Ž .Econometrica, Vol. 69, No. 4 July, 2001 , 1113�1119

CONVEX POTENTIALS WITH AN APPLICATIONTO MECHANISM DESIGN

BY VIJAY KRISHNA AND ELIOT MAENNER1

1. INTRODUCTION

THIS PAPER ESTABLISHES A GENERAL FORM of the ‘‘payoff equivalence’’ result in mecha-nism design theory: under certain conditions, the utility of any type in an incentive-com-patible mechanism is determined up to an additive constant by the allocation rule alone.

ŽWhen types are single-dimensional the result is well known see, for instance, MyersonŽ ..1981 . When types are multi-dimensional the result follows from the FundamentalTheorem of Calculus once sufficient smoothness is assumed. We obtain a more generalresult by using an extension of the Fundamental Theorem to nonsmooth convex func-tions and more generally, to the class of regular Lipschitzian functions.

2. PRELIMINARIES

2.1. Mechanisms

Let X denote the set of social alternatives with typical element x. There are I agentsand each agent i has a K-dimensional type t �RK. Let T denote the set of possiblei itypes for i. The set T�Ł T is the product of the sets of types with typical elementj� I j

Ž .t� t , t , . . . , t . We denote by t the vector of types of agents other than i and by1 2 I �iŽ .s , t the vector t with its ith component replaced by s .i �i i

Ž .Agent i’s payoff function takes the quasi-linear form, u x, t �� , where x is thei ialternative chosen by a planner or central agency and � is a monetary transfer made byii to the planner.

The types t are assumed to be independently distributed across agents according to aiprobability measure with full support on T .i

Ž .A direct mechanism is a pair � , � where � : T�X is an allocation rule andI Ž .� : T�R is a payment rule. Thus, given reports s�T , � s is the chosen alternative

Ž .and � s is the transfer payment made by i.iGiven a mechanism, the expected payoff to agent i from reporting s when his type is ti i

and all other agents are reporting truthfully is

Ž . Ž .U s , t �m si i i i i

where

Ž . Ž . � Ž Ž . .�1 U s , t �E u � s , t , ti i i t i i �i�i

is the expected utility of agent i from reporting s when his type is t , andi i

Ž . Ž . � Ž .�2 m s �E � s , ti i t i i � i�i

is the expected payment of i when reporting s .i1 We thank the referees and the editor for helpful suggestions. This research was supported by

Ž .the National Science Foundation SBR 9618726 .

1113

V. KRISHNA AND E. MAENNER1114

Ž .The mechanism � , � is incenti�e compatible if for all i and t :i

Ž . Ž . Ž . Ž .3 V t �U t , t �m ti i i i i i i

� Ž . Ž .4� sup U s , t �m s .i i i i is i

It may be useful to think of V as an indirect utility function.iThe payoff equivalence principle is that the indirect utility function V is determinedi

up to an additive constant by the allocation rule alone. If V is continuously differen-iŽ .tiable, then the envelope theorem implies that � V t depends only on the allocationi i

Ž .rule. Now the multi-dimensional Fundamental Theorem of Calculus implies that V alsoidepends only on the allocation rule up to an additive constant and hence payoffequivalence holds. However, the supremum operation that defines V does not preservei

Ž .smoothness in general. We impose conditions on the utility functions u x, t and theiŽ .mechanism � , � that ensure that V is a regular Lipschitzian function. Results oni

generalized differentiability for nonsmooth functions are then applied to obtain payoffequivalence.

2.2. Con�ex and Lipschitzian Functions

Suppose F : C�R is a convex function where C�Rn. Recall that a vector x*�Rn isa subgradient of F at x�C if for all y�C,

Ž . Ž . Ž .F y �F x x* � y�x .

Ž . Ž .The set of subgradients of F at x is denoted by � F x and we know that � F x ��. TheŽ . nsubdifferential of F is the set valued mapping � F : x�� F x . A function f : C�R is a

Ž . Ž .selection from � F if for all x�C, f x �� F x . We then write f�� F.The one-sided directional deri�ati�e of F at x with respect to a vector y�Rn is defined

as

Ž . Ž .F x� y �F xŽ .F� x ; y � lim .

���0

Ž . Ž .Now from Rockafellar 1970, Theorem 23.2 we know that if x*�� F x , then forall y,

Ž . Ž . Ž .4 �F� x ; �y x* �yF� x ; y .

A function : S�R is Lipschitzian relative to S if there exists an L such that for all� Ž . Ž . � � �x, y�S, y � x L y�x .

