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This paper studies the problem on two-dimensional mechanism design where the buyer’s taste and budget are his private information. The paper investigates the problem by the method of dimension-reduction, i.e. , by focusing only on the buyer’s budget and constructing an indirect mechanism: function of one variable, the buyer’s budget. It is an approach quite antipodal to that by Kojima [1] where he focused on the buyer’s taste instead of his budget. It is shown that the seller does not lose any money by adopting the indirect mechanism of this paper. In other words, the seller’s revenue-maximizing direct mechanism is implemented by such an indirect mechanism

The present paper studies the problem in which the principal sells a variety of qualities of a commodity to agent-buyers who have a two-dimensional type, taste (or valuation) and budget. Two-dimensional private information involves inevitable technical difficulties inherent in multi-dimensional mechanism design pointed out by Armstrong [

In analysis of the two-dimensional mechanism of our context, Che and Gale [

The present paper examines yet another indirect mechanism on the lines of Kojima [^{1}.

^{1}Both in Cheand Gale [

^{2}The examples are numerous: cars, computers, mobile phones, or more generally, electronic goods and so forth.

^{3}In reality, not a few businesses offer tariffs only related to age such as students or senior person discount.

Recall that Wilson [^{2}. In those contexts, past purchase or customer data are of little use and one can only resort to socio-demographic data such as age, sex, income, etc. and often they can be complemented through surveys or questionnaires. In such a situation, one is bound to resort to an indirect mechanism instead of a direct mechanism^{3} which refers to data available on hand as a message space.

Besides the advantage of data collection, our indirect mechanism is rather intuitive with only one of the two pieces of information being a variable of a mechanism. Suppose that some, say technological firm launches a new product. A marketing and sales team, and a tech team get together to decide on a product line. They discuss how much customers are ready to spend for the product― budget―and what qualities to provide. Most likely, they would not talk about consumers’ taste (valuation) for quality. They would just match consumers’ budgets to qualities and then decide prices. Agents’ budgets are quite easy to obtain as socio-demographic data or through questionnaires. It has, actually, been getting easier and easier through data collection by Internet companies such as social media and e-commerce business―Amazon, Google, Facebook, Alibaba, Baidu and so forth. By contrast, customers’ taste (valuation) is much harder to measure or evaluate, especially in monetary (numeric) terms―the utility function is quasi-linear in our context.

The seller’s ultimate objective is to earn the greatest revenue, in which respect its interest is how much the buyer can afford to pay. The buyer’s taste is not as important inasmuch as by ignoring it, the seller does not lose money.

This article shows that there exists a one-dimensional indirect mechanism described above that brings the seller as great a revenue as an optimal direct mechanism. Therefore, a technological firm launching a product range in our given example will not lose any money in their business planning.

There are a seller and a continuum of buyers both risk-neutral. The seller has one unit of an indivisible commodity to sell of quality q ∈ Q : = [ 0 , 1 ] for normalization. Alternatively, one can interpret that the seller has one unit of a divisible commodity and q is a quantity. The buyer purchases either one unit of the commodity of quality q or none. The seller values the commodity at zero. The buyer has taste t for the commodity as well as a budget w. The couple ( t , w ) takes a value in the non-empty space T × W with T : = [ 0 , t ¯ ] and W : = [ 0, w ¯ ] , which is the buyer’s private information unknown to the seller. The pair ( t , w ) is referred to the buyer’s type from now onwards.

The buyer’s utility function is of quasi-linear form: taste t buyer obtains utility t q − p when buying quality q and paying price p.

In the two-dimensional context of the present paper, the direct mechanism is defined by:

( q ( t , w ) , p ( t , w ) ) : T × W → Q × R .

The direct mechanism has to satisfy the following condition so that it induces the agent’s truthful revelation.

^{4}“S” in the tags SBC and SIC stands for strong.

