More precisely, problem $1$ is as follows:
\begin{eqnarray} &\max_{1{\le}i{\le}N}\min_{[\gamma_m^i]}\left[\lambda_i - \sum_{m=1}^M\phi_m\gamma_m^iF_m^i\right] \\ &\mbox{subject to} \sum_{i=1}^N\gamma_m^i = 1,\mbox{ for all }m=1,\cdots,M \\ &\gamma_m^i \ge 0,\mbox{ for all }i=1,\cdots,N\mbox{ and }m. \end{eqnarray}
More precisely, Problem $2$ is as follows:
\begin{eqnarray} &\min_{[\gamma_m^i]}\max_{1{\le}i{\le}N}\left[\lambda_i - \sum_{m=1}^M\phi_m\gamma_m^iF_m^i\right] \\ &\mbox{subject to} \sum_{i=1}^N\gamma_m^i = 1,\mbox{ for all }m=1,\cdots,M \\ &\gamma_m^i \ge 0,\mbox{ for all }i=1,\cdots,N\mbox{ and }m. \end{eqnarray}
where $\boldsymbol{\phi}$ is a probability vector in $\textbf{P}_M$, $F_m^i{\in}\Re, \forall{i}, \forall{m}$. According to Sion's maxmin theorem, are these two problems equivalent? I would appreciate if I would get some hints.