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Fascinating article, and a brilliant illustration to the Theory of Moves.

My only issue was that I took exception to your explanation of rule 5 (ii), as I believed that a couple of crucial points had been missed.

Firstly, should we find that it is the player who made the first move, who would choose to shift the state back to it's origin, the opponent would then have the option of moving or staying. Not restricting to a 2x2 board, the second player may well have alternatives than the first move that the original player used, which could lead the game in a different direction.

However in the full rules of the game this is not a plausible situation (least not with 2 players).

In a game, a player can only move one dimensionally. Whilst implied, this isn't explicitly stated in the rules, causing my brief confusion. However since a player can only move one dimensionally (on any sized grid), with each player alternating in moves, there must be an even number of moves to return the game back to it's original state.

Secondly, is what happens when the grid is expanded. As you have said, if player one knows that his original move will result in the same set of moves as occurred before, then he will not choose it. On a 2x2 grid, his original move was his only move from the original state.

As you hinted at earlier in the article, options are usually more wide ranging (especially in a situations such as the Cuban Missile Crisis). Suppose then Player 1 may want to consider the possibility of taking a different route? This move may be his "second best" choice to the first move he could've made, (thereby explaining why it wasn't chosen to begin with) and results (as per rule 5(i)) with a state better than the one he has started with. Given the implications of Rule 5(ii), the game would still finish at this point and negotiations would not continue, and you can have no "Plan B", which is not usually the case. I'd be interested to know if there is an argument against an amendment to Rule 5(ii) for such a situation.

However when you do expand to show progression of possible moves, in the Cuban Missile Crisis example, knowing the reactions of what the other side would choose in this case, and logically working through all possible paths, you will come to the same conclusion as before. Any alternatives not mentioned, that had been options to the US or the Soviet Union, would have led to unfavourable consequences to both sides, or back circular again to the original starting point, and hence then why they then chose the moves that they did.