Ž .Suppose F : C�R is Lipschitzian relative to C. Following Clarke 1983 , the general-ized directional deri�ati�e of F at x in the direction y�Rn is defined by

Ž . Ž .F z� y �F z0 Ž .F x ; y � lim sup .

�z�x��0

Then x*�Rn is a generalized subgradient of F at x�C if for all y�Rn,

0 Ž .x* �yF x ; y .

Ž .The set of generalized subgradients of F at x is also denoted by � F x and we knowŽ . Ž .� F x ��. The generalized subdifferential of F is the set valued mapping � F : x�� F x .

CONVEX POTENTIALS 1115

A function F : C�R is said to be regular at x if for all y the directional derivativeŽ . Ž . 0Ž .F� x; y exists and F� x; y �F x; y ; F is said to be regular if it is regular for all x�C.

Ž ŽEvery convex function over a compact domain is regular Lipschitzian Clarke 1983, p...40 . Every continuously differentiable function over a compact domain is also regular

Ž Ž ..Lipschitzian Clarke 1983, pp. 32 and 40 .

3. PAYOFF EQUIVALENCE

We provide two separate formulations of the mechanism design problem. Togetherthese encompass most of the existing formulations introduced in a variety of settingsincluding auction design, nonlinear pricing, and optimal taxation. The first set ofconditions guarantees that V is convex. The second set of conditions guarantees that Vi iis regular Lipschitzian.

Ž .HYPOTHESIS I: For each i, the set of types T is con�ex and u x, � , t is a con�exi i �ifunction.

This hypothesis is satisfied in all standard auction design problems. There are a finitenumber of indivisible objects to be allocated and therefore the set of possible allocations

� 4X is finite, say, X� x , x , . . . , x . Since an agent’s utility may depend on the whole1 2 KŽ Ž ..allocation, this allows for externalities in consumption Jeheil et al. 1999 . And since an

agent’s utility may also depend on the entire vector of types, this is general enough toallow for ‘‘interdependent values.’’ In the standard private values setting, we can writeŽ . ku x , t � t , the kth component of t so that the convexity assumption on u isi k i i i

automatically satisfied.The formulation also encompasses many nonlinear pricing models. In the multi-prod-

Ž .uct nonlinear pricing model, Armstrong 1996 directly assumes that consumers’ utilityŽ .functions u are convex in t . Rochet and Chone 1998 assume that u is of the additively´i i i

Ž . K k kŽ k.separable form u x, t �Ý t u x and is thus linear in the type t .i i k�1 i i i

Ž . 2HYPOTHESIS II: The mechanism � , � is regular Lipschitzian and for each i, u isiregular Lipschitzian and monotonically increasing in all its arguments.

Ž .This formulation accommodates the nonlinear pricing model of Wilson 1993, p. 317 .

Ž .PROPOSITION 1 Payoff Equivalence : Suppose that either Hypothesis I or II is satisfied.Ž .If the mechanism � , � is incenti�e compatible, then the expected payoff function V isi

determined by � up to an additi�e constant. For all s , t �T , and any smooth path i i ijoining s to t in T ,i i i

Ž . Ž .V t �V s Q �dHi i i i i

Ž . Ž .where Q t is a generalized subgradient of U t , � at t .i i i i i

2 A vector function is said to be regular Lipschitzian if every component of the function is regularLipschitzian.

V. KRISHNA AND E. MAENNER1116

The proof of the proposition relies on the following theorem which shows that everyŽ .convex function more generally, every regular Lipschitzian function is the integral of

Ž . 3any selection from its generalized subdifferential along any smooth path in its domain.

Ž .THEOREM 1: If F : C�R is a con�ex regular Lipschitzian function defined on an openŽ . n Ž .con�ex connected set C�R and f is any measurable selection from its generalized

subdifferential mapping � F, then for any smooth path joining a to b in C,

Ž . Ž .f �d�F b �F a .H

PROOF OF PROPOSITION 1: Hypothesis I: Since the integral of a family of convexŽ .functions is convex, the fact that u is convex in t implies that the function U t , � : T �Ri i i i i

Ž . Ž .is also convex. For each t , let P t , � ��U t , � be a measurable selection from thei i i i iŽ .subdifferential of U t , � . Then, for all r , s , and t ,i i i i i

Ž . Ž . Ž . Ž . Ž .5 U t , s �U t , r P t , r � s � r .i i i i i i i i i i i

Ž Ž .The expected payoff function V is also convex from 3 it is the supremum of a family ofi.convex functions and for all s and t ,i i

