Partizan 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.

A partizan game played on a graph attached to the ground. Left removes blue edges and Right removes red edges; after a removal, all edges disconnected from the ground disappear. The player with no legal move loses under the normal-play convention.

A Hackenbush position whose graph is a single path attached to the ground at one endpoint. Finite red-blue stalks have dyadic rational game values; the initial monochromatic run determines the integer part and subsequent colors determine successive binary fractions.

A number game is a recursively defined normal-play partizan game whose options are number games and every left option is smaller than every right option. Its value is the simplest surreal number between those options. Finite number games have dyadic rational values.