In "Gregory" By Panos Ioannides, the captors’ dilemma is that they do not want to kill their prisoner, but they have been ordered to make an example of him. Since Gregory will not escape, they have to kill him.
A dilemma is defined as "a problem offering two possibilities, neither of which is practically acceptable" (enotes reference).
The guards at the prisoner of war camp do not want to kill Gregory. The narrator, the executioner, specifically does not want to kill him. This is because they have come to like Gregory and see him as a friend. They pity him, because “Gregory was such a miserable little creature, such a puny thing, such a nobody.” Gregory has total faith that they won’t hurt him. He’s not “a bad fellow” except that he talks too much.
“Those who eat from the same mess tins and drink from the same water canteen,” he said, “remain good friends no matter what.”
Gregory is a “silly fool” because he does not run when the guards give him the chance. He asks them where he would go in the cold weather. The guards tease him about the wonderful conditions in the prison camp, and even ask him about his girl to encourage him to try to escape.
In the end, the guards agree that it’s either his skin or theirs, and they execute him. Instead of hanging his body as an example, they give him a proper burial because they respect him and did not want to kill him.
IOANNIDES, PANOS (2011-08-15). GREGORY AND OTHER STORIES . (Kindle location 163-4). Armida Publications. Kindle Edition.