The Complexity of Continuous Contract Design over Polymatroids
We study optimal contracts when an agent allocates continuous effort subject to polymatroid capacity constraints. Effort produces a linear expected reward and incurs a cost. With binary outcomes and limited liability, the principal chooses a single reward share. For linear costs, an optimal contract can be computed in polynomial time. For quadratic costs, the agent has a unique best response computable by polynomially many submodular minimizations, yet optimizing the principal's share is NP-hard even for explicitly represented cut-based rank functions. With $n$ activities, the best-response path can have $2^{n-O(\log n)}$ breakpoints even when each reward coefficient and each rank value is an integer representable in $O(n\log n)$ bits. The construction also answers a question of Allman, Lo, and McCormick on polynomial bit length for source-sink monotone two-parameter minimum cuts. Despite this complexity, an optimal contract for quadratic costs always exists with a share of at most $1/2$. Consequently, $O(1/\varepsilon)$ best-response computations suffice for a $(1-\varepsilon)$-approximation. For rational quadratic costs on explicitly described rational polytopes containing zero, this gives a fully polynomial-time approximation scheme. More generally, a share of at most $1/p$ suffices for continuous convex costs homogeneous of degree $p>1$, on any compact convex action set containing zero. For general convex costs with an exact value and bounded-subgradient oracle, we obtain an oracle approximation scheme using approximate best responses and a supplied lower bound on positive welfare.