Optimal Delegation with Multi-Dimensional Decisions ∗ Fr´ed´eric K OESSLER

†

David M ARTIMORT

‡

January 26, 2012

Abstract This paper investigates optimal mechanisms in a principal-agent framework with a twodimensional decision space, quadratic payoffs and no monetary transfers. If the conflicts of interest between the principal and the agent are different on each dimension, then delegation is always strictly valuable. The principal can better extract information from the agent by using the spread between the two decisions as a costly screening device. Delegation sets no longer trade off pooling intervals and intervals of full discretion but instead take more complex shapes. We use advanced results from the calculus of variations to ensure existence of a solution and derive sufficient and necessary conditions for optimality. The optimal mechanism is continuous and deterministic. The agent’s informational rent, the average decision and its spread are strictly monotonic in the agent’s type. The comparison of the optimal mechanism with standard one-dimensional mechanisms shows how cooperation between different principals controlling various dimensions of the agent’s activities facilitates information revelation. K EYWORDS: mechanism; Delegation; Mechanism Design; Multi-Dimensional Decision. JEL C LASSIFICATION: D82; D86.

1 Introduction Consider a principal who contracts with an agent who is privately informed. When the principal’s and the agent’s interests are conflicting, the principal may want to exert some ex ante control on the agent by restricting the decision set from which the agent picks actions. Examples of such constrained delegation abound across all fields of economics and political science. CEOs control division managers by designing capital budgeting rules and allocating decision rights among unit managers.1 Many different aspects of corporate decisions involving product design and quality, prices, or polluting emissions are scrutinized by regulators who may impose various limits on those variables. Lastly, Congress Committees exert ex ante control on better informed regulatory agencies by designing various administrative procedures and rules that limit bureaucratic drift and constrain the agencies’ discretion.2 ∗

We thank Thomas Palfrey for helpful discussions at the early stage of this project, Ricardo Alonso, Wouter Dessein, Jeffrey Ely, Larry Samuelson, Aggey Semenov, Shan Zhao, two anonymous referees and an associate Editor for useful comments and suggestions. The usual disclaimer applies. † Paris School of Economics – CNRS. Email: [email protected]. ‡ Paris School of Economics – EHESS. Email: [email protected]. 1 Harris and Raviv (1996) and Alonso, Dessein, and Matouschek (2008). 2 McCubbins, Noll, and Weingast (1987), Huber and Shipan (2002), Epstein and O’Halloran (1999).

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These examples share the common feature that principals make little use of monetary transfers to control their agents. Following the seminal works of Holmstrom ¨ (1977, 1984) and Melumad and Shibano (1991), these settings are fruitfully analyzed as mechanism design problems in which the principal commits to a decision rule but cannot use monetary transfers to implement that rule.3 With no transfers and when actions lie in a one-dimensional set, optimal mechanisms look rather simple. Quite intuitively, the principal finds it hard to induce information revelation and align conflicting objectives when he controls only a single action of the agent. In a one-dimensional setting, an optimal mechanism balances the flexibility gains of letting the agent freely choose this action according to his own private information and the agency cost deriving from the fact that the principal and the agent might have conflicting objectives. The first major result provided by the existing literature highlights the trade-off between “rules and discretion” that arises in such contexts. Inflexible rules allow the principal to choose his most preferred policy from an ex ante viewpoint, i.e., in the absence of any information. This is so because those rules make no use of the agent’s private information. Leaving full discretion to the agent, on the other hand, allows to implement state-dependent actions, but these choices now reflect only the agent’s preferences and not those of the principal. The second important result advanced by the literature is that the optimal mechanism (when continuous) can be implemented by means of interval delegation sets which set bounds on the agent’s action. This is an important theoretical insight because it reduces the design of the mechanism to a simple exercise consisting in finding those bounds. This simplification is also of great value when it comes to implementing the 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 the agent’s activities can be controlled by a principal or, equivalently, when several principals, each being endowed with the same bargaining power and each controlling a single decision of the agent, can cooperate in designing a common mechanism. First, one may wonder how the trade-off between rules and discretion is modified. Clearly, screening possibilities are now improved and pooling certainly seems less attractive. Second, in a multi-dimensional context, the agency problem between the principal and his agent may not only be related to their average conflict of interest over all dimensions but also to the distribution of conflicts across the different dimensions. The extent to which this is the case must also be clarified. These are highly relevant issues not only from a pure theoretical viewpoint but also because many real-world problems are multi-dimensional. For instance, when designing vertical restraints with his retailers, a manufacturer may not only leave them discretion on how to fix retail prices but also on some other dimensions like after-sales services. An economic regulator may put a stringent cap on prices charged by regulated firms while leaving more discretion in choosing environmental quality. In these contexts, it is important to understand whether and how treating each dimension separately excludes important screening 3

See Armstrong (1994), Baron (2000), Martimort and Semenov (2006), Alonso and Matouschek (2008), Goltsman, Horner, ¨ Pavlov, and Squintani (2009), Kovac and Mylovanov (2009), and Anesi and Seidmann (2011), among others.

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possibilities. As a first approach to these questions, this paper investigates the form of optimal mechanisms in a simple two-dimensional setting with quadratic and separable payoffs, a uniform distribution of types, and constant biases in players’ ideal actions. In such context, the optimal mechanism restricts the agent’s choices on each dimension through a smooth and deterministic delegation set. When the conflicts of interest between the principal and the agent on both dimensions differ, delegation is always strictly valuable. The optimal delegation set never includes the agent’s ideal points, nor does it exhibit any pooling. The interval delegation sets used in the one-dimensional case, which trade off pooling intervals and intervals with full discretion, are thus suboptimal. To evaluate incentive distortions in a multi-dimensional context, the two important factors are the average decision that the principal would like to implement and the spread, i.e., how far apart the principal would like to set the levels of the different activities. This spread plays a similar, albeit somewhat different, role to that played by transfers in standard models with quasilinear preferences and monetary payments. Like transfers in standard screening models, using a type-dependent spread as a screening device facilitates information revelation. To see how, let us assume that the agent would ideally like to choose the same decision on dimensions 1 and 2 of his activity but, on average, prefers lower levels of those decisions compared to the principal. The principal wants to avoid information manipulation aiming to implement such lower levels of activity. To limit the agent’s incentives to claim for lower average decisions, the principal might increase the spread between decisions 1 and 2 following such claims. By so doing, he imposes a cost on the agent and reduces his temptation to manipulate information in that direction. Conversely, the principal reduces this spread when the agent’s report induces higher levels of activity. Punishments and rewards are obtained by playing on the spread. The optimal spread is strictly positive and monotonic in the agent’s type whenever the conflicts of interest between the principal and the agent differ along each dimension. Moreover, we show that this spread is not a compromise between the principal’s and the agent’s ideal spread and it could be strictly greater. Compared with a setting where monetary transfers are available, utility is no longer transferable between the principal and the agent. Implementing a spread in the decisions also introduces complex costs and benefits for the principal. Let us assume that the principal would ideally prefer one more unit of activity 2 than of activity 1 for any realization of the agent’s private information, so that his ideal spread is just one, and let us also assume that both activities give more return to the principal than to the agent. In that case, the agent may want to pretend that lower average activities should be implemented. Increasing the spread between the two activities above 1 for low average activity levels and reducing it below 1 for higher activity levels is of course costly for the principal, but it certainly facilitates screening. From a technical viewpoint, the nonlinearity due to the absence of monetary payments makes

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the characterization of the optimal mechanism quite complex.4 We use results from the calculus of variations (Clarke, 1990) to ensure the existence of a solution and derive sufficient and necessary conditions for optimality. Related Literature. Melumad and Shibano (1991) provided a significant analysis of the delegation problem with quadratic payoffs and a uniform distribution of types in contexts where no transfers are available and where the uninformed party (the principal) commits to a mechanism with the informed party (the agent). Martimort and Semenov (2006) and Alonso and Matouschek (2008) characterized settings where simple connected delegation sets are optimal, a feature that was a priori assumed in Holmstrom ¨ (1984), Armstrong (1994) and Baron (2000) for instance. Alonso and Matouschek (2007) applied the standard delegation model to a dynamic context where the principal and the agent interact repeatedly. Focusing on dominant strategy to get a sharp characterization of the set of incentive feasible allocations, Martimort and Semenov (2008) extended this mechanism design approach to the case of multiple privately-informed agents (lobbyists) dealing with a single principal (a legislature) in a political economy context where the principal chooses a one-dimensional policy.5 Farrell and Gibbons (1989) and Goltsman and Pavlov (2011) analyzed private and public mechanism with a single informed agent and two decision-makers. As in our model, the decisions on each dimension enter separately into the agent’s payoff function and are strategically independent across the two decision-makers. None of these papers has addressed the design of multi-dimensional mechanism with commitment. In that respect, the closest paper to ours may be Ambrus and Egorov (2010). These authors introduce the possibility that the principal “burns money” or imposes costly activities for the agent in an otherwise standard delegation set-up. Money burning constitutes a second instrument that facilitates screening, but the impact of these new screening possibilities on the principal’s payoff is different to that of our paper. Also related is a recent paper by Che, Dessein, and Kartik (2011) that analyzes a kind of dual problem to ours. They solve for optimal mechanisms in an environment without transfers in which the decision space is finite but the type space is multi-dimensional. Organization of the paper. Section 2 presents the model and the by-now standard result where a single activity of the agent is controlled by the principal. Section 3 presents some preliminary results on incentive compatibility and assesses the performances of simple and intuitive mechanisms that illustrate the new screening possibilities available in multi-dimensional environments. Section 4 is the core of the paper. We formulate the design problem using advanced tools of the calculus of variations and derive the optimal mechanism in this multi-dimensional context. Some robustness checks are provided in Section 5. Section 6 concludes and paves the way for future research. 4

The absence of monetary transfers makes the screening problem look like those that are found for instance in the optimal taxation literature (Mirrlees, 1971), where utility functions are also not quasi-linear. However, the techniques of this literature cannot be directly imported into our framework. Even with quadratic payoffs, the principal’s objective function may not be everywhere Lipschitz-continuous, contrary to what is assumed in the optimal taxation literature. 5 Austen-Smith (1993), Battaglini (2002, 2004), Krishna and Morgan (2001), Levy and Razin (2007), and Ambrus and Takahashi (2008), on the other hand, considered cheap talk settings with multiple privately-informed senders.

