Real-world contracts are often ambiguous. While recent work by Dütting, Feldman, Peretz, and Samuelson (EC 2023, Econometrica 2024) demonstrates that ambiguous contracts can yield large gains for the principal, their optimal solutions often require deploying an impractically large menu of contracts. This paper investigates \emph{succinct} ambiguous contracts, which are restricted to consist of at most $k$ classic contracts. By letting $k$ range from $1$ to $n-1$, this yields an interpolation between classic contracts ($k=1$) and unrestricted ambiguous contracts ($k=n-1$). This perspective enables important structural and algorithmic results. First, we establish a fundamental separability property: for any number of actions $n$ and any succinctness level $k$, computing an optimal $k$-ambiguous contract reduces to finding optimal classic contracts over a suitable partition of the actions, up to an additive balancing shift acting as a base payment. Second, we show bounds on the principal's loss from using a $k$-ambiguous rather than an unrestricted ambiguous contract, which uncover a striking discontinuity in the principal's utility regarding contract size: lacking even a single contract option may cause the principal's utility to drop sharply by a multiplicative factor of $2$, a bound we prove to be tight. Finally, we characterize the tractability frontier of the optimal $k$-ambiguous contract problem. Our separability result yields a poly-time algorithm whenever the number of partitions of $n-1$ actions into $k$ sets is polynomial, recovering and extending known results for classic and unrestricted ambiguous contracts. We complement this with a tight hardness result, showing that the problem is \textsf{NP}-hard whenever the number of partitions is super-polynomial. Moreover, when $k \approx n/3$, the problem is even hard to approximate.

翻译：现实世界中的合同往往具有模糊性。尽管Dütting、Feldman、Peretz和Samuelson（EC 2023, Econometrica 2024）的最新研究表明模糊合同能为委托方带来巨大收益，但其最优解决方案通常需要部署数量庞大到不切实际的合同菜单。本文研究\emph{简洁}模糊合同，这类合同被限制为最多包含$k$个经典合同。通过让$k$在$1$到$n-1$之间变化，这实现了经典合同（$k=1$）与无限制模糊合同（$k=n-1$）之间的插值。该视角催生了重要的结构性与算法性结果。首先，我们建立了一个基本可分离性：对于任意行动数量$n$和任意简洁度$k$，计算最优$k$-模糊合同可归结为在行动集的适当划分上寻找最优经典合同，直至作为基础支付的附加平衡偏移。其次，我们证明了委托方使用$k$-模糊合同而非无限制模糊合同的效用损失上界，揭示了合同规模对委托方效用的显著不连续性：即使仅缺失一个合同选项，也可能导致委托方效用以乘性因子$2$急剧下降，我们证明该上界是紧的。最后，我们刻画了最优$k$-模糊合同问题的可处理性边界。我们的可分离性结果导出了一个多项式时间算法，适用于将$n-1$个行动划分为$k$个集合的划分数目为多项式的情况，恢复并扩展了经典合同和无限制模糊合同的已知结果。我们通过紧的硬度结果对此进行补充，证明当划分数目为超多项式时该问题是\textsf{NP}-难的。此外，当$k \approx n/3$时，该问题甚至难以近似求解。