Given $n$ agents, and $m$ items where $v_i(g) \geq 0$ is the value of item $g$ for agent $i$, does there always exist a partition $A_1, ..., A_n$ of the $m$ items into $n$ sets s.t. for all $i, j \in \{1, ..., n\}$: $$ \sum_{g \in A_i} v_i(g) \geq \left(\sum_{g \in A_j} v_i(g)\right) - \left(\min_{g \in A_j} v_i(g)\right) $$ Where if $A_j$ is empty, the min is taken as $0$. In essence this is a fairness property where we allocate all the items to the agents s.t. every agent believes his bundle is worth at least as much as every other player's bundle when removing his least valued item. I want to know whether this always exists.
Extensive computer simulations have yet to yield a counterexample.
