Cooperative Multi-Agent Contracts

• Cooperative Multi-Agent Principal–Agent Contracts are formal incentive structures that align multiple agents’ actions to optimize overall outcomes while internalizing risk, externalities, and relative performance.
• They utilize continuous-time stochastic control methods, including BSDEs and Nash equilibria, to model and balance both competitive and cooperative dynamics among agents.
• Optimal contract design integrates relative performance metrics and cross-project incentives to ensure risk-sharing, manage inter-agent spillovers, and minimize the firm’s incentive costs.

A cooperative multi-agent principal–agent contract is a formal incentive structure through which a principal (or principals) induces multiple agents—whose actions can both directly and externally affect a set of interrelated outcomes—to act in ways that collectively optimize a desired objective, while internalizing individual incentives, risk preferences, externalities, and interactions. In such environments, agents may be motivated not only by their absolute rewards but also by competition (relative performance considerations), and their effort choices may exhibit multi-dimensional spillovers (both positive and negative) across multiple projects or tasks. The mathematical investigation of these contracts integrates continuous-time stochastic control, Nash/FBSDE and BSDE techniques, and explicit modeling of risk-sharing in the presence of competitive appetence.

1. Stochastic Framework and Contract Structure

The canonical setting involves a principal contracting with $N \geq 2$ agents, each managing a project whose terminal value is $X^{i}_T$, with dynamics governed by a multidimensional stochastic differential equation. Agents select action vectors $a_t \in \mathbb{R}^{N \times N}$: for each project $i$, the control $a_{i,j}$ quantifies how the $j$-th agent's effort impacts the $i$-th project's drift. This implementation explicitly permits both self-effort and cross-project effects.

The principal chooses contracts $\xi = (\xi_1, ..., \xi_N)$ where each $\xi_i$ is $\mathcal{F}_T$-measurable, often linear in the $X_T$ vector. Risk preferences are modeled exponentially with respective risk-aversion parameters $R_P$ (principal) and $R_A$ (agents). Reservation utilities $u_i^0$ are imposed for voluntary participation.

The optimal contract in the first-best (full-information) setting is shown to be linear in the vector of all terminal outputs (equation (3.6)): $\xi^*_i = C_i + K_i^\top X_T,$ where $K_i$ is a project-sharing coefficient vector; $C_i$ adjusts for reservation utility. The shares depend on structural parameters including risk-aversion, competitive appetence, and the inter-agent impact matrix.

2. Agent Interactions, Competition, and Relative Performance

Each agent's actions exert both direct and external (spillover) effects. The interaction matrix $a$ encodes the magnitude/sign of these effects: $a_{i,j} > 0$ models helping, $a_{i,j} < 0$ models negative externalities (“spoiling”). This interaction structure is fundamental, as agent $i$'s utility is not only a function of their own project, but also (via $T_i$) of how their project compares to others'—a representation of relative performance concern (equation (2.6)):

$T_i(x) = V_i(r_i - r_i^-)$

where $r_i$ is agent $i$’s project value, $r_i^-$ is the (average or sum of) other agents' project values.

This structure forces optimal contracts to “share the pie” (risk-pooling over all projects), integrating both self- and cross-project performance to manage envy and support cooperation.

3. Competitive vs. Cooperative Dynamics in Contract Design

Although each agent exhibits competition—seeking to outperform peers—the optimal contract induces nontrivial cooperation. Under the optimal risk-sharing rule, even agents with strong competitive appetence (high $V_i$) may receive help from less competitive agents via contractually enforced “cross-project” payments. The contract's linearity in all projects' outputs effectively neutralizes excessive envy and balances aggressive effort with support, avoiding scenarios where agent $i$ earns strictly more by outperforming all others.

Additionally, the contract structure can penalize low-competition agents when their project is not central to collective performance, while strong competitors are shielded from excessive risk via allocation to less volatile projects.

4. Nash Equilibria and BSDE Characterization

Under moral hazard (second-best), agents choose efforts non-cooperatively in response to the contract—generating a Nash equilibrium. The existence and explicit construction of this equilibrium rest on the solvability of a multidimensional quadratic backward stochastic differential equation (BSDE). The relevant BSDE is (see (4.2)):

$$Y_T = g(X_T) + \int_t^T f(s, Z_s, X_s) ds - \int_t^T Z_s \cdot dW_s.$$

Here, $f$ encodes both the marginal benefits (drift) and marginal costs (effort, risk aversion), and $Y, Z$ must belong to appropriate integrability spaces (e.g., BMO for the quadratic case). The critical insight is that the solution of this BSDE (under suitable well-posedness) is in one-to-one correspondence with a Nash equilibrium among the agents. The methodology provides both analytical and practical means to compute contracts in high-dimensional stochastic settings.

5. Compensation, Reservation Utilities, and Volatility Allocation

The contract ensures that for each agent $i$, the certainty equivalent utility:

$$U_A(\xi_i, a^*) = -\exp(-R_A \cdot \text{(linear function of } X_T))$$

matches the reservation utility $u^0_i$. Thus, apart from stochastic bonuses linked to project performance (and relative performance terms), the fixed parts of the contract guarantee minimal compensation compatible with voluntary participation—insuring against low outcomes.

A critical allocation result is that agents with high competitive appetence are assigned less volatile projects, decreasing the uncertainty in outcome rankings and reducing the cost of inducing effort. If needed for incentive compatibility, the contract prescribes additional support to balance increased risk, implemented via cross-project payment rules.

6. Principal's Utility and Design Implications

The analysis demonstrates that optimal hiring and contract design require heterogeneity in agents' competitive appetence. A principal employing agents of varying competitiveness can exploit complementarities—assigning resources such that highly competitive agents are supported on less volatile projects while less competitive agents provide stabilizing effort elsewhere. The linear contract ensures that overall incentive costs are minimized for a given level of firm performance, optimizing both risk-sharing and motivational alignment.

If the agent population is too homogeneous in competitive appetence, overall incentive costs for the firm rise—a strong argument for diversity in team composition.

7. Methodological and Practical Impact

This research advances the literature by formally connecting Nash-equilibrium contract design for multi-agent principal–agent systems to multidimensional quadratic BSDEs, thereby generalizing classical single-agent and linear principal–agent theory to competitive, high-dimensional stochastic environments. The optimal contracts derived are implementable (linear in observables), robust to agent competition and cross-effects, and incorporate individual constraints via explicit Lagrangian duality.

These formulations and solution principles inform design in corporate teams, executive compensation across divisions, and regulated industries where relative performance is contractible. The explicit link to risk preference, agent externalities, and the importance of agent diversity offers actionable insight for practitioners constructing contracts in environments characterized by moral hazard, competition, and cooperation.