Team incentives with imperfect mutual inference

Abstract

I study the optimal team incentive when the agents can coordinate private actions through repeated interaction with imperfect public monitoring. The agents are able to imperfectly infer each other’s private actions via the stochastically correlated measurements. Correlation of measurement noise, besides its risk sharing role in the conventional multiple-agent moral hazard problem, is crucial to the accuracy of each agent’s inference. The principal’s choice of performance pay to provide incentive via inducing competition or coordination among the agents thus exhibits the trade-off between risk sharing and mutual inference between the agents. I characterize the optimal form of performance pay with respect to the correlation of measurement noise. Whether the conventional theoretical prediction holds depends on how the agents form the mutual inference, based on an exogenous standard or formed endogenously.

Appendix

1.1 A Proof of Propositions

1.1.1 A.1 Proof of Lemma 1

Suppose that the agents coordinate to exert $(a_{i}^{0},a_{j}^{0})$ when the non-cooperative Nash equilibrium is $(a_{i}^{'},a_{j}^{'})$. Agent j detects agent i’s deviation correctly with probability

$$\begin{aligned} q_{i}^{d}={\left\{ \begin{array}{ll} Pr\left( x_{it}(a_{i}^{'})|x_{jt}(a_{j}^{0})<a_{it}^{0}+\sigma (x_{jt}-a_{jt}^{0})-s\sqrt{1-\sigma ^{2}}\right) =\Phi \left( \frac{a_{i}^{0}-a_{i}^{'}}{\sqrt{1-\sigma ^{2}}}-s\right) &{} for\,a_{i}^{0}>a_{i}^{'}\\ Pr\left( x_{it}(a_{i}^{'})|x_{jt}(a_{j}^{0})>a_{it}^{0}+\sigma (x_{jt}-a_{jt}^{0})+s\sqrt{1-\sigma ^{2}}\right) =1-\Phi \left( \frac{a_{i}^{0}-a_{i}^{'}}{\sqrt{1-\sigma ^{2}}}+s\right) &{} for\,a_{i}^{0}<a_{i}^{'} \end{array}\right. }, \end{aligned}$$

where $\Phi (\cdot )$ is the standard normal CDF. It is increasing in correlation as $\frac{\partial q_{i}^{d}}{\partial \sigma }=\phi (\cdot )\frac{\sigma |a_{i}^{0}-a_{i}^{'}|}{\sqrt{(1-\sigma ^{2})^{3}}}>0$, where $\phi (\cdot )$ is the standard normal PDF. Agent j mistakenly detects agent i of deviation with probability

$$\begin{aligned} q_{i}^{n}={\left\{ \begin{array}{ll} Pr\left( x_{it}(a_{i}^{0})|x_{jt}(a_{j}^{0})<a_{it}^{0}+\sigma (x_{jt}-a_{jt}^{0})-s\sqrt{1-\sigma ^{2}}\right) =\Phi (-s) &{} for\,a_{i}^{0}>a_{i}^{'}\\ Pr\left( x_{it}(a_{i}^{0})|x_{jt}(a_{j}^{0})>a_{it}^{0}+\sigma (x_{jt}-a_{jt}^{0})+s\sqrt{1-\sigma ^{2}}\right) =1-\Phi (s) &{} for\,a_{i}^{0}<a_{i}^{'} \end{array}\right. }, \end{aligned}$$

which is straightforward that $\frac{\partial q_{i}^{n}}{\partial \sigma }=0$. $\square$

1.1.2 A.2 Proof of Proposition 1

The optimal contract to implement $(a_{H},a_{H})$ and to deter collusion solves the expected transfer minimization problem $\min _{\alpha ,\beta ,\gamma }\mathbb {E}(\alpha x_{i}+\beta x_{j}+\gamma )$ subject to $(IR_{n})$, $(IC_{n})$, and (CP), $i=1,2$ and $i\ne j$. Given binding $(IR_{n})$, $(\alpha +\beta )a_{H}+\gamma =R(\alpha ,\beta )$, this contracting problem is reduced to

$$\begin{aligned} \mathscr {P}_{CP}:\qquad \min _{\alpha ,\beta }R(\alpha ,\beta )\equiv \frac{r}{2}(\alpha ^{2}+\beta ^{2}+2\sigma \alpha \beta ) \end{aligned}$$

subject to $(IC_{n})$ and (CP). $\Delta =\Delta _{s}$ with exogenous inference and $\Delta =\Delta ^{*}$ with endogenous inference.

