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.

翻译：现实世界中的合约常常具有模糊性。D\"utting等人（EC 2023, Econometrica 2024）的最新研究将模糊合约建模为一组经典合约的集合，其中代理人选择能最大化其最差情况效用的行动。在该模型中，最优模糊合约已被证明是“简单”的，因为它们由单结果支付（SOP）合约组成，并且可以在多项式时间内计算得出。然而，这种简单性可能因需要大量经典合约而受到挑战。受此启发，我们探索了\emph{简洁}模糊合约，其中模糊合约被限制为最多由$k$个经典合约组成。与无限制情况不同，简洁模糊合约不再仅由SOP合约构成，这使其结构和计算都变得更加复杂。我们证明，尽管增加了复杂性，最优简洁模糊合约遵循一个简单的分治原则，表明它们由针对行动的一个合适划分的“平移最小支付合约”组成。这一结构洞见意味着简洁模糊合约可实现性的一个特征刻画，并可被用于设计最优简洁模糊合约的算法。虽然该算法在$k$充分接近$n$时是多项式的，但对于较小的$k$值，该算法是指数级的，我们通过证明对于任何常数$k$，或$k=\beta n$（其中$\beta\in(0,1)$）的情况是NP难的，表明这不可避免（除非P=NP）。最后，我们引入了简洁性差距度量来量化因简洁性造成的损失，并给出了该差距的上界和下界。有趣的是，在仅缺少一个合约就能达到无限制情况效用的场景中，委托人的效用会下降至原来的$1/2$，并且这个界限是紧的。