  We study local regularity properties of value functions of time-dependent
tug-of-war games. For games with constant probabilities we get local Lipschitz
continuity. For more general games with probabilities depending on space and
time we obtain H\"older and Harnack estimates. The games have a connection to
the normalized $p(x,t)$-parabolic equation $(n+p(x,t))u_t=\Delta u+(p(x,t)-2)
\Delta_{\infty}^N u$.