Ž . Ž . Ž .V s �U t , s �m ti i i i i i i

Ž . Ž . Ž . Ž .�U t , t �m t P t , t � s � ti i i i i i i i i i

Ž . Ž . Ž .�V t P t , t � s � ti i i i i i i

where the first inequality follows from incentive compatibility and the second fromŽ . Ž . Ž .setting r � t in 5 . Thus we have shown that for each t , Q t �P t , t is ai i i i i i i i

subgradient of V at the point t , that is, Q �� V .i i i iObserve that, by definition, P and hence Q , depends only on U and hence on thei i i

allocation rule � alone and not on the payment rule �. Theorem 1 then immediatelyimplies V is determined by Q , and hence by � , up to an additive constant.i i

Ž .Hypothesis II: Hypothesis II is sufficient to ensure that the function U t , � : T �R isi i ialso regular Lipschitzian. This follows from two facts. First, a composition f�g � h oftwo regular Lipschitzian functions g and h is regular Lipschitzian provided that g is

Ž Ž ..monotonically increasing so that every p�� g is nonnegative Clarke 1983, p. 42 .Second, the integral of a family of regular Lipschitzian functions is regular LipschitzianŽ Ž ..Clarke 1983, p. 76 .

Ž . Ž .For each s and t , let P t , s be a generalized subgradient of the function U t , � : Ti i i i i i i i�R at the point s . Then for all y�RK

i

Ž . Ž . � Ž .6 P t , s �yU t , s ; 0, y .i i i i i i

3 � �A smooth path joining a to b in S is a continuous function : 0, 1 �S that is continuouslyŽ . Ž . Ž . 1 Ž Ž ..differentiable on 0, 1 and satisfies 0 �a and 1 �b. The line integral Hf �d equals H f r0

Ž .�D r dr.

CONVEX POTENTIALS 1117

The expected payoff function V is also regular Lipschitzian andi

Ž . Ž .V t � y �V ti i i i�Ž . Ž .7 V t ; y � limi i ���0

� Ž . Ž .� � Ž . Ž .�U t , t � y �m t � U t , t �m ti i i i i i i i i i� lim���0

Ž . Ž .U t , t � y �U t , ti i i i i i� lim���0

� Ž .�U t , t ; 0, yi i i

where the inequality results from incentive compatibility.Ž . Ž .Now using 6 when s � t we obtain from 7 that for all y,i i

Ž . � Ž .P t , t �yU t , t ; 0, yi i i i i i

� Ž .V t ; y .i i

Ž . Ž .Thus we have shown that for each t , Q t �P t , t is a generalized subgradient ofi i i i i iV at the point t , that is, Q �� V .i i i i

As before, P and hence Q , depends only on U and hence on the allocation rule �i i ialone and not on the payment rule �. Theorem 1, as generalized to regular Lipschitzianfunctions, now implies V is determined by Q , and hence by � , up to an additivei iconstant. Q.E.D.

Under Hypothesis I, the utility function u is convex in the type t and this immedi-i iately implies that both U and V are also convex, allowing us to invoke Theorem 1. Ini iparticular, no assumptions are made on the set of allocations X or the mechanismŽ .� , � . The assumption that u is convex in t can be replaced by the weaker assumptioni ithat u is regular Lipschitzian as a function of t only under the stronger assumptions oni iŽ .� , � given by Hypothesis II. Thus, even though the supremum of a family of regular

Ž .Lipschitzian even smooth functions need not be regular Lipschitzian, Hypothesis IIensures that this property is inherited by U and V .i i

4. POTENTIAL FUNCTIONS

Ž Ž ..The fundamental theorem of calculus for line integrals Apostol 1969, p. 334 statesthat if : S�R is a differentiable function with a continuous gradient � on an openconnected set S�Rn, then for any smooth path joining a to b in S,

Ž . Ž .��d� b � a .HRecall that if f��, then is called a potential function for f. Theorem 1 may be

rephrased as saying that a convex function F is a potential function for every measurableselection f�� F.

REMARK: Every convex function F is differentiable almost everywhere and at firstglance it may appear that the line integral of its gradient � F can be used to recover F.There may be paths, however, such that the function F is differentiable nowhere along

V. KRISHNA AND E. MAENNER1118

Ž .3the path. As a simple example, consider the convex function F : 0, 2 �R defined by

Ž . �Ž . Ž 2 2 2 .4F x , y , z �max xyz , x y z .

F is differentiable everywhere in its domain except in the two-dimensional set

3 2 2 2�Ž . Ž . 4S� x , y , z � 0, 2 : xyz�x y z .

