The sum of two games means a turn consists of playing in either game. Then by the mirror strategy, for any impartial game $G$, $G+G=0$, a second player win. That means that under this sum, equivalence classes of impartial games form an abelian group of exponent $2$, or an $F_2$-vector space. XOR describes the additions with respect to any basis, not just the standard one.

You can change the connection with XOR by changing the sum operation. If in a collection of games, you must play in at least one, but up to $k$ games, then the Grundy values of the components matter, but the second-player wins correspond to collections whose sums without carries have all digits divisible by $k+1$. This was observed by Moore where the games are Nim piles. Of course, this is related to your other question. There are other generalizations possible, where you are allowed to play in other collections of multiple games, and you don't simply XOR the Grundy-values of the components.

The sum of two games means a turn consists of playing in either game. Then by the mirror strategy, for any impartial game $G$, $G+G=0$, a second player win. That means that under this sum, equivalence classes of impartial games form an abelian group of exponent $2$, or an $F_2$-vector space. XOR describes the additions with respect to any basis, not just the standard one.

You can change the connection with XOR by changing the sum operation. If in a collection of games, you must play in at least one, but up to $k$ games, then the Grundy values of the components matter, but the second-player wins correspond to collections whose sums without carries have all digits divisible by $k+1$. This was observed by Moore where the games are Nim piles. Of course, this is related to your other question. There are other generalizations possible, where you are allowed to play in other collections of multiple games, and you don't simply XOR the Grundy-values of the components.

The sum of two games means a turn consists of playing in either game. Then by the mirror strategy, for any impartial game $G$, $G+G=0$, a second player win. That means that under this sum, equivalence classes of impartial games form an abelian group of exponent $2$, or an $F_2$-vector space. XOR describes the additions with respect to any basis, not just the standard one.

You can change the connection with XOR by changing the sum operation. If in a collection of games, you must play in at least one, but up to $k$ games, then the Sprague-Grundy values of the components matter, but the second-player wins correspond to collections whose sums without carries have all digits divisible by $k+1$. This was observed by Moore where the games are Nim piles. Of course, this is related to your other question. There are other generalizations possible, where you are allowed to play in other collections of multiple games, and you don't simply XOR the values of the components.

The sum of two games means a turn consists of playing in either game. Then by the mirror strategy, for any impartial game $G$, $G+G=0$, a second player win. That means that under this sum, equivalence classes of impartial games form an abelian group of exponent $2$, or an $F_2$-vector space. XOR describes the additions with respect to any basis, not just the standard one.

You can change the connection with XOR by changing the sum operation. If in a collection of games, you must play in at least one, but up to $k$ games, then the Grundy values of the components matter, but the second-player wins correspond to collections whose sums without carries have all digits divisible by $k+1$. This was observed by Moore where the games are Nim piles. Of course, this is related to your other question. There are other generalizations possible, where you are allowed to play in other collections of multiple games, and you don't simply XOR the Grundy-values of the components.