4

Proofs are relegated to the Appendix.

2 The Model A principal controls two actions, x1 and x2 , undertaken by a single agent on his behalf. We denote by (x1 , x2 ) the bi-dimensional vector of these actions. For simplicity, they lie in a compact set K = [−K, K] ⊆ R for K large enough. Utility functions are single-peaked and quadratic and the

differences between the principal’s and the agent’s ideal actions (the “biases”) are constant across

types. More precisely, the utility functions are respectively given for the principal and his agent by: 2

V (x1 , x2 , θ) = − and

1X (xi − θ − δi )2 , 2 2

U (x1 , x2 , θ) = − With these

(1)

i=1

preferences,6

1X (xi − θ)2 . 2

(2)

i=1

the agent’s ideal point on each dimension is xA (θ) = θ whereas the

principal has an ideal point located at xiP (θ) = θ + δi in dimension i, i = 1, 2. We denote by ∆ ≡ δ2 − δ1 the difference in biases between the two dimensions and by δ ≡

δ1 +δ2 2

the average

bias. We assume without loss of generality that δ1 ≤ δ2 and δ ≥ 0, but δ1 could be either positive

or negative.

The agent has private information on his ideal point (or type) θ. The agent’s type is drawn from a uniform distribution on Θ = [0, 1].7 The principal is uninformed about the agent’s type. The principal controls the whole vector of the agent’s activities (x1 , x2 ). From the Revelation Principle (Myerson, 1982), there is no loss of generality in restricting the analysis to direct mechanisms stipulating (maybe stochastic8 ) decisions as functions of the agent’s report on his type. Any deterministic mechanism is a mapping x(·) = {x1 (·), x2 (·)} : Θ → K2 . Remark 1 The model might also be viewed as describing a situation with a single common agent and two principals, P1 and P2 , whose utility functions are respectively: 1 Vi (xi , θ) = − (xi − θ − δi )2 , 2 6

i = 1, 2.

The choice of quadratic utility functions is a common restriction in the cheap talk and delegation literature. For instance, in an important paper, Alonso and Matouschek (2008) imposed this condition on the principal’s preferences while making no such explicit assumption on the agent. However, imposing this quadratic assumption on the agent’s preferences is without loss of generality in a one-dimensional context. Indeed, it can be easily seen that incentive compatibility constraints take the same form with more general preferences, as long as the agent’s preferences are single-peaked and symmetric. In our multi-dimensional context, this quadratic restriction is important to the extent that it pins down the marginal rate of substitution between each decision in a very specific way. This is a key ingredient of the tractability of our model. 7 The characterization of the optimal mechanism would be untractable if we were to assume other distributions. Section 5 nevertheless extends some of our qualitative results to more general distributions. 8 Analysis of stochastic mechanisms is deferred to Subsection 5.1, where we prove their suboptimality.

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Under a non-cooperative design with private mechanism between the agent and each principal, principals independently choose their mechanism spaces with the agent and design their own mechanism. Since there is no externality between principals (i.e., each principal Pi ’s utility only depends on the decision xi and the agent’s utility function is separable in the decisions controlled by each principal), each principal offers the same mechanism as if he alone was contracting with the agent. If principals cooperate in designing a common mechanism and have equal bargaining powers when designing the mechanism, the merged principal objective function is exactly that in Equation (1). Our analysis below thus reveals the gains from such cooperation between principals with equal bargaining power compared with a non-cooperative design.9 Let us now consider the case where the principal controls a single decision xi , for some i ∈

{1, 2} or, with the interpretation above, principals independently choose their mechanism spaces

with the agent and design their own mechanism.

Proposition 1 (One-Dimensional Activity; Holmstrom, ¨ 1977) In the one-dimensional case, the optimal mechanism xO i (·) is given by:   max {θ, 2δi } , O xi (θ) = min {θ, 1 + 2δi } ,   1/2 + δi ,

if δi ∈ [0, 1/2], if δi ∈ [−1/2, 0], if |δi | ≥ 1/2.

(3)

When a principal controls a one-dimensional activity of the agent, the optimal mechanism has a simple cut-off structure. For example, when δi ∈ (0, 1/2), the optimal action corresponds

to the agent’s ideal point if it is large enough and is otherwise independent of the agent’s type. This outcome can be easily achieved by means of an interval delegation set. Instead of using a direct revelation mechanism and communicating with the agent, the principal could also offer a menu of options Di = [2δi , +∞) and let the agent freely choose within this set. When the floor 2δi is not binding, the agent is not constrained by the principal and it is just as if he had full

discretion in choosing his own ideal point. When the floor is reached, the agent is constrained and can no longer choose his bliss point, which is too low compared with what the principal would implement himself. The optimal mechanism trades off the benefits of flexibility (the agent sometimes choosing a state-dependent action) against a loss of control (this state-dependent action being different from the principal’s ideal point). Setting a floor or a cap limits the agent’s discretion and reduces the loss of control. Clearly, this floor (resp. cap) increases (resp. decreases) with δi , meaning that a less rigid rule is chosen when the conflict of interest between the principal and the agent is less 9

The assumption of equal bargaining power between cooperating principals is innocuous when monetary transfers are feasible between those principals. There are instances where such assumption is quite natural. Think of the case of two regulatory agencies, one concerned by regulating prices, the other being interested in controlling pollution emissions. Although direct regulatory transfers with firms might be banned, cooperating agencies would merge their budgets and behave collectively as having a common objective as described in Equation (1).

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pronounced.10 When the conflict of interest is significant (|δi | > 1/2) the principal simply chooses

a pooling allocation at his expected ideal points. Delegation is not valuable.

3 Preliminary Results In the multi-dimensional case, incentive compatibility constraints can be written as: 2

θ ∈ arg max − ˆ θ∈Θ

2 1 X ˆ xi (θ) − θ . 2 i=1

Lemma 1 The necessary and sufficient condition for incentive compatibility is that

P2

i=1 xi (θ)

is non-

decreasing in θ and thus a.e. differentiable in θ. At any point of differentiability, we have: 2 X

x˙ i (θ) ≥ 0,

(4)

x˙ i (θ)(xi (θ) − θ) = 0.

(5)

i=1

2 X i=1

In this multi-dimensional world, the principal can now use both x1 (·) and x2 (·) to screen the agent’s preferences. To understand how this can be so, it is useful to observe that the principal could at least offer the optimal one-dimensional mechanisms that he would choose if he was controlling each dimension alone, namely the pair of mechanisms described in Proposition 1. Although mechanisms that would satisfy (3) for i = 1, 2 also satisfy (4) and (5), more mechanisms are now incentive compatible. By trading off distortions along each dimension or by choosing actions that vary in opposite directions on each dimension as the agent’s type changes, the principal might for instance induce countervailing incentives that facilitate information revelation.11 This characterization of incentive-compatible allocations already gives some powerful insights into the properties of mechanisms when we examine a couple of simple two-dimensional mechanisms. Example 1 Consider the linear mechanism {xα1 (θ), xα2 (θ)}θ∈Θ such that xα1 (θ) = θ − α and xα2 (θ) =

θ + α, where α is a fixed number. This mechanism is incentive compatible since it satisfies both (4) and (5). The best of such mechanisms maximizes the principal’s profit, i.e., α should be optimally

chosen so that any concession made by the principal on x1 by moving this decision closer to the 10

As shown in Alonso and Matouschek (2008), this might not be the case when the bias δi is state-dependent. In addition, the delegation set might have several intervals of full discretion. 11 The literature on countervailing incentives (Lewis and Sappington, 1989a,b and Laffont and Martimort, 2002, Chapter 3 among others) has been developed in settings with monetary transfers. Sometimes those models generate pooling as an optimal response to simultaneous incentives to over- and under-report types as in Lewis and Sappington (1989b). In our model, on the contrary, pooling is never an issue, as we show below.

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agent’s own ideal point is compensated by an equal shift in x2 in the direction of the principal’s ideal point. Typically, α = arg min α

Z

0

1

∆ 2

does the trick, since: !

2 X (xαi (θ) − θ − δi )2 i=1

dθ = arg min(α + δ1 )2 + (α − δ2 )2 = α

∆ . 2

In sharp contrast to the one-dimensional case, this mechanism induces full separation of types, never gives the agent his ideal action (as long as ∆ 6= 0) and already achieves the first best when principals have opposite biases, i.e., δ2 = −δ1 .

Still, when δ > 0, the principal can find that the above decisions are too close to the agent’s ideal points. The mechanism {xα1 (θ), xα2 (θ)}θ∈Θ can be improved by introducing a pooling area

over the lower tail of the distribution as in the one-dimensional case. This is illustrated in the next example. Example 2 (Simple Mechanism) Consider the incentive compatible mechanism {˜ x1 (θ), x ˜2 (θ)}θ∈Θ

defined as:

( θ˜ − x ˜1 (θ) = θ−

∆ 2 ∆ 2

if θ ≤ θ˜ and x ˜2 (θ) = otherwise

( θ˜ + θ+

∆ 2 ∆ 2

if θ ≤ θ˜ otherwise.

