Credibly Confidential Contracts

A.1: Proof of Proposition 1

Disutility of identity exposure does not depend on the content of the contract, so the contract proposed to agent i with probability p _i , C i * = t 1 i * , t 0 i * implementing e i * , solves the reduced problem

subject to

and

For any t_1i > t_0i, the incentive compatibility constraint (IC _i ) can be replaced by the local incentive compatibility q′(e _i |v _i ) ⋅ (u(t_1i) − u(t_0i)) − c = 0. Incentive compatibility only depends on the difference in transfer. The individual rationality constraint (IR _i ) is binding. For any pair of transfer (t_0i, t_1i) such that (IR _i ) is slacking, there is an alternative pair t 0 i ′ , t 1 i ′ < ( t 0 i , t 1 i ) that generates u t 0 i ′ = u ( t 0 i ) − ε and u t 1 i ′ = u ( t 1 i ) − ε such that q ( e i | v i ) ⋅ ( u ( t 1 i ) − ε ) + ( 1 − q ( e i | v i ) ) ⋅ ( u ( t 0 i ) − ε ) − c ⋅ e i − η i ⋅ ( p i − α i ) 2 ≥ 0 and that local incentive compatibility is unaffected. The principal clearly prefers t 0 i ′ , t 1 i ′ to (t_0i, t_1i). By binding (IR _i ) and local incentive compatibility, the optimal transfer schedule t 1 i * , t 0 i * to implement effort e i * has

With q e i * | v i > 0 and q ′ e i * | v i > 0 for any e i * > 0 , the optimal transfer schedule satisfies t 1 i * > t 0 i * .

◼ Risk-Neutral Agent with u(t) = t.

Binding (IR _i ) implies that q e i * | v i ⋅ t 1 i * + 1 − q e i * | v i ⋅ t 0 i * − c ⋅ e i * = η i ⋅ ( p i − α i ) 2 > 0 . The excessive transfer beyond the cost of effort is to motivate agent i to participate when he is averse to identity exposure, defined as the confidentiality premium m * = q e i * | v i ⋅ t 1 i * + 1 − q e i * | v i ⋅ t 0 i * − c ⋅ e i * = η i ⋅ ( p i − α i ) 2 .

With binding individual rationality, the effort e i * implemented by t 1 i * , t 0 i * solves

The expected transfer is a sum of the total cost of effort and confidentiality premium, with the latter being independent of effort choice. The contract thus implements the efficient level of effort e i * = e ̄ i ∈ arg max e q ( e | v i ) − c ⋅ e .

◼ Risk-Averse Agent with u′′(t) < 0.

Denote u 0 i * ≡ u t 0 i * and u 1 i * ≡ u t 1 i * . With binding individual rationality, the effort e i * implemented by t 1 i * , t 0 i * solves

The first-order optimality condition has q ′ ( e | v i ) = q ′ ( e | v i ) ⋅ u − 1 u 1 i * − u − 1 u 0 i * + q ( e | v i ) ⋅ 1 u ′ t 1 i * ⋅ ∂ u 1 i * ∂ e + ( 1 − q ( e | v i ) ) ⋅ 1 u ′ t 0 i * ⋅ ∂ u 0 i * ∂ e , which can be rearranged to q ′ ( e | v i ) = q ′ ( e | v i ) ⋅ u − 1 u 1 i * − u − 1 u 0 i * + 1 u ′ t 1 i * − 1 u ′ t 0 i * ⋅ − q ( e | v i ) ⋅ ( 1 − q ( e | v i ) ) ⋅ q ′ ′ ( e | v i ) ⋅ c ( q ′ ( e | v i ) ) 2 . At η _i  = 0, the solution corresponds to the second-best effort in a standard contracting problem with a risk-averse agent. The difference in transfers u − 1 u 1 i * − u − 1 u 0 i * is increasing in the disutility of identity exposure as ∂ u − 1 u 1 i * − u − 1 u 0 i * ∂ d ( μ i | η i , α i ) = 1 u ′ t 1 i * − 1 u ′ t 0 i * > 0 , given a strictly concave utility of transfer and t 1 i * > t 0 i * . The difference in inverse marginal utilities 1 u ′ t 1 i * − 1 u ′ t 0 i * are also increasing in the disutility of identity exposure if ∂ u ′ t 1 i * − 1 − u ′ t 0 i * − 1 ∂ d ( μ i | η i , α i ) = − u ′ ′ t 1 i * u ′ t 1 i * 3 + u ′ ′ t 0 i * u ′ t 0 i * 3 > 0 , which holds if the absolute risk aversion is not sharply decreasing in transfer. Therefore, the right-hand-side of the first-order condition is increasing in d(μ _i |η _i , α _i ). The marginal contractual cost of effort for the principal is increasing in the disutility of identity exposure. The optimal effort to implement is further distorted downwards from the second-best.

