# The Complexity of Contracts

• Paul Duetting
• Tim Roughgarden
• Inbal Talgam-Cohen
SIAM Journal on Computing, vol. 50(1) (2021), pp. 211-254

## Abstract

We initiate the study of computing (near-)optimal contracts in succinctly representable principal-agent settings. Here optimality means maximizing the principal's expected payoff over all incentive-compatible contracts---known in economics as second-best'' solutions. We also study a natural relaxation to \emph{approximately} incentive-compatible contracts.

We focus on principal-agent settings with succinctly described (and exponentially large) outcome spaces. We show that the computational complexity of computing a near-optimal contract depends fundamentally on the number of agent actions. For settings with a constant number of actions, we present a fully polynomial-time approximation scheme (FPTAS) for the separation oracle of the dual of the problem of minimizing the principal's payment to the agent, and use this subroutine to efficiently compute a $\delta$-incentive-compatible ($\delta$-IC) contract whose expected payoff matches or surpasses that of the optimal IC contract.

With an arbitrary number of actions, we prove that the problem is hard to approximate within any constant $c$. This inapproximability result holds even for $\delta$-IC contracts where $\delta$ is a sufficiently rapidly-decaying function of~$c$. On the positive side, we show that simple linear $\delta$-IC contracts with constant~$\delta$ are sufficient to achieve a constant-factor approximation of the first-best'' (full-welfare-extracting) solution, and that such a contract can be computed in polynomial time.