There are an uncountable number of smooth paths joining two points a and b in SŽ Ž .. Ž . � �that lie entirely in S. For any such path, � F r �D r is not defined for all r� 0, 1

and thus the line integral H� F �d cannot be evaluated.The content of Theorem 1 is that even in these circumstances, given any measurable

selection f from the subdifferential mapping � F, the integral Hf �d is well defined andcan be used to recover F.

PROOF OF THEOREM 1: Suppose is a smooth path joining a to b in C. Let H�C beŽ� �.the convex hull of the compact set 0, 1 . Since H is compact we know from

Ž .Rockafellar 1970, Theorem 10.4 that the convex function F : H�R is Lipschitzianrelative to H.

� � Ž . Ž Ž ..First, note that the composite function � : 0, 1 �R defined by � r �F r is also� � Ž .Lipschitzian relative to 0, 1 . Moreover, from Clarke 1983, Theorem 2.3.10 we know

that it is also regular.Ž .Second, from Rockafellar 1982, Theorem 2 and Corollary 2 we know that the

Ž . Ž . Ž .directional derivatives �� r ; 1 and �� r ; �1 are well defined for every r� 0, 1 andthat

1 1Ž . Ž . Ž . Ž .� 1 �� 0 � �� r ; 1 dr�� �� r ; �1 dr .H H0 0

Now, since a regular Lipschitzian function is differentiable almost everywhere, we canwrite:

1Ž . Ž . Ž . Ž .8 � 1 �� 0 � �� r dr .H0

Finally, it is routine to verify that almost everywhere

Ž . Ž . Ž Ž . Ž ..9 �� r �F� r ; D r

and also that

Ž . Ž . Ž Ž . Ž ..10 �� r ��F� r ; �D r .

Ž . Ž . Ž .Using 9 and 10 , 8 can be rewritten as

1 1Ž . Ž . Ž . Ž Ž . Ž .. Ž Ž . Ž ..11 F b �F a � F� r ; D r dr�� F� r ; �D r dr .H H0 0

Ž .Now 4 implies that for all measurable selections f�� F:

Ž Ž . Ž .. Ž Ž .. Ž . Ž Ž . Ž ..�F� r ; �D r f r �D r F� r ; D r

and hence

1 1 1Ž Ž . Ž .. Ž Ž .. Ž . Ž Ž . Ž ..� F� r ; �D r dr f r �D r dr F� r ; D r dr .H H H0 0 0

CONVEX POTENTIALS 1119

Ž .The result now follows from 11 .The convexity of F ensures first, that the subdifferential � F is well-defined and

second, that F is Lipschitzian relative to any compact subset of its domain. Since theseare the only properties of a convex function that are used in the proof above, ageneralization of the result to regular Lipschitzian functions is immediate. Q.E.D.

Dept. of Economics, Kern 419, Penn State Uni�ersity, Uni�ersity Park, PA 16802, U.S.A.;�krishna@psu.edu; http:��econ.la.psu.edu� �krishna˜

andDept. of Economics, Kern 604, Penn State Uni�ersity, Uni�ersity Park, PA 16802, U.S.A.;

eam179@psu.edu

Manuscript recei�ed September, 1998; final re�ision recei�ed June, 2000.

REFERENCES

Ž . Ž .APOSTOL, T. M. 1969 : Calculus, Vol. II 2nd edition . New York: Wiley.Ž .ARMSTRONG, M. 1996 : ‘‘Multiproduct Nonlinear Pricing,’’ Econometrica, 64, 51�77.

Ž .CLARKE, F. H. 1983 : Optimization and Nonsmooth Analysis. New York: Wiley.Ž .JEHEIL, P., B. MOLDOVANU, AND E. STACCHETTI 1999 : ‘‘Multidimensional Mechanism Design for

Auctions with Externalities,’’ Journal of Economic Theory, 85, 258�293.Ž .MYERSON, R. 1981 : ‘‘Optimal Auction Design,’’ Mathematics of Operations Research, 6, 58�73.

Ž .ROCHET, J.-C., AND P. CHONE 1998 : ‘‘Ironing, Sweeping, and Multidimensional Screening,’’´Econometrica, 66, 783�826.

Ž .ROCKAFELLAR, T. 1970 : Con�ex Analysis. Princeton: Princeton University Press.Ž .��� 1982 : ‘‘Favorable Classes of Lipschitz Continuous Functions in Subgradient Optimization,’’

in Progress in Nondifferentiable Optimization, ed. by E. Nurminski. Laxenburg, Austria: IIASA,125�143.

Ž .ROYDEN, H. 1988 : Real Analysis, 3rd edition. New York: Macmillan.Ž .WILSON, R. 1993 : Nonlinear Pricing. Oxford: Oxford University Press.