optimal delegation with multi-dimensional decisions

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Optimal delegation with multi-dimensional decisions

Frdric Koessler a,, David Martimort b

a Paris School of Economics CNRS, 48 Boulevard Jourdan, 75014 Paris, Franceb Paris School of Economics EHESS, 48 Boulevard Jourdan, 75014 Paris, France

Received 14 September 2010; final version received 26 January 2012; accepted 8 February 2012

Available online 22 May 2012

bstract

This paper investigates optimal mechanisms in a principalagent framework with a two-dimensionalcision space, quadratic payoffs and no monetary transfers. If the conflicts of interest between the principald the agent are different on each dimension, then delegation is always strictly valuable. The principal cantter extract information from the agent by using the spread between the two decisions as a costly screeningvice. Delegation sets no longer trade off pooling intervals and intervals of full discretion but instead takeore complex shapes. We use advanced results from the calculus of variations to ensure existence of alution and derive sufficient and necessary conditions for optimality. The optimal mechanism is continuousd deterministic. The agents informational rent, the average decision and its spread are strictly monotonicthe agents type. The comparison of the optimal mechanism with standard one-dimensional mechanismsows how cooperation between different principals controlling various dimensions of the agents activitiescilitates information revelation.2012 Elsevier Inc. All rights reserved.

L classification: D82; D86ywords: Mechanism; Delegation; Mechanism design; Multi-dimensional decision

Introduction

Consider a principal who contracts with an agent who is privately informed. When the prin-pals and the agents interests are conflicting, the principal may want to exert some ex antentrol on the agent by restricting the decision set from which the agent picks actions. Examples

Corresponding author.E-mail addresses: [email protected] (F. Koessler), [email protected] (D. Martimort).

22-0531/$ see front matter 2012 Elsevier Inc. All rights reserved.

tp://dx.doi.org/10.1016/j.jet.2012.05.019

F. Koessler, D. Martimort / Journal of Economic Theory 147 (2012) 18501881 1851

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1such constrained delegation abound across all fields of economics and political science. CEOsntrol division managers by designing capital budgeting rules and allocating decision rightsong unit managers [26,2]. Many different aspects of corporate decisions involving product

sign and quality, prices, or polluting emissions are scrutinized by regulators who may imposerious limits on those variables. Lastly, Congress Committees exert ex ante control on betterformed regulatory agencies by designing various administrative procedures and rules that limitreaucratic drift and constrain the agencies discretion [41,29,19].These examples share the common feature that principals make little use of monetary transferscontrol their agents. Following the seminal works of Holmstrm [27,28] and Melumad andibano [42], these settings are fruitfully analyzed as mechanism design problems in which theincipal commits to a decision rule but cannot use monetary transfers to implement that rule.1ith no transfers and when actions lie in a one-dimensional set, optimal mechanisms look rathermple. Quite intuitively, the principal finds it hard to induce information revelation and alignnflicting objectives when he controls only a single action of the agent. In a one-dimensionaltting, an optimal mechanism balances the flexibility gains of letting the agent freely chooseis action according to his own private information and the agency cost deriving from the factat the principal and the agent might have conflicting objectives.The first major result provided by the existing literature highlights the trade-off between rulesd discretion that arises in such contexts. Inflexible rules allow the principal to choose hisost preferred policy from an ex ante viewpoint, i.e., in the absence of any information. This isbecause those rules make no use of the agents private information. Leaving full discretion to

e agent, on the other hand, allows to implement state-dependent actions, but these choices nowflect only the agents preferences and not those of the principal. The second important resultvanced by the literature is that the optimal mechanism (when continuous) can be implementedmeans of interval delegation sets which set bounds on the agents action. This is an important

eoretical insight because it reduces the design of the mechanism to a simple exercise consistingfinding those bounds. This simplification is also of great value when it comes to implementinge optimal mechanism, and it clearly echoes contractual arrangements found in practice.The objective of this paper is to study how optimal mechanisms are modified when several of

e agents activities can be controlled by a principal or, equivalently, when several principals,ch being endowed with the same bargaining power and each controlling a single decision ofe agent, can cooperate in designing a common mechanism. First, one may wonder how theade-off between rules and discretion is modified. Clearly, screening possibilities are now im-oved and pooling certainly seems less attractive. Second, in a multi-dimensional context, theency problem between the principal and his agent may not only be related to their averagenflict of interest over all dimensions but also to the distribution of conflicts across the differentmensions. The extent to which this is the case must also be clarified. These are highly relevantsues not only from a pure theoretical viewpoint but also because many real-world problemse multi-dimensional. For instance, when designing vertical restraints with his retailers, a man-acturer may not only leave them discretion on how to fix retail prices but also on some othermensions like after-sales services. An economic regulator may put a stringent cap on pricesarged by regulated firms while leaving more discretion in choosing environmental quality. Inese contexts, it is important to understand whether and how treating each dimension separatelycludes important screening possibilities.See [7,11,39,3,24,32], and [6], among others.

1852 F. Koessler, D. Martimort / Journal of Economic Theory 147 (2012) 18501881

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beAs a first approach to these questions, this paper investigates the form of optimal mech-isms in a simple two-dimensional setting with quadratic and separable payoffs, a uniformstribution of types, and constant biases in players ideal actions. In such context, the optimalechanism restricts the agents choices on each dimension through a smooth and deterministiclegation set. When the conflicts of interest between the principal and the agent on both dimen-

ons differ, delegation is always strictly valuable. The optimal delegation set never includes theents ideal points, nor does it exhibit any pooling. The interval delegation sets used in the one-mensional case, which trade off pooling intervals and intervals with full discretion, are thusboptimal.To evaluate incentive distortions in a multi-dimensional context, the two important factors

e the average decision that the principal would like to implement and the spread, i.e., howr apart the principal would like to set the levels of the different activities. This spread plays amilar, albeit somewhat different, role to that played by transfers in standard models with quasi-near preferences and monetary payments. Like transfers in standard screening models, using ape-dependent spread as a screening device facilitates information revelation. To see how, letassume that the agent would ideally like to choose the same decision on dimensions 1 and 2his activity but, on average, prefers lower levels of those decisions compared to the principal.

he principal wants to avoid information manipulation aiming to implement such lower levelsactivity. To limit the agents incentives to claim for lower average decisions, the principal

ight increase the spread between decisions 1 and 2 following such claims. By so doing, heposes a cost on the agent and reduces his temptation to manipulate information in that direc-

on. Conversely, the principal reduces this spread when the agents report induces higher levelsactivity. Punishments and rewards are obtained by playing on the spread. The optimal spreadstrictly positive and monotonic in the agents type whenever the conflicts of interest betweene principal and the agent differ along each dimension. Moreover, we show that this spread ist a compromise between the principals and the agents ideal spread and it could be strictlyeater.Compared with a setting where monetary transfers are available, utility is no longer transfer-

le between the principal and the agent. Implementing a spread in the decisions also introducesmplex costs and benefits for the principal. Let us assume that the principal would ideally prefere more unit of activity 2 than of activity 1 for any realization of the agents private information,that his ideal spread is just one, and let us also assume that both activities give more returnthe principal than to the agent. In that case, the agent may want to pretend that lower averagetivities should be implemented. Increasing the spread between the two activities above 1 forw average activity levels and reducing it below 1 for higher activity levels is of course costlyr the principal, but it certainly facilitates screening.From a technical viewpoint, the nonlinearity due to the absence of monetary payments makes

e characterization of the optimal mechanism quite complex.2 We use results from the calcu-s of variations [16] to ensure the existence of a solution and derive sufficient and necessarynditions for optimality.

The absence of monetary transfers makes the screening problem look like those that are found for instance in thetimal taxation literature [44], where utility functions are also not quasi-linear. However, the techniques of this literaturennot be directly imported into our framework. Even with quadratic payoffs, the principals objective function may not

everywhere Lipschitz-continuous, contrary to what is assumed in the optimal taxation literature.


