I have the following problem.
There is a right stochastic matrix $A\in \mathbb{R}^{\ell \times m}$. Denote with $A_j$ be the $j$th row of $A$. There is a vector $\psi\in \mathbb{R}_{\geq 0}^{\ell}.$ Further, there are $n$ convex domains $V^1,V^2,\dots,V^n\subseteq \mathbb{R}^m_{\geq 0}$.
I wish to know (in polynomial time) if there exists a function $f\colon[n]\times V^1\times\dots\times V^n \rightarrow \mathbb{R}^m_{\geq 0}$ s.t. $\forall v^1,\dots,v^n \in V^1,\dots,V^n$ the following two conditions hold: $$ \tag{a} k=\operatorname*{arg max}_{j\in [\ell]}\bigg\{\sum_{i=1}^nA_jv^i-\psi_j\bigg\} \implies k \in \operatorname*{arg max}_{j\in [\ell]}\bigg\{\sum_{i=1}^n A_jf(i,v^1,\dots,v^n)-\psi_j\bigg\}. $$ (I assume $\operatorname*{arg max}_{j\in [\ell]}\{\sum_{i=1}^nA_jv^i-\psi_j\}$ is a singelton). $$ \tag{b}\forall q\in [n]\,\,\min_{\bar{v}^q\in V^q}\max_{j\in [\ell]}\bigg\{\sum_{i=1, i\neq q}^nA_jv^i+A_j\bar{v}^q-\psi_j\bigg\} \geq \sum_{\substack{i=1\\ i\neq q}}^nA_jv^i+A_jf(q,v^1,\dots,v^n)-\psi_j. $$ I think that this problem should be intractable, but I am struggling to find a reduction and I can't find any relevant literature. Can anyone point me towards something I can read about or help me with proving it is NP-hard?