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Dominance of contributions monitoring in teams

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Abstract

In team problems it has been previously argued that there is no loss to the principal from monitoring team output compared to monitoring of individual contributions, a result known as monitoring equivalence. Optimal output monitoring, however, sometimes required up front payment from the agents to the principal. By introducing limited liability (LL) on the part of agents that rules out positive monetary transfers to the principal, it is shown that the principal strictly benefits by monitoring individual contributions. Positive rent of the lowest type under output monitoring with LL implies there will be a dominating contributions monitoring contract that further transfers some of this rent to the principal. Thus, unlimited agent liability is necessary for the equivalence result.

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Notes

  1. Exceptions could be law firms or a group of medical doctors in private practices where junior partners may have to pledge compensation at the start in case the firm (or the group) does not perform well. Simon Grant suggested this example.

  2. Vander Veen (1995) observed that McAfee–McMillan’s equivalence result should break down if the agents are risk averse.

  3. There are other differences as well that will become clear in the model section.

  4. See the discussion of Holmström (1982) conjectures below.

  5. Thus, our limited liability restriction is weaker: ex post participation constraint implies limited liability but not necessarily the other way around.

  6. Strausz (2006) had argued that in most principal-agent applications, focusing on deterministic mechanisms can be justified so long as the optimal (deterministic) mechanism satisfies a “no-bunching” condition. To our knowledge no such result is available in the principal-multi-agent setting, which is our focus.

  7. Note that the payment to agent i is restricted to depend only on i’s contribution. Admittedly this weakens the principal’s hand but given that ultimately we are going to show dominance of contributions monitoring, allowing a more general payment function that depends on other agents’ contributions as well would retain the dominance result if not strengthen it further.

  8. Note that there are multiple equilibria for each type of contract, all leading to the same contributions and principal’s payoffs. In particular, we do find optimal contracts such that payments under contributions monitoring are all positive. As can be clearly seen later, the optimal contract under contributions monitoring in Table 2 is also optimal without imposing the limited liability constraint.

  9. Similar to contributions monitoring, there are multiple solutions for the optimal payment under output monitoring without limited liability. Multiplicity of optimal payments will also be observed under contributions and output monitoring when limited liability applies, but only one set of optimal contributions is obtained for each case.

  10. A more natural mechanism would be to give a non-negative reward so long as i’s contribution is at least \({\bar{y}}_{i}({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\) and zero reward otherwise. All our analysis will hold for this alternative mechanism.

  11. Note that the actual implementation of the punishing contract involves all non-negative ex post payments.

  12. After setting the lower bound of the team production technology at 0, e.g., when all team members put in 0 effort output is low (with probability 1), we ran more simulations which show that the dominance result still holds. Thus, the lower bound assumption is not necessary for our dominance result.

  13. In fact, it is easy to see that in our model the lowest type of every agent would earn a positive rent under limited liability, so long as one justifiably ignores the extreme contract in which an agent receives zero payments for all output realizations. See the discussion following Assumption 1 and Proposition 3. Thus, a stronger version of Proposition 3\(^\prime \) can be established.

  14. In fact, for any feasible output monitoring contract, except for the ‘null contract’, we can find a dominating contributions monitoring contract. Thus, the existence result is not essential for our dominance result.

  15. Unlike in contributions monitoring, under output monitoring the principal has to penalize all following a deviation because he won’t be able to tell who has deviated.

  16. The latter requirement is fulfilled so long as the downward-adjusted payment does not fall below the cost of the suggested contribution \(y^c_{i}({\hat{z}}_{i}^\prime , \mathbf{z }_{-i})\) by the true type \({\bar{z}}_i\).

  17. Note that the reductions in type \({{\bar{z}}}_i\)’s deviation payment that were carried out earlier for \(({\hat{z}}_{i}^\prime ,z_{-i})\) reported profiles, the same reductions will happen when the true type is \({{\hat{z}}}_i^\prime \) and the reported profiles are the same \(({\hat{z}}_{i}^\prime , \mathbf{z }_{-i})\). That is, the same reductions apply to both sides of (A.8).

References

  • Che Y-K, Yoo S-W (2001) Optimal incentives for teams. Am Econ Rev 91:525–541

    Article  Google Scholar 

  • Gershkov A, Winter E (2015) Formal vs. informal monitoring in teams. Am Econ J Microecon 72:27–44

    Article  Google Scholar 

  • Holmström B (1982) Moral hazard in teams. Bell J Econ 13:324–340

    Article  Google Scholar 

  • Khalil F, Lawarrée J (1995) Input versus output monitoring: who is the residual claimant? J Econ Theory 66:139–157

    Article  Google Scholar 

  • Laffont JJ, Martimort D (2002) The theory of incentives: the principal-agent model. Princeton University Press, Princeton, ISBN 0-691-09183-8

  • McAfee RP, McMillan J (1991) Optimal contracts for teams. Int Econ Rev 32:561–577

    Article  Google Scholar 

  • Myerson R (1982) Optimal coordination mechanisms in generalized principal-agent problems. J Math Econ 10:67–81

    Article  Google Scholar 

  • Ollier S, Thomas L (2013) Ex post participation constraint in a principal-agent model with adverse selection and moral hazard. J Econ Theory 148:2383–2403

    Article  Google Scholar 

  • Rahman D (2012) But who will monitor the monitor? Am Econ Rev 102:2267–2297

    Article  Google Scholar 

  • Raith M (2008) Specific knowledge and performance measurement. Rand J Econ 39:1059–1079

    Article  Google Scholar 

  • Strausz R (2006) Deterministic versus stochastic mechanisms in principal-agent models. J Econ Theory 128:306–314

    Article  Google Scholar 

  • Vander Veen TD (1995) Optimal contracts for teams: a note on the results of McAfee and McMillan. Int Econ Rev 36:1051–1056

    Article  Google Scholar 

  • Varian H (1990) Monitoring agents with other agents. J Inst Theor Econ 146:153–174

    Google Scholar 

Download references

Acknowledgements

We thank two anonymous referees, Murali Agastya, Indranil Chakraborty, Simon Grant, Bing Liu, Jingfeng Lu and Satoru Takahashi for helpful comments and suggestions, and Tat How Teh for his generosity with research assistance. Any remaining mistakes are ours.

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Correspondence to Peng Wang.

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Appendix

Appendix

Proof of Lemma 1

First, we can see that the contract with the payment function \(p_{i}(x,\hat{\mathbf{z }})=0\), \(\forall i,x,\hat{\mathbf{z }}\) and equilibrium contribution level \(y_{i}(\hat{\mathbf{z }})=0\), \(\forall i,\hat{\mathbf{z }}\) satisfies all the feasibility constraints. Let \({\tilde{V}}\) denote the principal’s expected profit under such contract.

Next, we are going to show the existence of optimal solution.

