Joint Dynamic Programming: Group Activity
If I understand correctly the temporal dimension is merely a distraction as agents face a separate optimization problem every round with no intertemporal budget constraint. Hence
\begin{align}
&amp; \max_{h_i} h^{\alpha_i}l_i^{1-\alpha_i} \\
&amp; \text{s.t.} \quad h_i + l_i = \bar{T} \\
&amp; h = \min\left\{h_i, h_{-i}\right\} 
\end{align}
A continuous best response function exists for each agent:
$$
BR_i(h_{-i}) = \min\left(\alpha_i \cdot \bar{T}; h_{-i} \right)
$$
The equilibrium is
$$
h_1 = h_2 = \min_i \alpha_i \cdot \overline{T}.
$$