Schmidt's Game and Vitali Sets

While many types of non-measurable sets are never $(α, β)$-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 $α, β$ one can construct a Vitali set which is $(α, β)$-winning, while for other values of $α,β$ every Vitali set is $(α,β)$-losing. We also investigate the $(α,β)$-Schmidt game for various other types of pathological sets, highlighting their differences from Vitali sets.