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Incentive Design with Spillovers
Krishna Dasaratha, Benjamin Golub, Anant Shah
arXiv:2411.08026v3econ.THcs.GT
TL;DR
The paper asks how to design outcome-based incentives when team members’ efforts spill over to one another. It applies a multi-agent first-order approach using network methods, characterizing balanced incentives and showing how production structure shapes pay.
Problem
How should outcome-based team incentives account for production functions when agents’ efforts create incentive spillovers?
Source: §1 · p. 2Method
The paper uses a first-order approach and network methods to characterize optimal incentive allocation under arbitrary contracts.
Source: §1 · pp. 3–4Results
Optimal contracts require equalizing across agents receiving incentive pay the product of marginal productivity, centrality, and responsiveness to incentives.
Source: §6 · p. 28Takeaways & Limitations
Pay dispersion depends on substitution elasticity: substitutable efforts favor high-productivity agents, while strong team-level complementarities favor more equal pay.
Source: §1 · p. 5The analysis assumes outcomes depend on actions only through one-dimensional team performance.
Source: §6.4 · p. 29
Abstract
from arXiv · showhide
A principal uses payments conditioned on stochastic outcomes of a team project to elicit costly effort from the team members. We develop a multi-agent generalization of a classic first-order approach to contract optimization by leveraging methods from network games. The main results characterize the optimal allocation of incentive pay across agents and outcomes. Incentive optimality requires equalizing, across agents, a product of (i) individual productivity (ii) organizational centrality and (iii) responsiveness to monetary incentives. We specialize the model to explore several applied questions, including whether compensation should reward individual ability or collaborativeness and how the strength of complementarities shapes pay dispersion.
1. Introduction
The paper studies how to design outcome-based team incentives when agents’ efforts create spillovers. It characterizes optimal incentive allocation and applies the characterization to network and team-level complementarities.
Related literature.
- The study addresses limited understanding of incentive spillovers in multi-agent contract design, building on the single-agent Holmström (1979) model.The related literature includes network studies of optimal incentives with spillovers and algorithmic contract theory for finite-action settings.Source: §1 · p. 2, Related literature · pp. 5–6
1. Introduction
- Optimal contracts balance marginal productivity, organizational centrality, and responsiveness to incentives equally across agents receiving positive incentive pay.The characterization applies when the binding incentive constraints are local.Source: §1 · p. 3
- The framework uses a first-order approach and network-game methods to characterize optimal incentive allocation across agents and outcomes under arbitrary contracts.The production function matters through its first derivatives and Hessian at the optimal contract.Source: §1 · pp. 3–4
- With concave money utility, agents with higher productivity-times-centrality are paid more in every outcome; payments also favor outcomes with greater incentive power.The outcome ranking extends Holmström’s (1979) single-agent result to multiple agents.Source: §1 · p. 4
- In rank-one settings, optimal pay combines standalone productivity and complementarity; stronger spillovers shift compensation toward complementarity, and neglecting large spillovers can forgo most potential profits.The rank-one structure maps each agent’s complementarity parameter monotonically to centrality.Source: §1 · p. 4
- With general complementarity networks and equal standalone productivities, stronger complementarities can weaken an agent’s network position and pay as the principal rebalances the contract and equilibrium actions.Centrality need not increase monotonically with complementarity.Source: §1 · pp. 4–5
- For team-level complementarities, elasticity of substitution governs pay dispersion: substitutable efforts favor high-productivity agents, while strong complementarity favors more equal pay.These applications use Cobb–Douglas and CES production functions and yield closed-form solutions.Source: §1 · p. 5
2. Model
The model has a principal who offers nonnegative outcome-contingent payments to a team whose costly efforts determine stochastic project outcomes through team performance. It then specializes to success-or-failure and linear-quadratic settings.