Constraint $(IC_{n})$ can be reduced to $\alpha \ge \eta$, where $\eta \equiv \frac{c_{H}-c_{L}}{a_{H}-a_{L}}$. If $\beta \ge 0$, $\alpha +h(\delta ,\Delta )\beta \ge \alpha \ge \eta$ as $0\le h(\delta ,\Delta )\equiv \frac{\delta \Delta }{1-\delta +\delta \Delta }\le 1$. $(IC_{n})$ is binding while (CP) slacks except at $\beta =0$. With binding $(IC_{n})$, ignoring (CP) and $\beta \ge 0$, the problem is that of the non-cooperative benchmark, in which the optimal $\beta _{n}=\arg \min _{\beta }\eta ^{2}+\beta ^{2}+2\sigma \eta \beta =-\sigma \eta <0$. Among all $\beta \ge 0$, $\beta _{CP}=0$ is thus optimal to deter collusion. If $\beta <0$, $\alpha >\alpha +h(\delta ,\Delta )\beta \ge \eta$. (CP) is the only binding constraint, i.e. $\alpha +h(\delta ,\Delta )\beta =\eta$. Solving the first order condition, $\left( h(\delta ,\Delta )\beta -\eta \right) h(\delta ,\Delta )+\beta +\sigma \left( \eta -2h(\delta ,\Delta )\beta \right) =0$, the optimal contract with exogenous inference has $\beta _{CP}=\frac{(h(\delta ,\Delta _{s})-\sigma )\eta }{1+h(\delta ,\Delta _{s})^{2}-2\sigma h(\delta ,\Delta _{s})}<0$ if $h(\delta ,\Delta _{s})<\sigma$, and $\beta _{CP}=0$ if $h(\delta ,\Delta _{s})\ge \sigma$. Rearranging $h(\delta ,\Delta _{s})<\sigma$ yields $\delta <\delta _{CP}\equiv \frac{\sigma }{(1-\sigma )\Delta _{s}+\sigma }$.

The optimal contract to implement $(a_{H},a_{H})$ and to induce coordination solves the expected transfer minimization problem $\min _{\alpha ,\beta ,\gamma }\sum _{i=1,2}\left( \mathbb {E}(\alpha x_{i}+\beta x_{j}+\gamma )\right)$ subject to $(IR_{c})$, $(IC_{c})$, and (CI). Given binding $(IR_{c})$, $(\alpha +\beta )a_{L}+\gamma =R(\alpha ,\beta )$, this contracting problem is reduced to

$$\begin{aligned}&\mathscr {P}_{CI}:\quad \min _{\alpha ,\beta }(\alpha +\beta )(a_{H}-a_{L})+R(\alpha ,\beta )\equiv (\alpha +\beta )(a_{H}-a_{L})\\&\qquad \quad \,\,\quad +\frac{r}{2}(\alpha ^{2}+\beta ^{2}+2\sigma \alpha \beta ) \end{aligned}$$

subject to $(IC_{c})$ and (CI). $\Delta =\Delta _{s}$ with exogenous inference and $\Delta =\Delta ^{*}$ with endogenous inference.