Definition 1 (Strong implementability). The direct mechanism ( q ( t , w ) , p ( t , w ) ) is strongly implementable^{4} if and only if

p ( t , w ) ≤ w for any ( t , w ) ∈ T × W , (SBC)

t q ( t , w ) − p ( t , w ) ≥ t q ( t ˜ , w ˜ ) − p ( t ˜ , w ˜ ) for any ( t , w ) and ( t ˜ , w ˜ ) ∈ T × W such that p ( t ˜ , w ˜ ) ≤ w . (SIC)

The direct mechanism assigns a quality-price pair to a buyer of each taste and budget. The condition (SBC) ensures that the agent can indeed afford the quality assigned to him. The condition (SIC) ensures that among pairs affordable to him, ( t , w ) buyer will indeed choose a pair assigned to his type.

The participation constraint assures that the agent will actually purchase the quality assigned.

t q ( t , w ) − p ( t , w ) ≥ 0 on T × W . (SIR)

The revelation principle ensures that the outcome of an indirect mechanism is brought into effect by a direct mechanism; hence justification that most of mechanism design literature focuses on a direct mechanism. The opposite to this fact, however, is not true in general. From the point of view of revenue earning, it implies that resorting to an indirect mechanism, the seller might as well make less revenue.

This paper aims to show that there does exist an indirect mechanism which does the seller as well as a direct mechanism. In particular, the optimal direct mechanism-direct mechanism bringing in the greatest revenue-is implemented by an indirect mechanism. The indirect mechanism to be examined to that end is the following: the weak mechanism is defined as a map

( q , p ) : W → Q × R .

That is, the principal constructs an indirect mechanism by only taking account of the agents’ budgets while ignoring differences in taste. Later below, we will construct a concrete weak mechanism to realize our objective. It is, however, clear that direct comparison between a direct and a weak mechanism is rather cumbersome considering that a strongly implementable mechanism carries conditions, (SBC) and (SIC). The paper gets around the difficulty via a non-linear price scheme. It is shown first―Che and Gale [

Lemma 1. Given the direct mechanism ( q ( t , w ) , p ( t , w ) ) satisfying (SBC), (SIC) and (SIR), there exists a non-linear price scheme, i.e. τ : Q → R such that

it is continuous, strictly increasing, convex and τ ( 0 ) = 0 , (1)

and further that for all ( t , w ) ,

τ ( y ) ≥ p ( t , w ) (2)

where

y ∈ arg max x s .t . τ ( x ) ≤ w t x − τ ( x ) .

Proof. See Lemma 1 in Che and Gale [

,

The revelation principle asserts that the outcome of a non-linear price scheme is realized by a direct mechanism since a non-linear price scheme is an indirect mechanism.

Let a direct mechanism ( q ( t , w ) , p ( t , w ) ) satisfying (SBC), (SIC) and (SIR) be given as well as the corresponding non-linear price scheme τ ( x ) in the above lemma satisfying (1). One can construct a “weak” mechanism which implements quality-price choices by all types of agents faced with the corresponding non-linear price scheme in the following way.

Let us define the “weak mechanism” by

q ( w ) : = τ − 1 ( w ) , p ( w ) : = τ ( q ( w ) ) = w . (3)

q ( w ) is obviously concave, strictly increasing and continuous due to (1) and thus it follows easily that

q ( W ) = τ − 1 ( W ) = [ 0 , q ( w ¯ ) ] = { x | τ ( x ) ≤ w ¯ } (4)

Whatever t an agent is, the qualities that he can buy faced with the non-linear price scheme τ are of necessity in { x | τ ( x ) ≤ w ¯ } = q ( W ) . Otherwise, they are beyond his budget. It is, thus, clear that the weak mechanism provides all the qualities that the agent can afford with the non-linear price scheme. In addition, the price which an agent pays for a quality with the weak mechanism is the same for that quality with the non-linear price scheme, by virtue of the definition (3).

