A family of games $G = G(\sigma, u)$ is defined such that (a) for each $\sigma$ the set of all $u$ for which Player I can force a win in $G(\sigma, u)$ is a Borel set $B(u)$ and (b) every Borel set is a $B(u)$ for some $u$.
"Borel Sets Via Games." Ann. Probab. 9 (2) 321 - 322, April, 1981. https://doi.org/10.1214/aop/1176994474