When Contracts Get Complex: Information-Theoretic Barriers

TL;DR

This paper proves that designing optimal contracts in complex combinatorial settings with submodular and additive functions is computationally infeasible, requiring exponentially many demand queries, even with advanced communication protocols.

Contribution

It establishes demand query hardness for contract optimization with submodular and additive functions, extending previous value query limitations and introducing the concept of sparse demand.

Findings

Demand query complexity is exponential for submodular $f$ and additive $c$.

Introduces the sparse demand property to strengthen hardness results.

Establishes exponential communication complexity for combinatorial functions.

Abstract

In the combinatorial-action contract model (D\"utting et al., FOCS'21) a principal delegates the execution of a complex project to an agent, who can choose any subset from a given set of actions. Each set of actions incurs a cost to the agent, given by a set function $c$, and induces an expected reward to the principal, given by a set function $f$. To incentivize the agent, the principal designs a contract that specifies the payment upon success, with the optimal contract being the one that maximizes the principal's utility. It is known that with access to value queries no constant-approximation is possible for submodular $f$ and additive $c$. A fundamental open problem is: does the problem become tractable with demand queries? We answer this question to the negative, by establishing that finding an optimal contract for submodular $f$ and additive $c$ requires exponentially many demand queries. We leverage the robustness of our techniques to extend and strengthen this result to different combinations of submodular/supermodular $f$ and $c$; while allowing the principal to access $f$ and $c$ using arbitrary communication protocols. Our results are driven by novel equal-revenue constructions when one of the functions is additive, immediately implying value query hardness. We then identify a new property -- sparse demand -- which allows us to strengthen these results to demand query hardness. Finally, by augmenting a perturbed version of these constructions with one additional action, thereby making both functions combinatorial, we establish exponential communication complexity.