Constraint $(IC_{c})$ can be reduced to $\alpha +\beta \ge \eta$. If $\beta \le 0$, $\alpha +h(\delta ,\Delta )\beta \ge \alpha +\beta \ge \eta$ as $0\le h(\delta ,\Delta )\equiv \frac{\delta \Delta }{1-\delta +\delta \Delta }\le 1$. $(IC_{c})$is binding while (CI) slacks except at $\beta =0$. With binding $(IC_{c})$, ignoring (CI) and $\beta \le 0$, the problem is reduced to that of the cooperative benchmark in terms of the choice of $(\alpha ,\beta )$, in which the optimal $\beta _{c}=\arg \min _{\beta }\eta ^{2}-2(1-\sigma )\left( \eta -\beta \right) \beta =\frac{\eta }{2}>0$. Among all $\beta \le 0$, $\beta _{CI}=0$ is thus optimal to induce coordination. If $\beta >0$, $\alpha +\beta >\alpha +h(\delta ,\Delta )\beta \ge \eta$. (CI) is the only binding constraint, $\alpha +h(\delta ,\Delta )\beta =\eta$. Solving the first order condition, $(a_{H}-a_{L})\left( 1-h(\delta ,\Delta )\right) +r\left( \left( h(\delta ,\Delta )\beta -\eta \right) h(\delta ,\Delta )+\beta +\sigma \left( \eta -2h(\delta ,\Delta )\beta \right) \right) =0$, the optimal contract with exogenous inference has $\beta _{CI}=\frac{r(h(\delta ,\Delta _{s})-\sigma )\eta -(1-h(\delta ,\Delta _{s}))(a_{H}-a_{L})}{r(1+h(\delta ,\Delta _{s})^{2}-2\sigma h(\delta ,\Delta _{s}))}>0$ if $h(\delta ,\Delta _{s})-\sigma >\frac{(1-h(\delta ,\Delta _{s}))(a_{H}-a_{L})}{r\eta }$, and $\beta _{CI}=0$ if otherwise. Rearranging $h(\delta ,\Delta _{s})-\sigma >\frac{(1-h(\delta ,\Delta _{s}))(a_{H}-a_{L})}{r\eta }$ yields $\delta >\delta _{CI}\equiv \frac{r\sigma \eta +(a_{H}-a_{L})}{r((1-\sigma )\Delta _{s}+\sigma )\eta +(a_{H}-a_{L})}$.

As $\frac{\sigma }{(1-\sigma )\Delta _{s}+\sigma }<1$, $\delta _{CI}>\delta _{CP}$. For $\delta <\delta _{CP}$, the optimal collusion-proof RPE is preferred to an IPR and the optimal coordination-inducing contract has IPR. Collusion-proof RPE is thus preferred to coordination-inducing JPE for $\delta <\delta _{CP}$. For $\delta >\delta _{CI}$, the optimal coordination inducing JPE is preferred to an IPR and the optimal collusion-proof contract has IPR. Coordination-inducing JPE is thus preferred to collusion-proof RPE for $\delta >\delta _{CI}$. For $\delta _{CI}\ge \delta \ge \delta _{CP}$, coordination is induced with IPR and collusion is deterred with IPR; IPR is optimal. $\square$

1.1.3 A.3 Proof of Proposition 2

It is easy to verify that $\delta \gtreqless \delta _{CI}$ is merely a rearrangement of $h(\delta ,\Delta _{s})\gtreqless \varphi (\sigma )$ and $\delta \lesseqgtr \delta _{CP}$ is a rearrangement of $h(\delta ,\Delta _{s})\lesseqgtr \sigma$. Proposition 2 holds if there exists parameters s and $a_{H}-a_{L}$ such that $\delta _{CI}$ and $\delta _{CP}$ are diminishing in $\sigma$ for $\sigma _{0}<\sigma <\sigma ^{*}$, and are increasing in $\sigma$ for $\sigma _{1}>\sigma >\sigma ^{*}$, where $(\sigma _{0},\sigma _{1})\subseteq [0,1]$. Differentiate $\delta _{CI}\equiv \frac{r\sigma \eta +(a_{H}-a_{L})}{r((1-\sigma )\Delta _{s}+\sigma )\eta +(a_{H}-a_{L})}$ with respect to $\sigma$ finds $\frac{\partial \delta _{CI}}{\partial \sigma }\gtreqless 0$ if $\frac{\partial \Delta _{s}}{\partial \sigma }\frac{1-\sigma }{\Delta _{s}}\lesseqgtr \frac{r\eta +(a_{H}-a_{L})}{\sigma r\eta +(a_{H}-a_{L})}\equiv \frac{1}{\varphi (\sigma )}$, or $\frac{\partial \Delta _{s}}{\partial \sigma }\frac{\sigma }{\Delta _{s}}\lesseqgtr \frac{1}{\varphi (\sigma )}\frac{\sigma }{1-\sigma }$ for $\sigma \in (0,1)$. Differentiate $\delta _{CP}\equiv \frac{\sigma }{((1-\sigma )\Delta _{s}+\sigma )}$ with respect to $\sigma$ finds $\frac{\partial \delta _{CP}}{\partial \sigma }\gtreqless 0$ if $\frac{\partial \Delta _{s}}{\partial \sigma }\frac{1-\sigma }{\Delta _{s}}\lesseqgtr \frac{1}{\sigma }$, or $\frac{\partial \Delta _{s}}{\partial \sigma }\frac{\sigma }{\Delta _{s}}\lesseqgtr \frac{1}{1-\sigma }$ for $\sigma \in (0,1)$. For all $\sigma \in (0,1)$ such that $\delta _{CI}$ is increasing in $\sigma$, $\delta _{CP}$ is increasing in $\sigma$ as well, and for all $\sigma \in (0,1)$ such that $\delta _{CP}$ is diminishing in $\sigma$, $\delta _{CI}$ is diminishing in $\sigma$ as well, because $\frac{\sigma }{\varphi (\sigma )}<1$. To see the non-monotonicity result, it is sufficient to focus on the existence of parameters such that $\frac{\partial \delta _{CI}}{\partial \sigma }>0$ for sufficiently high correlations and $\frac{\partial \delta _{CP}}{\partial \sigma }<0$ for some intermediate correlations.

