A sequence $(X_n)$ is said to be a game fairer with time if for every $\varepsilon > 0$ we have $\lim_n\sup_{m\geq n}P(\| E_n(X_m)-X_n\| > \varepsilon)=0$. It is known that every $L^1$-bounded Banach space-valued game fairer with time has a unique Riesz-Talagrand decomposition: $X_n=M_n+P_n$, where $(M_n)$ is a uniformly integrable martingale and $(P_n)$ converges to zero in probability. The aim of this note is to give a classification of a class of martingale-like sequences considerably more general than games fairer with time for which the above Riesz-Talagrand decomposition still holds.