One accordingly obtains that along with lemma 1 and the revelation principle:

Proposition 1. Given a direct mechanism ( q ( t , w ) , p ( t , w ) ) satisfying (SBC), (SIC) and (SIR), there exists a weak mechanism defined by (3) which provides a weakly greater revenue to the seller than a direct mechanism. In particular, there exists a weak mechanism which provides the same revenue as a profit-maximizing optimal direct mechanism.

One caution that one should bear in mind is that since the weak mechanism is an indirect mechanism, an agent-buyer with budget w will not necessarily select ( q ( w ) , p ( w ) ) because his taste t is totally left out of consideration in construction of the weak mechanism.

Let us see, then, what quality an agent of type ( t , w ) indeed chooses. Given some ( t , w ) , consider:

K ( w ) : = arg max x s .t . τ ( x ) ≤ w t x − τ ( x ) = q ( w ˜ )

where w ˜ : = arg max w ′ ≤ w t q ( w ′ ) − p ( w ′ ) = arg max w ′ ≤ w t q ( w ′ ) − w ′ .

^{5}For maximization, see Rockafellar [

^{6 ∂ q ( w ) } denotes a subdifferential of q at w.

^{7 ∂ q ( w ′ ) } is generally a set. The notation that 0 > t ∂ q ( w ′ ) − 1 signifies that 0 > t x − 1 for any x ∈ ∂ q ( w ′ ) .

There are three cases for this maximization^{5}.

The first two cases result from^{6}

∃ w ′ ≤ w s .t . 0 ∈ t ∂ q ( w ′ ) − 1

In other words, the two cases are those of boundary solutions.

(1) Type ( t , w ) chooses ( q ( 0 ) , 0 ) = ( 0 , 0 ) ^{7}:

For ∀ w ′ ≤ w s .t . 0 > t ∂ q ( w ′ ) − 1 ;

this leads, due to concavity, to

0 > t ∂ q ( 0 ) − 1.

(2) Type ( t , w ) chooses ( q ( w ) , w ) :

For ∀ w ′ ≤ w s .t . 0 < t ∂ q ( w ′ ) − 1 ;

once again, due to concavity, it follows that

0 < t ∂ q ( w ) − 1.

The third case arises:

∃ w ′ ≤ w s .t . 0 ∈ t ∂ q ( w ′ ) − 1

^{8}Such w ′ might not be unique.

(3) This is the case in which the agent of ( t , w ) might not choose ( q ( w ) , p ( w ) ) . In other words, it is when the agent is well off enough to effectively buy a quality that his taste parameter t wishes, without the budget hampering such purchase^{8}. It leads to:

1 t ∈ ∂ q ( w ′ ) if q ˙ − 1 ( 0 ) < t < q ˙ − 1 ( w ) (5)

where ∂ q is written as derivative q ˙ since q is almost everywhere differentiable and also q − 1 ( 0 ) and q − 1 ( 0 ) are each reciprocals.

In view of the preceding paragraph, the principal’s optimization problem is to find q maximizing the revenue:

∫ q ˙ − 1 ( w ) t ¯ d t ∫ 0 w ¯ d w w f ( t , w ) + ∫ q ˙ − 1 ( 0 ) q ˙ − 1 ( w ) d t ∫ 0 w ¯ d w q ˙ i n v ( 1 t ) f ( t , w )

where q ˙ i n v is an inverse function of q ˙ .

Unfortunately, it is not straightforward to find such an optimal q analytically. In all liklihood, one needs a numerical procedure to approximate the optimal indirect mechanism.

The paper studied an indirect mechanism in the context where agent-buyers have a one-dimensional taste parameter and a budget as private information. In addition to two indirect mechanisms by Che and Gale [

This work was supported by JSPS KAKENHI Grant Number JP16K03545.

Kojima, N. (2017) Two-Dimensional Mechanism Design and Implementability by an Indirect Mechanism. Theoretical Economics Letters, 7, 1595- 1601. https://doi.org/10.4236/tel.2017.76107