Since $\frac{1}{\varphi (\sigma )}\ge 1$ with equality at $\sigma =1$, and $\lim _{\sigma \rightarrow 1}\Delta _{s}=1$ if s is finite, $\frac{\partial \Delta _{s}}{\partial \sigma }\frac{\sigma }{\Delta _{s}}<\frac{1}{\varphi (\sigma )}\frac{\sigma }{1-\sigma }$ and $\frac{\partial \delta _{CI}}{\partial \sigma }>0$ for sufficiently correlated measurement noise. If $a_{H}-a_{L}$ and s are such that there is sufficiently high percentage improvement in the accuracy of inference with respect to higher correlation, captured by $\frac{\partial \Delta _{s}}{\partial \sigma }\frac{\sigma }{\Delta _{s}}>\frac{1}{1-\sigma }$, $\frac{\partial \delta _{CP}}{\partial \sigma }<0$ for some correlation. Given these parameters, $\frac{\partial \delta _{CP}}{\partial \sigma }<0$ and $\frac{\partial \delta _{CI}}{\partial \sigma }<0$ for $\sigma \in (\sigma _{0},\sigma ^{*})$, with $\sigma ^{*}$ satisfying $\frac{\partial \Delta _{s}}{\partial \sigma }\frac{\sigma }{\Delta _{s}}=\frac{1}{1-\sigma }$. There then exists $\delta \in \left( \delta (\sigma ^{*}),\,\min \left\{ \delta (\sigma _{0}),\delta (\sigma _{1})\right\} \right)$ at which $h(\delta ,\Delta _{s})>\varphi (\sigma )$ for $\sigma \in \left( \widehat{\sigma },\overline{\sigma }\right)$ and $h(\delta ,\Delta _{s})<\varphi (\sigma )$ otherwise, where $\left( \widehat{\sigma },\overline{\sigma }\right) \subseteq \left( \sigma _{0},\sigma _{1}\right)$. Examples of such parameters are easily found, Figure 1 as one of which. $\square$

1.1.4 A.4 Proof of Lemma 2

Given a contract that deters collusion with accuracy of inference $\Delta <\max _{s}\Delta$, the agents have incentive to adjust s to raise the accuracy of inference to $\widetilde{\Delta }>\Delta$ such that (EF) is satisfied at $\widetilde{\Delta }$, which allows them to benefit from collusion. A contract is collusion-proof only if (EF) is weakly violated at $\max _{s}\Delta$. To induce coordination, the principal is better off to implement a higher $\Delta$ that satisfies (EF). If the contract $\mathbb {C}$ is such that (EF) holds with equality at $\Delta <\max _{s}\Delta$, the principal has incentive to lower $\beta$, anticipating that the agents will have incentive to adjust s to raise the accuracy of inference to $\widetilde{\Delta }>\Delta$ such that (EF) is satisfied at $\widetilde{\Delta }$. Coalition would be enforced to exert $(a_{H},a_{H})$ at a lower transfer. If it is optimal for her to induce coordination, it is optimal for her to implement the level of trust such that $\Delta$ is maximized. Thus, the optimal coordination-inducing contract is where (EF) is binding at $\max _{s}\Delta$.

