Definition (Time consistent)

Let $D:=\{ J,\mathbf{\Gamma},\mathcal{T} \}$ be a dynamic team which admits a solution $\underline{\gamma}^{*}\in\mathbf{\Gamma}$. Let $t>1$ be an arbitrary point in $\mathcal{T}$, and consider the decision problem $D^{\beta}_{[t,T]}$ which is derived from $D$ by setting $\underline{\gamma}_{[1,t-1]}=\underline{\beta}_{[1,t-1]}$, for an arbitrary $\underline{\beta}_{[1,t-1]}\in\mathbf{\Gamma}_{[1,t-1]}$. Then:

1. The solution $\underline{\gamma}^{*}\in\mathbf{\Gamma}$ is strongly time consistent (STC) if the subpolicy $\underline{\gamma}^{*}_{[t,T]}$ constitutes a solution to the dynamic team $D^{\beta}_{[t,T]}$, this being so for every $t\in\mathcal{T},t>1$, and every permissible $\underline{\beta}_{[1,t-1]}\in\mathbf{\Gamma}_{[1,t-1]}$.
2. The solution $\underline{\gamma}^{*}\in\mathbf{\Gamma}$ is weakly time consistent (WTC) if the subpolicy $\underline{\gamma}^{*}_{[t,T]}$ constitutes a solution to the dynamic team $D^{\beta}_{[t,T]}$ when $\underline{\beta}_{[1,t-1]}=\underline{\gamma}^{*}_{[1,t-1]}$