Is there a game-theoretic interpretation of nimber multiplication? There is such for addition (a single move in a+b is either a move in a or a move in b).



Nimber multiplication
As Alex says, there's no good one. One way to see the problem is to compare nimbers with (surreal) numbers. Addition in both is just game addition, so you can consider nimbers and numbers together and add them consistently. But there is no consistent multiplication. (What would the unit be?) A good game theoretic multiplication would explain how to multiply arbitrary games.


Instead, we have nimber multiplication, which is constructed ad hoc and based on the algebraic structure of the nimbers: nimbers include all additive groups of characteristic 2, and the multiplication makes them into the universal field of characteristic 2 (in the sense that any field of characteristic 2 embeds in the nimbers); any totally ordered field embeds in the surreal numbers.


