Up game (source code)
= Up game
{title2=$\uparrow$}
= Conway's up
{c}
{synonym}
The <up game> is the short <partizan game> $\uparrow=\{0\mid *\}$, where $*=\{0\mid0\}$ 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, $\downarrow=\{*\mid0\}=-\uparrow$.
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