Partizan game ($G=\{G^L\mid G^R\}$)

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.

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

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 stalk is a Hackenbush position whose graph is a single path attached to the ground at one endpoint. Read its blue and red graph edges upward from the ground; Left may cut blue edges and Right red edges, with every edge above the cut falling away. Finite red-blue stalks are number games with dyadic rational values.

Here is the explicit value rule. Give blue sign $+1$ and red sign $-1$. If the initial monochromatic run has length $m$ and sign $\epsilon$, it contributes $\epsilon m$. If more edges remain, edge $j>m$, of sign ${\epsilon }_j$, contributes ${\epsilon }_j/2^{j-m}$. ThusFor example, a blue-red stalk has value $1-1/2=1/2$, blue-red-red has value $1/4$, and blue-blue-red has value $3/2$. The recursive number game rule verifies the first example directly: its Left option is $0$, its Right option is $1$, and the simplest surreal number between them is $1/2$. Repeating this comparison after successive cuts gives the dyadic rule.

In combinatorial game theory, a cold game is a position with no gain from taking the next move, as illustrated by number games. For $G=\{0\mid 1\}=1/2$, Left can only lower the value to zero, while Right can only raise it to one; either player would prefer to leave the position untouched if another component offered a useful move.

This contrasts with a hot position such as $\{1\mid -1\}$, where either player benefits from moving first. The distinction matters when choosing which component of a disjunctive sum to play. Exact definitions at zero temperature depend on the adopted cooling convention; a cold numerical component need not be a losing position by itself.

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.