The Complexity of Continuous Contract Design over Polymatroids
Max Klimm, Paul Lippke, Philipp Warode
原文摘要
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.
cs.GT