Up game ($\uparrow$)

The up game is the short partizan game $\uparrow =\{0\mid \ast \}$, where $\ast =\{0\mid 0\}$ is the star game. If Left starts, moving to zero wins. If Right starts, the only move leaves star to Left, who again moves to zero and wins. Thus up is strictly positive in the game ordering.

Its value is nevertheless smaller than every positive real number: it is an infinitesimal combinatorial game. This makes it useful for separating numerical advantage from a tiny advantage that can decide an otherwise balanced disjunctive sum. Reversing the players gives down, $\downarrow =\{\ast \mid 0\}=-\uparrow$.