(6)

˜ In what follows, This new mechanism is obtained by imposing floors on each dimension for θ ≤ θ.

the optimal so-called simple mechanism within this class is denoted as {˜ x∗1 (θ), x ˜∗2 (θ)}θ∈Θ . It is such that:

˜∗

θ = arg min θ˜

Z

0

1

2 X i=1

2

(˜ xi (θ) − θ − δi )

!

  1 dθ = min 2δ, + δ . 2

This mechanism has a non-trivial pooling area when 0 < 2δ < 1. This pooling area is an average between the pooling areas that the principal would choose when designing an optimal mechanism on each dimension separately. ˜∗2 (θ)}θ∈Θ is instructive because it stresses two features of optiThe simple mechanism {˜ x∗1 (θ), x

mal mechanisms that our more general analysis will confirm. First, the principal trades off distor-

tions on each dimension by introducing a strictly positive spread between x1 and x2 for every type θ, so the agent never gets his ideal actions. Second, decision rules are rather flat on the lower tail of the distribution. However, contrary to what happens with this simple mechanism, the optimal mechanism will never exhibit any pooling, the spread between the two dimensions will not be constant with the agent’s type, and one action might be decreasing on the upper and lower tails of the distribution without conflicting with incentive compatibility.

4 Optimal Multi-Dimensional Mechanism To characterize the optimal mechanism, it is useful to re-parameterize our problem with a new set of variables. This transformation not only brings new insights into the nature of the economic 8

problem but it will also subsequently facilitate the proof that stochastic or discontinuous mechanisms are not optimal. Consider thus the following two extra auxiliary variables which are the average decision and a measure of the spread of those decisions: 2

2

i=1

i=1

1X 1 1X xi (θ) and t(θ) ≡ (xi (θ) − x(θ))2 = (x2 (θ) − x1 (θ))2 . x(θ) ≡ 2 2 4

(7)

Under complete information, the principal would like to choose an average decision xP (θ) = θ + δ and an optimal spread tP (θ) =

∆2 4 .

These two quantities differ from those that would be ideally

chosen by the agent on his own, namely, xA (θ) = θ and tA (θ) = 0. Solving the system of equations (7) for x1 (θ) and x2 (θ) yields immediately:12 x1 (θ) = x(θ) −

p

t(θ) and x2 (θ) = x(θ) +

p

t(θ).

(8)

We now define the agent’s non-positive information rent U (θ) as: ! 2 2 1 X ˆ xi (θ) − θ . U (θ) ≡ max − ˆ 2 θ∈Θ i=1

Using (8) and incentive compatibility, we rewrite: ˆ − θ)2 − t(θ). ˆ U (θ) = −(x(θ) − θ)2 − t(θ) = max −(x(θ) ˆ θ∈Θ

(9)

With this formulation, the agent’s rent now only depends on the screening variables through the average decision x(θ) and the spread t(θ). This utility function becomes “quasi-linear” with the spread or “transfer” t(θ) ≥ 0 measuring the cost for the agent of choosing different decisions along each dimension. Varying that cost with the realization of his private information certainly

eases screening. The technical difficulty that we will face in what follows comes from the fact that this transfer does not enter linearly into the principal’s objective. The average decision x(θ) has an impact on the agent’s marginal utility, which depends on his type. It can thus be used as a screening variable as in standard screening models. Clearly, an agent of type θ may be tempted to lie ”downwards” to move the average decision closer to his own ideal point. The principal can make that strategy less attractive by increasing the spread between decisions for the lowest types.13 As usual in screening problems with quasi-linear utility functions, the incentive compatibility conditions (4) and (5) can be restated in terms of the properties of the pair (U (θ), x(θ)). 12

The other solution, x1 (θ) = x(θ) +

p

t(θ) ≥ x2 (θ) = qx(θ) −

p

t(θ), is not optimal for the principal because his loss

˙2 in Equation (13) below would be L∆ (U (θ), U˙ (θ)) + 2∆ −U (θ) − U 4(θ) instead of L∆ (U (θ), U˙ (θ)). 13 The principal-agent literature has stressed that a principal can use the agent’s risk-aversion to ease incentives (see, e.g., Arnott and Stiglitz, 1988) by for instance using stochastic mechanisms. Introducing some spread in the agent’s decisions in a model with quadratic payoffs has a similar flavor. Subsection 5.1 shows that stochastic mechanisms are suboptimal in our framework.

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Lemma 2 U (θ) is absolutely continuous with a first derivative defined almost everywhere and, at any point of differentiability: U˙ (θ) = 2(x(θ) − θ).

(10)

The average decision x(θ) is non-decreasing and thus almost everywhere differentiable with, at any point of differentiability: x(θ) ˙ =

¨ (θ) U + 1 ≥ 0. 2

(11)

Note that the non-negativity of the spread entails: t(θ) = −U (θ) −

U˙ 2 (θ) ≥ 0, 4

(12)

with an equality only when x1 (θ) = x2 (θ) = x(θ), i.e., when both decisions are equal. With the new set of variables, we rewrite the principal’s loss in each state of nature θ as: s 2 U˙ 2 (θ) ∆2 1X (xi (θ) − θ − δi )2 ≡ L∆ (U (θ), U˙ (θ)) = −U (θ) − δU˙ (θ) − ∆ −U (θ) − + δ2 + . (13) 2 4 4 i=1

From this, we get the following expression of the principal’s relaxed problem, neglecting for the time being the monotonicity condition on x(θ) that will be verified ex post: (P∆ ) :

min

Z

1

U ∈W 1,1 (Θ) 0

L∆ (U (θ), U˙ (θ))dθ,

where W 1,1 (Θ) denotes the set of absolutely continuous arcs on Θ.14 We now proceed as follows. First, we prove existence of an optimal arc in W 1,1 (Θ). Second, we characterize this arc by means of a second-order Euler-Lagrange equation. Third, a first quadrature tells us that this solution solves a first-order differential equation known up to a constant. Finally, we impose conditions on that constant so that the monotonicity condition (11) always holds. Lemma 3 A solution U ∗ (·) to (P∆ ) exists. Once this solution is known, the pair of decision rules {x∗1 (·), x∗2 (·)} is recovered using the

formulae:

U˙ ∗ (θ) x∗1 (θ) = θ + − 2

s

(U˙ ∗ (θ))2 U˙ ∗ (θ) −U ∗ (θ) − and x∗2 (θ) = θ + + 4 2

14

s

−U ∗ (θ) −

(U˙ ∗ (θ))2 . (14) 4

In the parlance of the calculus of variations, (P∆ ) is actually a Bolza problem with free end-points (see Clarke, 1990, Chapter 4). It is non-standard because even though the functional L∆ (s, v) is continuous and strictly convex in (s, v), 2 it is not everywhere differentiable (or even Lipschitz), especially at points where −s − v4 = 0 if any such point exists on an admissible curve where v(θ) = U˙ (θ) and s(θ) = U (θ).

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Proposition 2 An optimal arc U ∗ (·) is such that: • The following Euler-Lagrange equation holds at any interior point of differentiability:   d ∂L∆ ∗ ∂L∆ ∗ (U (θ), U˙ ∗ (θ)) = (U (θ), U˙ ∗ (θ)) ; ∂U dθ ∂ U˙

(15)

• The following free end-point conditions hold on the boundaries of the interval [0, 1]: ∂L∆ ∗ ∂L∆ ∗ (U (θ), U˙ ∗ (θ))|θ=0 = (U (θ), U˙ ∗ (θ))|θ=1 = 0; ∂ U˙ ∂ U˙

(16)

• U ∗ (θ) is continuously differentiable, and thus x∗1 (θ) and x∗2 (θ) are continuous. The next proposition investigates the nature of the solution to the second-order ordinary differential equation (15) by obtaining a first quadrature parameterized by some integration constant λ ∈ R. This constant must be non-positive to ensure that the second-order condition (11) holds. Proposition 3 For each solution U (θ, λ) to (15) which is everywhere negative and satisfies (11), there exists λ ∈ R− such that:15 s

U˙ (θ, λ) = 2

−U (θ, λ) − ∆2



U (θ, λ) U (θ, λ) + λ

2

,

(17)

(U (θ, λ) + λ)2 + ∆2 U (θ, λ) > 0, for all θ ∈ Θ.

(18)

and

Reminding the reader that if δ = 0 the principal can achieve the first-best while, if ∆ = 0, the optimal mechanism is the twice-replica of the one-dimensional solution, we are now ready to characterize the optimal mechanism in the multi-dimensional case for the other cases relevant for our analysis. Theorem 1 (Two-Dimensional Activity) For all (δ, ∆) ∈ R2++ , the optimal mechanism entails the

following properties.

• Optimal decisions on each dimension are never equal to the agent’s ideal points: x∗1 (θ) = x∗ (θ) − 2

2

∆U ∗ (θ) ∆U ∗ (θ) ∗ ∗ < x (θ) = x (θ) + . 2 U ∗ (θ) + λ∗ U ∗ (θ) + λ∗

with λ∗ ∈ (− ∆4 − δ2 , − ∆4 ) and x∗ (θ) = θ +

(19)

U˙ ∗ (θ) 2 ;

15

It is important to note that the differential equation (17) may a priori have a singularity and more than one solution going through a given point. This might be the case when, for this solution, there exists θ0 such that U (θ0 , λ) +  2 0 ,λ) ∆2 UU(θ(θ = 0. The right-hand side of ( 17) fails to be Lipschitz at such a point. It turns out that this possibility ,λ)+λ 0 does not arise for the optimal mechanism described below, because a careful choice of λ ensures that the condition (18) holds everywhere on the optimal path.

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• The rent profile U ∗ (θ) is everywhere negative, strictly increasing and solves (17) for λ∗ . • There is no pooling area. Monotonicity conditions are satisfied everywhere: x˙ ∗ (θ) > 0. Preliminary remarks. In the limiting case where ∆ = 0, we already know that the optimal mechanism will coincide with that described in Proposition 1 for the one-dimensional problem, with U0∗ (θ) = −(min{θ − 2δ, 0})2 ,

x∗0 (θ) = max{θ, 2δ}

and

λ∗ = 0,

(20)

when δ ≤ 1/2.16 Exactly as in Examples 1 and 2, when δ1 = δ2 , there is no gain for the principal

in trading off distortions on each dimension, because there is no conflict of interest between the principal and the agent concerning their ideal spread. It is therefore costly for the principal to use a spread on decisions. On the contrary, when ∆ > 0, the double replica of the one-dimensional mechanism is suboptimal. This is particularly illuminating when biases on each dimensions are just opposite, i.e., δ = 0, since then the principal can get his first-best xi (θ) = θ + δi with a constant spread. Beyond the special cases where δ = 0 or ∆ = 0, several features of the optimal mechanism are worth emphasizing when the principal and the agent have conflicting preferences. Comparative Statics. The third result of the previous theorem shows that, contrary to existing models of delegation (see for example Alonso and Matouschek, 2008), pooling is never optimal, whatever the conflicts of interest (δ1 and δ2 could be arbitrarily large), as long as δ1 6= δ2 . Therefore,

when conflicts of interest between the principal 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 preferences (the average conflict δ and the dispersion of the biases, ∆) for our multi-dimensional environment. For this purpose, let us denote the principal’s expected loss with preferences (δ, ∆) at the optimal mechanism by: ∗

L (δ, ∆) =

Z

1

L∆ (U ∗ (θ), U˙ ∗ (θ))dθ.

0

The next proposition extends the standard “Ally Principle”17 to our multidimensional environment by showing that the principal prefers to appoint an agent with a smaller average bias, at least when the biases are not too large (∆2 + 4δ2 < 1).18 When δ ≥ 1/2 we have U0∗ (θ) = −(1/2 + δ − θ)2 and x∗0 (θ) = 1/2 + δ. Huber and Shipan (2006) survey the political science literature on this topic. Since the seminal work of Crawford and Sobel (1982), the literature on mechanism and delegation in settings with private information and conflicting interests has flourished by pursuing the analysis of this “Ally Principle” (see Alonso and Matouschek, 2008, Gilligan and Krehbiel, 1987 and Dessein, 2002 for instance). 18 We conjecture that this property is always valid, but we have no analytical proof for ∆2 + 4δ 2 > 1. 16

17

12

Proposition 4 If ∆2 + 4δ2 < 1 and δ > 0, then a greater average conflict strictly increases the principal’s expected loss:

∂L∗ (δ, ∆) > 0. ∂δ

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Comparative statics with respect to ∆ are more difficult to obtain because increasing the spread between the principal’s and the agent’s ideal points has two effects. On the one hand, it facilitates screening. On the other hand, it also increases the divergence between the principal’s and the agent’s preferences, thereby increasing the cost of rewarding the agent for truthful revelation by letting him choose points close to his ideal ones. Note, however, that the simple mechanism (Example 2) is available for any ∆ and yields a loss worth δ2 −

4δ3 3

when δ ≤ 1/2 and 1/12 when

δ ≥ 1/2, which is independent of ∆. This value corresponds to the optimal loss when ∆ = 0.

More generally, the following proposition shows that L∗ (δ, ∆) is strictly decreasing in the

spread, at least when the average bias and the spread are not too large.19 Proposition 5 For all (δ, ∆) ∈ R2++ , the following property holds: L∗ (δ, ∆) < L∗ (δ, 0).

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If in addition δ ≤ 1/2, then the principal’s expected loss L∗ (δ, ∆) is strictly decreasing with ∆ when ∆ is

small enough.

Altogether, Propositions 4 and 5 show that there always exist some directions in which preferences can be changed to improve the principal’s payoff. In particular, when 2δ > 1, we already know from the study of one-dimensional mechanisms that not using the spread leads to an allocation with full pooling. On the contrary, increasing even slightly the spread on biases is always beneficial 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 the optimal average decision and spread. Average decision. Contrary to what happens in the one-dimensional case, even when the agent’s ideal point is close to θ, the contract has no pooling; x∗ (θ) is everywhere monotonically increasing. This stands in sharp contrast to the one-dimensional case. The next corollary further shows that the average decision lies systematically in a greater interval than if the principal was restricted to ˜∗2 (θ)}θ∈Θ of Example 2 or the one-dimensional mechanism offering the simple mechanism {˜ x∗1 (θ), x

O {xO 1 (θ), x2 (θ)}θ∈Θ of Proposition 1. It also shows that the principal raises the average decision

further away from the agent’s ideal points. 19

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

13

x 1.1 1.0 0.9 0.8 0.7 0.6 Θ 0.2

0.4

0.6

0.8

1.0

Figure 1: Average decision x∗ (θ) when δ = 0.3, and ∆ = 0.6 (dotted line), ∆ = 0.2 (dashed line) and ∆ = 0 (continuous line, which coincides with x ˜∗ (θ) and xO (θ)). U Θ 0.2

0.4

0.6

0.8

1.0

-0.1 -0.2 -0.3 -0.4 -0.5

Figure 2: Agent’s information rent U ∗ (θ) when δ = 0.3, and ∆ = 0.6 (dotted line), ∆ = 0.2 (dashed line), and ∆ = 0 (continuous line). Corollary 1 For all ∆ > 0 and δ ∈ (0, 1/2), we have [2δ, 1] ( [x∗ (0), x∗ (1)], and there exists θ ∗ (∆) ∈ (0, 2δ) such that:

x∗ (θ) < 2δ if and only if θ ≤ θ ∗ (∆). Moreover: x∗ (θ) > θ for all θ ∈ Θ. These features are illustrated in Figure 1, which compares the average of the one-dimensional O decision rules xO (θ) = 21 (xO 1 (θ) + x2 (θ)) = max{θ, 2δ} (which coincides with the average simple

˜∗2 (θ))) with the optimal average decision rule x∗ (θ) for a fixed avx∗1 (θ) + x decision rule x ˜∗ (θ) = 21 (˜ erage bias δ and different values of ∆. The agent’s information rent under the optimal mechanism, which is strictly increasing when ∆ > 0, is represented in Figure 2. In sharp contrast to the one-dimensional case, the agent’s ideal points are never chosen at the optimal mechanism. When ∆ > 0, the principal always induces truth-telling without leaving 14

xi 1.0 0.8 0.6 0.4 0.2 Θ 0.2

0.4

0.6

0.8

1.0

Figure 3: Decisions x∗i (θ) (dashed lines), x ˜∗i (θ) (continuous lines) and xO i (θ) (dotted line), for i = 1, 2, when δ1 = 0.2 and δ2 = 0.4. full discretion to the agent. However, distortions on each dimension become quite complex, as illustrated by Figure 3 for δ1 = 0.2 and δ2 = 0.4. While x∗2 (θ) is always strictly greater than θ, it is not always increasing, while x∗1 (θ) is strictly increasing over [0, 1] but not always greater than θ. In addition, for the incentive compatibility constraint (5) to be satisfied, x∗2 (θ) should be strictly decreasing if and only if x∗1 (θ) is larger than θ. This feature of the optimal mechanism is general, and is summarized in the next corollary. Corollary 2 For all (δ, ∆) ∈ R2++ , the following properties hold: 1. For every θ ∈ [0, 1], we have x˙ ∗1 (θ) > 0 and x∗2 (θ) > θ; 2. For every θ ∈ [0, 1], we have x∗1 (θ) > θ if and only if x˙ ∗2 (θ) < 0; 3. If δ1 ≥ 0, then x∗1 (0) ≥ 0 and x˙ ∗2 (0) ≤ 0 (with strict inequalities when δ1 > 0); 4. If δ1 ≥ 0, then x∗1 (1) ≥ 1 and x˙ ∗2 (1) ≤ 0 (with strict inequalities when δ1 > 0). Optimal spread. While the principal distorts decisions on each dimension as in Examples 1 and 2, Figure 3 also shows that, contrary to what happens with those simple mechanisms, the optimal spread is no longer constant. It is actually strictly decreasing in θ whenever ∆ > 0, as shown in the next corollary. This monotonicity facilitates information revelation. Indeed, the principal imposes a cost on the agent by making activities on both dimensions more dispersed for lower types. When the agent claims the state is lower (and he is biased to do so to get a smaller average decision x∗ (θ)), he is thus eventually punished by having to implement dispersed actions as well.20 20

Note that the spread might also be decreasing with the one-dimensional mechanisms of Proposition 1 (as illustrated by the dotted lines in Figure 3), but only for intermediate values of θ and when the conflict of interest is not too large (δ1 , δ2 < 1/2).

15

When ∆ = 0, the principal and the agent find it equally costly to make decisions on each dimension more divergent. There is no longer any conflict of interest concerning the ideal spread (both the principal and the agent want x2 (θ) = x1 (θ) even though their most preferred values for that decision diverge). Making decisions on each dimension more divergent is no longer useful. The optimal mechanism is simply the double replica of the one-dimensional mechanism. Remark 2 Taking a broader perspective, let us think of two different principals as each controlling a single dimension of the agent’s activity, say xi (i = 1, 2), and having the objective Vi (xi , θ) = P − 12 2i=1 (xi − θ − δi )2 . The intuition of our result then becomes quite obvious. There is no gain in

jointly maximizing the sum of the principals’ payoffs when their preferences coincide (δ1 = δ2 ).

