a maxim: "What our agent does we do ourselves." Historically it is quite likely that the centurion sent elders and/or friends to make the requests -- under the ancient rules of patronage and honor, a person of high social status (like the centurion) never made a request of one of lower status unless they were desperate. To have actually come out to Jesus physically would have been exceptional.

Why does Matthew shorten the story using the principle of agency? The answer lies in his addition of the material about Abraham, Isaac, and Jacob, and his goal to compile the teachings of Jesus and his lesser emphasis on action. But Luke, whose concern is more to express the universality of the Gospel and is less concerned with specific parties, prefers to report the interaction between the centurion and the Jewish leaders, thus exemplifying the cooperation to be demonstrated by those who come to Jesus with no concern for ethnic barriers.