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Incentive Design with Spillovers

Krishna Dasaratha, Benjamin Golub, Anant Shah

Abstract

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.

Incentive Design with Spillovers

Abstract

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.

Paper Structure

This paper contains 40 sections, 28 theorems, 202 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Simple success-or-failure environments
  4. Remarks on the model
  5. Optimal Contracts
  6. Key objects
  7. Balance condition across agents
  8. Differentiability of equilibrium
  9. Comparisons across agents and outcomes
  10. Ranking agents at the optimal contract
  11. Payments across outcomes
  12. Application to a canonical network game
  13. Equilibrium characterization
  14. Optimal contract
  15. An explicit characterization of the optimal contract
  16. ...and 25 more sections

Key Result

Theorem 1

Suppose $\bm{\tau}^{*}$ is an optimal contract and $Y^*$ is the induced team performance. There exist constants $\lambda_{s}$ such that for any agent $i$ receiving a positive payment under an outcome $s$, we have Moreover, the outcome-dependent constants $\lambda_s$ satisfy $\lambda_{s} \propto \frac{P_{s}(Y^*)}{P_{s}'(Y^*)}.$

Figures (2)

  • Figure 1: Three agent weighted graph with weights $G_{12}, G_{13},$ and $G_{23}$.
  • Figure 2: The optimal payments and resulting equilibrium payoffs as a function of the weight $G_{23}$. Here $G_{13}=0.8$ while $P(Y) = \min\{0.5Y,1\}$ (the kink is not relevant for the principal's problem). In both diagrams, the curve corresponding to agent $1$ is the topmost (solid blue) one; the curve corresponding to agent $2$ is the second from the top (dashed red); and the curve corresponding to agent $3$ is the lowest (dotted orange) one.

Theorems & Definitions (45)

  • Theorem 1
  • Lemma 1
  • Corollary 1
  • Proposition 1
  • Corollary 2
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • proof : Proof of \ref{['t:optsharescharacterization']}
  • Corollary 3
  • ...and 35 more