 Partizan game

A partizan game gives the two players, traditionally Left and Right, potentially different legal moves from the same position. Its recursive notation is $G=\{G^L\mid G^R\}$, listing Left's and Right's options. An impartial game is the special case where those option sets agree at every reachable position.

For example $1=\{0\mid \}$ lets Left move to zero while Right has no move. The game $-1=\{\mid 0\}$ reverses the roles, and their disjunctive sum is zero in game value: responding in the other component leaves the previous player winning. Such comparisons concern behavior in sums, rather than simply counting moves available to one player.