Given standard normal distribution of the measurement noise,

$$\begin{aligned} \Delta =\left\{ \begin{array}{cc} \Phi \left( \frac{a_{i}^{0}-a_{i}^{'}}{\sqrt{1-\sigma ^{2}}}-s\right) -\Phi (-s) &{} \;for\,a_{i}^{0}>a_{i}^{'}\\ -\Phi \left( \frac{a_{i}^{0}-a_{i}^{'}}{\sqrt{1-\sigma ^{2}}}+s\right) +\Phi (s) &{} \;for\,a_{i}^{0}<a_{i}^{'} \end{array}\right. , \end{aligned}$$

where the agents collude/coordinate to exert $(a_{i}^{0},a_{j}^{0})$ when the non-cooperative Nash equilibrium is $(a_{i}^{'},a_{j}^{'})$. A contract that deters collusion implements $a_{i}^{'}>a_{i}^{0}$ while a contract that induce coordination implements $a_{i}^{0}>a_{i}^{'}$. In the former case, $\frac{\partial \Delta }{\partial s}\gtreqless 0$ if $\phi (s)\gtreqless \phi \left( s-\frac{a_{H}-a_{L}}{\sqrt{1-\sigma ^{2}}}\right)$, which holds if $s\lesseqgtr \frac{a_{H}-a_{L}}{2\sqrt{1-\sigma ^{2}}}$. In the latter case, $\frac{\partial \Delta }{\partial s}\gtreqless 0$ if $\phi (-s)\gtreqless \phi \left( \frac{a_{H}-a_{L}}{\sqrt{1-\sigma ^{2}}}-s\right)$, which holds if $s\lesseqgtr \frac{a_{H}-a_{L}}{2\sqrt{1-\sigma ^{2}}}$. The optimal threshold of believed unilateral deviation satisfies $s^{*}\sqrt{1-\sigma ^{2}}=\frac{a_{H}-a_{L}}{2}$ , at which the maximum accuracy of mutual inference is $\Delta ^{*}=\Phi \left( \frac{a_{H}-a_{L}}{2\sqrt{1-\sigma ^{2}}}\right) -\Phi \left( -\frac{a_{H}-a_{L}}{2\sqrt{1-\sigma ^{2}}}\right)$. It is then straightforward that the optimal threshold of believed unilateral deviation $s^{*}$ and the maximum accuracy of mutual inference $\Delta ^{*}$ are both increasing in the correlation of measurement noise. $\square$

1.1.5 A.5 Proof of Proposition 3

The enforcement constraint is rearranged as $\Delta ^{*}\ge \frac{1-\delta }{\delta }\left( \frac{\eta -\alpha }{\alpha +\beta -\eta }\right)$ when collusion is deterred or when coordination is induced to implement $(a_{H},a_{H})$ in equilibrium. Rearranging (CP) and (CI) given $\Delta ^{*}$ results in the same constraint. The optimal team incentive is then straightforward from Lemma 2 and the proof of Proposition 1 in A.2, with $\widehat{\beta }_{CP}=\frac{(h(\delta ,\Delta ^{*})-\sigma )\eta }{1+h(\delta ,\Delta ^{*})^{2}-2\sigma h(\delta ,\Delta ^{*})}<0$ when $\delta <\widehat{\delta }_{CP}\equiv \frac{\sigma }{(1-\sigma )\Delta ^{*}+\sigma }$ and $\widehat{\beta }_{CI}=\frac{r(h(\delta ,\Delta ^{*})-\sigma )\eta -(1-h(\delta ,\Delta ^{*}))(a_{H}-a_{L})}{r(1+h(\delta ,\Delta ^{*})^{2}-2\sigma h(\delta ,\Delta ^{*}))}$ when $\delta >\widehat{\delta }_{CI}\equiv \frac{r\sigma \eta +(a_{H}-a_{L})}{r((1-\sigma )\Delta ^{*}+\sigma )\eta +(a_{H}-a_{L})}$, IPR is optimal if otherwise. As $\Delta ^{*}=\max _{s}\Delta \ge \Delta _{s}$ for any $\sigma$ by definition, $h(\delta ,\Delta ^{*})>h(\delta ,\Delta _{s})$, so $\widehat{\beta }_{CP}>\beta _{CP}$ and $\widehat{\beta }_{CI}>\beta {}_{CI}$. In addition, $\widehat{\delta }_{CI}\le \delta _{CI}$ and $\widehat{\delta }_{CP}\le \delta _{CP}$ as $\Delta ^{*}\ge \Delta _{s}$. $\square$