2. Model
- Agents’ efforts jointly determine a smooth, increasing one-dimensional team performance that governs probabilities over a finite set of outcomes.The principal values realized outcomes and maximizes expected revenue net of promised transfers.Source: §2 · pp. 6–7
- The principal commits to nonnegative payments contingent on observable outcomes, and agents simultaneously choose costly efforts in a pure-strategy Nash equilibrium.Efforts and team performance are not contractible; agents may have multiple equilibria, with the principal-favorable equilibrium selected.Source: §2 · pp. 6–7
2.1. Simple success-or-failure environments.
- In success-or-failure environments, the principal optimally pays nothing upon failure, so the contract can be represented by success-state payments.This follows because incentives depend on the utility difference between success and failure payments, allowing failure payments to be shifted to zero.Source: §2.1 · p. 8
2.1.1. Linear-quadratic production functions.
- Linear-quadratic production uses standalone productivities and a symmetric nonnegative complementarity matrix, whose off-diagonal entries represent bilateral complementarities.With concave success probability, a fixed contract induces a unique equilibrium; its structure resembles network games with linear best responses, with incentive spillovers depending on endogenous incentives and success probability.Source: §2.1.1 · p. 9
- The rank-one complementarity case represents each agent’s complementarity with a parameter, while a scalar governs overall complementarity strength.The model notes that general networks and rank-one structures both matter for applications.Source: §2.1.1 · p. 9
2.2. Remarks on the model.
- The analysis assumes effort affects outcomes only through one-dimensional performance and that output is the only contractible consequence of effort.The model otherwise allows nonnegative-payment contracts beyond simpler forms such as equity contracts, and equilibrium existence may depend on the environment.Source: §2.2 · pp. 9–10
3. Optimal Contracts
The section characterizes optimal incentive contracts through a balance condition across paid agents, then derives implications for how payments rank across agents and outcomes.
3.1. Key objects.
- An agent’s marginal productivity is the effect of their action on team performance, while centrality captures the total effect of changing their incentives through spillovers.Centrality weights spillover paths by the productivity of agents whose incentives are affected.Source: §3.1 · pp. 10–11, §3.2.1 · p. 15
3.2. Balance condition across agents.
- At each outcome, optimality requires a balance across paid agents involving productivity, centrality, and responsiveness to monetary incentives.The outcome-specific constants are proportional to the probability of that outcome.Source: §3.2 · p. 12
- The balance condition also holds when the principal minimizes transfers to implement a chosen performance level, before optimizing over outcome distributions.With multiple agents, this implementation step must allocate incentives across agents as well as outcomes.Source: §3.2 · p. 13
3.2.1. What centrality captures.
- The spillover matrix tracks how effort changes others’ marginal returns, and its powers capture successive rounds of strategic responses.Its centrality measure depends endogenously on the contract and equilibrium.Source: §3.2.1 · p. 15
3.2.2. Further comments on the result and the modeling behind it.
- The first-order approach extends to this multi-agent setting under conditions made more tractable by the one-dimensional team-performance aggregator.That structure lets the authors omit second derivatives of outcome probabilities when assessing incentive redistribution across agents.Source: §3.2.2 · p. 15
3.3. Comparisons across agents and outcomes.
- The section derives corollaries for ranking agent payments and allocating payments across outcomes.The outcome allocation result generalizes a single-agent result of Holmström (1979).Source: §3.3 · p. 16, §3.3.2 · p. 17
3.3.1. Ranking agents.
- With identical strictly concave utility functions, agents can be ranked by productivity times centrality, with higher-ranked agents receiving stronger incentives.The ranking applies across outcomes when agents share positive payments; identical utility functions yield a complete ranking.Source: §3.3.1 · pp. 16–17
3.3.2. Allocation across outcomes.
- For an agent paid under two outcomes, marginal utility is proportional to outcome probability divided by its marginal change with team performance.A risk-neutral agent is paid only under outcomes maximizing the marginal probability-to-probability ratio described in the passage.Source: §3.3.2 · p. 17
4. Implications: Whom to incentivize in teams?
The applications show how optimal compensation shifts between standalone productivity and complementarity as spillovers strengthen, then examine how general network structure shapes incentives and payoffs. They also show that stronger links can benefit the principal while reducing some agents’ payoffs.
4. Implications: Whom to incentivize in teams?
- The applications vary agents’ standalone productivities and complementarity traits, then examine general networks with equal standalone productivities to isolate network structure.The rank-one setting permits heterogeneity in both traits, while the general-network application assumes equal standalone productivities and risk neutrality.Source: §4 · p. 18, §4.2 · p. 22
4.1.1. Characterization of optimal contracts.
- Optimal pay in the rank-one setting is a quadratic form in standalone productivity k_i and complementarity β_i, with endogenous coefficients determining their relative weights.The relative weight shifts toward β terms in industries or teams with stronger spillovers.Source: §4.1.1 · pp. 18–19, §4.1.2 · p. 19
4.1.2. Spillover regimes.
- As spillovers strengthen, optimal pay shifts from favoring high-standalone-productivity agents to favoring high-complementarity agents, with payments nearly equal at intermediate strengths.In the numerical example, T1 agents receive about sixteen times T2 agents’ payments when ρ is near zero; at large ρ, T2 agents receive much more.Source: §4.1.2 · pp. 19–20, Figure 1 · p. 20
4.1.3. The importance of spillovers in contract design.
- The optimal contract significantly outperforms a benchmark that ignores spillovers when failures are not too rare, and its profit advantage grows large as success probability steepens with ξ.Figure 2 compares optimal-contract profits with profits under the spillover-ignoring benchmark.Source: Figure 2 · p. 21, §4.1.3 · p. 21
4.2. General networks.
- The rank-one analysis allows individual heterogeneity but restricts complementarity structure; explicit characterization is unavailable here for general networks with heterogeneous standalone productivities.Source: §4.2 · p. 22
- With equal standalone productivities and risk neutrality, active agents have equal productivities and centralities, while their link-weighted neighboring actions and equity are balanced.The explicit compensation index based on row sums of the inverse active-subnetwork matrix differs from Katz–Bonacich centrality: stronger links need not raise it.Source: §4.2 · pp. 22–23
4.2.1. Comparative statics.
- Strengthening G23 can lower agent 2’s payment and the payoffs of agents 1 and 2, even though the principal’s payoff is weakly increasing in any edge weight.This can create a tension: agents may resist links that benefit the principal or team.Source: §4.2.1 · pp. 23, 24, 26, Figure 4 · p. 25
5. Production functions with broader complementarities
The balance condition extends to Cobb-Douglas and CES production, yielding explicit optimal-contract characterizations and implications for pay dispersion. Under CES, optimal compensation is more unequal when efforts are more substitutable.
5. Production functions with broader complementarities
- The balance condition also yields simple optimal-contract characterizations for classical production functions with team-level complementarities beyond bilateral network links.The analysis establishes sufficient conditions for the first-order approach, including concavity of P(·) and specified parameter restrictions.Source: §5 · pp. 26, 28
- Monotone transformations preserve the optimal contract and equilibrium actions while making team performance additively separable, simplifying the calculation of productivities and centralities.The transformations apply to the Cobb-Douglas and CES cases discussed here.Source: §5 · p. 27
- For Cobb-Douglas production, the principal pays more to agents with higher factor shares.Compensation is linear in the factor shares γ_i.Source: §5 · pp. 27–28
- More substitutable efforts produce more unequal optimal compensation, while strong complementarities produce little pay dispersion.In the Leontief limit, all agents are paid equally; in the Cobb-Douglas case, compensation is linear in factor shares.Source: §5 · pp. 26–28
- The CES characterization makes pay dispersion depend on the elasticity of substitution, whereas relative compensation does not depend on returns to scale κ.The total payment when the project succeeds, and thus absolute individual payments, does depend on κ.Source: §5 · pp. 27–28
6. Concluding discussion
The paper concludes that optimal incentives must balance across agents receiving positive pay, and discusses extensions and boundaries of that result. Participation constraints, noisy spillover estimates, alternative principal objectives, and multidimensional outcome information each affect how the framework would apply.
