Optimal Linear Contracts in Principal-Agent Models

• Optimal linear contracts are incentive schemes that offer a fixed fraction of output to agents, balancing risk and reward in principal-agent models.
• They provide tractable analytical solutions and robust approximation guarantees across settings such as dynamic, multi-agent, and uncertain environments.
• Learning methods like empirical utility maximization and regret minimization enable efficient estimation of contract parameters with tight statistical bounds.

Optimal linear contracts are a canonical class of incentive schemes in principal–agent theory, specifying reward shares such that an agent (or multiple agents) receives a fixed linear fraction of the observed project outcome. These contracts are both analytically tractable and widely used in practice, forming the theoretical basis of commission schemes, profit-sharing arrangements, and many implementations of delegated decision-making under uncertainty and asymmetric information. Research over recent years has established both their optimality in several foundational models and tight approximation guarantees even in the face of complex uncertainty, dynamic information, learning, and multi-agent interactions.

1. Mathematical Formulation and Characterization

In the classical hidden-action principal–agent model, the agent privately chooses an unobservable action from a finite set $A$ at a cost $c_i$. Each action $i$ induces a distribution over observed outcomes, with the principal's expected reward denoted as $R_i$ (Dütting et al., 2018). A linear contract offers the agent a share $\alpha \in [0,1]$ of the outcome, or more generally, $t_j = \alpha r_j$ for outcome $j$ with reward $r_j$. The agent's utility is $\alpha R_i - c_i$, and the principal's utility is $(1-\alpha)R_i$ if the agent selects action $i$.

In the dynamic continuous-time Holmström–Milgrom framework and its generalizations, including memory effects, risk aversion, and adverse selection, the optimal contract remains (affine-)linear in terminal output:

$\xi^* = A + B X_T$

with explicit formulas for $A$ and $B$ depending on agent risk aversion, cost structure, and model primitives (Jaber et al., 2022, Packham, 2018). For multi-agent settings, the principal assigns shares $(\rho_i)_{i\in N}$ with feasibility $\sum_i \rho_i \leq 1$; the minimal shares needed to induce a coalition $S$ are determined by local incentive compatibility:

$$\rho_S(i) = \frac{c_i}{f(i:S)}, \quad f(i:S) = f(S \cup \{i\}) - f(S \setminus \{i\})$$

where $f$ is the project's value function (Aharoni et al., 26 Apr 2025, Duetting et al., 2022).

2. Robustness and Optimality under Minimal Information

Linear contracts are provably optimal in min–max robustness frameworks when only first-moment (mean) reward information is available. The principal cannot improve her worst-case payoff beyond what is available through a pure-commission contract, regardless of the agent’s hidden action set or action–outcome mapping (Dütting et al., 2018, Peng et al., 2024, Liu, 2022). Formally, for ambiguous instances constrained by $\mu_i = \mathbb{E}[r | a_i]$, a linear contract achieves the optimal guarantee:

$$\max_{\alpha \in [0,1]} \min_{F_a : \mathbb{E}_{F_a}[r]=\mu_a} \max_{a \text{ chosen}} [R_a - \alpha R_a]$$

This robust property extends to the multitask/continuous-effort field, where even under complex ambiguity in effort-to-outcome mappings, the minimax contract is linear with a uniform slope determined solely by the moment information and the homogeneity of the cost function (Zuo, 2024).

In dynamic settings—including time models with exogenous shutdown, default, or sequential learning—linear contracts remain robustly optimal at each stage under worst-case ambiguity, as shown by induction across periods and envelope arguments (Liu, 2022, Martin et al., 2021).

3. Learning Optimal Linear Contracts: Statistical and Algorithmic Guarantees

When model parameters (reward distributions, agent types, cost structures) are unknown ex ante, the problem becomes one of learning from data. For offline learning, empirical utility maximization (EUM) over a fine discretization of the contract parameter space achieves optimal uniform convergence rates:

$$n = O\left( \frac{\log(1/\delta)}{\varepsilon^2} \right)$$

samples suffice for an $\varepsilon$-optimal contract with probability $1-\delta$. The critical structural property enabling sharp chaining and Rademacher complexity bounds is the monotonicity of the agent's expected reward in the linear-share parameter, even though principal-utility is piecewise-linear and discontinuous (Høgsgaard, 4 Jan 2026).

For online/repeated contract allocation, regret minimization using algorithms exploiting the piecewise-linear, monotone structure—such as Recursive Jump-Identification with Optimistic Shrinking (RJI–OS)—achieves minimax optimal regret bounds:

$\text{Regret}_T = \widetilde{O}(\sqrt{nT})$

where $n$ is the number of action segments ("pieces") in the utility function and $T$ is the number of rounds. This improves over previous $\widetilde{O}(T^{2/3})$ bounds and is tight for $n \leq T^{1/3}$ (Bacchiocchi et al., 3 Mar 2025).

In the multitask case with observable signals per task, instrumental variable regression allows for statistically efficient estimation of the optimal linear contract parameters. Explore-then-commit and pure-exploitation (when design diversity exists) strategies yield regret bounds $O(m\sqrt{T})$ or even $O(m)$ (for $m$ tasks), matching lower bounds for high-dimensional incentive alignment (Zuo, 2024).

4. Approximation Guarantees, Complexity, and Tractability

When complete information about outcome distributions is available, the revenue/welfare gap between optimal (possibly highly nonlinear) and linear contracts can be tightly bounded:

• For $n$ actions, linear contracts achieve a worst-case $n$-approximation; i.e., principal's utility via the best linear contract is at least $1/n$ of the optimum (Dütting et al., 2018, Alon et al., 2022).
• Under cost/reward range constraints, the factor tightens to $O(\log C)$ or $O(\log H)$, with $C$ and $H$ the maximal cost and reward, respectively.
• For hidden-type settings with $T$ types, linear contracts provide a $\Theta(n \log T)$-approximation to first-best welfare, and optimal menus of linear contracts cannot improve on this factor (Guruganesh et al., 2020).
• In Bayesian agency with a finite number of types, linear contracts are provably optimal among all tractable (polynomial-time computable) contracts up to a small multiplicative and exponentially small additive loss (Castiglioni et al., 2021).
• With sufficient Bayesian uncertainty (i.e., agent types/costs are well spread), linear contracts recover a constant fraction of first-best welfare, and this ratio converges to $1$ as uncertainty increases (Alon et al., 2022).

Computing the optimal linear contract is always a low-dimensional (often univariate) optimization. In multi-agent settings with gross substitutes value functions or tractable demand oracles, enumeration of $O(n^2)$ critical points suffices (Dütting et al., 2023, Duetting et al., 2022), and FPTAS schemes are possible even when the underlying (arbitrary) contract design is APX-hard (Castiglioni et al., 2024).

5. Extensions: Dynamic, Multiagent, and Combinatorial Environments

Dynamic and non-Markovian environments admit explicit optimal linear contracts via stochastic control and martingale methods, even under Gaussian processes with memory and under Poisson shutdown risks (Jaber et al., 2022, Martin et al., 2021). The linearity of incentive schemes is preserved, although an additional affine component (insurance or deposit scheme) may be required when randomness affects project termination (shutdown).

In multi-agent models with complement-free (submodular, XOS, and subadditive) project value functions, optimal linear contracts maximize the principal's utility over feasible agent coalitions. A central result is that for submodular and XOS value functions, the principal's optimal contract utility is always a constant fraction of welfare maximum. However, the welfare-utility gap diverges beyond XOS (subadditive), demonstrating a sharp frontier in contract performance (Aharoni et al., 26 Apr 2025). Algorithms for welfare-maximization under linear contracts are polynomial time, yielding constant-factor approximations in these classes (Duetting et al., 2022). For submodular value, the ratio is at most $5$; for XOS, explicit constants (e.g., $188$) are established.

In combinatorial contract design for single or multiple agents, the optimal contract is again linear or affine if only first moments are known, and tractability is strongly tied to the gross substitutes property; beyond this, the number of critical values can be exponential, but FPTAS is possible for bounded-integer/budget additive cases (Duetting et al., 2021, Dütting et al., 2023).

6. Randomization and Further Robustness

Recent results demonstrate that randomized linear contracts (randomizing the incentive slope) can strictly outperform deterministic linear contracts in worst-case robust settings. The optimal randomized contract is a distribution over slope parameters with a closed-form cumulative distribution function supported on $[0, b^*]$, where $b^*$ solves

$$b^* = \arg\max_{b \in [0,1)} \frac{\underline{u}(b)}{ - \ln(1-b) }$$

with $\underline{u}(b)$ the agent's guaranteed utility. The principal’s robust payoff under this scheme can exceed the deterministic optimum arbitrarily, even when she only knows one possible agent action (Peng et al., 2024).

The construction and optimality of randomized team-linear contracts generalize to multi-agent scenarios, where randomization over the sum of shares (along the corresponding contract "ray") is matched to worst-case utility via variational and Myerson-ironing arguments.

7. Applications and Open Directions

Optimal linear contracts are foundational in mechanism and contract theory, spanning applications in delegated search and exploration, labor and multitasking, token-incentivized multi-agent coordination, crowdsourcing, dynamic pricing, and robust mechanism design. Their tractable structure, provable robustness, and strong approximation guarantees have cemented linear schemes as both a theoretical and practical mainstay in economics and algorithmic mechanism design (Dütting et al., 2018, Duetting et al., 2022, Hoefer et al., 2024).

Open questions persist regarding the phase transition between constant- and superconstant-approximate performance as reward functions cross the complement-free hierarchy, the potential for tractable improvements beyond linear contracts in subadditive cases, and the full computational boundary of robust contract design under multi-dimensional and dynamic uncertainty. Additionally, tightening the empirical and algorithmic convergence rates for learning linear contracts in high-dimensional, multitask, or active-control regimes remains an ongoing research focus.