Such a joint design only replicates what each of them individually would like to do. We now summarize some further properties of the optimal spread t∗ (θ). Corollary 3 For all (δ, ∆) ∈ R2++ , the optimal spread t∗ (θ) = 41 (x∗2 (θ) − x∗1 (θ))2 is non-negative, contin-

uous and strictly decreasing in θ with

∆2 > t∗ (0) >

∆2 > t∗ (1) > 0. 4

One might think that the optimal spread lies somewhere in between the ideal spreads of the principal (tP (θ) =

∆2 4 )

and the agent (tA (θ) = 0), to achieve some kind of compromise between

the two players’ objectives. The above corollary shows that this is actually not the case. Although the optimal spread is positive, which always hurts the agent, it may be significantly beyond the principal’s ideal spread for θ close enough to zero. For θ close to zero, the principal “overshoots” and is ready to push the optimal spread beyond his own ideal one just to better reward information revelation. Instead, for θ close enough to one, increasing the spread above the agent’s ideal point while still keeping it lower than tP (θ) relaxes the agent’s incentive constraint but also tends to increase the principal’s payoff. Delegation sets. One interpretation of the optimal mechanisms in the one-dimensional case is that “interval delegation” might be optimal. In our setting, leaving full discretion to the agent in choosing within a given delegation set bounded by a floor is optimal; in others a cap may prevail. In our multi-dimensional context, this benefit of delegation carries over although delegation sets are more complex than simple caps and floors. Indeed, it is still true that a version of the Taxation Principle21 holds in our context. The principal can implement the optimal mechanism by offering an indirect mechanism, i.e., a (continuous) curve in the (x1 , x2 ) space constructed from the parametrization {x∗1 (θ), x∗2 (θ)}θ∈Θ and leaving the agent free to pick any point on this curve.

Figure 4 represents such a curve, corresponding to the optimal mechanism, in the (x1 , x2 ) space. At the same time, this figure also features the indirect mechanisms corresponding to the simple 21

Rochet (1985).

16

x2 1.1 1.0 0.9 0.8 0.7 0.4

0.5

0.6

0.7

0.8

0.9

1.0

x1

Figure 4: Delegation sets for the optimal mechanism (dashed line), the simple mechanism O (˜ x∗1 (·), x˜∗2 (·)) (continuous line), and the one-dimensional mechanism (xO 1 (·), x2 (·)) (dotted line) when δ1 = 0.2 and δ2 = 0.4. O mechanism {˜ x∗1 (θ), x˜∗2 (θ)}θ∈Θ and the one-dimensional mechanism {xO 1 (θ), x2 (θ)}θ∈Θ . Note that

the slope of the optimal indirect mechanism is lower than the slope of the simple indirect mech-

anism. This is a general feature that can be directly deduced from Corollary 3. The slope of the simple curve is one, while the slope of the optimal curve is strictly smaller than one when the spread is strictly decreasing. More generally, simple duality arguments give us a little bit more information about the shape of those delegation sets. Let us define T (x) as T (x) = t∗ (θ) for x = x∗ (θ). T (·) is thus the nonlinear “tax” paid by the agent in terms of spread decisions when the average decision is itself x. By definition of the agent’s optimality conditions, we have: U ∗ (θ) = max 2θx − T (x) x

where

U ∗ (θ)

=

U ∗ (θ) + θ 2

and T (x) = T (x) + x2 . From those definitions, we immediately get that

U ∗ (θ) is convex as a maximum of linear functions. Therefore U ∗ (θ) is the difference between two

convex functions.

By duality, we also have T (x) = max 2θx − U ∗ (θ). θ

Hence, T (x) is also convex as a maximum of linear functions. The nonlinear “tax” T (x) is itself

the difference between two convex functions.

Simple mechanisms. Since the design of the optimal mechanism looks rather complex, one may wonder whether the one-dimensional or the simple mechanisms perform well enough. The simple ˜∗2 (θ)}θ∈Θ of Example 2 is particularly appealing since, as already noted, it is mechanism {˜ x∗1 (θ), x

also approximately optimal when δ is small (and achieves the first best when δ = 0), while the O mechanism {xO 1 (θ), x2 (θ)}θ∈Θ that replicates the one-dimensional mechanisms of Proposition 1 is

17

only optimal when ∆ = 0. The intuition is that, although the optimal mechanism requires full separation of types, this is only marginally the case on the lower tail of the distribution of types, and one may not lose so much by using simple mechanisms. Our next proposition shows that both these mechanisms perform quite well when ∆ is small enough. When two principals each control a dimension of the agent’s activity, this result means that the gain for cooperation between the principals is only significant for a large enough difference between their ideal actions. ˜∗2 (θ)}θ∈Θ Proposition 6 If |δ1 |, |δ2 | < 1/2, then the principal’s loss from using the simple mechanism {˜ x∗1 (θ), x O ∗ ∗ or the one-dimensional mechanism {xO 1 (θ), x2 (θ)}θ∈Θ instead of the optimal mechanism {x1 (θ), x2 (θ)}θ∈Θ

is of order at most 2 in ∆: Z Z

1 0

1 0

˜˙ ∗ (θ))dθ − L∆ (U˜ ∗ (θ), U

Z

Z

1

L∆ (U (θ), U˙ O (θ))dθ − O

1

0

0

L∆ (U ∗ (θ), U˙ ∗ (θ))dθ ≤

3∆2 , 4

L∆ (U ∗ (θ), U˙ ∗ (θ))dθ ≤ (1 − δ)∆2 .

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

5.1 Non-Optimality of Stochastic Mechanisms Kovac and Mylovanov (2009) showed that the restriction to deterministic mechanisms is without loss of generality in the case of quadratic payoffs, a constant bias, a one-dimensional activity, and when the type distribution is uniform as here.22 This insight carries over in our framework when ∆ = 0 mutatis mutandis. However, in our multi-dimensional context, the principal has more tools to screen the agent’s type when ∆ > 0. As such, using stochastic mechanisms is less attractive even beyond the case of a uniform distribution. Proposition 7 Suppose that θ is distributed according to any density f (θ) on Θ. The optimal deterministic mechanism cannot be improved by stochastic mechanisms. To relax incentive compatibility, the principal could a priori use random allocations and play on the variance of each decision, i.e., choose how decisions move around their expected values to threaten the agent with some risk if he reports low values of θ. Of course, the principal can still play on how decisions are spread, as in our analysis of deterministic mechanisms. The second of 22 They also provide conditions for more general distributions. Alonso and Matouschek (2008) give an example with a type-dependent bias in which the optimal mechanism is stochastic.

18

these strategies has already been shown to be useful above. The first is suboptimal. The intuition is straightforward: the principal and the agent are both equally averse to such random allocations, and there is no gain from using this randomness that could not be achieved by playing only on the spread between decisions.

5.2 Not Leaving Full Discretion is Generic Our result that the agent never receives his ideal points is highly robust. The next proposition shows that the principal never finds it optimal to leave full discretion to the agent, letting him choose his ideal points on a subset I with a non-empty interior, whatever the everywhere-positive density f (θ) on Θ. Proposition 8 Suppose that θ is distributed according to any density f (θ) everywhere-positive on Θ and bounded. If ∆ > 0, then the optimal deterministic mechanism is never such that x∗1 (θ) = x∗2 (θ) = θ on any subset of Θ with a non-empty interior. The intuition is straightforward. Let us assume the contrary. The principal could, as in Example 1, move down x1 and up x2 by the same small amount on that interval, still keeping the same average decision so that incentives for truth-telling are unchanged. Doing so yields a strict benefit to the principal, who prefers more divergent decisions than his agent does.

6 Conclusion Optimal multi-dimensional mechanisms are quite different from the simple delegation sets found in the one-dimensional case. The principal and the agent may differ not only on their most preferred average decision but also on the distribution of those decisions. The possibility of trading off distortions along each dimension of the agent’s activities eases screening and leads to fully separating allocations. If the conflicts of interest between the principal and the agent are different on each dimension, then delegation is always strictly valuable, there is no pooling of types, and the agent never chooses his ideal points. The spread on decisions that is necessary to induce cheaper information revelation is a decreasing function of the average decision taken by the agent. One-dimensional mechanisms, taking the form of interval(s) delegation sets (pooling intervals or intervals of full discretion) are optimal only when the biases between the principal and agent’s ideal actions are exactly the same along all dimensions. Such extended possibilities for screening provide a strong reason to merge principals who control different dimensions of the agent’s activities and let them jointly design mechanisms. At worst, the analysis of the cooperative contracting design undertaken in this paper characterizes the benefits of cooperation in settings where divided control is otherwise often pervasive. Regulation by different agencies and bureaucratic oversights by different legislative committees are 19

of course two examples in order. From a theoretical viewpoint, this comparison may depend on the fine details of the contracting possibilities available under a non-cooperative design. For instance, the non-cooperative outcome may depend on whether or not principals observe decisions that they do not directly control 23 and whether the agent’s messages towards each principal are private or public, the latter case being a priori closer to the cooperative outcome developed in this paper.24 It would also be worth investigating optimal mechanisms in more complex environments allowing more general utility functions, more than two decisions and more general distributions of types. Some relatively easy extensions would be to investigate optimal mechanisms when the principal and the agent value differently the losses on each dimension, while keeping quadratic utility functions. We conjecture that the simple decomposition in terms of average decisions and spread would generalize and would still be useful in characterizing optimal mechanisms in such contexts. More dispersion in decisions is certainly needed when the agent makes decisions that go counter to what the principal would like on average. Finally, it would also be interesting to extend our approach by allowing for multi-dimensional preferences, as in the cheap talk framework developed in Battaglini (2002, 2004) and Chakraborty and Harbaugh (2007): the agent’s bliss points on each dimension of his activity being not necessarily perfectly correlated. This extension is also likely to meet strong technical difficulties, but certainly deserves some attention.25 For all those cases, we conjecture that the decomposition between the average decision and its spread will play a crucial role in contract design.