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Melumad and Shibano [42] provided a significant analysis of the delegation problem withadratic payoffs and a uniform distribution of types in contexts where no transfers are availabled where the uninformed party (the principal) commits to a mechanism with the informed party

he agent). Martimort and Semenov [39] and Alonso and Matouschek [3] characterized settingshere simple connected delegation sets are optimal, a feature that was a priori assumed in [28,7]d [11] for instance. Alonso and Matouschek [1] applied the standard delegation model to anamic context where the principal and the agent interact repeatedly. Focusing on dominant

rategy to get a sharp characterization of the set of incentive feasible allocations, Martimortd Semenov [40] extended this mechanism design approach to the case of multiple privately-formed agents (lobbyists) dealing with a single principal (a legislature) in a political economyntext where the principal chooses a one-dimensional policy.3 Farrell and Gibbons [21] andoltsman and Pavlov [25] analyzed private and public mechanism with a single informed agentd two decision-makers. As in our model, the decisions on each dimension enter separately intoe agents payoff function and are strategically independent across the two decision-makers.one of these papers has addressed the design of multi-dimensional mechanism with commit-ent. In that respect, the closest paper to ours may be [4]. These authors introduce the possibilityat the principal burns money or imposes costly activities for the agent in an otherwise stan-rd delegation set-up. Money burning constitutes a second instrument that facilitates screening,t the impact of these new screening possibilities on the principals payoff is different to that ofr paper. Also related is a recent paper by Che, Dessein and Kartik [15] that analyzes a kind ofal problem to ours. They solve for optimal mechanisms in an environment without transfers in

hich the decision space is finite but the type space is multi-dimensional.

2. Organization of the paper

Section 2 presents the model and the by-now standard result where a single activity of theent is controlled by the principal. Section 3 presents some preliminary results on incentivempatibility and assesses the performances of simple and intuitive mechanisms that illustratee new screening possibilities available in multi-dimensional environments. Section 4 is the corethe paper. We formulate the design problem using advanced tools of the calculus of variationsd derive the optimal mechanism in this multi-dimensional context. Some robustness checkse provided in Section 5. Section 6 concludes and paves the way for future research. Proofs arelegated to Appendix A.

The model

A principal controls two actions, x1 and x2, undertaken by a single agent on his behalf. Wenote by (x1, x2) the bi-dimensional vector of these actions. For simplicity, they lie in a compactt K= [K,K] R for K large enough. Utility functions are single-peaked and quadratic ande differences between the principals and the agents ideal actions (the biases) are constantross types. More precisely, the utility functions are respectively given for the principal and hisent by:

Austen-Smith [10], Battaglini [12,13], Krishna and Morgan [33], Levy and Razin [35], and Ambrus and Taka-

shi [5], on the other hand, considered cheap talk settings with multiple privately-informed senders.


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fereV (x1, x2, ) = 122

i=1(xi i)2, (1)

d

U(x1, x2, ) = 122

i=1(xi )2. (2)

ith these preferences,4 the agents ideal point on each dimension is xA() = whereas theincipal has an ideal point located at xiP () = + i in dimension i, i = 1,2. We denote by 2 1 the difference in biases between the two dimensions and by 1+22 the average

as. We assume without loss of generality that 1 2 and 0, but 1 could be either positivenegative.The agent has private information on his ideal point (or type) . The agents type is drawn

om a uniform distribution on = [0,1].5 The principal is uninformed about the agents type.The principal controls the whole vector of the agents activities (x1, x2). From the Revelation

rinciple [45], there is no loss of generality in restricting the analysis to direct mechanismsipulating (maybe stochastic6) decisions as functions of the agents report on his type. Anyterministic mechanism is a mapping x() = {x1(), x2()} : K2.

emark 1. The model might also be viewed as describing a situation with a single commonent and two principals, P1 and P2, whose utility functions are respectively:

Vi(xi, ) = 12 (xi i)2, i = 1,2.

nder a non-cooperative design with private mechanism between the agent and each principal,incipals independently choose their mechanism spaces with the agent and design their ownechanism. Since there is no externality between principals (i.e., each principal Pi s utility onlypends on the decision xi and the agents utility function is separable in the decisions controlledeach principal), each principal offers the same mechanism as if he alone was contracting with

e agent. If principals cooperate in designing a common mechanism and have equal bargainingwers when designing the mechanism, the merged principal objective function is exactly thatEq. (1). Our analysis below thus reveals the gains from such cooperation between principals

ith equal bargaining power compared with a non-cooperative design.7

The choice of quadratic utility functions is a common restriction in the cheap talk and delegation literature. Forstance, in an important paper, Alonso and Matouschek [3] imposed this condition on the principals preferences whileaking no such explicit assumption on the agent. However, imposing this quadratic assumption on the agents preferenceswithout loss of generality in a one-dimensional context. Indeed, it can be easily seen that incentive compatibilitynstraints take the same form with more general preferences, as long as the agents preferences are single-peaked andmmetric. In our multi-dimensional context, this quadratic restriction is important to the extent that it pins down thearginal rate of substitution between each decision in a very specific way. This is a key ingredient of the tractability ofr model.

The characterization of the optimal mechanism would be untractable if we were to assume other distributions. Sec-n 5 nevertheless extends some of our qualitative results to more general distributions.Analysis of stochastic mechanisms is deferred to Section 5.1, where we prove their suboptimality.The assumption of equal bargaining power between cooperating principals is innocuous when monetary transfers are

asible between those principals. There are instances where such assumption is quite natural. Think of the case of two

gulatory agencies, one concerned by regulating prices, the other being interested in controlling pollution emissions.


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haLet us now consider the case where the principal controls a single decision xi , for some {1,2} or, with the interpretation above, principals independently choose their mechanismaces with the agent and design their own mechanism.

roposition 1 (One-dimensional activity). (See [27].) In the one-dimensional case, the optimalechanism xOi () is given by:

xOi () =

max{,2i}, if i [0,1/2],min{,1 + 2i}, if i [1/2,0],1/2 + i, if |i | 1/2.

(3)

When a principal controls a one-dimensional activity of the agent, the optimal mechanisms a simple cut-off structure. For example, when i (0,1/2), the optimal action correspondsthe agents ideal point if it is large enough and is otherwise independent of the agents type.

his outcome can be easily achieved by means of an interval delegation set. Instead of using arect revelation mechanism and communicating with the agent, the principal could also offer aenu of options Di = [2i,+) and let the agent freely choose within this set. When the floori is not binding, the agent is not constrained by the principal and it is just as if he had fullscretion in choosing his own ideal point. When the floor is reached, the agent is constrainedd can no longer choose his bliss point, which is too low compared with what the principalould implement himself.The optimal mechanism trades off the benefits of flexibility (the agent sometimes choosing a

ate-dependent action) against a loss of control (this state-dependent action being different frome principals ideal point). Setting a floor or a cap limits the agents discretion and reduces thess of control. Clearly, this floor (resp. cap) increases (resp. decreases) with i , meaning thatless rigid rule is chosen when the conflict of interest between the principal and the agent isss pronounced.8 When the conflict of interest is significant (|i | > 1/2) the principal simplyooses a pooling allocation at his expected ideal points. Delegation is not valuable.

Preliminary results

In the multi-dimensional case, incentive compatibility constraints can be written as:

arg max

12

2i=1

(xi( )

)2.

emma 1. The necessary and sufficient condition for incentive compatibility is that 2i=1 xi()non-decreasing in and thus a.e. differentiable in . At any point of differentiability, we have:

2i=1

xi () 0, (4)

lthough direct regulatory transfers with firms might be banned, cooperating agencies would merge their budgets andhave collectively as having a common objective as described in Eq. (1).

As shown in [3], this might not be the case when the bias i is state-dependent. In addition, the delegation set might

ve several intervals of full discretion.