Since the domain of \(p_{i}(x,\hat{\mathbf{z }})\) and \(y_{i}(\hat{\mathbf{z }})\) are finite, choosing the set of such functions will be the same as choosing a finite number of vectors. Let V be the space of \(n\times m^{n}\times (1+|X|)\) dimensional vectors which satisfy all the feasibility constraints and yields principal the expected payoff at least \({\tilde{V}}\). Thus, each element \(v_{k}\in V\) can be represented as

$$\begin{aligned} v_{k}(x,\hat{\mathbf{z }})&=(y_{k}(\hat{\mathbf{z }}), p_{k}(x,\hat{\mathbf{z }}))\\&=(y_{1k}(\hat{\mathbf{z }}), y_{2k}(\hat{\mathbf{z }}),\ldots ,y_{nk}(\hat{\mathbf{z }}), p_{1k}(x,\hat{\mathbf{z }}), p_{2k}(x,\hat{\mathbf{z }}),\ldots ,p_{nk}(x,\hat{\mathbf{z }})). \end{aligned}$$

Claim 1: The objective function in the program\([{\mathcal {P}}_{\text{ out }}]\)is continuous onV.

This is trivial since the objective function only consists of simple arithmetic operators, so the proof is omitted.

Claim 2: V is bounded.

Since there are finite number of output x and type profiles \(\mathbf{z }\), there always exists \({\bar{x}}\) and \(\bar{\mathbf{z }}\) such that \(U(x,\mathbf{z })\le U({\bar{x}},\bar{\mathbf{z }})\), \(\forall x, \mathbf{z }\). As \(p_{i}(x,\mathbf{z })\ge 0\), \(\forall i,x,\mathbf{z }\), the principal’s maximum possible expected profit is \(U({\bar{x}},\bar{\mathbf{z }})\). Since \(\Pr (z_{i}=\theta _{j})=q_{j}>0\)\(\forall j\), there always exists a q̱ \(\in \{q_{1},q_{2},\ldots ,q_{m}\}\) such that \(q_{j}\ge \) q̱. Thus, the principal will choose a payment function such that \(p_{i}(x,\mathbf{z }) \le \frac{U({\bar{x}},\bar{\mathbf{z }}) - {\tilde{V}}}{a{}^{n}}\)\(\forall i,x,\mathbf{z }\). Otherwise, his expected profit will be strictly smaller than \({\tilde{V}}\). By (LL(out)), we know that \(p_{i}(x,\mathbf{z })\ge 0\)\(\forall i,x,\mathbf{z }\). Thus, the choice set of the function \(p_{i}(x,\mathbf{z })\) is bounded.

Since \(c(y_{i},z_{i})\rightarrow \infty \) as \(y_{i}\rightarrow \infty \), \(p_{i}(x,\mathbf{z })\) is bounded uniformly by \(\frac{U({\bar{x}},\bar{\mathbf{z }})-{\tilde{V}} }{a^{n}}\) and \(0<f(x|\mathbf{y })\le 1\), we have \(E_{-i}\big [\sum \limits _{x} p_{i}(x,z_{i}, \mathbf{z }_{-i})f(x|\mathbf{y }(z_{i},\mathbf{z }_{-i} ))-c(y_{i}(z_{i}, \mathbf{z }_{-i}), z_{i})\big ]\rightarrow -\infty \) as \(y_{i}(z_{i}, \mathbf{z }_{-i})\rightarrow \infty \) for any \(z_i\) and \(\mathbf{z }_{-i}\). So if \(y_{i}(z_{i}, \mathbf{z }_{-i})\) is too large, then (PC(out)) cannot be satisfied. Thus, for every type profile \(\mathbf{z }\), there exists an upper bar \({\bar{y}}_{i}(\mathbf{z })\) such that the original unconstrained maximization problem is equivalent to the problem with the addition of the constraint \(y_{i}(\mathbf{z }) \le {\bar{y}}_{i}(\mathbf{z })\). Therefore, the choice set of the function \(y_{i}(\mathbf{z })\) of the original problem is bounded.

Claim 3: V is closed.

The proof is also trivial since all the constraints are weak inequalities.

Claim 4: The optimal solution exists.

Since V is closed and bounded, V is compact. Since the objective function is continuous on V, the optimal output monitoring contract exists. \(\square \)

Proof of Proposition 1

Fix any feasible contributions monitoring contract \(\{\bar{p}_i(y_i,{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}), {{\bar{y}}}_i ({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\}\). Consider the proposed punishing contract with the target level of contribution \(y_i^c({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})={{\bar{y}}}_i ({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\) and payment function

$$\begin{aligned} \breve{p}_{i}(y_{i},{\hat{z}}_{i},\hat{\mathbf{z }}_{-i})=\left\{ \begin{array}[c]{ll} {{\bar{p}}}_i({{\bar{y}}}_i({\hat{z}}_{i},\hat{\mathbf{z }}_{-i}),{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}), &{} \text {if }\;\; y_{i}={\bar{y}}_{i}({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\\ 0, &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

We are going to show that the proposed contract will implement the same output at an identical cost to the principal as the given contributions monitoring contract.

Clearly, \(\breve{p}_i({\bar{y}}_{i}({\hat{z}}_{i},\hat{\mathbf{z }}_{-i}),{\hat{z}} _{i},\hat{\mathbf{z }}_{-i})\ge 0\)\(\forall i, {\hat{z}}_{i}, \hat{\mathbf{z }}_{-i}\) since \({\bar{p}}_i (y_{i},{\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\ge 0\)\(\forall i, y_{i},{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}\). Thus, (LL(con)) is satisfied.

Also, (PC(con)) is satisfied as each agent’s (ex-ante) equilibrium payoff (when he reports truthfully and follows the recommendation) is the same as the given contributions monitoring contract.

We now proceed to prove that (IC(con)) is also satisfied. Under the given contract \(\{{{\bar{p}}}_i(y_i,{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}), \bar{y}_i({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\}\), agent i’s interim deviation payoff (after reporting \({\hat{z}}_i \ne z_i\) and receiving recommendation \({{\bar{y}}}_i({\hat{z}}_{i},{\mathbf{z }}_{-i})\)) when he contributes \(y_i\) is

$$\begin{aligned} \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|\bar{y}_i({\hat{z}}_{i},\mathbf{z }_{-i}),{\hat{z}}_{i})\, \bar{p}_i(y_i,{\hat{z}}_{i},\mathbf{z }_{-i}) - c(y_i,z_i), \end{aligned}$$

where \(\Pr (\mathbf{z }_{-i}|{{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i}),{\hat{z}}_{i})\) is his updated belief about other agents’ types based on the received recommendation and his reported type.

