Delegation and Information Disclosure with Unforeseen Contingencies

A.1 Proof of Proposition 1

Following Alonso and Matouschek (2008), when b < ( θ ̄ − θ ̲ ) / 2 the delegation is valuable and the (minimal) optimal delegation set is [ θ ̲ + 2 b , θ ̄ + b ] ; when b > ( θ ̄ − θ ̲ ) / 2 , the investor chooses her optimal project y * = θ ̄ + θ ̲ 2 + b in the (minimal) optimal delegation set. By Assumption A1, the delegation can be valuable if the investor is aware of a relatively large set of states.

As we focus on the maximal delegation set, in equilibrium the investor delegates all projects above the threshold y H = θ ̲ + min { 2 b , θ ̄ − θ ̲ 2 + b } . Moreover, she also delegates those projects below the threshold y _L satisfying

The reason is that she believes these lower states will never be implemented by the expert. The lower cutoff is y L = θ ̲ − min { 2 b , θ ̄ − θ ̲ 2 + b } . To sum up, the investor’s optimal delegation choice is:

where Δ = min 2 b , θ ̄ − θ ̲ 2 + b .

A.2 Proof of Lemma 2

By Proposition 1, the delegation set is characterized by the gap Δ = min { 2 b , θ ̄ − θ ̲ 2 + b } . Therefore, the expert has incentives to make the investor aware of one singleton to minimize the measure of the undelegated projects; that is, θ ̲ = θ ̄ .

The expert’s expected utility depends on the interception of the set of undelegated options and his preferred options, 0,1 . Suppose that the expert’s revelation choice is θ′ = 0. Then the expert’s expected utility would be

Suppose that the revelation choice is θ″ ∈ (0, 1). Then the intersection would be (θ″ − b, θ″ + b) ∩ [0, 1]. When b ≤ θ″ ≤ 1 − b, the expert’s expected utility would be

When θ ″ < b , the expert’s expected utility would be

The same case holds for θ″ > 1 − b. Therefore, the expert’s optimal revelation choice would be Θ ̂ = { 0 } or symmetrically Θ ̂ = { 1 } .

A.3 Proof of Proposition 3

If θ ₂ − θ ₁ ≥ 2b, the gap Δ * = min 2 b , θ ̄ − θ ̲ 2 + b = 2 b as θ ̄ − θ ̲ 2 ≥ θ 2 − θ 1 2 = b . Let the new awareness set be [ θ ̲ , θ ̄ ] . The set of undelegated options is ( θ ̲ − 2 b , θ ̲ + 2 b ) .

First, the expert would choose θ ̄ = 1 as we focus on the maximal awareness set. Second, a lower θ ̲ benefits the expert as more of his preferred options are delegated. The optimal choice is hence θ ̲ = 0 .

A.4 Proof of Proposition 4

Let the awareness set before revelation be [θ ₁, θ ₂]. The expert expands the awareness set to Θ ̂ = [ θ ̲ , θ ̄ ] with θ ̲ ≤ θ 1 ≤ θ 2 ≤ θ ̄ . Denote the expert’s choice by [ θ ̲ * , θ ̄ * ] .

By Proposition 3, if in the solution θ ̄ * − θ ̲ * ≥ 2 b then the expert must choose full revelation: [ θ ̲ * , θ ̄ * ] = [ 0,1 ] . Then we focus on θ ̄ − θ ̲ < 2 b . In this case, the set of undelegated options take the form of ( θ ̲ − Δ * , θ ̲ + Δ * ) where Δ * = θ ̄ − θ ̲ 2 + b . Clearly, the set of undelegated options expands when θ ̄ increases. Therefore, in the solution θ ̄ * = θ 2 must hold when the expert does not choose full revelation.

On the other hand, decreasing θ ̲ by one unit, the agent decreases the lower bound and upper bound of the set ( θ ̲ − Δ * , θ ̲ + Δ * ) by 3/2 and 1/2 units respectively.^[8] As a result, lowering θ ̲ strictly benefits the expert when θ ̲ is close to 0, and he would choose θ ̲ * = 0 in that situation. When revealing more awareness is beneficial to the expert, he would choose Θ ̂ = [ 0 , θ 2 ] .

We identify the conditions under which the expert is willing to reveal extra awareness by comparing his expected utility in the two cases: [ θ ̲ , θ ̄ ] = [ 0 , θ 2 ] and [ θ ̲ , θ ̄ ] = [ θ 1 , θ 2 ] . Denote by ℓ the length of ( θ ̲ − Δ * , θ ̲ + Δ * ) ∩ [ 0,1 ] . Due to the symmetrical form of the utility function, the expert is better off if and only if ℓ is smaller.

1. Suppose θ 1 > θ 2 − θ 1 2 + b . Without awareness revelation, ℓ = θ ₂ − θ ₁ + 2b. With partial revelation, ℓ = θ ₂/2 + b. The expert would reveal nothing if and only if θ 2 / 2 + b > θ 2 − θ 1 + 2 b ⇔ θ 1 + θ 2 / 2 < 3 b .

2. Suppose θ 1 < θ 2 − θ 1 2 + b . Without awareness revelation, ℓ = θ 1 + θ 2 − θ 1 2 + b > θ 2 2 + b . The expert is always willing to expand awareness in this case.

Last, note that the expert would choose full revelation ( Θ ̂ = [ 0,1 ] ) instead of partial revelation ( Θ ̂ = [ 0 , θ 2 ] ) if θ ₂ ≥ 2b. To sum up, if θ 2 − θ 1 < 2 b , the expert’s optimal revelation strategy is:

A.5 Proof of Proposition 5

In this case, the investor’s delegation choice, given her interim awareness set [ θ ̲ , θ ̄ ] , is

where y * = θ ̲ + θ ̄ 2 + b is the investor’s most preferred option. It follows that if b < θ 2 − θ 1 2 , the expert will reveal all states to induce the largest delegation set. Next we focus on the case of interest when b > θ 2 − θ 1 2 .

When b > θ 2 − θ 1 2 , the expert might choose full, partial, or no revelation. Moreover, when the expert’s optimal choice is not full revelation, the delegation set must contain only one point. Otherwise, the investor would find the delegation valuable and the expert should have chosen full revelation as argued above. Therefore, the delegation set can take two forms:

1. The delegation set with partial or no revelation is {y*}, and the expert’s expected utility is

   U NR = ∫ 0 1 − 1 2 ( y * − θ ) 2 d θ = − 1 6 ( 1 − 3 y * + 3 ( y * ) 2 ) .

2. The delegation set with full revelation is [2b, 1], and the expert’s expected utility is

   U FR = ∫ 0 2 b − 1 2 ( 2 b − θ ) 2 d θ = − 4 3 b 3 .

Note that U NR ≤ − 1 24 with the equality at y* = 1/2. Therefore, the inequality U _FR > U _NR always holds as long as b < 1 32 3 ≈ 1 3.17 . So the expert always chooses full revelation with a relatively small b. When b > 1 32 3 , the expert has incentives to reveal lower states if y* > 1/2 while has incentives to reveal higher states if y * < 1 / 2 . Two observations are followed. First, when y* = 1/2, the expert’s optimal choice is no revelation. Second, when y* ≠ 1/2 and b is relatively large, the expert will reveal partially to make the induced action more close to 1/2.^[9] The revealed states can be high or low, depending on the sign of y* minus 1/2.