What kind of image does Sir Gawain and the Green Knight try to convey through the beheading game?
Since we know that both Fled Bricrenn and Sir Gawain and the Green Knight contain a plot of the beheading game, what kind of image does Sir Gawain and the Green Knight try to convey through the beheading game?
Both stories mentioned have a great deal in common. There is the "beheading game" in both, as well as a hero that honorably responds to the challenge made by the giant/churl. (Fled Bricrenn or is found in Celtic mythology, while the Gawain tale is set in the court of King Arthur.)
I believe the image being conveyed in Sir Gawain and the Green Knight is one that reflects the integrity and chivalry of a true hero.
While the killing of the giant seems an easy task to the untried Sir Gawain, when he does not win as he expects—because the giant has the magic ability to pick up his missing head and walk out—instead of fleeing in a panic, Gawain remains steadfast.
Sir Gawain accepts that as a true knight he must answer the summons of the Green Knight as they had agreed prior to the "beheading game." So one year later, he packs up and travels to answer to his end of the bargain, a "swing for a swing."
The image here is that of assuming the responsibility for one's actions, or keeping one's word, because it is a question of duty, not of one's personal preference. An average man might run away, as is the case in the Celtic version of the story, but Gawain, not only a true knight, but a member of Arthur's court, must fulfill his promise—it is the only honorable choice.