A.2: Proof of Lemma 1, Proposition 2, and Corollary 1

Anticipating the optimal contracts proposed with probability p = (p₁, p₂, …, p _n ) in Proposition 1, the principal contracts with agent i with probability p i * that

subject to ∑ _i p _i ≤ 1. Denote λ as the Lagrange multiplier to this constraint and λ^* as its equilibrium measure.

By the first-order optimality conditions, q e i * | v i − c ⋅ e i * − η i ⋅ ( p i − α i ) 2 − p i ⋅ η i ⋅ 2 ⋅ ( p i − α i ) − η 0 ⋅ 2 ⋅ ( p i − α 0 ) − λ * ≤ 0 , p i ⋅ q e i * | v i − c ⋅ e i * − η i ⋅ ( p i − α i ) 2 − p i ⋅ η i ⋅ 2 ⋅ ( p i − α i ) − η 0 ⋅ 2 ⋅ ( p i − α 0 ) − λ * = 0 , and (1 − ∑ _i p _i ) ⋅ λ^* = 0. By the second order conditions, −η _i ⋅ (3 ⋅ p _i  − 2 ⋅ α _i ) − η₀ < 0 if η₀ > 2 ⋅ η _i ⋅ α _i for all p _i or p i > 2 3 α i − 1 3 ⋅ η 0 η i . For any agent i to whom the principal proposes a contract with a positive probability p i * > 0 , it satisfies q e i * | v i − c ⋅ e i * − η i ⋅ ( p i * − α i ) 2 − p i * ⋅ η i ⋅ 2 ⋅ ( p i * − α i ) − η 0 ⋅ 2 ⋅ ( p i * − α 0 ) = λ * . It is optimal to propose a contract to agent −i with probability p − i * = 0 if q e − i * | v − i − c ⋅ e − i * − η − i ⋅ α − i 2 + η 0 ⋅ 2 ⋅ α 0 < λ * .

Denote the surplus in the contractual relationship with agent i as s ( v i ) ≡ q e i * | v i − c ⋅ e i * . Suppose that among all agents to whom the principal proposes a contract with a positive probability, agent h has the highest matching value. Any other agent l ≠ h with l > 0 has v _l  < v _h . In equilibrium, p h * , p l * > ( 0,0 ) satisfies s ( v h ) − η h ⋅ ( p h * − α h ) 2 − p h * ⋅ η h ⋅ 2 ⋅ ( p h * − α h ) − η 0 ⋅ 2 ⋅ ( p h * − α 0 ) = s ( v l ) − η l ⋅ ( p l * − α l ) 2 − p l * ⋅ η l ⋅ 2 ⋅ ( p l * − α l ) − η 0 ⋅ 2 ⋅ ( p l * − α 0 ) from the optimality conditions in Lemma 1, which can be rearranged to s ( v h ) − s ( v l ) − η 0 ⋅ 2 ⋅ ( p h * − p l * ) = η h ⋅ ( p h * − α h ) ⋅ ( 3 ⋅ p h * − α h ) − η l ⋅ ( p l * − α l ) ⋅ ( 3 ⋅ p l * − α l ) . The left-hand side of this equation is the difference in surplus and the difference in the principal’s disutility of identity exposure, while the right-hand side is the difference in the marginal confidentiality premium.

To see conditions under which the equilibrium has p l * > p h * , evaluate each side of the equation at p l = p h * . It is more likely for the principal to contract with agent l who has a lower matching value if and only if s ( v h ) − s ( v l ) < η h ⋅ ( p h * − α h ) ⋅ ( 3 ⋅ p h * − α h ) − η l ⋅ ( p h * − α l ) ⋅ ( 3 ⋅ p h * − α l ) . The right-hand side of this inequality is the difference in marginal expected confidentiality premium if the principal proposes a contract to agent h and agent l with the same probability p h * , denoted as Δ α h , α l , p h * ≡ η h ⋅ ( p h * − α h ) ⋅ ( 3 ⋅ p h * − α h ) − η l ⋅ ( p h * − α l ) ⋅ ( 3 ⋅ p h * − α l ) . If s ( v h ) − s ( v l ) < Δ α h , α l , p h * , q e l * | v l − c ⋅ e l * − η l ⋅ ( p h * − α l ) 2 − p h * ⋅ η l ⋅ 2 ⋅ ( p h * − α l ) − η 0 ⋅ 2 ⋅ ( p h * − α 0 ) > λ * , the principal has incentive to raise p l * above p h * . If p l * > p h * , by the optimality conditions, q e l * | v l − c ⋅ e l * − η l ⋅ ( p h * − α l ) 2 − p h * ⋅ η l ⋅ 2 ⋅ ( p h * − α l ) − η 0 ⋅ 2 ⋅ ( p h * − α 0 ) > λ * = q e h * | v h − c ⋅ e h * − η h ⋅ ( p h * − α h ) 2 − p h * ⋅ η h ⋅ 2 ⋅ ( p h * − α h ) − η 0 ⋅ 2 ⋅ ( p h * − α 0 ) , which can be rearranged to s ( v h ) − s ( v l ) < Δ α h , α l , p h * .

