The Complexity of Contracts

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 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 δ-incentive-compatible (δ-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 δ-IC contracts where δ is a sufficiently rapidly-decaying function of c. On the positive side, we show that simple linear δ-IC contracts with constant δ are sufficient to achieve a constant-factor approximation of the "firstbest" (full-welfare-extracting) solution, and that such a contract can be computed in polynomial time.

Contracts under Moral Hazard and Adverse Selection

“…Our work is most closely related (and in some sense is a direct successor) to the work of Dütting et al ( [7,8]), which is one of the first papers we are aware of to study contract theory with the tools of approximation algorithms and computational complexity. In [8], the authors show that linear contracts are "robust" in a certain sense and that they can achieve an ( )-approximation to welfare.…”

“…The earlier work of Balmaceda et al [4] also examines how well contracts could approximate the welfare, including linear contracts. In [7], the authors show that there exist natural principal-agent problems with exponentially-sized (but succinctly representable) outcome spaces for which finding the optimal contract is computationally hard. Some earlier research studies specific variants of contract problems from a computational perspective.…”

“…Cost-stability, when applied to the Opt-Single, is related to Dütting et al 's notion of -incentive compatible contracts [7]. Namely, we define cost-stability to measure how much a benchmark can change (increase) under these shifts, aiming to prevent contracts from suddenly becoming better when proving negative results.…”

Contracts with Private Cost per Unit-of-Effort

“…Further afield, some works consider hidden types of principals rather than agents [e.g., 3]. Additional works on contract design through the computational lens (but without private types) include, e.g., [1,9,16,19].…”

“…Contracts are a main tool for effort allocation since they use payments (monetary or other) to determine which actions agents will take. While the computer science community has largely ignored contract theory, driven by increased practical demand, there is a growing momentum and a recent set of papers have started to explore applications of contract theory from a computational viewpoint [e.g., 1,8,9,15].…”