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The significance of the White (Peacekeeper) King is to prevent their victory being inevitable. If White takes the Red King and Black takes the White King that is a draw, although it could be argued that it should be a Black win. I agree that what should happen to an army losing its King to White.
From the description of the goals of the individual players: 'For the two Foreigns, the goal is to capture the other country's king, while the goal of the Peacekeeper is to capture the two Foreign kings', I do not see why the peacekeeping force should include a king at all. It is not necessary for the resolution of the game. Also, the author did not specify what happens when the Peacekeeper captures its first king; let's say they captured the Red king. Game is not over: at this point it is still possible for the Peacekeepers to win. But it is no longer possible for Black to win, even though they still have a king standing! On the other hand, it is still possible for Red to win -- by capturing the Black king before the Peacekeepers get to it. Somehow, this doesn't make sense. Third, once a king (say, the Red king) is eliminated, can Red continue playing? If not, then it's no longer possible for anyone but the Peacekeepers to win. This practically reduces the winning condition for the Peacekeepers to capturing one king -- but then there is no call for making the forces uneven. Furthermore, the author does not specify what happens to the eliminated player's pieces -- are they removed from the board? Do they pass to the victor? On the other hand, if Red can continue playing kingless, it leads to the paradox outlined above.
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