Allan Wechsler writes: In response to Jim Propp's original question: Permitting double moves
doesn't affect the basic nature of the two classes of combinatorial games with the most complete theories, namely impartial games and partisan games. That is, "doubling" an impartial game gives another impartial game, still amenable to Sprague-Grundy analysis, and "doubling" a partisan game gives another partisan game, amenable to Berlekamp-Conway-Guy analysis.
Yes and no. There are still game trees, P-positions, and N-positions. But the whole apparatus of disjunctive sums and equivalence (sometimes called equality) falls apart when you try to make it work in this context, so nimbers and surreal numbers can't be applied. That's because the induction arguments that make the theory work are sensitively tuned to the fact that when you make a move in a disjunctive sum, you make a move in exactly ONE of the summands --- whereas, when you are faced with the game G+H and you want to make a double-move, you can make a double-move in G, a double-move in H, or a single move in each of the two sub-games. Which is not to say things are hopeless; after all, BC&G do consider (and prove things about) conjunctive sums, in which a player must move in EVERY component. But that's a different theory. If anyone thinks I'm missing something, I hope they'll tell me how to apply Sprague-Grundy theory to Nim when doubled moves are allowed! Jim Propp
The vast ocean of games between these two extreme islands remains vast and oceanic upon doubling :)
On Tue, Dec 30, 2014 at 9:24 AM, John Aspinall <j@jkmfamily.org> wrote:
In Go, ko-threats are effectively saying "I dare you to give me two moves in succession here", i.e. with the first of the two specified.
On 12/30/2014 08:02 AM, James Propp wrote:
Has anyone studied variants of standard combinatorial games (such as Nim) in which each player makes two standard moves in succession on each turn instead of just one?
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