Define

$$\begin{aligned} y_i^{\max }= \arg \max _{y_i} \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|{{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i})\, {{\bar{p}}}_i(y_i, {\hat{z}}_{i}, \mathbf{z }_{-i}) - c(y_i,z_i). \end{aligned}$$

Consider two cases:

(i) Suppose \(y_i^{\max }={{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i})\). Then \(y_i^{\max }\) should still be the best deviation contribution under the punishing contract, since

$$\begin{aligned}&\sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|{{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i})\, {{\bar{p}}}_i(y_i^{\max }, {\hat{z}}_{i}, \mathbf{z }_{-i}) - c(y_i^{\max },z_i)\\&\qquad \ge \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|\bar{y}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i})\, \bar{p}_i(\delta _i,{\hat{z}}_{i}, \mathbf{z }_{-i}) - c(\delta _i,z_i), \;\;\; \forall \delta _i \ne {{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i})\\&\qquad \ge \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|\bar{y}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}) \times 0 - c(\delta _i,z_i), \;\;\; \forall \delta _i \ne {{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i}). \end{aligned}$$

(ii) Suppose \(y_i^{\max } \ne {{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i})\). Then the interim deviation payoff under the punishing contract will be smaller. This is because

$$\begin{aligned}&\sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|{{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i})\, {{\bar{p}}}_i(y_i^{\max }, {\hat{z}}_{i}, \mathbf{z }_{-i}) - c(y_i^{\max },z_i)\\&\quad \ge \max \left\{ \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|\bar{y}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i})\, {{\bar{p}}}_i(\bar{y}_i({\hat{z}}_{i}, \mathbf{z }_{-i}),{\hat{z}}_{i}, \mathbf{z }_{-i}) - c({{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i}),z_i),\right. \\&\left. \qquad \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|\bar{y}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i})\, {{\bar{p}}}_i(\delta _i, {\hat{z}}_{i}, \mathbf{z }_{-i}) - c(\delta _i,z_i)\right\} , \;\;\; \forall \delta _i \ne {{\bar{y}}}_i({\hat{z}}_{i},\mathbf{z }_{-i})\\&\quad \ge \max \left\{ \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|\bar{y}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i})\, {{\bar{p}}}_i(\bar{y}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}, \hat{\mathbf{z }}_{-i}) - c({{\bar{y}}}_i({\hat{z}}_{i}, \mathbf{z }_{-i}),z_i),\right. \\&\left. \qquad \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|\bar{y}_i({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}) \times 0 - c(\delta _i,z_i)\right\} , \;\;\; \forall \delta _i \ne \bar{y}_i({\hat{z}}_{i}, \mathbf{z }_{-i}). \end{aligned}$$

Therefore, the interim deviation payoff under the punishing contract is always smaller than that under the given contributions monitoring contract, so does the ex-ante deviation payoff. Thus, (IC(con)) is satisfied.

Hence, the proposed punishing contract implements the same outcomes in the ex-ante as well as interim stages as in the contributions monitoring contract. \(\square \)

Proof of Proposition 2

Fix any feasible output monitoring contract \(\{p_i^{\prime }(x,{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}),y_i^{\prime } ({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\}\). Consider the proposed punishing contract with payment function \(\breve{p}_{i}(y_{i},{\hat{z}}_{i} ,\hat{\mathbf{z }}_{-i})\) and the target contribution \(y_{i}^c({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})=y_{i}^{\prime }({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\)\(\forall i\). Next, following the same procedure as in the proof of Proposition 1, we are going to show that the punishing contract and the output monitoring contract are outcome equivalent.

Since \({p}_{i}^{\prime }(x,{\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\ge 0,\forall i,x, {\hat{z}}_{i},\hat{\mathbf{z }}_{-i}\), we have

$$\begin{aligned} \sum \limits _{x}{p}_{i}^{\prime }(x,{\hat{z}}_{i},\hat{\mathbf{z }}_{-i})f(x|\mathbf{y }^{\prime } ({\hat{z}}_{i}, \hat{\mathbf{z }}_{-i}))\ge 0, \quad \forall i,{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}. \end{aligned}$$

Thus,

$$\begin{aligned} \breve{p}_{i}(y_{i},{\hat{z}}_{i}, \hat{\mathbf{z }}_{-i})\ge 0,\;\;\;\forall i,y_{i},{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}, \end{aligned}$$

i.e., (LL(con)) is satisfied.

Also, (PC(con)) is satisfied as each agent’s (ex-ante) equilibrium payoff (when he reports truthfully and follows the recommendation) is the same as the given output monitoring contract.

We now proceed to prove that (IC(con)) is also satisfied. Under the given contract \(\{p_i^{\prime }(x,{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}),y_i^{\prime } ({\hat{z}}_{i},\hat{\mathbf{z }}_{-i})\}\), agent i’s interim deviation payoff (after reporting \({\hat{z}}_i \ne z_i\) and receiving recommendation \(y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i})\)) when he contributes \(y_i\) is

$$\begin{aligned} \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}),{\hat{z}}_{i})\left[ \sum \limits _{x}\bar{p}_{i}^{\prime }(x,{\hat{z}}_{i}, \mathbf{z }_{-i})f(x|y_i, \mathbf{y }_{-i}^{\prime } ({\hat{z}}_{i}, \mathbf{z }_{-i}))\right] - c(y_i,z_i), \end{aligned}$$

where \(\Pr (\mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i})\) is his updated belief about other agents’ type profile based on the received recommendation and his reported type.

Define

$$\begin{aligned} y_i^{\max }= & {} \arg \max _{y_i} \left\{ \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}| y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}) \left[ \sum \limits _{x}\bar{p}_{i}^{\prime }(x,{\hat{z}}_{i}, \mathbf{z }_{-i})f(x|y_i, \mathbf{y }_{-i}^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}))\right] \right. \\&\qquad \qquad \quad \left. - c(y_i,z_i)\right\} . \end{aligned}$$

Consider two cases:

(i) If \(y_i^{\max }=y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i})\). Then \(y_i^{\max }\) should still be the best deviation contribution under the contributions monitoring contract since

$$\begin{aligned}&\sum _{\mathbf{z }_{-i}} \Pr \left( \mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}),{\hat{z}}_{i}\right) \, \breve{p}_i\left( y_i^{\max },{\hat{z}}_{i}, \mathbf{z }_{-i}\right) - c\left( y_i^{\max },z_i\right) \\&\qquad = \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}) \left[ \sum _{x} p_i^{\prime }(x,{\hat{z}}_{i}, \mathbf{z }_{-i})f\left( x|y_i^{\max }, \mathbf{y }_{-i}^{\prime }\left( {\hat{z}}_i, \mathbf{z }_{-i}\right) \right) \right] - c(y_i^{\max },z_i)\\&\qquad \ge \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}) \left[ \sum _{x} p_i^{\prime }(x, {\hat{z}}_{i}, \mathbf{z }_{-i})f\left( x|\delta _i, \mathbf{y }_{-i}^{\prime }\left( {\hat{z}}_i, \mathbf{z }_{-i}\right) \right) \right] \\&\qquad \quad - c(\delta _i,z_i), \quad \forall \delta _i \ne y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i})\\&\qquad \ge \sum _{\mathbf{z }_{-i}} \Pr \left( \mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}\right) \times 0 - c(\delta _i,z_i), \quad \forall \delta _i \ne y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}). \end{aligned}$$

