Succinct Ambiguous Contracts

Real-world contracts are often ambiguous. While recent work by D\"utting, 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.