Appendix Proof of Proposition 1. See Holmstrom ¨ (1977) and Melumad and Shibano (1991). ˆ ∈ Θ2 , incentive compatibility entails that: Proof of Lemma 1. Necessity: For all pairs (θ, θ) 2 2 2 2 X X X X ˆ − θ) ˆ 2. ˆ2≥ ˆ − θ)2 ≥ (xi (θ) (xi (θ) − θ) (xi (θ) − θ)2 and (xi (θ) i=1

i=1

i=1

(A.1)

i=1

Summing those inequalities yields: 2 X ˆ ˆ ≥ 0. (xi (θ) − xi (θ))(θ − θ)

(A.2)

i=1

23

Martimort (2007) described this situation as a case of public agency. A previous version of this paper (Koessler and Martimort, 2008) analyzed public mechanism with two principals (see also Goltsman and Pavlov, 2011 for further analysis of the cheap talk setting combining both public and private messages). It was shown that there is no non-cooperative equilibrium with continuous and deterministic action rules. The characterization of the equilibrium mechanisms with public mechanism remains an interesting open problem. 25 The literature on multi-dimensional screening has already stressed that pooling allocations are pervasive in nonlinear pricing environments (Armstrong, 1996; Fang and Norman, 2008; Rochet and Chon´e, 1998). 24

20

Hence,

P2

i=1 xi (θ)

is non-decreasing in θ. Therefore, it is almost everywhere differentiable with,

at any point of differentiability, a derivative such that (4) holds. At such a point, an incentive compatible mechanism must also satisfy the first-order condition of the agent’s revelation problem, namely (5). Moreover, using (A.1), we get: 2 X i=1

Hence,

P2

2 i=1 xi (θ)

x2i (θ) −

2 X

ˆ x2i (θ)

i=1

≥ 2θˆ

is non-decreasing in θ when

2 X i=1

xi (θ) −

P2

i=1 xi (θ)

2 X

!

ˆ . xi (θ)

i=1

is itself non-decreasing.

P2

2 i=1 xi (θ)

is

thus almost everywhere differentiable. P Sufficiency: That 2i=1 xi (θ) is non-decreasing in θ is then also a sufficient condition for optimalP P ity.26 Indeed, since 2i=1 x2i (θ) and 2i=1 xi (θ) are both non-decreasing in θ and thus almost ev-

erywhere differentiable with, at any point of differentiability, a derivative which is measurable, Theorem 3 in Royden (1988, p. 100) entails: 2 Z 2 2 X X X 2 2 ˆ (xi (θ) − θ) ≥ (xi (θ) − θ) −

=

2 Z X i=1

i=1

i=1

i=1

θ

θˆ

x˙ i (s)(xi (s) − s + s − θ)ds =

θ

2 Z X i=1

θˆ

x˙ i (s)(xi (s) − θ)ds θˆ

θ

x˙ i (s)(s − θ)ds ≥ 0,

where the last equality follows from (5) and the last inequality from (4). Proof of Lemma 2. The proof is standard and follows Milgrom and Segal (2002). Proof of Lemma 3. We proceed along the lines of Clarke (1990, Chapter 4). Let us first define the extended-value Lagrangian L∗∆ (s, v)

( L∆ (s, v) = +∞

2

if s ≤ − v4 , otherwise.

As required in Clarke (1990, p. 167), we observe that: 1. L∗∆ (s, v) is B-measurable where B denotes the σ−algebra of subsets of R × R; 2. L∗∆ (s, v) is lower-semi continuous; 3. L∗∆ (s, v) is convex in v. 2

We now define the Hamiltonian as H(s, p) = supv∈R {pv−L∗∆ (s, v)}. When s ≤ − v4 , L∗∆ (s, v) =

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

∂L∆ (s, v). ∂v

26

(A.3)

Garcia (2005) provides an analysis of the multi-dimensional adverse selection model in a framework with quasilinear utility functions, but focuses a priori on differentiable mechanisms.

21

This yields the maximand r v ∗ = 4(p + δ)

−s , 4(p + δ)2 + ∆2

which gives ( p s + −s(4(p + δ)2 + ∆2 ) − δ2 − H(s, p) = −∞

∆2 4

if s ≤ 0, otherwise.

Note that H(s, p) is differentiable on (−∞, 0) × R. We get the following inequality: H(s, p) ≤ |s| + ∆ p

|s| 2

p

p ∆2 . |s| + 2|p + δ| |s| − δ2 − 4

and |p + δ| ≤ |p| + δ, we finally obtain:     ∆2 ∆ ∆ H(s, p) ≤ ∆ + 2δ − δ2 − + 2|p|+ |s| 1 + δ + + |p| ≤ 2+ 2|p|+ |s| 1 + δ + + |p| . (A.4) 4 2 2

Now using

|s| ≤ 1 +

This is a “growth” condition on the Hamiltonian as required in Clarke (1990, Theorem 4.1.3). Lemma 4 Clarke (1990). If L∗∆ (·) satisfies conditions 1. to 3. above, H(·) satisfies the “growth” equation R1 (A.4 ) and 0 L∗∆ (U0 (θ), U˙ 0 (θ))dθ is finite for at least one admissible arc U0 (θ), then problem P∆ has a solution.

It remains for us to show that

R1 0

L∗∆ (U0 (θ), U˙ 0 (θ))dθ is finite for at least one admissible arc

U0 (θ) . Take U0 (θ) = 0, which corresponds to decisions x10 (θ) = x20 (θ) = θ. This arc does the job R1 2 and yields 0 L∗∆ (U0 (θ), U˙ 0 (θ))dθ = δ2 + ∆4 . Proof of Proposition 2. Preliminaries: We say that H satisfies the strong Lipschitz condition near an

arc U if there exists ǫ > 0 and a constant k such that for all p ∈ R and for all (s1 , s2 ) ∈ T(U, ǫ) the tube of radius ǫ centered on the arc U , the following inequality holds: |H(s1 , p) − H(s2 , p)| ≤ k(1 + |p|)|s1 − s2 |.

(A.5)

This property holds in our context when there exists η > 0 such that U (θ) < −η for all θ (i.e., U (θ)

is bounded away from zero, which will be the case for the solution we exhibit below). Over the relevant range where si ≤ 0, we have:

p √ √ |H(s1 , p) − H(s2 , p)| = |s1 − s2 + ( −s1 − −s2 ) ∆2 + 4(p + δ)2 |.

√ √ 1 −s2 | √ Note that | −s1 − −s2 | = |s for some s0 ∈ T(U, ǫ) from the Mean-Value Theorem. Therefore, 2 −s0 √ √ |s1 −s2 | | −s1 − −s2 | ≤ 2√η−ǫ for ǫ small enough. Hence, we get: |H(s1 , p) − H(s2 , p)| ≤ |s1 − s2 | 1 +

p

∆2 + 4(p + δ)2 √ 2 η−ǫ

!

  ∆ + 2(δ + |p|) √ ≤ |s1 − s2 | 1 + 2 η−ǫ

  1 ∆ + 2δ ,√ ≤ max 1 + √ |s1 − s2 |(1 + |p|), 2 η−ǫ η−ǫ 22

n which is (A.5) with k = max 1 +

∆+2δ 1 √ , √η−ǫ 2 η−ǫ

o

.

Euler equation and boundaries conditions: From Clarke (1990, Theorem 4.2.2, p.169), and since L∗∆ (·) satisfies conditions 1., 2., and 3. above and H(·) satisfies the strong Lipschitz condition (A.5), there exists an absolutely continuous arc p(·) such that the following conditions hold for the optimal arc U ∗ (θ). • Optimality conditions for the Hamiltonian H(·): ∂H ∗ (U (θ), p(θ)), ∂s

(A.6)

∂H ∗ (U (θ), p(θ)). U˙ ∗ (θ) = ∂p

(A.7)

p(0) = p(1) = 0.

(A.8)

−p(θ) ˙ =

• Boundary conditions:

∂L∆ (U ∗ (θ), U˙ ∗ (θ)). Differentiating ∂ U˙ ∂L∆ ∂H ∗ ∗ ˙∗ ∂s (U (θ), p(θ)) = − ∂U (U (θ), U (θ))

Using (A.3) yields p(θ) =

(A.6) and observing that p(θ) = ∂L∆ (U ∗ (θ), U˙ ∗ (θ)) yields (16).

with respect to θ, inserting into yields (15). Finally, again using

∂ U˙

Continuity: First observe that, a.e. on Θ, we have by definition H(U ∗ (θ), p(θ)) = p(θ)U˙ ∗ (θ)−L∆ (U ∗ (θ), U˙ ∗ (θ)) ≥ p(θ)v−L∆ (U ∗ (θ), v),

p ∀ v ≤ 2 −U ∗ (θ). (A.9)

If U˙ is not continuous at some θ0 ∈ (0, 1), there exists an increasing sequence θn− and a decreasing

sequence θn+ (n ≥ 1) both converging towards θ0 , such that (A.9) applies at θn− , θ0 and θn+ , and (using monotonicity to get the strict inequality):

lim U˙ ∗ (θn− ) = U˙ ∗ (θ0− ) < U˙ ∗ (θ0+ ) = lim U˙ ∗ (θn+ ).