(ii) If \(y_i^{\max } \ne y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i})\). Then the interim deviation payoff under the punishing contract will be smaller. This is because

$$\begin{aligned}&\sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}) \left[ \sum _{x} p_i^{\prime }(x, {\hat{z}}_{i}, \mathbf{z }_{-i})f(x|y_i^{\max }, y_{-i}^{\prime }({\hat{z}}_i, \mathbf{z }_{-i}))\right] - c(y_i^{\max },z_i)\\&\qquad \ge \max \bigg \{\sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}) \left[ \sum _{x} p_i^{\prime }(x,{\hat{z}}_{i}, \mathbf{z }_{-i})f\big (x|y_i^{\prime }({\hat{z}}_i, \mathbf{z }_{-i}), y_{-i}^{\prime }({\hat{z}}_i, \mathbf{z }_{-i})\big )\right] \\&\qquad \quad - c(y_i^{\prime }({\hat{z}}_i, \mathbf{z }_{-i}), z_i),\\&\sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}) \left[ \sum _{x} p_i^{\prime }(x, {\hat{z}}_{i}, \mathbf{z }_{-i})f\big (x|\delta _i, \mathbf{y }_{-i}^{\prime }({\hat{z}}_i, \mathbf{z }_{-i})\big )\right] - c(\delta _i,z_i)\bigg \} \\&\qquad \ge \max \bigg \{\sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}),{\hat{z}}_{i}) \breve{p}_i(y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}, \mathbf{z }_{-i}) - c(y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}),z_i),\\&\qquad \quad \sum _{\mathbf{z }_{-i}} \Pr (\mathbf{z }_{-i}|y_i^{\prime }({\hat{z}}_{i}, \mathbf{z }_{-i}), {\hat{z}}_{i}) \times 0 - c(\delta _i,z_i)\bigg \}, \;\;\; \forall \delta _i \ne y_i^{\prime }({\hat{z}}_i, \mathbf{z }_{-i}). \end{aligned}$$

Therefore, the interim deviation payoff under the punishing contract is always smaller than that under the given output monitoring contract, so does the ex-ante deviation payoff. Thus, (IC(con)) is satisfied.

Hence, the proposed punishing contract implements the same outcomes in the ex-ante as well as interim stages as in the output monitoring contract. \(\square \)

Proof of Proposition 3

First, we know that a higher type should earn no less payoff than a lower type, since the higher type can always mimic as the lower type and he has (weakly) lower cost. Thus, if the lowest type of agent i earns strictly positive information rent, the higher type of him must also earn strictly positive information rent.

Next, suppose for a feasible output monitoring contract \(\{p_{i}(x,z_{i} ,\mathbf{z }_{-i}),y_{i}(z_{i},\mathbf{z }_{-i})\}\), there exists an agent i such that the lowest type of him earns zero information rent, i.e.,

$$\begin{aligned} E_{-i}\left[ \sum \limits _{x} p_{i}(x,\theta _{1},\mathbf{z }_{-i})f(x|\mathbf{y }(\theta _{1}, \mathbf{z }_{-i}))- c(y_{i}(\theta _{1}, \mathbf{z }_{-i}),\theta _{1})\right] =0. \end{aligned}$$
(A.1)

By (IC(out)), we have

$$\begin{aligned}&E_{-i}\left[ \sum \limits _{x} p_{i}(x,{\hat{z}}_{i}, \mathbf{z }_{-i})f(x|\delta _{i}(y_i({\hat{z}}_{i}, \mathbf{z }_{-i}), \theta _1, {\hat{z}}_i), \mathbf{y }_{-i}({\hat{z}}_{i}, \mathbf{z }_{-i}))\right. \\&\left. \quad - c(\delta _{i}(y_i({\hat{z}}_{i} , \mathbf{z }_{-i}), \theta _1, {\hat{z}}_i), \theta _{1})\right] \le 0,\quad \forall {\hat{z}}_{i}\ne \theta _{1},\, \delta _{i}(y_i({\hat{z}}_{i}, \mathbf{z }_{-i}), \theta _1,{\hat{z}}_i). \end{aligned}$$

Thus,

$$\begin{aligned}&E_{-i}\left[ \sum \limits _{x} p_{i}(x,{\hat{z}}_{i}, \mathbf{z }_{-i})f(x|0,\mathbf{y }_{-i}({\hat{z}}_{i}, \mathbf{z }_{-i})) - c(0,\theta _{1})\right] \le 0, \nonumber \\&\qquad \quad \forall {\hat{z}}_{i}\ne \theta _{1}. \end{aligned}$$
(A.2)

Since \(p_{i}(x,{\hat{z}}_{i}, \mathbf{z }_{-i})\ge 0\)\(\forall x,{\hat{z}}_{i}, \mathbf{z }_{-i}\) due to (LL(out)), we have

$$\begin{aligned} \sum \limits _{x} p_{i}(x,{\hat{z}}_{i}, \mathbf{z }_{-i})f(x|0,\mathbf{y }_{-i}({\hat{z}}_{i}, \mathbf{z }_{-i})) - c(0,\theta _{1})\ge 0, \;\;\forall {\hat{z}}_{i}\ne \theta _{1}, \mathbf{z }_{-i}, \end{aligned}$$

that, together with (A.2) and the fact that \(\Pr (\mathbf{z }_{-i})>0\) for all \(\mathbf{z }_{-i}\), yields:

$$\begin{aligned} \sum \limits _{x}p_{i}(x,{\hat{z}}_{i}, \mathbf{z }_{-i})f(x|0,\mathbf{y }_{-i}({\hat{z}}_{i}, \mathbf{z }_{-i})) - c(0,\theta _{1})=0, \;\;\forall {\hat{z}}_{i}\ne \theta _{1}, \mathbf{z }_{-i}, \end{aligned}$$

implying \(p_{i}(x,{\hat{z}}_{i}, \mathbf{z }_{-i})=0\), \(\forall x,{\hat{z}}_{i},\mathbf{z }_{-i}\), where \({\hat{z}}_{i}\ne \theta _{1}\). Also, from (A.1) it follows that \(p_{i}(x,\theta _1, \mathbf{z }_{-i})=0\)\(\forall x, \mathbf{z }_{-i}\).

For agent i whose true type is \({\hat{z}}_{i}\), given that \(p_{i}(x,{\hat{z}}_{i}, \mathbf{z }_{-i})=0\), \(\forall x,{\hat{z}}_{i}, \mathbf{z }_{-i}\), his best contribution level is 0, and thus, his interim and ex-ante expected payoffs are 0. Thus, if the lowest type of agent i earns zero information rent, every type of him earns zero information rent. \(\square \)

Proof of Lemma 2

Take a feasible punishing contract \(\{\breve{p}_i(y_{i},{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}), y_i^c({\hat{z}}_{i}, \hat{\mathbf{z }}_{-i})\}\). Suppose there exists an agent k with true type \({\tilde{z}}_{k}\) such that all of his (IC-type*(con)) and (PC(con)) constraints are non-binding, i.e.,

$$\begin{aligned}&E_{-k}{\bar{\pi }}_{k}({\tilde{z}}_{k}, {\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), \mathbf{z }_{-k}) \\&\qquad \quad> E_{-k}{\bar{\pi }}_{k}({\tilde{z}}_{k},{\hat{z}}_{k},{\hat{\delta }}_{k}^{con}(y_k^c({\hat{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_k, {\hat{z}}_k), \mathbf{z }_{-k}),\quad \forall {\hat{z}}_{k} \ne {\tilde{z}}_{k},\\&\qquad \quad \text {and } E_{-k}{\bar{\pi }}_{k}({\tilde{z}}_{k},{\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), \mathbf{z }_{-k}) >0. \end{aligned}$$

The second inequality above implies that, without loss of generality we can pick one type profile, say \(\check{\mathbf{z }}_{-k}\), so that \({\bar{\pi }}_{k}({\tilde{z}}_{k},{\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}),\check{\mathbf{z }}_{-k})>0\). This also implies \(\breve{p}_{k}(y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}),{\tilde{z}}_{k}, \check{\mathbf{z }}_{-k})~>~0\).

Also define

$$\begin{aligned} \delta\equiv & {} \min \limits _{{\hat{z}}_{k} \ne {\tilde{z}}_{k}} \bigg \{E_{-k}{\bar{\pi }}_{k}\left( {\tilde{z}}_{k},{\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), \mathbf{z }_{-k}\right) \\&\quad - E_{-k}{\bar{\pi }}_{k}\left( {\tilde{z}}_{k},{\hat{z}}_{k} ,{\hat{\delta }}_{k}^{con}(y_i^c({\hat{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_k, {\hat{z}}_k\right) , \mathbf{z }_{-k})\bigg \} > 0, \end{aligned}$$

which is the smallest difference between agent k’s expected equilibrium payoff and expected deviation payoffs obtained from among all non-truthful type reports, and

$$\begin{aligned} \epsilon\equiv & {} \frac{1}{2} \min \Big \{\delta ,{\bar{\pi }}_{k}\left( {\tilde{z}}_{k},{\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k},\check{\mathbf{z }}_{-k}), \check{\mathbf{z }}_{-k}\right) , \\&\qquad \qquad E_{-k}{\bar{\pi }}_{k}\left( {\tilde{z}}_{k}, {\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), \mathbf{z }_{-k}\right) \Big \} > 0. \end{aligned}$$

Consider the new contract \(\{\breve{p}_{i}^{\prime }(y_{i},{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}), y_i^c({\hat{z}}_{i}, \hat{\mathbf{z }}_{-i})\}\) with the following payment rule:

$$\begin{aligned}&\breve{p}_{k}^{\prime }(y_{k},{\hat{z}}_{k},\hat{\mathbf{z }}_{-k})\\&\qquad =\left\{ \begin{array}[c]{ll} \breve{p}_{k}(y^c_{k}({\tilde{z}}_{k},\check{\mathbf{z }}_{-k}),{\hat{z}}_{k}, \hat{\mathbf{z }}_{-k}) - \epsilon , &{} \text {if} ({\hat{z}}_{k}, \hat{\mathbf{z }}_{-k}) = ({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k})\text { and }y_{k}=y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k})\\ 0, &{} \text {if} ({\hat{z}}_{k},\hat{\mathbf{z }}_{-k}) = ({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}) \text { and } y_{k}\ne y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k})\\ \breve{p}_{k}(y_{k}, {\hat{z}}_{k}, \hat{\mathbf{z }}_{-k}), &{} \text{ otherwise }, \end{array} \right. \end{aligned}$$

and \(\breve{p}_{\ell }^{\prime }(y_{\ell },{\hat{z}}_{\ell },\hat{\mathbf{z }}_{-\ell }) = \breve{p}_{\ell }(y_{\ell },{\hat{z}}_{\ell }, \hat{\mathbf{z }}_{-\ell })\) for \(\ell \ne i\).

Under this new contract, only the payment to agent k is reduced by a fixed amount \(\epsilon \) if the reported profile is \(({\tilde{z}}_{k},\check{\mathbf{z }}_{-k})\) and agent k chooses \(y_{k} = y^c_{k}({\tilde{z}}_{k},\check{\mathbf{z }}_{-k})\). Now, check that under the new contract, (LL(con)) is still satisfied, i.e., \(\breve{p}_{k}^{\prime }(y_{k},{\hat{z}}_{k},\hat{\mathbf{z }}_{-k})\ge 0\). Since \(\breve{p}_{k}(y_{k},{\hat{z}}_{k},\hat{\mathbf{z }}_{-k})\) is feasible by assumption, \(\breve{p}_{k}(y_{k}, {\hat{z}}_{k}, \hat{\mathbf{z }}_{-k})\ge 0\). Also,

$$\begin{aligned}&\breve{p}_{k}(y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}),{\tilde{z}}_{k}, \check{\mathbf{z }}_{-k})-\epsilon \\&\qquad \quad = \breve{p}_{k}(y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}),{\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}) - \frac{1}{2}\min \left\{ \delta ,{\bar{\pi }}_{k}({\tilde{z}}_{k} ,{\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}),\check{\mathbf{z }}_{-k}),\right. \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\qquad \qquad \qquad \qquad \qquad \qquad \qquad \left. E_{-i}{\bar{\pi }}_{k}({\tilde{z}}_{k},{\tilde{z}}_{k}, y^c_{k}(\tilde{z}_{k}, \mathbf{z }_{-k}), \mathbf{z }_{-k})\right\} \\&\qquad \quad \ge \breve{p}_{k}(y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}),{\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}) - \frac{1}{2}{\bar{\pi }}_{k}({\tilde{z}}_{k},{\tilde{z}}_{k},{y}^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}), \check{\mathbf{z }}_{-k})\\&\qquad \quad = \breve{p}_{k}(y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}),{\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}) - \frac{1}{2}\left[ \breve{p}_{k}(y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}),{\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}) - c({y}^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}), {\tilde{z}}_{k})\right] \\&\qquad \quad = \frac{1}{2}\left[ \breve{p}_{k}(y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}), {\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}) + c(y^c_{k}({\tilde{z}}_{k}, \check{\mathbf{z }}_{-k}),{\tilde{z}}_{k})\right] > 0. \end{aligned}$$

Thus, the new contract \(\{\breve{p}_{i}^{\prime }(y_{i},{\hat{z}}_{i},\hat{\mathbf{z }}_{-i}), y_i^c({\hat{z}}_{i}, \hat{\mathbf{z }}_{-i})\}\) satisfies (LL(con)).

Next, check that under this new contract, (PC(con)) is still satisfied if agent i reports truthfully and follows the recommendation:

$$\begin{aligned}&\sum _{\mathbf{z }_{-k}} \Pr (\mathbf{z }_{-k}| y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_k)\breve{p}_{k}^{\prime }(y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_{k}, \mathbf{z }_{-k}) - c(y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_k)\\&\qquad = \sum _{\mathbf{z }_{-k}}\Pr (\mathbf{z }_{-k}| y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_k)\breve{p}_{k}(y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_{k}, \mathbf{z }_{-k}) \\&\qquad \qquad - \Pr (\check{\mathbf{z }}_{-k}| y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_k)\epsilon - c(y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_k)\\&\qquad \ge E_{-k}{\bar{\pi }}_{k}({\tilde{z}}_{k}, {\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), \mathbf{z }_{-k}) \\&\qquad \qquad - \Pr (\check{\mathbf{z }}_{-k}| y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), \tilde{z_k})E_{-k}{\bar{\pi }}_{k}({\tilde{z}}_{k}, {\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), \mathbf{z }_{-k})\\&\qquad = \left[ 1-\Pr (\check{\mathbf{z }}_{-k}| y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_k)\right] E_{-k}{\bar{\pi }}_{k}({\tilde{z}}_{k}, {\tilde{z}}_{k}, y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), \mathbf{z }_{-k}) \ge 0. \end{aligned}$$

Finally, check that under this new contract, (IC-type*(con)) is also satisfied:

$$\begin{aligned}&E_{-k}\left[ \breve{p}_{k}^{\prime }(y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_{k}, \mathbf{z }_{-k}) - c(y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}),{\tilde{z}}_k)\right] \\&\qquad \qquad - E_{-k}\left[ \breve{p}_{k}^{\prime }\left( {\hat{\delta }}_k^{con}(y^c_{k}({\hat{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_{k}, {\hat{z}}_k), {\hat{z}}_{k}, \mathbf{z }_{-k}\right) \right. \\&\left. \qquad \qquad - c\left( {\hat{\delta }}_k^{con}\left( y^c_{k}({\hat{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_{k}, {\hat{z}}_k\right) , {\tilde{z}}_k\right) \right] ,\;\;\;\forall {\hat{z}}_{k} \ne {\tilde{z}}_{k}\\&\qquad = E_{-k}\Big [\breve{p}_{k}(y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_{k}, \mathbf{z }_{-k})\\&\qquad \qquad - c(y^c_{k}({\tilde{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_k)\Big ] - \Pr (\check{\mathbf{z }}_{-k})\epsilon \\&\qquad \qquad -E_{-k}\left[ \breve{p}_{k}({\hat{\delta }}_k^{con}(y^c_{k}({\hat{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_{k}, {\hat{z}}_k), {\hat{z}}_{k}, \mathbf{z }_{-k}) \right. \\&\left. \qquad \qquad - c\left( {\hat{\delta }}_k^{con}(y^c_{k}({\hat{z}}_{k}, \mathbf{z }_{-k}), {\tilde{z}}_{k}, {\hat{z}}_k), {\tilde{z}}_k\right) \right] , \quad \forall {\hat{z}}_{k} \ne {\tilde{z}}_{k}\\&\qquad \ge \delta - \Pr (\check{\mathbf{z }}_{-k})\frac{1}{2} \delta \ge \frac{1}{2}\delta > 0. \end{aligned}$$

Therefore, the new contract \(\{\breve{p}_{i}^{\prime }(y_{i}, {\hat{z}}_{i}, \hat{\mathbf{z }}_{-i}), y_i^c({\hat{z}}_{i}, \hat{\mathbf{z }}_{-i})\}\) is feasible and induces same amount of contributions with less expected cost. Thus, the principal’s ex-ante expected profit will be higher, given our assumption that each type profile occurs with positive probability (and thus the occurrence of \(({{\tilde{z}}}_k, \check{\mathbf{z }}_{-k})\) is a non-negligible event). \(\square \)

Proof of Proposition 4

For the replicated punishing contract \(\{\breve{p} _{i}(y_{i},z_{i},\mathbf{z }_{-i}),y^c_{i}(z_{i}, \mathbf{z }_{-i})\}\), without loss of generality, suppose the lowest type of agent i earns strictly positive information rent (by Proposition \(3^\prime \)). Thus, every type of agent i should earn strictly positive ex-ante expected payoff: the higher type can always mimic as the lowest type, choose the target contribution for the lowest type at a weakly lower cost, and earn a weakly higher payoff than the lowest type’s equilibrium payoff.

Note that for any type \(z_{i}\) of agent i, his (PC(con)) is non-binding. If there is a particular type such that all of his (IC-type*(con)) are non-binding, by Lemma 2, we know that there exists another feasible punishing contract which generates a strictly higher expected profit for the principal. Thus, the strict dominance of contributions monitoring easily follows. So suppose for any type \(z_{i}\), at least one of his (IC-type*(con)) is binding. This implies that for every\(z_i\), at least one of his ex-ante deviation payoff is positive. Denote R to be the smallest positive ex-ante deviation payoff across all type pairs \((z_{i},{\hat{z}}_{i})\) (where \(z_{i}\) is the true type and \({\hat{z}}_{i}\ne z_{i}\) is the reported type). That is,

$$\begin{aligned} R\equiv & {} \min \limits _{z_{i}, {\hat{z}}_{i}}\bigg \{\,E_{-i}{\bar{\pi }}_{i}(z_{i}, {\hat{z}}_{i}, {\hat{\delta }}_i^{con}(y_i^c({\hat{z}}_i, \mathbf{z }_{-i}), z_i, {\hat{z}}_i), \mathbf{z }_{-i}): \\&\quad ~~~~~~~~~~~~~~~~~~~~~ E_{-i}{\bar{\pi }}_{i}(z_{i},\hat{z}_{i},{\hat{\delta }}_i^{con}(y_i^c({\hat{z}}_i, \mathbf{z }_{-i}), z_i, {\hat{z}}_i), \mathbf{z }_{-i}) > 0\bigg \}. \end{aligned}$$

For any type \({\hat{z}}_{i}\), we know that when some other type \(z_{i}\) reports as type \({\hat{z}}_{i}\), either

$$\begin{aligned} E_{-i}{\bar{\pi }}_{i}(z_{i}, {\hat{z}}_{i}, {\hat{\delta }}_i^{con}(y_i^c({\hat{z}}_i, \mathbf{z }_{-i}), z_i, {\hat{z}}_i), \mathbf{z }_{-i})\ge R, \end{aligned}$$

or

$$\begin{aligned} E_{-i}{\bar{\pi }}_{i}\left( z_{i}, {\hat{z}}_{i}, {\hat{\delta }}_i^{con}(y_i^c({\hat{z}}_i, \mathbf{z }_{-i}), z_i, {\hat{z}}_i), \mathbf{z }_{-i}\right) = 0. \end{aligned}$$

Next, observe that the following inequalities are always satisfied due to the monotonicity of cost of any contribution with respect to types:

$$\begin{aligned}&E_{-i}{\bar{\pi }}_{i}\left( z_{i}, z_{i}, y^c_{i}(z_{i}, \mathbf{z }_{-i}), \mathbf{z }_{-i}\right) \nonumber \\&\quad \ge E_{-i}{\bar{\pi }}_{i}({\tilde{z}}_{i}, z_{i}, {\hat{\delta }}_i^{con}(y_{i}^{c}(z_{i}, \mathbf{z }_{-i}), {\tilde{z}}_{i}, z_i), \mathbf{z }_{-i}), \quad \forall i,\text { if } z_{i} > {\tilde{z}}_{i}; \end{aligned}$$
(A.3)
$$\begin{aligned}&E_{-i}{\bar{\pi }}_{i}\left( z_{i}, {\tilde{z}}_{i}, {\hat{\delta }}_i^{con}\left( y_{i}^{c}({\tilde{z}}_{i}, \mathbf{z }_{-i}), z_{i}, {\tilde{z}}_i\right) , \mathbf{z }_{-i}\right) \nonumber \\&\quad \ge E_{-i}{\bar{\pi }}_{i}({{\tilde{z}}}_{i}, {{\tilde{z}}}_{i}, y^c_{i} ({\tilde{z}}_{i}, \mathbf{z }_{-i}), \mathbf{z }_{-i}), \quad \forall i,\text { if }z_{i}>{\tilde{z}}_{i}; \end{aligned}$$
(A.4)
$$\begin{aligned}&\text{ and }\;\;E_{-i}{\bar{\pi }}_{i}(z_{i}, {\hat{z}}_{i}, {\hat{\delta }}_i^{con}(y_{i}^{c}({\hat{z}}_{i}, \mathbf{z }_{-i}), z_i, {\hat{z}}_{i}), \mathbf{z }_{-i})\nonumber \\&\qquad \quad \ge E_{-i}{\bar{\pi }}_{i}({\tilde{z}}_{i}, {\hat{z}}_{i}, {\hat{\delta }}_i^{con}(y_{i}^{c}({\hat{z}}_{i}, \mathbf{z }_{-i}), {\tilde{z}}_i, {\hat{z}}_{i}), \mathbf{z }_{-i}), \forall i,\; {{\hat{z}}}_i\ne z_i,{{\tilde{z}}}_i,\text { if }z_{i}>{\tilde{z}}_{i}.\nonumber \\ \end{aligned}$$
(A.5)

The first inequality says for each agent i, the higher type of him can always obtain a weakly higher equilibrium payoff than the deviation payoff of any lower type who wants to mimic him. The second inequality says for each agent i, when the higher type mimics a lower type, the higher type can always obtain a weakly higher deviation payoff than the equilibrium payoff of the lower type. The last inequality says for each agent i, the deviation payoff of higher type is always weakly higher than that of the lower type if they mimic as some other common type.

The rest of the proof is divided into two steps.

Step 1. Choose one \({\hat{z}}_{i}^\prime \) for which there exists some \(z_i\) such that

$$\begin{aligned} E_{-i}{\bar{\pi }}_{i}(z_{i}, {\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}(y_i^c({\hat{z}}_i^{\prime }, \mathbf{z }_{-i}), z_i, {\hat{z}}_i^\prime ), \mathbf{z }_{-i})\ge R. \end{aligned}$$

The existence of such \({\hat{z}}_{i}^\prime \) is guaranteed by our earlier argument above.

Let

$$\begin{aligned} {\bar{z}}_{i} = \min \bigg \{z_i\,\, :\, E_{-i}{\bar{\pi }}_{i}(z_{i}, {\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}(y_i^c({\hat{z}}_i^\prime , \mathbf{z }_{-i}), z_i, {\hat{z}}_i^\prime ), \mathbf{z }_{-i})\ge R\bigg \}. \end{aligned}$$

Thus, for all types \(z_{i}<{\bar{z}}_{i}\),

$$\begin{aligned} E_{-i}{\bar{\pi }}_{i}(z_{i}, {\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}({y}_i^c({\hat{z}}_i^\prime , \mathbf{z }_{-i}), z_i, {\hat{z}}_i^\prime ), \mathbf{z }_{-i})=0, \end{aligned}$$
(A.6)

and for all types \(z_{i}^{\prime }>{\bar{z}}_{i}\), we know by (A.5) that

$$\begin{aligned}&E_{-i}{\bar{\pi }}_{i}(z_{i}^{\prime }, {\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}(y_i^c({\hat{z}}_i^\prime , \mathbf{z }_{-i}), z_i^{\prime }, {\hat{z}}_i^\prime ), \mathbf{z }_{-i}) \nonumber \\&\quad \ge E_{-i}{\bar{\pi }}_{i}({{\bar{z}}}_{i}, {\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}(y_i^c({\hat{z}}_i^\prime , \mathbf{z }_{-i}), {\bar{z}}_i, {\hat{z}}_i^\prime ), \mathbf{z }_{-i}) \ge R. \end{aligned}$$
(A.7)

Consider the following two cases:

Case (i)\({\bar{z}}_{i}<{\hat{z}}_{i}^\prime \).

Given that \(E_{-i}{\bar{\pi }}_{i}({\bar{z}}_{i},{\hat{z}}_{i}^\prime ,{\hat{\delta }}_i^{con}({y}_i^c({\hat{z}}_i^{\prime }, \mathbf{z }_{-i}), {\bar{z}}_i,{\hat{z}}_i^{\prime }), \mathbf{z }_{-i}) \ge R\), identify all \(z_{-i}\) profiles such that \({\bar{\pi }}_{i}({\bar{z}}_{i},{\hat{z}}_{i}^\prime , y^c_{i}({\hat{z}}_{i}^\prime , \mathbf{z }_{-i}), \mathbf{z }_{-i})>0\), which also implies \(\breve{p}_{i}(y^c_{i}({\hat{z}}_{i}^\prime ,z_{-i}),{\hat{z}}_{i}^\prime , \mathbf{z }_{-i})>0\). Now reduce the above payments \(\breve{p}_{i}(y^c_{i}({\hat{z}}_{i}^\prime , \mathbf{z }_{-i}), {\hat{z}}_{i}^\prime , \mathbf{z }_{-i})\) such that the expected reductions in the ex-ante deviation payoff of agent \({{\bar{z}}}_i\) is R, and his optimal contribution response following misreporting as \({{\hat{z}}}_i^\prime \) remains the same as before the reduction.Footnote 16