Given s(v _h ) > s(v _l ), s ( v h ) − s ( v l ) < Δ α h , α l , p h * only if Δ α h , α l , p h * > 0 . Several cases are possible for Δ α h , α l , p h * > 0 : (i) p h * > max { α h , α l } or p h * < min { α h 3 , α l 3 } , and η _l < η _h , (ii) p h * ∈ max { α h 3 , α l 3 } , min { α h , α l } , and η _l > η _h , (iii) α _h > α _l and p h * ∈ α l 3 , min { α h 3 , α l } , or α _h < α _l and p h * ∈ max { α l 3 , α h } , α l . To conclude, it is more likely for the principal to contract with agent l, who has a lower matching value, if only if the surplus loss from contracting with agent l is lower than the reduced contracting cost in the form of marginal confidentiality premium, i.e. s ( v h ) − s ( v l ) < Δ α h , α l , p h * , which occurs only if one of the above three scenarios hold.

In the special scenario where only the principal cares about identity exposure, η₀ > 0 = η _i for all agent i, optimality conditions in Lemma 1 implies s ( v h ) − s ( v l ) − η 0 ⋅ 2 ⋅ ( p h * − p l * ) = 0 . Therefore, p h * > p l * for any v _h  > v _l .

A.3: Example

Let q ( e | v i ) = v i ⋅ e , where e ∈ [0, 1] and v _i  ∈ (0, 1]. The efficient level of effort is e i * ∈ arg max e v i ⋅ e − c ⋅ e = v i 2 ⋅ c 2 . The maximum surplus in the contractual relationship with agent i is thus s ( v i ) = v i 4 ⋅ c . By Proposition 1, the efficient level of effort is implemented within a contractual relationship. We focus on the equilibrium distribution of contractual relationships.

If the agents prefer being completely off the radar, i.e. α = 0, and the principal is not averse to identity exposure, i.e. η₀ = 0, only case (i) in Proposition 2 applies, and the optimality conditions in Lemma 1 have

Plugging the second equation into the first, we find

If η _h  < η _l , the optimality conditions hold only at p h * > 1 2 > p l * . Inefficient organization occurs only if η _h  > η _l . Both equality hold at p h = p l = 1 2 if and only if v h − v l 12 ⋅ c = η h − η l 4 . If v h − v l 12 ⋅ c > η h − η l 4 > 0 , the optimality conditions hold at p h * > 1 2 > p l * . If 0 < v h − v l 12 ⋅ c < η h − η l 4 , the optimality conditions hold at p h * < 1 2 < p l * . The equilibrium distribution of contractual relationships is inefficient if and only if v h − v l 3 ⋅ c < η h − η l .

A.4: Risk-Neutrality with Limited Liability

Suppose that the agents are risk-neutral with limited liability in the sense that the ex-post payoff from the contract (excluding the disutility of identity exposure) must not be negative. The contract proposed to agent i with probability p _i , C i l l = t 1 i l l , t 0 i l l implementing e i l l , solves the reduced problem

subject to

and

With η _i  = 0, the problem corresponds to the standard contracting model with limited liability. (LL₀) is binding and (IR _i ) is strictly satisfied. Binding limited liability and local incentive compatibility jointly imply t 0 i l l = c ⋅ e i l l and t 1 i l l − t 0 i l l = c q ′ e i l l | v i . The principal implements e i l l ∈ arg max e q ( e | v i ) − c ⋅ e − q ( e | v i ) ⋅ c q ′ e i l l | v i .

With a sufficiently low disutility of identity exposure η i ≤ η ̲ = def q e i l l | v i q ′ e i l l | v i ⋅ c ( p i − α i ) 2 , the above contract does not violate (IR _i ). A minor confidentiality concern does not affect the contract and the induced effort. With η i > η ̲ , the above contract violates (IR _i ), so the optimal contract satisfies binding (IR _i ). With a sufficiently high disutility of identity exposure η i ≥ η ¯ = def q e i * | v i q ′ e i * | v i ⋅ c ( p i − α i ) 2 , where e i * is the solution to Proposition 1, limited liability constraints are strictly satisfied. The optimal contract is the same as that in Section 3 with risk neutrality. The productive effort is restored to the efficient level. As e i * > e i l l and q(e _i |v _i ) being increasing and concave, it is straightforward that η ̄ > η ̲ . For intermediate disutility of identity exposure, the optimal contract and the implemented effort solve binding (IR _i ), local incentive compatibility, and (LL₀).