Information suppression in multi-agent contracting

Abstract

This paper examines the principal’s preferences over reporting systems in multi-agent settings. In multi-agent settings, the principal’s contract offer depends on the credibility of the agents’ commitment not to collude on the terms of the contract. If the agents can credibly rule out collusion, then the principal prefers that the agents observe all performance measures and she prefers an accounting system that releases detailed instead of aggregated information. To the contrary, when the principal cannot preclude the agents from writing side contracts, it can be efficient to suppress certain information signals for contracting purposes. Specifically, restricting the information dissemination within organizations or releasing aggregated instead of detailed information are two efficient avenues to suppress information.

Appendix A

1.1 Proof of propositions

Lemma 1

Substituting (7b) and (11b) into Π ^†(η ^g) − Π ^‡(η ^g) and simplifying, yields

$$ \Uppi^{\dag } (\eta^{g} ) - \Uppi^{\ddag } (\eta^{g} ) { } = \frac{{m^{2} r[m^{2} \rho + r(1 - \rho^{2} )]^{2} }}{{[m^{2} + r(1 - \rho^{2} )][m^{2} + 2r(1 - \rho )][(m^{2} + r)(m^{2} + r(1 - \rho^{2} )) - m^{2} r(1 - \rho^{2} )]}} \ge \, 0. $$

Differentiating (11b) with respect to ρ and setting ρ = 0 yields

$$ \partial \Uppi^{\ddag } (\eta^{g} )/\partial \rho |_{\rho = 0} = - \,\,\,\,\frac{{2m^{2} r^{2} (m^{2} + r)}}{{[m^{2} + 2r]^{2} [(m^{2} + r)^{2} - m^{2} r]}} < \, 0. $$

Proposition 1

Substituting (11b) and (13b) into Π ^‡(η ^g) > Π ^‡(η ^s), rearranging, and simplifying yields

$$ r^{ 2} ( 1- \rho ) { [} - r( 1- \rho )( 1 { } + { 2}\rho ) - 2m^{ 2} \rho ] { } > \, 0. $$

(A.1)

The left hand side of (A.1) is equal to zero for ρ ∈ {1, ρ _gs , (2 m ² + r)/(4r) + [½ + (2 m ² + r)²/(4r)²]^½}, with ρ _gs  = (2 m ² + r)/(4r) − [½ + (2 m ² + r)²/(4r)²]^½, where the latter two elements are the roots to −r (1 − ρ)(1 + 2ρ) − 2 m ² ρ = 0. The upper root is larger than 1. Also, Π ^‡(η ^g) = Π ^‡(η ^s) if ρ = 1, ∂(Π ^‡(η ^g) − Π ^‡(η ^s))/∂ρ| _ρ=1 > 0, and $\partial({\varPi ^{\ddagger}}(\eta^{g}) - {\varPi ^{\ddagger}} (\eta^{s}))/\partial\rho\vert_{\rho = \rho_{gs}} < 0.$ Since ρ ∈ [−1,1], strict preference for η ^g requires ρ < ρ _gs . Inspection of ρ _gs reveals that ρ _gs ∈ (−½,0). The preference reverses for ρ > ρ _gs .

Lemma 2

Substituting (17a) and (17c) into v ^‡₁₁  − v ^‡₂₂ and simplifying, yields

$$ v_{ 1 1}^{\ddag } - v_{ 2 2}^{\ddag } = D^{ - 1} mr^{ 2} [\rho^{ 2} ( 2m^{ 2} + r ) { } + r ] { } > \, 0 $$

since D > 0.

Using (17b), v ^‡₁₂  < 0 if

$$ \rho [(m^{ 2} + r)^{ 2} + { (1} + \rho )r^{ 2} ] \, > { (}m^{ 2} + r )^{ 2} . $$

(A.2)

Observe that, with ρ ∈ [−1,1], the expression in brackets of the left hand side in (A.2) is positive. Thus, (A.2) holds only if ρ > 0. For ρ > 0, the left hand side of (A.2) increases monotonically in ρ, and the condition is surely satisfied for ρ = 1. Hence, there exists a single ρ ∈ (0,1) such that (A.2) is true for ρ > ρ.

Proposition 2

Substituting (13b) and (17d) into Π ^‡(η ^p) > Π ^‡(η ^s) and simplifying, yields

$$ ((m^{ 2} + r)( 2m^{ 2} + r) \, + { 2}r^{ 2} )\rho^{ 2} - 2m^{ 2} (m^{ 2} + r)\rho > r(m^{ 2} + r). $$

(A.3)

Note that for (A.3) to hold true, the absolute value of ρ must exceed a certain threshold, which is given by ρ ^t _sp , t = l,u, where ρ ^t _sp are the roots that solve (A.3) as an equality. In particular, (A.3) does not hold if ρ = 0. Given uncorrelated reports, the principal strictly prefers single-agent reporting to partial information dissemination. Inspection of ρ ^t _sp , t = l,u reveals that ρ ^l _sp ∈ (−1,0) and ρ ^u _sp ∈ (0,1).

Corollary 1

Using Propositions 1 and 2, the principal strictly prefers partial information dissemination for ρ ∈ (ρ ^u _sp ,1], and she strictly prefers single-agent reporting for ρ ∈ (max{ρ _gs ,ρ ^l _sp },ρ ^u _sp ). Comparing ρ ^l _sp with ρ _gs reveals ρ ^l _sp  < ρ _gs . Finally, $\Pi ^{\ddagger}(\eta^{g})\vert_{ \rho = \rho_{gs}}> \Pi^{\ddagger}(\eta^{p})\vert_{ \rho = \rho_{gs}}\,and\,\partial\varPi ^{\ddagger}(\eta^{g}\vert \rho)/\partial\rho > \partial\varPi^{\ddagger}(\eta^{p}\vert \rho)/\partial\rho$ for ρ < 0. Hence, for ρ < ρ _gs the principal has strict preferences for global reporting.

Proposition 3

Substituting (11b) and (18b) into Π ^‡(η ^a) > Π ^‡(η ^g) and simplifying, yields

$$ 2\rho (r \, \rho - m^{ 2} ) > m^{ 2} + 2r. $$

(A.4)

(A.4) is satisfied as an equality for

$$ \rho \equiv \frac{1}{2r}\left( {m^{2} \pm \sqrt {(m^{2} + 2r)^{2} - 2m^{2} r} } \right). $$

(A.5)

Inspection of the two roots reveals that the upper root exceeds 1. While (A.4) does not hold for ρ ∈ [0,1], for ρ = −1, (A.4) simplifies to

$$ 2 { (}m^{ 2} + r ) { } > m^{ 2} + { 2}r, $$

which is always true. Hence, Π ^‡(η ^a) > Π ^‡(η ^g) if ρ < ρ _ag , where ρ _ag is the lower root in (A.5). Inspection of ρ _ag reveals that ρ _ag ∈ (−1, − ½). Finally, the preference reverses for ρ > ρ _ag .

Corollary 2

Extending Corollary 1, it is straightforward to show that ρ _ag  < ρ _gs . Substituting the two closed-form solutions for ρ _ag and ρ _gs and simplifying completes the proof.