contract designers

Abstract: We study the computational complexity of optimal contract design in the multi-agent binary-action model, focusing on gross-substitutes reward functions and related classes. While additive rewards admit an FPTAS and general submodular rewards admit only constant-factor approximation, the complexity within the intermediate class of gross substitutes has remained largely open. We uncover a fine-grained approximation landscape within this class. We first show that the optimal contract problem is APX-complete even for OXS rewards - a strict subclass of gross substitutes rewards, ruling out a PTAS for gross substitutes unless $\mathsf{P}=\mathsf{NP}$. In contrast, for weighted matroid rank functions (WMRFs) - another natural subclass of gross substitutes - we obtain an EPTAS and show that no randomized FPTAS exists in general. We further identify the special case of partition (weighted) matroid rank functions, for which we obtain an FPTAS. This stands in contrast to the multi-agent multi-action setting, where no PTAS exists even for unweighted partition matroid rank functions. Finally, we consider the broader class of ultra reward functions. While ultra rewards retain the tractability of gross substitutes in a related combinatorial contract model with a single agent, we show a sharp contrast in the multi-agent model: no polynomial-time randomized algorithm using value queries can achieve a $2^{o(n)}$-approximation in expectation. Together, our results reveal several qualitatively distinct computational regimes within and beyond gross substitutes, and identify submodularity as a crucial ingredient for the approximability of multi-agent contracts.

Abstract: We study strategic disclosure of the action space in principal-agent contracting, where an agent selects a disclosed action set to shape the principal's perception of her capabilities before contract design. Unaware of the strategic disclosure, the principal designs a revenue-optimal contract as if the disclosed action set were complete and accurate. We consider two variants distinguished by cost verifiability. When costs are unverifiable, the agent can extract the entire first-best surplus, leaving the principal with zero revenue. When costs are verifiable, we characterize the agent's optimal disclosure strategy in binary-outcome settings and, more generally, when the principal is restricted to linear contracts, reducing the agent's problem to a two-variable convex optimization problem. We prove that the agent can secure utility of at least a $1/e$ fraction of the first-best surplus, which also yields a $1/e$ welfare guarantee under optimal disclosure. While the principal's revenue can be arbitrarily small compared to the first-best surplus, when the ratio of maximum to minimum expected reward among non-null base actions is at most $L$, we establish a revenue guarantee of $Θ(1/\log L)$ relative to the first-best surplus. We also compare utilities and welfare under strategic disclosure with their counterparts in the canonical model. Finally, we extend the agent's $1/e$ utility guarantee to general outcome spaces without restricting the principal to linear contracts. Our results show how strategic action-space disclosure changes the distribution of surplus while preserving a constant-factor welfare guarantee under the agent's optimal disclosure.