Up (game theory)

In combinatorial game theory, up, written as $\uparrow$, is the value of the game where Left can only move to a zero game, while Right can only move to a star game. It is greater than zero and less than all positive numbers, hence it is an infinitesimal. $\uparrow$ may be represented by the surreal form$$\uparrow =\{0\mid *\}.$$Because $\uparrow >0$, Left always wins a game with value $\uparrow$. Its additive inverse, down, written as $\downarrow$, may be represented as$$\downarrow =\{*\mid 0\}.$$Similarly, $\downarrow$ is less than zero and greater than all negative numbers.

$\uparrow$ and $\downarrow$ are equivalent to $+_{0}$ and $-_{0}$, tiny-0 and miny-0.

Value-$\uparrow$ games

Hackenbush

A 1-petaled blue flower, often called a delphinium. It has value $\uparrow +*$.

In Hackenbush, a 1-petaled delphinium, that is, a blue edge on top of a green edge, has value $\{0,*\mid 0\}=\uparrow +*$. Hence, a green edge along with a 1-petaled delphinium has value $\uparrow +*+*=\uparrow$.

Col

Because a Col position can have only values $z$ or $z+*$ for some number $z$, no Col position can have value $\uparrow$.

Toads and Frogs

In Toads and Frogs, $\uparrow$ may be represented by the position $$T\square TFF=\{\square TTFF\mid TFT\square F\}=\{0\mid \{TF\square TF\mid TFTF\square \}\}=\{0\mid \{0\mid 0\}\}=\uparrow .$$

Confusion with nimbers

The only nimber confused with $\uparrow$ is $*$. This is easily illustrated by the Hackenbush example, as a blue delphinium and a green edge along with a $*2$ position is not fuzzy.

See also

• Star
• Tiny and miny