Since \({\hat{z}}_{i}^\prime > {\bar{z}}_{i}\), by (A.3), we know

$$\begin{aligned}&E_{-i}{\bar{\pi }}_{i}({\hat{z}}_{i}^\prime , {\hat{z}}_{i}^\prime , y^c_{i}({\hat{z}}_{i}^\prime , \mathbf{z }_{-i}), \mathbf{z }_{-i})\nonumber \\&\ge E_{-i}{\bar{\pi }} _{i}({\bar{z}}_{i},{\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}({y}_{i}^{c}({\hat{z}}_{i}^{\prime }, \mathbf{z }_{-i}), {\bar{z}}_{i}, {\hat{z}}_i^{\prime }), \mathbf{z }_{-i}). \end{aligned}$$
(A.8)

Thus, the expected reduction in the ex-ante equilibrium payoff for type \({\hat{z}}_{i}^\prime \) agent is also R.Footnote 17 Before the reduction, we know \(E_{-i}{\bar{\pi }}_{i}({\hat{z}}_{i}^\prime , {\hat{z}}_{i}^\prime , y^c_{i}({\hat{z}}_{i}^\prime , \mathbf{z }_{-i}), \mathbf{z }_{-i})\ge R\), so the ex-ante expected payoff for type \({\hat{z}}_{i}^\prime \) agent is still non-negative after the reduction, i.e., (PC(con)) is still satisfied.

Case (ii)\({\bar{z}}_{i}>{\hat{z}}_{i}^\prime \).

Given that \(E_{-i}{\bar{\pi }}_{i}({\hat{z}}_{i}^\prime ,{\hat{z}}_{i}^\prime , y^c_{i}({\hat{z}}_{i}^\prime , \mathbf{z }_{-i}), \mathbf{z }_{-i})\ge R\) (due to the fact that R is the smallest positive deviation payoff), identify all \(z_{-i}\) profiles such that \({\bar{\pi }}_{i}({\hat{z}}_{i}^\prime ,{\hat{z}}_{i}^\prime , y^c_{i}({\hat{z}}_{i}^\prime , \mathbf{z }_{-i}), \mathbf{z }_{-i})>0\), which also implies \(\breve{p}_{i}(y^c_{i}({\hat{z}}_{i}^\prime , \mathbf{z }_{-i}), {\hat{z}}_{i}^\prime , \mathbf{z }_{-i})>0\). Now reduce the above payments \(\breve{p}_{i}(y^c_{i}({\hat{z}}_{i}^\prime , \mathbf{z }_{-i}), {\hat{z}}_{i}^\prime , \mathbf{z }_{-i})\) such that the expected reductions in the ex-ante equilibrium payoff of agent \({{\hat{z}}}_i^\prime \) is R, and yet his optimal contribution remains the same as before the reduction.

Since \({\bar{z}}_{i} > {\hat{z}}_{i}^\prime \), by (A.4), we know

$$\begin{aligned} E_{-i}{\bar{\pi }}_{i}({\bar{z}}_{i}, {\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}(y_{i}^{c}({\hat{z}}_{i}^{\prime }, \mathbf{z }_{-i}), {\bar{z}}_{i}, {\hat{z}}_i^{\prime }), \mathbf{z }_{-i})\ge E_{-i}{\bar{\pi }}_{i}({\hat{z}}_{i}^\prime , {\hat{z}}_{i}^\prime , y^c_{i}({\hat{z}}_{i}, \mathbf{z }_{-i}), \mathbf{z }_{-i}). \end{aligned}$$

Thus, the expected reduction in the ex-ante deviation payoff for type \({\bar{z}}_{i}\) agent when reporting as type \({\hat{z}}_{i}^\prime \) is also R. ||

For all types \(z_{i}<{\bar{z}}_{i}\), initially \(E_{-i}{\bar{\pi }}_{i}(z_{i},{\hat{z}}_{i}^\prime ,{\hat{\delta }}_i^{con}(y_{i}^{c}({\hat{z}}_{i}^{\prime }, \mathbf{z }_{-i}), z_i, {\hat{z}}_{i}^{\prime }), \mathbf{z }_{-i})=0\) (by (A.6)), and after the reduction in payment, those ex-ante deviation payoff of agent \(z_i\) when he reports as \({{\hat{z}}}_i^\prime \) is still 0.

For all types \(z_{i}^{\prime }>{\bar{z}}_{i}\), initially \(E_{-i}{\bar{\pi }}_{i}(z_{i}^{\prime }, {\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}(y_{i}^{c}({\hat{z}}_{i}^{\prime }, \mathbf{z }_{-i}), z_i^{\prime }, {\hat{z}}_{i}^{\prime }), \mathbf{z }_{-i})\ge R\) (by (A.7)). By (A.5), we know

$$\begin{aligned}&E_{-i}{\bar{\pi }}_{i}(z_{i}^{\prime }, {\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}(y_{i}^{c}({\hat{z}}_{i}^{\prime }, \mathbf{z }_{-i}),z_i^{\prime }, {\hat{z}}_{i}^{\prime }), \mathbf{z }_{-i})\\&\ge E_{-i}{\bar{\pi }}_{i}({\bar{z}}_{i}, {\hat{z}}_{i}^\prime , {\hat{\delta }}_i^{con}(y_{i}^{c}({\hat{z}}_{i}^{\prime }, \mathbf{z }_{-i}), {\bar{z}}_i,{\hat{z}}_{i}^{\prime }), \mathbf{z }_{-i}). \end{aligned}$$

Thus, the expected reduction in the deviation payoff of agent \(z_i^\prime \) when he reports as \({{\hat{z}}}_i^\prime \) is R.

Step 2. Repeat such reductions as in Step 1 for each possible type \({\hat{z}}_{i}^{\prime \prime }\) for which there exists \(z_{i}\) such that

$$\begin{aligned} E_{-i}{\bar{\pi }}_{i}(z_{i}, {\hat{z}}_{i}^{\prime \prime }, {\hat{y}}_{i}^{con}(z_{i},{\hat{z}}_{i}^{\prime \prime }, \mathbf{z }_{-i}), \mathbf{z }_{-i})\ge R. \end{aligned}$$

Thus, we know that each such \({\hat{z}}_{i}^{\prime \prime }\) type’s ex-ante equilibrium payoff is reduced by R, and the payoff is still non-negative. Also, for the ex-ante deviation payoff that is initially 0, it is still 0 after the reduction in payment; for the ex-ante deviation payoff that is initially positive, it is also reduced by R after the reduction in payment. Thus, every type’s (IC-type*(con)) is still satisfied.

We have thus shown the existence of a feasible punishing contract which generates a higher ex-ante expected profit for the principal (by maintaining the same contributions level with a lower payment) than the replicated punishing contract \(\{\breve{p}_{i}(y_{i}, z_{i}, \mathbf{z }_{-i}), y^c_{i}(z_{i}, \mathbf{z }_{-i})\}\). \(\square \)

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Bag, P.K., Wang, P. Dominance of contributions monitoring in teams. Soc Choice Welf 53, 467–495 (2019). https://doi.org/10.1007/s00355-019-01193-7

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