Succinct Ambiguous Contracts
Paul Duetting, Michal Feldman, Yarden Rashti
TL;DR
This work advances the theory of contract design under ambiguity by introducing succinct (k-ambiguous) contracts, i.e., ambiguous contracts restricted to at most k classic contracts. It proves a divide-and-conquer structure: optimal IC k-ambiguous contracts can be composed of shifted min-pay contracts on a partition of actions, enabling a constructive algorithm that solves the problem for given partitions and target actions. The authors show that, in general, the optimal succinct contract problem is NP-hard for broad ranges of k, via a reduction from Makespan Minimization, while also deriving bounds on the Succinctness Gap that quantify the principal’s utility loss from restricting contract size. They provide a general framework linking implementability under k-ambiguous contracts to partitions of actions, and supply a polynomial-time solution in regimes where k is near n. These results illuminate the trade-off between contract simplicity and performance, with implications for practical contract design where ambiguity is leveraged under explicit size constraints.
Abstract
Real-world contracts are often ambiguous. Recent work by Dütting et al. (EC 2023, Econometrica 2024) models ambiguous contracts as a collection of classic contracts, with the agent choosing an action that maximizes his worst-case utility. In this model, optimal ambiguous contracts have been shown to be ``simple" in that they consist of single-outcome payment (SOP) contracts, and can be computed in polynomial-time. However, this simplicity is challenged by the potential need for many classic contracts. Motivated by this, we explore \emph{succinct} ambiguous contracts, where the ambiguous contract is restricted to consist of at most $k$ classic contracts. Unlike in the unrestricted case, succinct ambiguous contracts are no longer composed solely of SOP contracts, making both their structure and computation more complex. We show that, despite this added complexity, optimal succinct ambiguous contracts are governed by a simple divide-and-conquer principle, showing that they consist of ``shifted min-pay contracts" for a suitable partition of the actions. This structural insight implies a characterization of implementability by succinct ambiguous contracts, and can be leveraged to devise an algorithm for the optimal succinct ambiguous contract. While this algorithm is polynomial for $k$ sufficiently close to $n$, for smaller values of $k$, this algorithm is exponential, and we show that this is inevitable (unless P=NP) by establishing NP-hardness for any constant $k$, or $k=βn$ for some $β\in(0,1)$. Finally, we introduce the succinctness gap measure to quantify the loss incurred due to succinctness, and provide upper and lower bounds on this gap. Interestingly, in the case where we are missing just a single contract from the number sufficient to obtain the utility of the unrestricted case, the principal's utility drops by a factor of $2$, and this is tight.