Schmidt's Game and Vitali Sets

While many types of non-measurable sets are never $(\alpha ,\beta )$-winning in the sense of Schmidt’s game, we show that this is not the case for certain Vitali sets. Our main theorems show that for certain values of $\alpha ,\beta$ one can construct a Vitali set which is $(\alpha ,\beta )$-winning, while for other values of $\alpha ,\beta$ every Vitali set is $(\alpha ,\beta )$-losing. We also investigate the $(\alpha ,\beta )$-Schmidt game for various other types of pathological sets, highlighting their differences from Vitali sets.