6. Concluding discussion
- Optimal contracts must satisfy a balance condition across agents receiving positive incentive pay.The paper studies a team whose members contribute through noncontractible effort.Source: §6 · p. 28
6.1. Participation constraints.
- With standard participation constraints, binding constraints would add agent-specific multipliers to the balance condition, a case left for future work.The constraints would need to remain respected under contract perturbations.Source: §6.1 · p. 28
6.2. Robustness to measurement error.
- When spillovers are moderate, balance is robust to sufficiently small measurement error; with strong spillovers and weakly connected subteams, even arbitrarily small error can mislead.A spectral gap bounded away from zero rules out the latter issue.Source: §6.2 · pp. 28–29
6.3. Other objectives.
- For other principal objectives, incentive spillovers remain relevant when first-order conditions depend only on local equilibrium comparative statics.The paper gives maximizing a weighted sum of workers’ utilities as an example.Source: §6.3 · p. 29
6.4. The role of a one-dimensional outcome.
- Because actions affect outcomes only through one-dimensional team performance, extending the framework to outcome distributions that depend on the full action profile remains an open direction.Such distributions may reveal more about individual effort, requiring balance conditions to incorporate that information alongside centralities and productivities.Source: §6.4 · p. 29
Appendix A. Omitted proofs
Appendix A contains proofs for results from Sections 2, 3, and 4; proofs from Section 5 are in the Online Appendix.
Appendix A. Omitted proofs
- Appendix A proves results from Sections 2, 3, and 4, while Section 5 proofs appear in the Online Appendix.Source: Appendix A · p. 32
A.1. Proof of Fact 1.
The proof establishes that an optimal contract attains the supremum of attainable principal payoffs.
A.1. Proof of Fact 1.
- Compactness, upper-hemicontinuity of equilibrium, and payoff continuity establish that a limit contract attains the supremum.The proof takes a convergent subsequence of contracts and corresponding equilibria.Source: §A.1 · p. 32
A.2. Proof of Lemma 1.
At an optimal contract, an agent exerts zero effort exactly when the contract makes no payment to that agent at any outcome. The proof then studies how perturbing transfers changes effort and team performance.
A.2. Proof of Lemma 1.
- At an optimal contract, an agent exerts zero effort if and only if the agent receives no payment at any outcome.Source: §A.2 · p. 32
- An agent who exerts zero effort can be paid at no outcome under an optimal contract: removing a positive payment that leaves the action at zero preserves equilibrium and reduces expected payments.Source: §A.2 · p. 32
- The transfer perturbation analysis focuses on agents with positive equilibrium actions, since an agent exerting zero effort has a unique best response of zero.Source: §A.2 · p. 33
A.3. Proof of Theorem 1.
The proof uses the principal’s first-order condition for each positive transfer, then substitutes Lemma 1 to obtain the stated result.
A.3. Proof of Theorem 1.
- For any positive transfer at an optimal contract, the principal’s first-order condition balances its marginal effect against the expected additional transfer required.Source: §A.3 · p. 35
- Substituting Lemma 1 into the first-order condition yields the result, with the relevant sum nonzero because each outcome has positive probability.Source: §A.3 · p. 35
- The proof concludes that the resulting constant λ_s is independent of agent i.Source: §A.3 · p. 35
A.4. Proof of Corollary 1.
The proof applies Theorem 1 to obtain the claimed equality, using a nonzero constant λ_s that does not vary by agent.
A.4. Proof of Corollary 1.
- Theorem 1 implies that a nonzero constant λ_s yields the equality claimed in the corollary.Source: §A.4 · p. 35
- The proof’s conclusion relies on λ_s being independent of agent identity.Source: §A.4 · p. 35
A.5. Proof of Proposition 1.
The proof establishes conditions on outcomes where agents are paid, then uses them to compare payments for agents with identical strictly concave utility functions.
A.5. Proof of Proposition 1.
- At every outcome where an agent receives a positive payment, the outcome probability’s derivative with respect to performance is positive.Source: §A.5 · p. 35
- If two agents each receive a positive payment, possibly at different outcomes, there must be an outcome where both receive a positive payment.Source: §A.5 · pp. 36–37
- For agents with identical strictly concave utility functions, the proof establishes a weak payment ordering at outcomes where both agents are paid.Source: §A.5 · pp. 37–38
- The proof rules out an outcome where one agent is paid and the other is not, when the latter is paid at another outcome.Source: §A.5 · pp. 38–39
A.6. Proof of Corollary 2.
The proof recalls Theorem 1, takes a ratio of its equations for two outcomes, and then concludes the statement is proved.
A.6. Proof of Corollary 2.
