Optimal Common Contract with Heterogeneous Agents (1911.04146v1)

Abstract: We consider the principal-agent problem with heterogeneous agents. Previous works assume that the principal signs independent incentive contracts with every agent to make them invest more efforts on the tasks. However, in many circumstances, these contracts need to be identical for the sake of fairness. We investigate the optimal common contract problem. To our knowledge, this is the first attempt to consider this natural and important generalization. We first show this problem is NP-complete. Then we provide a dynamic programming algorithm to compute the optimal contract in $O(n^2m)$ time, where $n,m$ are the number of agents and actions, under the assumption that the agents' cost functions obey increasing difference property. At last, we generalize the setting such that each agent can choose to directly produce a reward in $[0,1]$. We provide an $O(\log n)$-approximate algorithm for this generalization.

• The paper establishes that designing an optimal common contract for heterogeneous agents is strongly NP-complete via a reduction from NAE3SAT.
• It introduces a dynamic programming solution leveraging the increasing differences property, achieving O(n²m) time complexity for discrete action cases.
• An approximation algorithm is provided for continuous action spaces, offering an O(log n) approximation for more generalized settings.

Optimal Common Contract with Heterogeneous Agents

This paper examines the principal-agent problem in scenarios with multiple heterogeneous agents, where the principal aims to design a single, uniform contract that incentivizes agents to exert effort. The paper demonstrates the problem's NP-completeness, introduces a dynamic programming solution for cases with increasing differences in agent costs, and provides an approximation algorithm for a generalized setting.

Problem Complexity

The authors address the complexity of designing an optimal common contract for heterogeneous agents. They establish that determining whether a payment profile exists that ensures the principal's payoff exceeds a given threshold is strongly NP-complete. The proof involves a reduction from the Not-All-Equal 3-Satisfiability (NAE3SAT) problem, demonstrating the computational intractability of the general problem. This result highlights the inherent difficulty in designing optimal common contracts when agents have diverse characteristics and costs.

Dynamic Programming Approach

To address the intractability, the paper identifies a specific condition under which the problem can be solved efficiently. Assuming that the agents' costs adhere to the increasing differences property, the authors present a dynamic programming algorithm with a time complexity of $O(n^2m)$, where $n$ is the number of agents and $m$ is the number of actions. This algorithm leverages the structure imposed by the increasing differences property to compute an optimal contract. The increasing difference property requires the agents to be ordered from weak to strong, $i_1,i_2,\ldots,i_n$, such that the cost for a stronger agent to produce a certain amount of reward is less than a weaker agent such that for each $j\in [m]$, $c_{i_1,j}>c_{i_2,j}>\cdots>c_{i_n,j}$. Additionally, the costs obey increasing differences when there exists a permutation $j_1,j_2,\ldots,j_m$ of $1,2,\ldots,m$ and a permutation $i_1,i_2,\ldots,i_n$ of $1,2,\ldots,n$ such that for any $k<k'$, $$0<c_{i_k,j_1}-c_{i_{k'},j_1}<c_{i_k,j_2}-c_{i_{k'},j_2}<\cdots<c_{i_k,j_m}-c_{i_{k'},j_m}$$. The subproblem is defined as $$\mathrm{OPT}(i,j)=\max_{0\le j_1\le j_2\le \cdots \le j_i\le j}\sum_{i'=1}^i\phi(i',j_{i'})$$ with the recursion formula $$\mathrm{OPT}(i,j+1)=\max_{0\le k\le i}\left(\mathrm{OPT}(k,j)+\sum_{i'=k+1}^i\phi(i',j+1)\right)$$ with $\mathrm{OPT}(i,0)=0$ for each $i$.

Generalization and Approximation

The paper extends the analysis to a more general scenario where each agent can directly produce a reward within the continuous range of $[0, 1]$. The authors demonstrate that this generalization is more complex than the original discrete-action version of the problem. They provide an $O(\log n)$-approximation algorithm for this generalized setting, offering a practical approach to contract design when agents have continuous action spaces. The algorithm first finds $y_i=\max_{x\in [0,1]} (x-c_i(x))$ and sorts them such that $y_1\ge y_2\ge\cdots\ge y_n$. It then lets $i^*\in\arg\max_{1\le i\le n} (n-i^*+1)y_{i^*}$ and outputs the payment function $t(x)=0$ if $0\le x\le y_{i^*}$ and $t(x)=x-y_{i^*}$ if $y_{i^*}< x\le 1$.

Conclusion

The paper provides valuable insights into the design of optimal common contracts for heterogeneous agents. It establishes the problem's complexity, offers an efficient solution for a specific case, and provides an approximation algorithm for a more general setting. These results contribute to the understanding of principal-agent problems and offer practical guidance for contract design in various economic scenarios.