Word Generator

$$\gdef\game#1#2{\left\{#1 \,\middle|\, #2\right\}}$$ $$\gdef\impgame#1{\left\{#1 \,\middle|\, #1\right\}}$$ $$\gdef\star{\mathord{\ast}}$$ $$\gdef\up{\mathord{\uparrow}}$$ $$\gdef\down{\mathord{\downarrow}}$$ $$\gdef\doubleup{\mathord{\Uparrow}}$$ $$\gdef\doubledown{\mathord{\Downarrow}}$$ $$\gdef\tiny#1{\pmb{\boldsymbol{+}}_{#1}}$$ $$\gdef\miny#1{\pmb{\boldsymbol{-}}_{#1}}$$