- The proof takes the ratio of the equation for any pair of outcomes s1 and s2 in S*_i.It then states that the claim is proved, without specifying the resulting claim in these passages.Source: §A.6 · p. 39
A.7. Proof of Lemma 2.
The proof derives expressions for agent productivity and centrality, then uses them to establish nonnegative constants in the balance condition.
A.7. Proof of Lemma 2.
- The proof derives an expression for agent productivity and uses nonnegative actions and spillover parameters to establish eB1 ≥ 0.The productivity derivation begins from its definition; the balance condition then yields nonnegative constants B1 and B2.Source: §A.7 · pp. 39–40
- Agent centrality is expressed as productivity plus a nonnegative spillover term.The proof uses the Sherman–Morrison formula; the denominator is strictly positive because the spillover matrix has spectral radius below one.Source: §A.7 · pp. 39–40
- For any agent receiving positive pay under an optimal contract, Theorem 1 supplies a common constant λ.Source: §A.7 · p. 40
A.8. Proof of Theorem 2.
The proof characterizes equilibrium actions under a contract and applies that characterization to the theorem’s asymptotic payment comparisons. It establishes uniqueness of equilibrium and reports different limiting behavior for an intermediate term as μ approaches zero versus one.
A.8. Proof of Theorem 2.
- A fixed contract has a unique pure Nash equilibrium, characterized by the action system and the consistency condition Y* = Y(a*).The proof establishes that this equilibrium system is equivalent to agents’ first-order conditions and that each action is a unique global best response.Source: §A.8 · pp. 40–41
- Agents receiving zero pay choose zero action, while agents receiving positive pay choose positive action in equilibrium.Source: §A.8 · p. 40
- For any contract, the equilibrium satisfies P′(Y*)ρ(TG) < 1.The proof rules out an equilibrium at or above this threshold using a positive left eigenvector of the spillover matrix.Source: §A.8 · p. 42
- As μ(m) → 0, B1(m) and B2(m) both converge to zero.The proof establishes convergence by showing ρ(m)P′(Y*(m))k⊤T(m)β converges to zero.Source: §A.8 · p. 43
- As μ(m) → 1, ρ(m)P′(Y*(m))k⊤T(m)β neither converges to zero nor becomes unbounded.Source: §A.8 · p. 44
A.9. Proof of Proposition 2.
This appendix derives conditions and implications for optimal incentive allocation in networked teams, covering equilibrium and payment properties, network effects and measurement error, and equity-pay restrictions. It also examines equilibrium-preserving contract changes and conditions for providing incentives across agents and outcomes.
A.9. Proof of Proposition 2.
- Under an optimal contract, α_i c_i is equal across active agents, which implies that α_i is also equal across them.Proposition 2’s proof further uses this result to show that (G a*)_i and (G τ*)_i are constant across active agents.Source: §A.9 · pp. 44–46
A.10. Proof of Proposition 3.
- Increasing one network edge weakly raises equilibrium team performance and the principal’s profit under the original optimal contract, while the new network’s optimal contract yields at least as much payoff.The comparison holds when an edge weight G_ij is increased by ε > 0.Source: §A.10 · p. 46
Appendix OA.B. Omitted Proofs
- For Cobb-Douglas production, best responses are strict when P is concave and γ_i ≤ 1, and a unique positive equilibrium differentiable in τ exists when Σ_i γ_i < 2.The all-zero action profile is also an equilibrium; the principal’s preferred equilibrium, when positive, is the unique positive one under the stated conditions.Source: Appendix OA.B. Omitted Proofs · pp. 47–48
- When P is concave, κ ∈ (0, 1], and ρ ≤ 1, best responses are unique; if κ < 2, equation (OA.3) has a unique positive solution Y*(τ) for each contract τ, differentiable in τ.Source: Appendix OA.B. Omitted Proofs · p. 49
- Under Cobb-Douglas production, optimal payments are proportional to factor shares; when ρ = 1, only agents with maximal γ_i receive positive payments.For the ρ < 1 case, equilibrium actions also satisfy (a*_i)^(1−ρ) proportional to γ_i.Source: Appendix OA.B. Omitted Proofs · pp. 50–51
Appendix OA.C. Applicability of the First-Order Approach
- The first-order approach is implied by an invertible utility Hessian at the optimal contract and a strict equilibrium; a balance condition can also yield insights without strictness.The appendix provides sufficient conditions for strictness in one subsection and an analogous balance condition requiring differentiability only in some directions in another.Source: Appendix OA.C. Applicability of the… · pp. 51–52
OA.C.1. Sufficient conditions for strictness of equilibrium.
