Principal–agent problem under the linear contract

Abstract

We consider a classical principal–agent model in the contract theory. A principal designs the payment $\mathbf {w}=\left\{ w_{0},w_{1},\ldots ,w_{n}\right\}$ to incentivize the agent to enter into the contract. Given the payment $\mathbf {w}$, the agent will take hidden actions from her strategy set $\mathbf {S_{t}^{n}}$ to finish it and from the perspective of the agent, she will select the best strategy to maximize her expected utility. Due to the hidden strategy set, the principal obtains the expected revenue $R(S_{t}^{n})$ from the agent. Furthermore, the principal has a non-decreasing revenue function r(k), which is common information, where k is the number of successful tasks in the total n independent tasks. The objective of the problem is to maximize the principal’s expected profit, i.e., $\max _{S,\mathbf {w}}\left\{ R(S_{t}^{n})-P(S_{t}^{n},\mathbf {w})\right\}$, where $P(S_{t}^{n},\mathbf {w})$ is the agent’s expected payment. The difficulty of this problem is due to the asymmetric information. If the principal knows all the information about the agent, then the optimal contract can be solved by linear programming. Based on Dütting et al. (in: Proceedings of the EC, pp 369–387, 2019), we consider the more general model. When information is asymmetric, we further analyze that the approximation ratio of the linear contract can reach $(1-\alpha _{N})/(1-\alpha _{N}^{N})$, which improves the results of Dütting et al. (in: Proceedings of the EC, pp 369–387, 2019), where $\alpha _{N}\in [0,1)$ is a given constant and the coefficient of the linear contract.