Let $p>1$. The Harnack inequality and Hölder continuity for $p$-harmonic functions in bounded domains in $\mathbb{R}^d$ are usually proved via Moser iteration. In 2013 Luiro, Parviainen and Saksman showed that tug-of-war games can also be used to derive these inequalities. We refine their analysis and obtain improved dependence on $p$ and the dimension $d$ by probabilistic methods. In particular, we show that for all $p>1$, the constant in Harnack's inequality is $O(\exp(C_p d\log d))$ as $d\rightarrow\infty$, which improves the constant derived from Moser iteration.