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xi ()(xi()

)= 0. (5)In this multi-dimensional world, the principal can now use both x1() and x2() to screen the

ents preferences. To understand how this can be so, it is useful to observe that the princi-l could at least offer the optimal one-dimensional mechanisms that he would choose if he wasntrolling each dimension alone, namely the pair of mechanisms described in Proposition 1. Al-ough mechanisms that would satisfy (3) for i = 1,2 also satisfy (4) and (5), more mechanismse now incentive compatible. By trading off distortions along each dimension or by choosing ac-ons that vary in opposite directions on each dimension as the agents type changes, the principalight for instance induce countervailing incentives that facilitate information revelation.9This characterization of incentive-compatible allocations already gives some powerful in-

ghts into the properties of mechanisms when we examine a couple of simple two-dimensionalechanisms.

xample 1. Consider the linear mechanism {x1 (), x2 ()} such that x1 () = and() = + , where is a fixed number. This mechanism is incentive compatible since ittisfies both (4) and (5). The best of such mechanisms maximizes the principals profit, i.e.,should be optimally chosen so that any concession made by the principal on x1 by movingis decision closer to the agents own ideal point is compensated by an equal shift in x2 in therection of the principals ideal point. Typically, = 2 does the trick, since:

arg min

10

( 2i=1

(xi () i

)2)d = arg min

( + 1)2 + ( 2)2 = 2 .

sharp contrast to the one-dimensional case, this mechanism induces full separation of types,ver gives the agent his ideal action (as long as = 0) and already achieves the first best whenincipals have opposite biases, i.e., 2 = 1.

Still, when > 0, the principal can find that the above decisions are too close to the agentseal points. The mechanism {x1 (), x2 ()} can be improved by introducing a pooling areaer the lower tail of the distribution as in the one-dimensional case. This is illustrated in thext example.

xample 2 (Simple mechanism). Consider the incentive-compatible mechanism {x1(), x2()}fined as:

x1() ={

2 if , 2 otherwise,

and x2() ={

+ 2 if , + 2 otherwise.

(6)

his new mechanism is obtained by imposing floors on each dimension for . In what fol-ws, the optimal so-called simple mechanism within this class is denoted as {x1 (), x2 ()} .is such that:

The literature on countervailing incentives ([36,37] and [34, Chapter 3] among others) has been developed in settingsith monetary transfers. Sometimes those models generate pooling as an optimal response to simultaneous incentives to

er- and under-report types as in [37]. In our model, on the contrary, pooling is never an issue, as we show below.


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his mechanism has a non-trivial pooling area when 0 < 2 < 1. This pooling area is an averagetween the pooling areas that the principal would choose when designing an optimal mechanismeach dimension separately.

The simple mechanism {x1 (), x2 ()} is instructive because it stresses two features oftimal mechanisms that our more general analysis will confirm. First, the principal trades offstortions on each dimension by introducing a strictly positive spread between x1 and x2 forery type , so the agent never gets his ideal actions. Second, decision rules are rather flat one lower tail of the distribution. However, contrary to what happens with this simple mechanism,e optimal mechanism will never exhibit any pooling, the spread between the two dimensionsill not be constant with the agents type, and one action might be decreasing on the upper andwer tails of the distribution without conflicting with incentive compatibility.

Optimal multi-dimensional mechanism

To characterize the optimal mechanism, it is useful to re-parameterize our problem with a newt of variables. This transformation not only brings new insights into the nature of the economicoblem but it will also subsequently facilitate the proof that stochastic or discontinuous mech-isms are not optimal. Consider thus the following two extra auxiliary variables which are theerage decision and a measure of the spread of those decisions:

x() 12

2i=1

xi() and t () 122

i=1

(xi() x()

)2 = 14(x2() x1()

)2. (7)

nder complete information, the principal would like to choose an average decision xP () =+ and an optimal spread tP () = 24 . These two quantities differ from those that would beeally chosen by the agent on his own, namely, xA() = and tA() = 0.Solving the system of Eqs. (7) for x1() and x2() yields immediately10:

x1() = x()

t () and x2() = x() +

t (). (8)We now define the agents non-positive information rent U() as:

U() max

12

( 2i=1

(xi()

)2).

sing (8) and incentive compatibility, we rewrite:U() = (x() )2 t () = max

(x() )2 t ( ). (9)

With this formulation, the agents rent now only depends on the screening variables throughe average decision x() and the spread t (). This utility function becomes quasi-linear with

The other solution, x1() = x() +

t () x2() = x()

t (), is not optimal for the principal because his loss 2Eq. (13) below would be L(U(), U ()) + 2 U() U ()4 instead of L(U(), U ()).


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tee spread or transfer t () 0 measuring the cost for the agent of choosing different decisionsong each dimension. Varying that cost with the realization of his private information certainlyses screening. The technical difficulty that we will face in what follows comes from the factat this transfer does not enter linearly into the principals objective.The average decision x() has an impact on the agents marginal utility, which depends on

s type. It can thus be used as a screening variable as in standard screening models. Clearly,agent of type may be tempted to lie downwards to move the average decision closer to

s own ideal point. The principal can make that strategy less attractive by increasing the spreadtween decisions for the lowest types.11As usual in screening problems with quasi-linear utility functions, the incentive compatibility

nditions (4) and (5) can be restated in terms of the properties of the pair (U(), x()).

emma 2. U() is absolutely continuous with a first derivative defined almost everywhere and,any point of differentiability:

U () = 2(x() ). (10)he average decision x() is non-decreasing and thus almost everywhere differentiable with, aty point of differentiability:

x() = U ()2

+ 1 0. (11)

Note that the non-negativity of the spread entails:

t () = U() U2()

4 0, (12)

ith an equality only when x1() = x2() = x(), i.e., when both decisions are equal. With thew set of variables, we rewrite the principals loss in each state of nature as:

12

2i=1

(xi() i

)2 L(U(), U())

= U() U()

U() U2()

4+ 2 +

2

4. (13)

rom this, we get the following expression of the principals relaxed problem, neglecting for theme being the monotonicity condition on x() that will be verified ex post:

(P): minUW 1,1()

10

L(U(), U ()

)d,

here W 1,1() denotes the set of absolutely continuous arcs on .12

The principalagent literature has stressed that a principal can use the agents risk-aversion to ease incentives (see,g., [9]) by for instance using stochastic mechanisms. Introducing some spread in the agents decisions in a model withadratic payoffs has a similar flavor. Section 5.1 shows that stochastic mechanisms are suboptimal in our framework.

In the parlance of the calculus of variations, (P) is actually a Bolza problem with free end-points (see [16, Chap-

r 4]). It is non-standard because even though the functional L(s, v) is continuous and strictly convex in (s, v), it is


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thcoWe now proceed as follows. First, we prove existence of an optimal arc in W 1,1(). Sec-d, we characterize this arc by means of a second-order EulerLagrange equation. Third, a firstadrature tells us that this solution solves a first-order differential equation known up to a con-

ant. Finally, we impose conditions on that constant so that the monotonicity condition (11)ways holds.

emma 3. A solution U() to (P) exists.

Once this solution is known, the pair of decision rules {x1 (), x2 ()} is recovered using thermulae:

x1 () = +U()

2

U() (U())24

and

x2 () = +U()

2+

U() (U())24

. (14)

roposition 2. An optimal arc U() is such that:

The following EulerLagrange equation holds at any interior point of differentiability:L

U

(U(), U()

)= dd

(L

U

(U(), U()

)); (15) The following free end-point conditions hold on the boundaries of the interval [0,1]:

L

U

(U(), U()

)=0 =

L

U

(U(), U()

)=1 = 0; (16)

U() is continuously differentiable, and thus x1 () and x2 () are continuous.

The next proposition investigates the nature of the solution to the second-order ordinary dif-rential equation (15) by obtaining a first quadrature parameterized by some integration constantR. This constant must be non-positive to ensure that the second-order condition (11) holds.

roposition 3. For each solution U(,) to (15) which is everywhere negative and satisfies (11),ere exists R such that13:

U (, ) = 2

U(,) 2(

U(,)

U(,) + )2

, (17)

d

t everywhere differentiable (or even Lipschitz), especially at points where s v24 = 0 if any such point exists on anmissible curve where v() = U () and s() = U().

It is important to note that the differential equation (17) may a priori have a singularity and more than onelution going through a given point. This might be the case when, for this solution, there exists 0 such that(0, ) + 2( U(0,)U(0,)+ )

2 = 0. The right-hand side of (17) fails to be Lipschitz at such a point. It turns out thatis possibility does not arise for the optimal mechanism described below, because a careful choice of ensures that the

ndition (18) holds everywhere on the optimal path.


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14(U(,) + )2 + 2U(,) > 0, for all . (18)

Reminding the reader that if = 0 the principal can achieve the first-best while, if = 0,e optimal mechanism is the twice-replica of the one-dimensional solution, we are now readycharacterize the optimal mechanism in the multi-dimensional case for the other cases relevantr our analysis.

heorem 1 (Two-dimensional activity). For all (,) R2++, the optimal mechanism entails thellowing properties.

Optimal decisions on each dimension are never equal to the agents ideal points:

x1 () = x() U()

U() + < x2 () = x() +

U()U() + , (19)

with (24 2,2

4 ) and x() = + U()2 ; The rent profile U() is everywhere negative, strictly increasing and solves (17) for ;

There is no pooling area. Monotonicity conditions are satisfied everywhere: x() > 0.

1. Preliminary remarks

In the limiting case where = 0, we already know that the optimal mechanism will coincideith that described in Proposition 1 for the one-dimensional problem, with

U0 () = (min{ 2,0})2, x0 () = max{,2} and = 0, (20)

hen 1/2.14 Exactly as in Examples 1 and 2, when 1 = 2, there is no gain for the principaltrading off distortions on each dimension, because there is no conflict of interest between theincipal and the agent concerning their ideal spread. It is therefore costly for the principal to usespread on decisions. On the contrary, when > 0, the double replica of the one-dimensionalechanism is suboptimal. This is particularly illuminating when biases on each dimensions arest opposite, i.e., = 0, since then the principal can get his first-best xi() = + i with anstant spread.Beyond the special cases where = 0 or = 0, several features of the optimal mechanism

e worth emphasizing when the principal and the agent have conflicting preferences.

2. Comparative statics

The third result of the previous theorem shows that, contrary to existing models of delegationee for example [3]), pooling is never optimal, whatever the conflicts of interest (1 and 2uld be arbitrarily large), as long as 1 = 2. Therefore, when conflicts of interest between theincipal and the agent are different on each dimension, delegation is always strictly valuable.Still, we can provide some interesting comparative statics concerning the conflict between

eferences (the average conflict and the dispersion of the biases, ) for our multi-dimensionalvironment. For this purpose, let us denote the principals expected loss with preferences (,)the optimal mechanism by:When 1/2 we have U0 () = (1/2 + )2 and x0 () = 1/2 + .


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0

L(U(), U()

)d.

he next proposition extends the standard Ally Principle15 to our multi-dimensional environ-ent by showing that the principal prefers to appoint an agent with a smaller average bias, atast when the biases are not too large (2 + 42 < 1).16

roposition 4. If 2 + 42 < 1 and > 0, then a greater average conflict strictly increases theincipals expected loss:

L

(,) > 0. (21)

Comparative statics with respect to are more difficult to obtain because increasing theread between the principals and the agents ideal points has two effects. On the one hand,facilitates screening. On the other hand, it also increases the divergence between the princi-ls and the agents preferences, thereby increasing the cost of rewarding the agent for truthfulvelation by letting him choose points close to his ideal ones. Note, however, that the simpleechanism (Example 2) is available for any and yields a loss worth 2 433 when 1/2d 1/12 when 1/2, which is independent of . This value corresponds to the optimal losshen = 0.More generally, the following proposition shows that L(,) is strictly decreasing in the

read, at least when the average bias and the spread are not too large.17

roposition 5. For all (,) R2++, the following property holds:L(,) < L(,0). (22)

in addition 1/2, then the principals expected loss L(,) is strictly decreasing with hen is small enough.

Altogether, Propositions 4 and 5 show that there always exist some directions in which pref-ences can be changed to improve the principals payoff. In particular, when 2 > 1, we alreadyow from the study of one-dimensional mechanisms that not using the spread leads to an allo-tion with full pooling. On the contrary, increasing even slightly the spread on biases is alwaysneficial for the principal and allows full screening, as we have seen from our general analysis.We now turn to a more precise analysis of the distortions in decisions, focusing on the role of

e optimal average decision and spread.

3. Average decision

Contrary to what happens in the one-dimensional case, even when the agents ideal pointclose to , the contract has no pooling; x() is everywhere monotonically increasing. This

Huber and Shipan [30] survey the political science literature on this topic. Since the seminal work of Crawford andbel [17], the literature on mechanism and delegation in settings with private information and conflicting interests hasurished by pursuing the analysis of this Ally Principle (see [3,23] and [18] for instance).We conjecture that this property is always valid, but we have no analytical proof for 2 + 42 > 1.

Numerical computations further suggest that L(,) is actually always strictly decreasing in .


Filin

Fi(c

stav

ofan

de

C

M

deplfim

thg. 1. Average decision x() when = 0.3, and = 0.6 (dotted line), = 0.2 (dashed line) and = 0 (continuouse, which coincides with x() and xO()).

g. 2. Agents information rent U() when = 0.3, and = 0.6 (dotted line), = 0.2 (dashed line), and = 0ontinuous line).

ands in sharp contrast to the one-dimensional case. The next corollary further shows that theerage decision lies systematically in a greater interval than if the principal was restricted tofering the simple mechanism {x1 (), x2 ()} of Example 2 or the one-dimensional mech-ism {xO1 (), xO2 ()} of Proposition 1. It also shows that the principal raises the averagecision further away from the agents ideal points.

orollary 1. For all > 0 and (0,1/2), we have [2,1] [x(0), x(1)], and there exists() (0,2) such that:

x() < 2 if and only if ().oreover:

x() > for all .

These features are illustrated in Fig. 1, which compares the average of the one-dimensionalcision rules xO() = 12 (xO1 ()+xO2 ()) = max{,2} (which coincides with the average sim-e decision rule x() = 12 (x1 () + x2 ())) with the optimal average decision rule x() for axed average bias and different values of . The agents information rent under the optimalechanism, which is strictly increasing when > 0, is represented in Fig. 2.In sharp contrast to the one-dimensional case, the agents ideal points are never chosen ate optimal mechanism. When > 0, the principal always induces truth-telling without leaving


Fi2

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(1(2(3(4

4.

alisne

a

Wdew

di(bthT

18

by(g. 3. Decisions xi() (dashed lines), x

i() (continuous lines) and xO

i() (dotted line), for i = 1,2, when 1 = 0.2 and

= 0.4.

ll discretion to the agent. However, distortions on each dimension become quite complex, asustrated by Fig. 3 for 1 = 0.2 and 2 = 0.4. While x2 () is always strictly greater than , it ist always increasing, while x1 () is strictly increasing over [0,1] but not always greater than .addition, for the incentive compatibility constraint (5) to be satisfied, x2 () should be strictlycreasing if and only if x1 () is larger than . This feature of the optimal mechanism is general,d is summarized in the next corollary.

orollary 2. For all (,) R2++, the following properties hold:

) For every [0,1], we have x1 () > 0 and x2 () > ;) For every [0,1], we have x1 () > if and only if x2 () < 0;) If 1 0, then x1 (0) 0 and x2 (0) 0 (with strict inequalities when 1 > 0);) If 1 0, then x1 (1) 1 and x2 (1) 0 (with strict inequalities when 1 > 0).

4. Optimal spread

While the principal distorts decisions on each dimension as in Examples 1 and 2, Fig. 3so shows that, contrary to what happens with those simple mechanisms, the optimal spreadno longer constant. It is actually strictly decreasing in whenever > 0, as shown in thext corollary. This monotonicity facilitates information revelation. Indeed, the principal imposescost on the agent by making activities on both dimensions more dispersed for lower types.hen the agent claims the state is lower (and he is biased to do so to get a smaller averagecision x()), he is thus eventually punished by having to implement dispersed actions asell.18When = 0, the principal and the agent find it equally costly to make decisions on each

mension more divergent. There is no longer any conflict of interest concerning the ideal spreadoth the principal and the agent want x2() = x1() even though their most preferred values forat decision diverge). Making decisions on each dimension more divergent is no longer useful.he optimal mechanism is simply the double replica of the one-dimensional mechanism.

Note that the spread might also be decreasing with the one-dimensional mechanisms of Proposition 1 (as illustratedthe dotted lines in Fig. 3), but only for intermediate values of and when the conflict of interest is not too large1, 2 < 1/2).


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itemark 2. Taking a broader perspective, let us think of two different principals as each con-olling a single dimension of the agents activity, say xi (i = 1,2), and having the objectivei (xi, ) = 12

2i=1(xi i)2. The intuition of our result then becomes quite obvious.

here is no gain in jointly maximizing the sum of the principals payoffs when their preferencesincide (1 = 2). Such a joint design only replicates what each of them individually would likedo.

