Of course, the there is a game theoretic reason to shoot the messenger. The whole point of doing so is to burn a bridge. The original meaning of the term is:

Originally in military sense of intentionally cutting off one's own retreat (burning a bridge one has crossed) to commit oneself to a course of action

Ancient battles, and probably to large extend in modern battles as well, were won or lost on moral. When a large part of your army panicked and ran your side was almost certain to loose. Furthermore, whoever was the last to run would be the first one killed when the enemy overran your position. Thus, if you were afraid the soldier next you would run, you were likely to run as well. Burning the bridge behind you was one way to resolve the game theoretic dilemma. Running cannot save your life, so you might as well hold the line.

Metaphorically burning a bridge by killing the messenger serves the same purpose. By publicly killing Sauron's messenger Aragorn is reassuring his allies that he's not going to betray them by cutting a deal with Sauron that leaves them out to dry.