1.1.6 A.6 Proof of Proposition 4

By the proof of Proposition 2 in A.3, $\widehat{\delta }_{CP}$ and $\widehat{\delta }_{CI}$ are increasing (decreasing) in $\sigma$ if $\frac{\partial \Delta ^{*}}{\partial \sigma }\frac{\sigma }{\Delta ^{*}}<(>)\frac{1}{1-\sigma }$ and $\frac{\partial \Delta ^{*}}{\partial \sigma }\frac{\sigma }{\Delta ^{*}}<(>)\frac{1}{\varphi (\sigma )}\frac{\sigma }{1-\sigma }$. Since $\frac{\sigma }{\varphi (\sigma )}<1$, for all $\sigma$such that $\widehat{\delta }_{CI}$ is increasing in $\sigma$, $\widehat{\delta }_{CP}$ is also increasing in $\sigma$. As $\lim _{\sigma \rightarrow 1}\Delta ^{*}=1$ and $\frac{1}{\varphi (\sigma )}\ge 1$ with equality at $\sigma =1$, $\frac{\partial \Delta ^{*}}{\partial \sigma }\frac{1-\sigma }{\Delta ^{*}}<\frac{1}{\varphi (\sigma )}$ and $\widehat{\delta }_{CI}$ is increasing for sufficiently high correlations. As $\lim _{\sigma \rightarrow 0}\left( \frac{\partial \Delta ^{*}}{\partial \sigma }\frac{1-\sigma }{\Delta ^{*}}\right) =0<\frac{1}{\varphi (0)}$, $\widehat{\delta }_{CI}$ is increasing for sufficiently low correlations as well.

For intermediate level of correlations, due to the special form of normal CDF, I proceed by fixing the exogenous threshold s and $\sigma$ at the levels such that $\delta _{CP}$ is decreasing in $\sigma$ and compare $\Delta _{\sigma }^{*}\equiv \frac{\partial \Delta ^{*}}{\partial \sigma }=\phi \left( \frac{a_{H}-a_{L}}{2\sqrt{1-\sigma ^{2}}}\right) \frac{\sigma (a_{H}-a_{L})}{2\sqrt{(1-\sigma ^{2})^{3}}}$ with $\Delta _{\sigma }^{s}\equiv \frac{\partial \Delta _{s}}{\partial \sigma }=\phi \left( \frac{a_{H}-a_{L}}{\sqrt{1-\sigma ^{2}}}-s\right) \frac{\sigma (a_{H}-a_{L})}{\sqrt{(1-\sigma ^{2})^{3}}}$. The ratio $\frac{\Delta _{\sigma }^{s}}{\Delta _{\sigma }^{*}}\gtreqless 1$ if $\frac{\phi \left( 2x-s\right) }{\phi \left( x\right) }\gtreqless \frac{1}{2}$ where $x\equiv \frac{a_{H}-a_{L}}{2\sqrt{1-\sigma ^{2}}}$. Define y such that $\phi (y)\equiv \frac{1}{2}\cdot \phi \left( \frac{a_{H}-a_{L}}{2\sqrt{1-\sigma ^{2}}}\right)$. For $(a_{H}-a_{L})$ such that $2x-s\in (-y,y)$, $\frac{\Delta _{\sigma }^{s}}{\Delta _{\sigma }^{*}}>1$, so that whenever $\delta _{CI}$ and $\delta _{CP}$ are increasing, $\widehat{\delta }_{CI}$ and $\widehat{\delta }_{CP}$ are increasing as well, and when $\delta _{CI}$ and $\delta _{CP}$ are decreasing, it is not necessary that $\widehat{\delta }_{CI}$ and $\widehat{\delta }_{CP}$ are decreasing as well. This condition rules out extreme cases in which $a_{H}-a_{L}$ is so large or so small that how inference is made becomes trivial. $\square$