We now summarize some further properties of the optimal spread t().

orollary 3. For all (,) R2++, the optimal spread t() = 14 (x2 () x1 ())2 is non-gative, continuous and strictly decreasing in with

2 > t(0) > 2

4> t(1) > 0.

One might think that the optimal spread lies somewhere in between the ideal spreads of theincipal (tP () = 24 ) and the agent (tA() = 0), to achieve some kind of compromise betweene two players objectives. The above corollary shows that this is actually not the case. Althoughe optimal spread is positive, which always hurts the agent, it may be significantly beyond theincipals ideal spread for close enough to zero. For close to zero, the principal overshootsd is ready to push the optimal spread beyond his own ideal one just to better reward informationvelation. Instead, for close enough to one, increasing the spread above the agents ideal pointhile still keeping it lower than tP () relaxes the agents incentive constraint but also tends tocrease the principals payoff.

5. Delegation sets

One interpretation of the optimal mechanisms in the one-dimensional case is that intervallegation might be optimal. In our setting, leaving full discretion to the agent in choosingithin a given delegation set bounded by a floor is optimal; in others a cap may prevail. In ourulti-dimensional context, this benefit of delegation carries over although delegation sets areore complex than simple caps and floors. Indeed, it is still true that a version of the Taxationrinciple [47] holds in our context. The principal can implement the optimal mechanism byfering an indirect mechanism, i.e., a (continuous) curve in the (x1, x2) space constructed frome parametrization {x1 (), x2 ()} and leaving the agent free to pick any point on this curve.ig. 4 represents such a curve, corresponding to the optimal mechanism, in the (x1, x2) space.t the same time, this figure also features the indirect mechanisms corresponding to the simpleechanism {x1 (), x2 ()} and the one-dimensional mechanism {xO1 (), xO2 ()} . Noteat the slope of the optimal indirect mechanism is lower than the slope of the simple indirectechanism. This is a general feature that can be directly deduced from Corollary 3. The slopethe simple curve is one, while the slope of the optimal curve is strictly smaller than one when

e spread is strictly decreasing.More generally, simple duality arguments give us a little bit more information about the shapethose delegation sets. Let us define T (x) as T (x) = t() for x = x(). T () is thus thenlinear tax paid by the agent in terms of spread decisions when the average decision is

self x. By definition of the agents optimality conditions, we have:U () = max

x2x T (x)


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4.

th{xapan

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en

then

P{xmg. 4. Delegation sets for the optimal mechanism (dashed line), the simple mechanism (x1 (), x2 ()) (continuous line),d the one-dimensional mechanism (xO1 (), xO2 ()) (dotted line) when 1 = 0.2 and 2 = 0.4.

here U() = U() + 2 and T (x) = T (x) + x2. From those definitions, we immediatelyt that U() is convex as a maximum of linear functions. Therefore U() is the differencetween two convex functions.By duality, we also have

T (x) = max

2x U().

ence, T (x) is also convex as a maximum of linear functions. The nonlinear tax T (x) is itselfe difference between two convex functions.

6. Simple mechanisms

Since the design of the optimal mechanism looks rather complex, one may wonder whethere one-dimensional or the simple mechanisms perform well enough. The simple mechanism1 (), x

2 ()} of Example 2 is particularly appealing since, as already noted, it is also

proximately optimal when is small (and achieves the first-best when = 0), while the mech-ism {xO1 (), xO2 ()} that replicates the one-dimensional mechanisms of Proposition 1 isly optimal when = 0. The intuition is that, although the optimal mechanism requires fullparation of types, this is only marginally the case on the lower tail of the distribution of types,d one may not lose so much by using simple mechanisms.Our next proposition shows that both these mechanisms perform quite well when is small

ough. When two principals each control a dimension of the agents activity, this result meansat the gain for cooperation between the principals is only significant for a large enough differ-ce between their ideal actions.

roposition 6. If |1|, |2| < 1/2, then the principals loss from using the simple mechanism1 (), x

2 ()} or the one-dimensional mechanism {xO1 (), xO2 ()} instead of the optimal

echanism {x1 (), x2 ()} is of order at most 2 in :1L(U(), U())d

1L(U(), U()

)d 3

2, (23)0 04


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5.

shchde

Ptix1

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19

ty10

L(UO(), UO()

)d

10

L(U(), U()

)d (1 )2. (24)

Extensions

This section develops some extensions of our basic framework and shows the robustness ofme of our results.

1. Non-optimality of stochastic mechanisms

Kovac and Mylovanov [32] showed that the restriction to deterministic mechanisms is withoutss of generality in the case of quadratic payoffs, a constant bias, a one-dimensional activity, andhen the type distribution is uniform as here.19 This insight carries over in our framework when= 0 mutatis mutandis. However, in our multi-dimensional context, the principal has more toolsscreen the agents type when > 0. As such, using stochastic mechanisms is less attractiveen beyond the case of a uniform distribution.

roposition 7. Suppose that is distributed according to any density f () on . The optimalterministic mechanism cannot be improved by stochastic mechanisms.

To relax incentive compatibility, the principal could a priori use random allocations and playthe variance of each decision, i.e., choose how decisions move around their expected values to

reaten the agent with some risk if he reports low values of . Of course, the principal can stillay on how decisions are spread, as in our analysis of deterministic mechanisms. The second ofese strategies has already been shown to be useful above. The first is suboptimal. The intuitionstraightforward: the principal and the agent are both equally averse to such random allocations,d there is no gain from using this randomness that could not be achieved by playing only one spread between decisions.

2. Not leaving full discretion is generic

Our result that the agent never receives his ideal points is highly robust. The next propositionows that the principal never finds it optimal to leave full discretion to the agent, letting himoose his ideal points on a subset I with a non-empty interior, whatever the everywhere-positivensity f () on .

roposition 8. Suppose that is distributed according to any density f () everywhere posi-ve on and bounded. If > 0, then the optimal deterministic mechanism is never such that() = x2 () = on any subset of with a non-empty interior.

The intuition is straightforward. Let us assume the contrary. The principal could, as in Ex-ple 1, move down x1 and up x2 by the same small amount on that interval, still keeping the

They also provide conditions for more general distributions. Alonso and Matouschek [3] give an example with a

pe-dependent bias in which the optimal mechanism is stochastic.


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prme average decision so that incentives for truth-telling are unchanged. Doing so yields a strictnefit to the principal, who prefers more divergent decisions than his agent does.

Conclusion

Optimal multi-dimensional mechanisms are quite different from the simple delegation setsund in the one-dimensional case. The principal and the agent may differ not only on theirost preferred average decision but also on the distribution of those decisions. The possibility ofading off distortions along each dimension of the agents activities eases screening and leadsfully separating allocations. If the conflicts of interest between the principal and the agent are

fferent on each dimension, then delegation is always strictly valuable, there is no pooling ofpes, and the agent never chooses his ideal points. The spread on decisions that is necessary toduce cheaper information revelation is a decreasing function of the average decision taken bye agent. One-dimensional mechanisms, taking the form of interval(s) delegation sets (poolingtervals or intervals of full discretion) are optimal only when the biases between the principald agents ideal actions are exactly the same along all dimensions.Such extended possibilities for screening provide a strong reason to merge principals who

ntrol different dimensions of the agents activities and let them jointly design mechanisms. Atorst, the analysis of the cooperative contracting design undertaken in this paper characterizese benefits of cooperation in settings where divided control is otherwise often pervasive. Reg-ation by different agencies and bureaucratic oversights by different legislative committees arecourse two examples in order. From a theoretical viewpoint, this comparison may depend on

e fine details of the contracting possibilities available under a non-cooperative design. For in-ance, the non-cooperative outcome may depend on whether or not principals observe decisionsat they do not directly control20 and whether the agents messages towards each principal areivate or public, the latter case being a priori closer to the cooperative outcome developed inis paper.21It would also be worth investigating optimal mechanisms in more complex environments al-

wing more general utility functions, more than two decisions and more general distributionstypes. Some relatively easy extensions would be to investigate optimal mechanisms when the

incipal and the agent value differently the losses on each dimension, while keeping quadraticility functions. We conjecture that the simple decomposition in terms of average decisions andread would generalize and would still be useful in characterizing optimal mechanisms in suchntexts. More dispersion in decisions is certainly needed when the agent makes decisions thatcounter to what the principal would like on average.Finally, it would also be interesting to extend our approach by allowing for multi-dimensional

eferences, as in the cheap talk framework developed in [12,13] and [14]: the agents bliss pointseach dimension of his activity being not necessarily perfectly correlated. This extension is alsoely to meet strong technical difficulties, but certainly deserves some attention.22 For all those

Martimort [38] described this situation as a case of public agency.A previous version of this paper [31] analyzed public mechanism with two principals (see also [25] for further analysis

the cheap talk setting combining both public and private messages). It was shown that there is no non-cooperativeuilibrium with continuous and deterministic action rules. The characterization of the equilibrium mechanisms withblic mechanism remains an interesting open problem.The literature on multi-dimensional screening has already stressed that pooling allocations are pervasive in nonlinearicing environments [8,20,46].