- Equilibrium actions are strict when outcome probabilities satisfy MLRP and CDFC and team performance is concave in each agent’s action.The team-performance concavity condition holds automatically in the network-games setting and for Cobb-Douglas and CES production when all factor shares are at most 1.Source: OA.C.1. Sufficient conditions for… · pp. 52–53
- Under Assumption OA.2, Fact 1’s conditions, and concavity of team performance in each action, an optimal contract exists that induces a strict equilibrium.The proof establishes strict concavity of agents’ utilities at an optimal solution to the relaxed problem, making its first-order conditions sufficient for equilibrium.Source: OA.C.1. Sufficient conditions for… · pp. 53, 55, 56
OA.C.2. Directional differentiability.
- For an optimal contract inducing a stable but non-strict equilibrium, balance conditions hold in payment-perturbation directions that maintain equilibrium, but need not hold in other directions.The theorem applies to directions that increase payments only where the optimal contract already makes a positive payment; its balance constant is independent of the direction and agent.Source: OA.C.2. Directional differentiability · p. 56
Appendix OA.D. Contract design under measurement error.
- With bounded spillovers and sufficiently small network measurement error, the principal can rely on the observed network’s balance condition to identify a profitable incentive shift.The sufficient condition bounds total spillover effects, and the permissible error bound can be chosen independently of the true network.Source: Appendix OA.D. Contract design under… · p. 60
- With very large spillovers, arbitrarily small network measurement errors can reverse the apparent centrality-based ranking of two disconnected subteams and mislead incentive reallocations.Strong within-team spillovers amplify measurement error; avoiding this case requires ruling out partitions into subteams connected only weakly to each other.Source: Appendix OA.D. Contract design under… · pp. 60–62
Appendix OA.D. Contract design under measurement error
- Under the stated spectral-gap condition, sufficiently small ∥G−G′∥2 preserves the sign of α_i c_i−α_j c_j, allowing profitable deviations to be checked in leading and residual network components.The leading-component contribution may be sensitive to mismeasurement in magnitude, but not sign; the spectral gap bounds spillovers in the residual component.Source: Appendix OA.D. Contract design under… · p. 62
Appendix OA.E. Sufficient conditions for positive payments
- With a nonzero optimal payment, a paid agent of positive centrality, Inada utility, complementary actions, and convex performance in own effort, every agent is paid at outcomes made more likely by increased team performance.In general, optimal contracts may exclude some agents; under these conditions, all agents are included, and payments go precisely to outcomes whose probability rises with team performance.Source: Appendix OA.E. Sufficient conditions… · p. 65
- At an optimal contract, every agent has strictly positive extended centrality, supported by a spillover matrix with spectral radius below one and nonnegative spillovers.This establishes positive centrality even for agents who initially receive no payment, a prerequisite for showing that the conditions can induce payments to all agents.Source: Appendix OA.E. Sufficient conditions… · pp. 65, 68, 69
- The perturbation result also holds for agents receiving zero payments at every outcome when perturbing at an outcome where P'_s(Y*) > 0.The proof identifies c_j with the centrality-vector term and invokes the Inada condition alongside positivity conditions.Source: Appendix OA.E. Sufficient conditions… · p. 71
Appendix OA.F. Optimal equity pay
- At an optimal equity contract, the total effect of each positively compensated agent’s equity perturbation on team performance is equal across those agents.Proposition OA.4 characterizes this balance condition for agents receiving positive equity.Source: Appendix OA.F. Optimal equity pay · p. 72
- Equity pay’s balance condition generally differs from unrestricted optimality because linking payments across outcomes sacrifices incentive power.The restriction imposes a linear relationship between payments at different outcomes, whereas unrestricted contracts can fine-tune payments outcome by outcome.Source: Appendix OA.F. Optimal equity pay · pp. 71–72