# The Complexity of Contracts

## Abstract

*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 we 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.

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**Submitted**: 24 February 2020

**Accepted**: 11 December 2020

**Published online**: 25 February 2021

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