  Constrained maximization of submodular functions poses a central problem in
combinatorial optimization. In many realistic scenarios, a number of agents
need to maximize multiple submodular objectives over the same ground set. We
study such a setting, where the different solutions must be disjoint, and thus,
questions of algorithmic fairness arise. Inspired from the fair division
literature, we suggest a simple round-robin protocol, where agents are allowed
to build their solutions one item at a time by taking turns. Unlike what is
typical in fair division, however, the prime goal here is to provide a fair
algorithmic environment; each agent is allowed to use any algorithm for
constructing their respective solutions. We show that just by following simple
greedy policies, agents have solid guarantees for both monotone and
non-monotone objectives, and for combinatorial constraints as general as
$p$-systems (which capture cardinality and matroid intersection constraints).
In the monotone case, our results include approximate EF1-type guarantees and
their implications in fair division may be of independent interest. Further,
although following a greedy policy may not be optimal in general, we show that
consistently performing better than that is computationally hard.
