An adapted sequence $(X_n)$ of Pettis integrable functions is said to be a game fairer with time iff for every $\varepsilon > 0$ there exists $p\in\mathbb N$ such that for all $n \geq q \geq p$ we have $P(\|E_q(X_n) − X_q\| > \varepsilon) < \varepsilon$. We prove some Pettis mean and almost sure convergence results for such games in Banach spaces without the Radon-Nikodym property.