n→+∞

n→+∞

Because L∆ (s, v) is continuous in (s, v) and U ∗ (θ) is absolutely continuous and thus continuous at θ0 , we have: L∆ (U ∗ (θ0 ), v) = lim L∆ (U ∗ (θn− ), v) and L∆ (U ∗ (θ0 ), U˙ ∗ (θ0− )) = lim L∆ (U ∗ (θn− ), U˙ ∗ (θn− )). n→+∞

n→+∞

(A.10) Taking θ = θn− into (A.9) and passing to the limit, using the continuity of p(θ), yields p(θ0 )U˙ ∗ (θ0− ) − L∆ (U ∗ (θ0 ), U˙ ∗ (θ0− )) ≥ p(θ0 )v − L∆ (U ∗ (θ0 ), v) Using similar arguments with the sequence θn+ , we also get p(θ0 )U˙ ∗ (θ0+ ) − L∆ (U ∗ (θ0 ), U˙ ∗ (θ0+ )) ≥ p(θ0 )v − L∆ (U ∗ (θ0 ), v)

p ∀ v ≤ 2 −U ∗ (θ). p ∀ v ≤ 2 −U ∗ (θ).

p Hence, the function v → p(θ0 )v − L∆ (U ∗ (θ0 ), v) defined for v ≤ 2 −U ∗ (θ0 ) achieves its maxima at both U˙ ∗ (θ0+ ) and U˙ ∗ (θ0− ). Since it is strictly concave, we get U˙ ∗ (θ0+ ) = U˙ ∗ (θ0− ). From this 23

contradiction, we conclude that any arbitrary θ ∈ Θ is contained in a relatively open interval on which U˙ ∗ is almost everywhere equal to a continuous function. U˙ ∗ and thus x∗1 and x∗2 are continuous on Θ. Proof of Proposition 3. Since the functional L∆ (·) does not depend on θ, we can obtain a first quadrature of (15) on any interval where U (θ) +

U˙ 2 (θ) 4

< 0 as:

∂L∆ ∆2 , L∆ (U (θ, λ), U˙ (θ, λ)) − U˙ (θ, λ) (U (θ, λ), U˙ (θ, λ)) = λ + δ2 + 4 ∂ U˙

(A.11)

where a priori λ ∈ R and where we make explicit the dependence of the solution on this parameter.

We obtain immediately:

U (θ, λ) + δU˙ (θ, λ) + ∆

s

Simplifying yields:

  ˙ (θ, λ) U˙ 2 (θ, λ) ∆ U  = −λ. − U˙ (θ, λ) δ − q −U (θ, λ) − 4 U˙ 2 (θ,λ) 4 −U (θ, λ) − 4 

Solving for U˙ (θ, λ) yields

U (θ, λ) 1 − q

∆ −U (θ, λ) −

U˙ (θ, λ) = −4 U (θ, λ) + ∆ 2

which requires −∆2



2 U (θ,λ) U (θ,λ)+λ

2

U˙ 2 (θ,λ) 4





 = −λ.

U (θ, λ) U (θ, λ) + λ

2 !

,

(A.12)

≥ U (θ, λ) or (U (θ, λ)+λ)2 +∆2 U (θ, λ) ≥ 0 given that U (θ, λ) ≤ 0,

since by definition the agent’s information rent is negative. Solving the second-order equation (A.12) and keeping only the positive root, 27 we get (17). When U˙ (θ, λ) > 0, differentiating ( 17) with respect to θ yields:   U (θ,λ)  U˙ (θ, λ) 1 + 2λ∆2 (U (θ,λ)+λ) 3 2 ¨ ¨ U (θ, λ) + r   = U (θ, λ) + 2 1 + 2λ∆ −U (θ, λ)

− ∆2

U (θ,λ) U (θ,λ)+λ

2

U (θ, λ) (U (θ, λ) + λ)3



=0

(A.13)

Hence, on any interval where U˙ (θ, λ) > 0, the second-order condition (11) can be written as: ¨ (θ, λ) + 2 = −4λ∆2 0≤U

U (θ, λ) . (U (θ, λ) + λ)3

(A.14)

Since U (θ, λ) ≤ 0 holds, λ ≤ 0 also entails U (θ, λ) + λ ≤ 0 and then (A.14) holds. This imposes

the requested restriction on the admissible solutions to (17). Finally, note that the second-order condition (11) obviously holds on any interval where U˙ (θ, λ) = 0 . 27

Since it corresponds to an average decision x(θ) biased towards the principal, namely x(θ) ≥ θ (see equation (10)).

24

Proof of Theorem 1. The structure of the proof is as follows. First, we derive from the necessary free end-point conditions (16) some properties of the boundary values of U ∗ that are used to find λ∗ . Sufficiency follows. Necessity: Define the function P (x) = degree polynomial (x +

λ)2

+

∆2 x

−x((x+λ)2 +∆2 x) . (x+λ)2

For x < 0, P (x) > 0 if and only if the second 2

is everywhere positive. This is so when λ < − ∆4 . When that

condition holds, the differential equation (17) is Lipschitz at any point where U (θ, λ) < 0 and

thus it has a single solution at any such point. Moreover, a solution U (θ, λ) is then everywhere increasing on the whole domain where U (θ, λ) < 0. As a result, the differential equation (17) is everywhere Lipschitz when U (1, λ) < 0, which turns out to be the case for the path we derive below. The necessary free end-point conditions (16) can be rewritten for an optimal path as:   ∗ ˙ U (θ) −δ + ∆ q  |θ=0,1 = 0. (U˙ ∗ (θ))2 ∗ 4 −U (θ) − 4

Using (17) to express U˙ ∗ (θ), these conditions can be simplified so that U ∗ (0) and U ∗ (1) solve the following second-order equation in U : (U + λ∗ )2 = −(∆2 + 4δ2 )U,

(A.15) 2

where λ∗ is the value of λ for the optimal arc U ∗ . Assuming now that λ∗ > − ∆4 − δ2 (a condition

verified below), (A.15) admits two solutions respectively given by:

 p 1 2 ∆ + 4δ2 + (∆2 + 4δ2 )2 + 4λ∗ (∆2 + 4δ2 ) , 2  p 1 2 ∆ + 4δ2 − (∆2 + 4δ2 )2 + 4λ∗ (∆2 + 4δ2 ) . U ∗ (1) = −λ∗ − 2 Note in particular that (A.15) implies that both U ∗ (0) and U ∗ (1) are negative. U ∗ (0) = −λ∗ −

2

(A.16) (A.17)

2

The last step is to show that there exists λ∗ ∈ (− ∆4 − δ2 , − ∆4 ) such that the corresponding

path U ∗ (θ) = U (θ, λ∗ ) solving (17) and starting from: U (0, λ) = −λ − reaches

 p 1 2 ∆ + 4δ2 + (∆2 + 4δ2 )2 + 4λ(∆2 + 4δ2 ) , 2

 p 1 2 ∆ + 4δ2 − (∆2 + 4δ2 )2 + 4λ(∆2 + 4δ2 ) . 2 ∗ This requires a solution λ to the equation U (1, λ) = −λ −

ϕ(λ) = ψ(λ),

(A.18)

p

(A.19)

with ϕ(λ) = U (1, λ) − U (0, λ) =

(∆2 + 4δ2 )2 + 4λ(∆2 + 4δ2 ),

25

and ψ(λ) =

Z

1

U˙ (θ, λ)dθ =

Z

0

0

1

s

2

−U (θ, λ) −

∆2



U (θ, λ) U (θ, λ) + λ

2

dθ,

where the path U (θ, λ) starts from the initial condition U (0, λ). Note that both ϕ(·) and ψ(·) are 2

continuous in λ. It is clear that ϕ(λ) is strictly increasing in λ with, for λ1 = − ∆4 − δ2 and 2

λ2 = − ∆4 ,

ϕ (λ1 ) = 0 < 2δ

On the other hand, note that

p

∆2 + 4δ2 = ϕ (λ2 ) .

ψ (λ1 ) > 0 = ϕ (λ1 ) ,

(A.20)

since the path U (θ, λ1 ) starting from U (0, λ1 ) is strictly increasing. Moreover, for λ2 , (17) can be rewritten as:

p |U (θ, λ2 ) − λ2 | U˙ (θ, λ2 ) = 2 −U (θ, λ2 ) . |U (θ, λ2 ) + λ2 |

(A.21)

The path solving (A.21) and starting at U (0, λ2 ) (note that U (0, λ2 ) < λ2 < U (1, λ2 )) is strictly increasing everywhere and cannot cross the boundary U = λ2 because the only solution to (A.21) such that U (θ1 ) = λ2 for a given θ1 > 0 is such that U (θ) = λ2 for all θ since the right-hand side of (A.21) satisfies a Lipschitz condition at any point U (θ, λ2 ) away from zero: a contradiction with U (0, λ2 ) < λ2 . From that, we deduce U (θ, λ2 ) < λ2 for all θ. Hence, the following sequence of inequalities holds: ψ (λ2 ) =

Z

0

1

U˙ (θ, λ2 )dθ < λ2 − U (0, λ2 ) = λ2 − U (1, λ2 ) + ϕ (λ2 ) .

Finally, we get: ψ (λ2 ) < ϕ (λ2 ) .

(A.22)

Combining Equations (A.20) and (A.22) yields the existence of λ∗ ∈ (λ1 , λ2 ) such that ϕ(λ∗ ) = 2

2

ψ(λ∗ ). For such λ∗ ∈ (− ∆4 − δ2 , − ∆4 ) we have U (1, λ∗ ) < 0 and thus U ∗ (θ) = U (θ, λ∗ ) < 0 for all ¨ ∗ (θ) + 2 and (A.14) θ. From Proposition 3 this entails U˙ ∗ (θ) > 0 for all θ. Finally, using 2x˙ ∗ (θ) = U with U ∗ (θ) < 0 and λ∗ < 0 we get x˙ ∗ (θ) > 0 for all θ.

