Implementability by a Canonical Indirect Mechanism of an Optimal Two-Dimensional Direct Mechanism

The present paper investigates the multi-dimensional mechanism design in which buyers have taste and budget as their private information. The paper shows an easy proof of a two-dimensional optimal direct mechanism by a one-dimensional indirect mechanism: A canonical mechanism in the traditional one-dimensional setting, i.e., function of one variable, the buyer’s taste. It also sheds light on where the difficulty lies implementability of a general direct mechanism—not optimal—by a canonical mechanism.


Introduction
In standard literature on adverse selection, an agent-buyer has his taste as private information.If the agent has, in addition, a budget constraint as his private information, the problem becomes one of two-dimensional mechanism design.It is well known that multi-dimensional mechanism design entails quite a few technical difficulties (see [1], [2]).One way to get around them is to reduce the dimension of private information [3] and thus to resort to an indirect mechanism.
In the setting of budget-constrained buyers, [4] invoked a non-linear price scheme while showing that this indirect mechanism realizes the equilibrium outcome of the optimal two-dimensional direct mechanism.[5], by only focusing on the agent's taste, reduced the whole setting to the well known one of a canonical one-dimensional mechanism, i.e., a map from the taste space to the quality-price space and showed that the optimal canonical mechanism achieves the same outcome as an optimal direct mechanism. 1The proof of equivalence, however, involved fancy theory of set-valued mappings not necessarily familiar to an audience in economics.
The above two works differ in perspective and motivation so that they are based on different assumptions as regards the distribution of agents' type.By adopting the assumptions of [4], this paper provides an elementary proof that their optimal non-linear price scheme is implemented by a canonical one-dimensional mechanism a la Kojima, while resorting to their structure of the optimal non-linear price scheme given by an integro-differential equation.
It turns out that a two-dimensional direct mechanism in general is not so straightforward to implement by a canonical mechanism.This article shows where this difficulty lies, which was not visible in the approach by theory of setvalued functions in [5].In one word, it consists in the lack of strict concavity of a non-linear price scheme implementing a direct mechanism.

The Model
A risk-neutral seller has one unit of an indivisible commodity to sell of quality [ ] : 0,1 q Q ∈ = .A continuum of buyers purchase one unit of the commodity of quality q or none.The seller values the commodity at zero.A buyer has taste t for the commodity and a budget w .The couple ( ) G w and the conditional distribution of t given w is ( ) the corresponding densities ( ) g w and ( ) The buyer of taste t obtains utility tq p − when buying quality q and paying price p .Suppose as in [4] that the seller incurs zero cost.The following is assumed, the reason for which is merely to assure that in Lemma 2, Equation ( 5) is always valid (see Th.1 in [4]).
The buyer of the highest taste t obtains utility t p − for the highest quality 1 q = , the highest acceptable price of which, reservation price, is t .The assumption ensures that the buyer with the highest budget can pay the price.

The Mechanism
The non-linear price scheme ( ) q τ is defined as a mapping of the space Q to the price space R .Note that it is an indirect mechanism since the type space here is T W × .
The canonical mechanism is defined so that it associates the quality and price only with the buyer's taste as in the standard setting while ignoring the budget constraint: 1 As a matter of course, a canonical mechanism here has no bearing on Maskin's one in literature.
N. Kojima 189 ( ) It for any , and , such that , .

tq t w p t w tq t w p t w t w t w T W p t w w
(SIC) induces truth-telling and (SIR) participation whereas (SBC) ensures that the buyer can actually purchase the quality assigned.
The seller's aim is to construct a strongly feasible mechanism which maximizes the expected revenue ( ) The revelation principle asserts that the outcome of an indirect mechanism is realized by a direct mechanism.Said contrariwise, an indirect mechanism might suffer sub-optimality compared to a direct mechanism [4], nonetheless, showed the following result and resorted to a non-linear price scheme instead of a direct mechanism to maximize the seller's revenue (see also [5] for the proof3 ).Lemma 1.Given any strongly feasible direct mechanism , , , q t w p t w there exists a non-linear price scheme it is continuous,strictly increasing,convex and 0 0 as well as 0 , t and further that for all ( ) where Note that the outcome of a non-linear price scheme given in the lemma is 2 We use the qualification of "weak" in order to avoid confusion with feasibility of a direct mechanism.
realized by a strongly feasible mechanism due to the revelation principle.The lemma, therefore, asserts that the revenue of an optimal5 strongly feasible mechanism is attained by a non-linear price scheme.
It shows that a weakly feasible mechanism can replicate the quality-price allocation of a strongly feasible mechanism, which basically amounts to proving that the former offers at least the same variety of qualities by the latter.In general, it is not such an easy task for a strongly feasible mechanism-which [5] deal with-but in the case of an optimal strongly feasible mechanism, one can construct a canonical mechanism which replicates the quality-price allocation of the former by way of the corresponding optimal non-linear price scheme.
This condition ensures that τ has a second derivative: hence we actually have a derivative τ τ for the optimal strong mechanism.A second observation is that τ is strictly convex since otherwise ( ) 0 x τ ′′ = at some x and Equation (5) leads to ( ) 0 x τ ′ = , which is contradictory to (2).
Let us now turn to the construction of a canonical mechanism which replicates the agent's quality (thus price) choice faced to the optimal direct mechanism.Suppose that the optimal direct mechanism is given as well as the corresponding non-linear price scheme as in lemma 2. Let us define the maximizer of utility: There are three possible cases for this maximization.
(2) Second case: If there exists x such that ( ) , it is unique due to strict convexity.Then, one defines ( ) : q t x * = .
Define now a canonical mechanism by , : , q t p t q t q t τ * * * * = .
Then, it satisfies (WIC) just by construction of ( ) q t * -through (6)-and also words, for any x , there is a buyer of taste t for whom x was a utility maximizer and an inner solution (Second case above).Thus, one obtains that continuously differentiable and positive on its non-empty support, the buyers' private information and the seller only knows the density ( ) , f t w .The pair ( ) , t w is referred to as the buyers' type from now onwards.The marginal distribution of w is ( )

(
is yet another indirect mechanism but very familiar to us in literature for decades.Let us define as usual: