Succinct Ambiguous Contracts

Real-world contracts are often ambiguous. Recent work by D\"utting 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 $k=\beta n$ for some $\beta\in(0,1)$. 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.