4.3 The Optimal Mechanism
4.3.1 Results
To prove our results, we require the following condition on the seller’s belief over the agent’s type. This condition is standard in the literature and satisfied by a large class of distributions.
Definition 4.3 We say that belief g satisfies Condition A iff, for all v,
3g(v) +v.∇g(v)≥0. (4.4)
Before we state our main result, we partition the type-space into three parts, V1 ={v ∈V :v1 = 1},
V2 ={v ∈V :v2 = 1, v1 <1}, V0 =V /{V1∪V2}.
Theorem 4.1 If g satisfies Condition A, then there is an optimal mechanism (f,p)¯ which is deterministic. Moreover, (f,p)¯ satisfies the following properties,
1. For every v ∈V1, there exists some κv such that, f(v, x) =
( 1 if x < κv, 0 if x≥κv. 2. For every v ∈V2, there exists some γv such that,
f(v, x) =
( 0 if x≤γv, 1 if x > γv. 3. For every v ∈V0, either
(f(v, x),p(v)) = (0,¯ 0) ∀x, or
there exists some u∈V1∪V2 such that (f(v, x),p(v)) = (f¯ (u, x),p(u))¯ ∀x.
The theorem shows that there exists an optimal mechanism, which isdeterministic. The first point states that for every type in V1 (the right line in Figure 4.1), if the attribute level is below a certain threshold (that corresponds to the type), the object is allocated with probability one; otherwise, the seller retains the object. In contrast, for every type in V2 (the top line in Figure 4.1), the object is given with probability one if the realized level of the attribute isabove a certain threshold that corresponds to the type. Otherwise, the seller retains the object. The third point states that every other type not included in the above two cases (all the interior points, left and bottom boundaries in Figure 4.1) is mapped to the outcome (0,0) or to one of the outcomes that are mapped to types in V1 orV2.
For an insight into the optimal mechanism, note that the types in V1 and V2 differ in terms of the inherent value of the object relative to that of the attribute. While types in V1 have a higher inherent value relative to the attribute, the opposite is true for the types in V2. The mechanism assigns the good to types inV2 for a higher realization of the attribute.
Whereas the types in V1 get the good for lower attribute realizations, thereby incentivizing from misreporting. More on this in Remark4.1 after the next Proposition.
Proof sketch.8 To solve for the optimal mechanism, we fix an IC and IR mechanism (f,p) and construct another indirect mechanism that is an improvement over this in terms¯
8The proof is in Appendix4.5.2.
0 1 1
0
v1
v2 1,x≤κv1 V:f(v,x)= 0,x>κv
0, x < γv
1, x≥γv
V2:f(v, x) ={
Figure 4.1: Optimal mechanism
of revenue. Given Condition A, Lemma4.2implies weakly increasing the revenue by keeping the utility of types in V1 and V2 constant while weakly reducing the utilities of types in the rest of the type-space. This exercise is done by manipulating the mechanism (f,p), first¯ by deleting the outcomes mapped to types in V0, and second, by adjusting the outcomes mapped toV1andV2 such that their utilities remain the same. Third, we add a null message to the mechanism and map it to the outcome (0,0), irrespective of the attribute realization.
Fourth we make sure that the utility of any type inV0 is weakly lower in the new mechanism by reporting as any type in V1 ∪V2 or the null message. Notice that the new mechanism is IR since the outcome (0,0) is present. Since we started with an arbitrary mechanism and reached a mechanism with the desired structure, we claim that the theorem is true. Note that this exercise does not imply that every optimal mechanism takes this form; we only claim that there exists at least one such optimal mechanism.
In the theorem, we did not discuss the payment function or the nature of the thresholds defining the mechanism. The following proposition states these two properties of the optimal mechanism.
Proposition 4.1 For the optimal mechanism (f,p)¯ defined in Theorem 4.1, the following statements are true.
1. For all v, v0 ∈V1 with v2 < v20, κv ≤κv0,
¯
p(v) = ¯p(1,0) + Z κv
κ(1,0)
ψ(x)dx+v2 Z κv
0
xψ(x)dx− Z v2
0
Z κ(1,t)
0
xψ(x)dx dt.
2. For all v, v0 ∈V2 with v1 < v10, γv ≥γv0,
¯
p(v) = ¯p(0,1) +
Z γ(0,1) γv
xψ(x)dx+v1 Z 1
γv
ψ(x)dx− Z v1
0
Z 1
γ(t,1)
ψ(x)dx dt.
The Proposition states for types in V1, V2, there is monotonicity of the thresholds de- scribed in Theorem4.1. The payment of any type inV1, V2 is pinned down by the thresholds, and payment by the respective lowest type - (1,0) forV1, (0,1) forV2. The result is similar to Myerson (1981)’s revenue equivalence result applied over V1 and V2 separately. This characterization of the payments will help determine the optimal thresholds, which will lead to a full description of the mechanism. The proof of the Proposition is in Appendix4.5.2.
Remark 4.1 The proposition shows that the utilities (and contingent allocations) are mono- tonic for types in V1 or V2. To see why, observe that for any v, v0 ∈ V1 with v2 < v20 we have κv < κv0. From the theorem and the equation 4.2, the utility of the type v = (1, v2) is
equals Z κv
0
ψ(x)dx+v2 Z κv
0
xψ(x)dx.
This is clearly less than the utility of the type v0 = (1, v20) which equals to Z κv0
0
ψ(x)dx+v02 Z κv0
0
xψ(x)dx,
since κv < κv0 and v2 < v02. A similar argument can be given for types inV2, that the utility of types in increasing as the v1 component is increasing, using the theorem and the proposi- tion together. This monotonicity of utilities with respect to the agent’s private information is due to incentive constraints and is true in other multidimensional models (for example see Rochet (1987)).
However, in the theorem we state that for types in V1, the allocation function f(v, x) is decreasing with respect to attribute realization x, which is public information. This is due to separation of types in V1 from V2 in the optimal mechanism. In the theorem, for the types
in V2, the allocation function is an increasing function with respect to attribute realization.
The example discussed in the Introduction section illustrates this, note that both mechanisms in Table 4.1 and 4.2 are incentive compatible. For the type w = (3,0) the allocation is decreasing in x, whereas for the type u = (1,10) the allocation is (weakly) increasing in x.
This is due to difference in relative weights attached inherent value and the attribute value across these types.