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utses, we conjecture that the decomposition between the average decision and its spread willay a crucial role in contract design.

cknowledgments

We thank Thomas Palfrey for helpful discussions at the early stage of this project, Ricardolonso, Wouter Dessein, Jeffrey Ely, Larry Samuelson, Aggey Semenov, Shan Zhao, two anony-ous referees and an associate Editor for useful comments and suggestions. The usual disclaimerplies.

ppendix A

roof of Proposition 1. See [27] and [42]. roof of Lemma 1. Necessity: For all pairs (, ) 2, incentive compatibility entails that:

2i=1

(xi()

)2 2i=1

(xi()

)2and

2i=1

(xi()

)2 2i=1

(xi()

)2. (A.1)

umming those inequalities yields:2

i=1

(xi() xi()

)( ) 0. (A.2)

ence,2

i=1 xi() is non-decreasing in . Therefore, it is almost everywhere differentiableith, at any point of differentiability, a derivative such that (4) holds. At such a point, ancentive-compatible mechanism must also satisfy the first-order condition of the agents rev-ation problem, namely (5). Moreover, using (A.1), we get:

2i=1

x2i () 2

i=1x2i ( ) 2

( 2i=1

xi() 2

i=1xi()

).

ence,2

i=1 x2i () is non-decreasing in when2

i=1 xi() is itself non-decreasing.2

i=1 x2i ()thus almost everywhere differentiable.Sufficiency: That 2i=1 xi() is non-decreasing in is then also a sufficient condition fortimality.23 Indeed, since

2i=1 x2i () and

2i=1 xi() are both non-decreasing in and thus

most everywhere differentiable with, at any point of differentiability, a derivative which iseasurable, Theorem 3 in [48, p. 100] entails:

2i=1

(xi()

)2 2i=1

(xi()

)2

2

i=1

xi(s)(xi(s)

)ds

Garcia [22] provides an analysis of the multi-dimensional adverse selection model in a framework with quasi-linear

ility functions, but focuses a priori on differentiable mechanisms.


w

P

Pex

A

123

L

T

w

N

N

T

Leqpr=2

i=1

xi(s)(xi(s) s + s

)ds =

2i=1

xi (s)(s ) ds 0,

here the last equality follows from (5) and the last inequality from (4). roof of Lemma 2. The proof is standard and follows [43]. roof of Lemma 3. We proceed along the lines of [16, Chapter 4]. Let us first define thetended-value Lagrangian

L(s, v) ={L(s, v) if s v24 ,+ otherwise.

s required in [16, p. 167], we observe that:

. L(s, v) is B-measurable where B denotes the -algebra of subsets of RR;

. L(s, v) is lower semi-continuous;

. L(s, v) is convex in v.

We now define the Hamiltonian as H(s,p) = supvR{pv L(s, v)}. When s v2

4 ,(s, v) = L(s, v) is strictly convex in v and the maximum above is achieved for

p = Lv

(s, v). (A.3)his yields the maximand

v = 4(p + )

s4(p + )2 + 2 ,

hich gives

H(s,p) ={s +s(4(p + )2 + 2) 2 24 if s 0, otherwise.

ote that H(s,p) is differentiable on (,0) R. We get the following inequality:

H(s,p) |s| + |s| + 2|p + ||s| 2 24

.

ow using|s| 1 + |s|2 and |p + | |p| + , we finally obtain:

H(s,p) + 2 2 2

4+ 2|p| + |s|

(1 + +

2+ |p|

)

2 + 2|p| + |s|(

1 + + 2

+ |p|). (A.4)

his is a growth condition on the Hamiltonian as required in [16, Theorem 4.1.3]. emma 4. (See [16].) If L() satisfies conditions 1 to 3 above, H() satisfies the growthuation (A.4) and 10 L(U0(), U0()) d is finite for at least one admissible arc U0(), then

oblem P has a solution.


U

jo

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T

TU

th

N

T

w

sa

ex

ar

inagIt remains for us to show that 1

0 L(U0(), U0()) d is finite for at least one admissible arc

0(). Take U0() = 0, which corresponds to decisions x10() = x20() = . This arc does theb and yields 10 L(U0(), U0()) d = 2 + 24 .roof of Proposition 2. Preliminaries: We say that H satisfies the strong Lipschitz conditionar an arc U if there exists > 0 and a constant k such that for all p R and for all (s1, s2) (U, ) the tube of radius centered on the arc U , the following inequality holds:H(s1,p) H(s2,p) k(1 + |p|)|s1 s2|. (A.5)his property holds in our context when there exists > 0 such that U() < for all (i.e.,() is bounded away from zero, which will be the case for the solution we exhibit below). Overe relevant range where si 0, we have:

H(s1,p) H(s2,p)= s1 s2 + (s1 s2 )2 + 4(p + )2 .ote that |s1 s2 | = |s1s2|2s0 for some s0 T(U, ) from the Mean-Value Theorem.herefore, |s1 s2 | |s1s2|2 for small enough. Hence, we get:H(s1,p) H(s2,p)

|s1 s2|(

1 +

2 + 4(p + )22

) |s1 s2|

(1 + + 2( + |p|)

2

)

max{

1 + + 22

,1

}|s1 s2|

(1 + |p|),

hich is (A.5) with k = max{1 + +22 , 1 }.Euler equation and boundaries conditions: From [16, Theorem 4.2.2, p. 169], and since L()

tisfies conditions 1, 2, and 3 above and H() satisfies the strong Lipschitz condition (A.5), thereists an absolutely continuous arc p() such that the following conditions hold for the optimalc U().

Optimality conditions for the Hamiltonian H():

p() = Hs

(U(),p()

), (A.6)

U() = Hp

(U(),p()

). (A.7)

Boundary conditions:

p(0) = p(1) = 0. (A.8)

Using (A.3) yields p() = LU

(U(), U()). Differentiating with respect to , insertingto (A.6) and observing that H

s(U(),p()) = L

U(U(), U()) yields (15). Finally,

Lain using p() = U

(U(), U()) yields (16).


Ifinan

Bat

m

Frinx2

Pqu

w

et

SiContinuity: First observe that, a.e. on , we have by definition

H(U(),p()

)= p()U() L(U(), U()) p()v L(U(), v),v 2U(). (A.9)

U is not continuous at some 0 (0,1), there exists an increasing sequence n and a decreas-g sequence +n (n 1) both converging towards 0, such that (A.9) applies at n , 0 and +n ,d (using monotonicity to get the strict inequality):

limn+ U

(n )= U(0 )< U(+0 )= limn+ U(+n ).ecause L(s, v) is continuous in (s, v) and U() is absolutely continuous and thus continuous0, we have:

L(U(0), v

)= limn+L

(U(n), v)

and

L(U(0), U

(0))= lim

n+L(U(n), U

(n))

. (A.10)

Taking = n into (A.9) and passing to the limit, using the continuity of p(), yieldsp(0)U

(0 ) L(U(0), U(0 )) p(0)v L(U(0), v) v 2U().Using similar arguments with the sequence +n , we also get

p(0)U(+0 ) L(U(0), U(+0 )) p(0)v L(U(0), v) v 2U().

