Infinitesimal combinatorial game

A game value lying strictly between $-r$ and $r$ for every positive rational number $r$, in the ordering of partizan games. Infinitesimal values can determine a sum whose numerical part is zero, while they cannot overturn a nonzero integer advantage.

The impartial game $\ast =\{0\mid 0\}$, so either player can move to zero. It is equivalent to a Nim heap of size one. It is incomparable with zero, and $\ast +\ast =0$ under the disjunctive sum of games.

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, $\downarrow =\{\ast \mid 0\}=-\uparrow$.