  Combinatorial contracts are emerging as a key paradigm in algorithmic
contract design, paralleling the role of combinatorial auctions in algorithmic
mechanism design. In this paper we study natural combinatorial contract
settings involving teams of agents, each capable of performing multiple
actions. This scenario extends two fundamental special cases previously
examined in the literature, namely the single-agent combinatorial action model
of [Duetting et al., 2021] and the multi-agent binary-action model of [Babaioff
et al., 2012, Duetting et al., 2023].
  We study the algorithmic and computational aspects of these settings,
highlighting the unique challenges posed by the absence of certain monotonicity
properties essential for analyzing the previous special cases. To navigate
these complexities, we introduce a broad set of novel tools that deepen our
understanding of combinatorial contracts environments and yield good
approximation guarantees.
  Our main result is a constant-factor approximation for submodular multi-agent
multi-action problems with value and demand oracles access. This result is
tight: we show that this problem admits no PTAS (even under binary actions). As
a side product of our main result, we devise an FPTAS, with value and demand
oracles, for single-agent combinatorial action scenarios with general reward
functions, which is of independent interest. We also provide bounds on the gap
between the optimal welfare and the principal's utility. We show that, for
subadditive rewards, perhaps surprisingly, this gap scales only logarithmically
(rather than linearly) in the size of the action space.