Hence, the function v p(0)v L(U(0), v) defined for v 2U(0) achieves itsaxima at both U(+0 ) and U(

0 ). Since it is strictly concave, we get U

(+0 ) = U(0 ).om this contradiction, we conclude that any arbitrary is contained in a relatively openterval on which U is almost everywhere equal to a continuous function. U and thus x1 andare continuous on .

roof of Proposition 3. Since the functional L() does not depend on , we can obtain a firstadrature of (15) on any interval where U() + U2()4 < 0 as:

L(U(,), U(, )

) U (, )LU

(U(,), U(, )

)= + 2 + 24

, (A.11)

here a priori R and where we make explicit the dependence of the solution on this param-er. We obtain immediately:

U(,) + U(, ) +

U(,) U2(, )

4

U (, )( U(,)

4

U(,) U2(,)4

)= .

mplifying yields:

U(,)

(1 2

)= .U(,) U (,)4


S

w

U

or

di

H

Sthco

Pfrfi

thU

theqpa

Uth

w

ve

24olving for U (, ) yields

U2(, ) = 4(U(,) + 2

(U(,)

U(,) + )2)

, (A.12)

hich requires 2( U(,)U(,)+ )

2 U(,) or (U(,) + )2 + 2U(,) 0 given that(, ) 0, since by definition the agents information rent is negative. Solving the second-der equation (A.12) and keeping only the positive root,24 we get (17). When U (, ) > 0,fferentiating (17) with respect to yields:

U (, ) +U (, )(1 + 22 U(,)

(U(,)+)3 )U(,) 2( U(,)

U(,)+ )2

= U (, ) + 2(

1 + 22 U(,)(U(,) + )3

)= 0. (A.13)

ence, on any interval where U (, ) > 0, the second-order condition (11) can be written as:

0 U(, ) + 2 = 42 U(,)(U(,) + )3 . (A.14)

ince U(,) 0 holds, 0 also entails U(,)+ 0 and then (A.14) holds. This imposese requested restriction on the admissible solutions to (17). Finally, note that the second-orderndition (11) obviously holds on any interval where U (, ) = 0.

roof of Theorem 1. The structure of the proof is as follows. First, we derive from the necessaryee end-point conditions (16) some properties of the boundary values of U that are used tond . Sufficiency follows.

Necessity: Define the function P(x) = x((x+)2+2x)(x+)2 . For x < 0, P(x) > 0 if and only if

e second degree polynomial (x + )2 + 2x is everywhere positive. This is so when 24 2 (a conditionrified below), (A.15) admits two solutions respectively given by:Since it corresponds to an average decision x() biased towards the principal, namely x() (see Eq. (10)).


N

pa

re

T

w

an

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ar

2

O

sibe

Tinsu

ofw

ofU(0) = 12(2 + 42 +

(2 + 42)2 + 4(2 + 42) ), (A.16)

U(1) = 12(2 + 42

(2 + 42)2 + 4(2 + 42) ). (A.17)

ote in particular that (A.15) implies that both U(0) and U(1) are negative.The last step is to show that there exists (24 2,

2

4 ) such that the correspondingth U() = U(,) solving (17) and starting from:

U(0, ) = 12(2 + 42 +

(2 + 42)2 + 4(2 + 42) ),

aches

U(1, ) = 12(2 + 42

(2 + 42)2 + 4(2 + 42) ).

his requires a solution to the equation

() = (), (A.18)ith

() = U(1, ) U(0, ) =(

2 + 42)2 + 4(2 + 42), (A.19)d

() =1

0

U(, ) d =1

0

2

U(,) 2

(U(,)

U(,) + )2

d,

here the path U(,) starts from the initial condition U(0, ). Note that both () and ()e continuous in . It is clear that () is strictly increasing in with, for 1 = 24 2 and= 24 ,

(1) = 0 < 2

2 + 42 = (2).n the other hand, note that

(1) > 0 = (1), (A.20)nce the path U(,1) starting from U(0, 1) is strictly increasing. Moreover, for 2, (17) can

rewritten as:

U (, 2) = 2U(,2) |U(,2) 2||U(,2) + 2| . (A.21)

he path solving (A.21) and starting at U(0, 2) (note that U(0, 2) < 2 < U(1, 2)) is strictlycreasing everywhere and cannot cross the boundary U = 2 because the only solution to (A.21)ch that U(1) = 2 for a given 1 > 0 is such that U() = 2 for all since the right-hand side(A.21) satisfies a Lipschitz condition at any point U(,2) away from zero: a contradiction

ith U(0, 2) < 2. From that, we deduce U(,2) < 2 for all . Hence, the following sequenceinequalities holds:

(2) =1U(, 2) d < 2 U(0, 2) = 2 U(1, 2) + (2).0


F

CFFw

RU

w

TU

th

P

U

B

Pfo

ex

T

B

w

painally, we get:

(2) < (2). (A.22)ombining Eqs. (A.20) and (A.22) yields the existence of (1, 2) such that () = ().or such (24 2,

2

4 ) we have U(1, ) < 0 and thus U() = U(,) < 0 for all .

rom Proposition 3 this entails U() > 0 for all . Finally, using 2x() = U()+2 and (A.14)ith U() < 0 and < 0 we get x() > 0 for all .

emark 3. Note that, when goes to zero, the solution to (A.18) converges to 0. Indeed,(1, ) (resp. U(0, )) converge towards 0 (resp. 2) as and go together to zero. Thus,() converges itself towards 2. At the same time () converges towards

10 2

U0() d ,here U0() is the solution to U () = U0() starting from lim,0 U(0, ) = 2.his solution is of course the solution to the one-dimensional problem; in particular, we have0(2) = 0 and

10 2

U0() d = 2.

Sufficiency: Sufficiency follows from [16, Chapter 4, Corollary, p. 179] since L() satisfiese convexity assumption and the function s H(s,p()) is concave in s. roof of Proposition 4. Note that:

L

(,) =1

0

(2 U())d = 2 (U(1) U(0)). (A.23)

sing Eq. (A.19) we get:L

(,) = 2 (

2 + 42)2 + 4(2 + 42). (A.24)ecause 24 , we finally obtain L

(,) 2(1 2 + 42 ) > 0 when2 + 42 < 1. roof of Proposition 5. Condition (22) follows from a remark in the text. The strict inequalityllows from the fact that the simple mechanism is never optimal when > 0.Let us denote by U(, ,) the solution of the principals problem where we now make

plicit the dependence of this function on the preference parameters (,). By the Envelopeheorem, we get from (13):

L

(,) =1

0

(

2

U(, ,) U2(, ,)

4

)d. (A.25)

y Eq. (17), this derivative can also be rewritten as:

L

(,) = 1

0

(12

U(, ,)U(, ,) + (,)

)d, (A.26)

here we also make explicit the dependence of the constant of integration on the preference

rameters (,).


Fr

w

OU

(

U

al

P

U

Fi

P(5x1

U4fo

P

TNote that the function (x) = xx+(,) is decreasing and concave since (,) 0. (We have

(x) = (,)(x+(,))2 0 and

(x) = 2(,)(x+(,))3 0 since U(, ,)+(,) 0 for all .)

om this, we deduce the following chain of inequalities:1

0

(U(, ,)

)d

( 10

U(, ,)d

)>

((,)

)= 12

(A.27)

here the last inequality holds if and only if:1

0

U(, ,)d < (,). (A.28)

bserve that this inequality is true when = 0. Indeed, in that case, we know that U(1, ,0) (0, ,0) = 42 (for 1/2) and this condition is compatible with (A.19) only when,0) = 0. We also know that 10 U(, ,0) d = 20 U(, ,0) d < 0. From Remark 3,(, ,) and (,) are such that (A.28) also holds for small enough. In turn, (A.27) holdsso for small enough. Inserting into (A.26) ends the proof. roof of Corollary 1. First, note that

x(0) = U(0)2

=U(0)((U(0) + )2 + 2U(0))

|U(0) + | .sing (A.16), we get:

x(0) = 2 |U(0)|

|U(0) + | < 2.

nally, we obviously have x() = U()2 > 0 for all . roof of Corollary 2. The first property follows from (19). Now, from the incentive constraint) and the first property of the corollary we get the second property. Next, using (19) we have(0) 0 if and only if

U(0)2

(

U(0)U(0) +

).

sing (17) and simplifying, we get 22U(0)(U(0) + )2, i.e., by (A.15), 22 2 +2, which is always satisfied when 1 0 (with a strict inequality when 1 > 0). x2 (0) 0

llows now from the second property of the corollary. The last property is proved similarly. roof of Corollary 3. From Theorem 1, Eq. (19), we have:

t() = 2(

U()U() +

)2.

herefore, we get:

2 U()t () = 2 U ()(U() + )3 < 0,


si

w

w

w

P

wnce < 0 and U() > 0 for all . Using (A.16), we get

t(0) = 2(

U(0)U(0) +

)2< 2

1 > U(0)

U(0) + = 1 +2

2 + 42 +(2 + 42)2 + 4(2 + 42) ,hich holds since < 0. Still using (A.16), we also get:

t(0) > 2

4

12

0, (A.29)

hich holds since > 24 2. Now using (A.17), we get:

t(1) = 2(

U(1)U(1) +

)2U(1)

U(1) + = 1 +2

2 + 42 (2 + 42)2 + 4(2 + 42) 2 + 42 + 4 24 2 and < 0. roof of Proposition 6. We have:

10

L(U(), U()

)d

=1

0

(L0(U(), U()

)

U() (U())2

4+

2

4

)d

1

0

(L0(U0 (), U0 ()

)

U() (U())2

4+

2

4

)d

= 2 + 2

4 4

3

3 2

10

U()U() + d,

here U0 () is given in (20) and the last inequality follows from (17). This entails:1L(U(), U()

)d 2 3

2 4

3.04 3


W

an

w

Pw

an

Tsu

deco

Ta

w

tin

w

Te also have:1

0

L(U(), U())d = 2 43

3,

d1

0

L(UO(), UO()

)d = 2 +

2

4 2

31

3 2

32

3= 2 +

2

4 4

3

3 2,

hich gives the required inequalities. roof of Proposition 7. A stochastic direct mechanism is a mapping (|) : (K K)here (K K) is the set of measures on K K. For further references, we define the meand variance of this stochastic mechanism as

xi ( ) =

KKxi d(x1, x2| ) and 2i ( ) =

KK

(xi xi ( )

)2d(x1, x2| ) 0.

he boundedness of K ensures that such moments exist. Note that deterministic mechanisms arech that 2i ( ) 0. For further references also, denote x( ) = 12

2i=1 xi ( ) the average (mean)

cision and y( ) = x2() x1() the spread of those mean decisions. In this context, incentivempatibility can be written as:

U() = max

KK

( 2i=1

12(xi )2

)d(x1, x2| ).

king expectations, we get:

U() = max

12

( 2i=1

(xi ( )

)2) z() = max

(x( ) )2 y2( )4

z( ),

here z() = 122

i=1 2i ( ) 0. From this, it immediately follows that U() is absolutely con-uous with a derivative defined almost everywhere as:

U () = 2(x( ) ), (A.31)ith

0 z() = U() U2()

4 y

2()

4. (A.32)

he principals expected loss with such a stochastic mechanism can be written as:1

0

( KK

( 2i=1

12(xi i)2

)d(x1, x2|)

)f () d

= 1 (

U() + U() + y() 2 2)

f () d.02 4


T

w

Cde

Pva

on

ea

plx

de

ou

w

Obe

Aco

byfux

b

the principal problem when stochastic mechanisms are allowed can be written as:

(Ps): min{UW 1,1(),z0}1

0

Ls(U(), U (), z()

)f () d,

here

Ls(U(), U(), z()

)= U() U()

U() U2()

4 z() + 2 +

2

4.

learly, the pointwise solution to this problem when > 0 is achieved for z() = 0, i.e., forterministic mechanisms.

roof of Proposition 8. Suppose that the optimal solution U is such that U() = 0 on an inter-l I with non-empty interior, i.e., x1 () = x2 () = (or equivalently x() = and t() = 0)I . Let [a, b] be any connected interval including I having this property (the proof extends

sily to the case of several such intervals). Note in particular that incentive compatibility im-ies continuity of U and thus U(a) = U(b) = 0 with x(a+) = limxa+ x() = a and(b) = limxb x() = b. Note also that U is increasing in the left-neighborhood of a andcreasing in the right-neighborhood of b.First assume that 0 < a < b < 1. Fix some > 0 small enough and consider a new (continu-s) utility profile U such that:

U () ={U() if [0,1]\[a u(), b + v()],2 if [a u(), b + v()], (A.33)

here u() and v() are such that

2 =a

au()U() d =

b+v()b

U() d. (A.34)

bserve that u() and v() are positive when is small enough, and converge to zero with

cause U() = 2(x() ) is bounded.The corresponding decisions (x(), t()) are such that

x() ={x() if [0,1]\[a u(), b + v()], if [a u(), b + v()], and

t () ={t() if [0,1]\[a u(), b + v()],

2 if [a u(), b + v()].

lthough (x(), t()) may not necessarily be continuous at a u() or b + v(), it is incentivempatible. Indeed, x() is monotonically increasing both on the interiors of [au(), b+v()]construction and on [0,1]\[a u(), b + v()] because it is equal to x() there and this

nction is increasing. Moreover, the monotonicity of x also implies that x(a u()) = a =(a+) x() for a u() and similarly x(b + v()) = = x(b) x() for + u() which proves monotonicity everywhere.We now compute the loss difference between mechanisms (x( ), t( ))

and (x(),()) as:


B

H

Fr

10

L(U (),

U()

)f () d

10

L(U(), U()

)f () d

=a

au()

(L(U(),

U()

) L(U(), U()))f () d

+b

a

(L(U (),

U()

) L(U(), U()))f () d

+b+v()b

(L(U(),

U()

) L(U(), U()))f () d. (A.35)Computing the second of those integrals, we get:

A() b

a

(L(U (),

U()

) L(U(), U()))f () d = (2 )(F(b) F(a)).Turning now to the first of those integrals, observe that on the interval [a u(), a]:

L(U (),

U()

) L(U(), U())

= 2 + U() + U() +

U() (U)2()4

ecause 0U()2 on this interval, we get:

L(U (),

U()

) L(U(), U())

2 + U() +

U() (U)2()4

2 + U() + U() 2 + U().ence,

B()=

aau()

(L(U(),

U()

) L(U(), U()))f () d

a

au()

(

2 + U())f () d.

om the definition of u() and the fact that U is increasing in the interval [a u(), a] for

small enough:


Hb

pane

po

bya

thw

R

[[

[[

[[[

[[

[1

[1[1[1[1[1[1[1[1[1[2

[2[2[2

[2aau()

(

2 + U())f () d

(

max

f ())(

2u() + a

au()U() d

)=(

max

f ())

2(u() + ).

ence, B() is of order 2 in . A similar upper bound applies to the third integral C() =b+v()

(L(U(),U())L(U(), U()))f () d . B() and C() are thus negligible com-

red to A(). The sign of the left-hand side of (A.35) is thus given by the sign of A(), i.e., it isgative when is small enough.Since the mechanism (x( ), t( ))

is incentive compatible, we get a contradiction with thesited optimality of (x(), t())

and the corresponding utility profile U.

The construction is similar when 0 < a < b = 1, by letting v() = 0 and replacing Eq. (A.34)

2 = a

au() U() d , in which case the integral C() is equal to zero; similarly, when 0 =

< b < 1, we let u() = 0 and replace Eq. (A.34) by 2 = b+v()b

U() d , in which casee integral B() is equal to zero. Finally, when [a, b] = [0,1], we spread the decisions on thehole interval as in Example 1, in which case both B() and C() are equal to zero. eferences

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Optimal delegation with multi-dimensional decisions1 Introduction1.1 Related literature1.2 Organization of the paper

2 The model3 Preliminary results4 Optimal multi-dimensional mechanism4.1 Preliminary remarks4.2 Comparative statics4.3 Average decision4.4 Optimal spread4.5 Delegation sets4.6 Simple mechanisms

5 Extensions5.1 Non-optimality of stochastic mechanisms5.2 Not leaving full discretion is generic

6 ConclusionAcknowledgmentsReferences

optimal delegation with multi-dimensional decisions

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