Remark 3 Note that, when ∆ goes to zero, the solution λ∗ to (A.18) converges to 0. Indeed, U (1, λ) (resp. U (0, λ)) converge towards 0 (resp. −2δ) as λ and ∆ go together to zero. Thus, R1 p ϕ(λ) converges itself towards 2δ . At the same time ψ(λ) converges towards 0 2 −U0 (θ)dθ, p where U0 (θ) is the solution to U˙ (θ) = −U0 (θ) starting from limλ,∆→0 U (0, λ) = −2δ. This solution is of course the solution to the one-dimensional problem; in particular, we have U0 (2δ) = 0 R1 p and 0 2 −U0 (θ)dθ = 2δ.

Sufficiency: Sufficiency follows from Clarke (1990, Chapter 4, Corollary p. 179) since L∆ (·) satisfies the convexity assumption and the function s → H(s, p(θ)) is concave in s.

26

Proof of Proposition 4. Note that: ∂L∗ (δ, ∆) = ∂δ

Z

1 0

(2δ − U˙ ∗ (θ))dθ = 2δ − (U ∗ (1) − U ∗ (0)).

(A.23)

Using Equation (A.19) we get: p ∂L∗ (A.24) (δ, ∆) = 2δ − (∆2 + 4δ2 )2 + 4λ∗ (∆2 + 4δ2 ). ∂δ   √ 2 ∗ 2 + 4δ 2 > 0 when ∆2 + 4δ 2 < 1. Because λ∗ ≤ − ∆4 , we finally obtain ∂L ∆ (δ, ∆) ≥ 2δ 1 − ∂δ

Proof of Proposition 5. Condition (22) follows from a remark in the text. The strict inequality follows from the fact that the simple mechanism is never optimal when ∆ > 0.

Let us denote by U (θ, δ, ∆) the solution of the principal’s problem where we now make explicit the dependence of this function on the preference parameters (δ, ∆). By the Envelope Theorem, we get from (13): ∂L∗ ∂∆

(δ, ∆) =

Z



1

∆ − 2

0

s

 2 ˙ U (θ, δ, ∆)  dθ. −U (θ, δ, ∆) − 4

(A.25)

By Equation (17), this derivative can also be rewritten as: ∂L∗ (δ, ∆) = ∆ ∂∆

Z

0

1

1 U (θ, δ, ∆) − 2 U (θ, δ, ∆) + λ(δ, ∆)



dθ,

(A.26)

where we also make explicit the dependence of the constant of integration λ on the preference parameters (δ, ∆). Note that the function ζ(x) = ζ ′ (x) =

λ(δ,∆) (x+λ(δ,∆))2

≤ 0 and ζ ′′ (x)

x x+λ(δ,∆) is decreasing and concave since λ(δ, ∆) ≤ 0. (We have 2λ(δ,∆) = − (x+λ(δ,∆)) 3 ≤ 0 since U (θ, δ, ∆) + λ(δ, ∆) ≤ 0 for all θ.) From

this, we deduce the following chain of inequalities: Z

0

1

ζ(U (θ, δ, ∆))dθ ≥ ζ

Z

1

U (θ, δ, ∆)dθ

0



> ζ(λ(δ, ∆)) =

1 2

(A.27)

where the last inequality holds if and only if: Z

1

U (θ, δ, ∆)dθ < λ(δ, ∆).

(A.28)

0

Observe that this inequality is true when ∆ = 0. Indeed, in that case, we know that U (1, δ, 0) − U (0, δ, 0) = 4δ2 (for δ ≤ 1/2) and this condition is compatible with (A.19) only when λ(δ, 0) = 0. R1 R 2δ We also know that 0 U (θ, δ, 0)dθ = 0 U (θ, δ, 0)dθ < 0. From Remark 3, U (θ, δ, ∆) and λ(δ, ∆)

are such that (A.28) also holds for ∆ small enough. In turn, (A.27) holds also for ∆ small enough. Inserting into (A.26) ends the proof.

27

Proof of Corollary 1. First, note that U˙ ∗ (0) = x (0) = 2 ∗

p

−U ∗ (0)((U ∗ (0) + λ∗ )2 + ∆2 U ∗ (0)) . |U ∗ (0) + λ∗ |

Using (A.16), we get: x∗ (0) = 2δ Finally, we obviously have x∗ (θ) − θ =

U˙ ∗ (θ) 2

|U ∗ (0)| < 2δ. |U ∗ (0) + λ∗ | > 0 for all θ.

Proof of Corollary 2. The first property follows from (19). Now, from the incentive constraint (5) and the first property of the corollary we get the second property. Next, using (19) we have x∗1 (0) ≥ 0 if and only if

  U˙ ∗ (0) U ∗ (0) ≥∆ . 2 U ∗ (0) + λ∗

Using (17) and simplifying, we get 2∆2 U ∗ (0) ≥ −(U ∗ (0) + λ∗ )2 , i.e., by (A.15), 2∆2 ≤ ∆2 + 4δ2 , which is always satisfied when δ1 ≥ 0 (with a strict inequality when δ1 > 0). x˙ ∗2 (0) ≤ 0 follows now from the second property of the corollary. The last property is proved similarly.

Proof of Corollary 3. From Theorem 1, Equation (19), we have:  2 U ∗ (θ) ∗ 2 t (θ) = ∆ . U ∗ (θ) + λ∗ Therefore, we get: t˙∗ (θ) = 2∆2 λU˙ ∗ (θ)

U ∗ (θ) < 0, (U ∗ (θ) + λ∗ )3

since λ∗ < 0 and U˙ ∗ (θ) > 0 for all θ. Using (A.16), we get  2 U ∗ (0) ∗ 2 t (0) = ∆ < ∆2 U ∗ (0) + λ∗ ⇔1>

U ∗ (0) 2λ∗ p , = 1 + U ∗ (0) + λ∗ ∆2 + 4δ2 + (∆2 + 4δ2 )2 + 4λ∗ (∆2 + 4δ2 )

which holds since λ∗ < 0. Still using (A.16), we also get: t∗ (0) >

1 U ∗ (0) 2λ∗ ∆2 p ⇔ < ∗ = 1 + 4 2 U (0) + λ∗ ∆2 + 4δ2 + (∆2 + 4δ2 )2 + 4λ∗ (∆2 + 4δ2 ) p ⇔ ∆2 + 4δ2 + 4λ∗ + (∆2 + 4δ2 )2 + 4λ∗ (∆2 + 4δ2 ) > 0,

(A.29)

2

which holds since λ∗ > − ∆4 − δ2 . Now using (A.17), we get: ∗

t (1) = ∆ ⇔

2



U ∗ (1) U ∗ (1) + λ∗

2

<

∆2 4

1 U ∗ (1) 2λ∗ p > ∗ = 1 + 2 U (1) + λ∗ ∆2 + 4δ2 − (∆2 + 4δ2 )2 + 4λ∗ (∆2 + 4δ2 ) 28

(A.30)

⇔ ∆2 + 4δ2 + 4λ∗ < 2

p

(∆2 + 4δ2 )2 + 4λ∗ (∆2 + 4δ2 )

which holds again since λ∗ > − ∆4 − δ2 and λ∗ < 0. Proof of Proposition 6. We have: Z

1

L∆ (U ∗ (θ), U˙ ∗ (θ))dθ =

Z

1

0

0

≥

Z

0

1



L0 (U ∗ (θ), U˙ ∗ (θ)) − ∆



L0 (U0∗ (θ), U˙0∗ (θ)) − ∆

s

∆2 4δ3 − − ∆2 =δ + 4 3

where

Z

1

0

U ∗ (θ) dθ, U ∗ (θ) + λ∗

is given in (20) and the last inequality follows from (17). This entails: Z

1

0

We also have:

L∆ (U ∗ (θ), U˙ ∗ (θ))dθ ≥ δ2 − Z

0

and

 ˙ ∗ (θ))2 ∆2 ( U  dθ −U ∗ (θ) − + 4 4

 2 2 ∗ ˙ ∆  (U (θ)) −U ∗ (θ) − + dθ 4 4

2

U0∗ (θ)

s

Z

1

1

3∆2 4δ3 − . 4 3

3 ˜ ∗ (θ), U ˜˙ ∗ (θ))dθ = δ2 − 4δ , L∆ (U 3

L∆ (U O (θ), U˙ O (θ))dθ = δ2 +

0

2δ3 ∆2 4δ3 ∆2 2δ13 − − 2 = δ2 + − − δ∆2 , 4 3 3 4 3

which gives the required inequalities. Proof of Proposition 7. A stochastic direct mechanism is a mapping µ(·|·) : Θ → ∆(K × K) where

∆(K × K) is the set of measures on K × K. For further references, we define the mean and variance

of this stochastic mechanism as Z Z 2 ˆ ˆ ˆ xi dµ(x1 , x2 |θ) and σi (θ) = x ¯i (θ) =

K×K

K×K

ˆ 2 dµ(x1 , x2 |θ) ˆ ≥ 0. (xi − x ¯i (θ))

The boundedness of K ensures that such moments exist. Note that deterministic mechanisms are ˆ ≡ 0. For further references also, denote x ˆ = 1 P2 x ˆ the average (mean) such that σ 2 (θ) ¯(θ) ¯i (θ) i

2

i=1

ˆ =x ˆ −x ˆ the spread of those mean decisions. In this context, incentive decision and y¯(θ) ¯2 (θ) ¯1 (θ) compatibility can be written as:

U (θ) = max ˆ θ∈Θ

Z

K×K

2 X i=1

! 1 ˆ − (xi − θ)2 dµ(x1 , x2 |θ). 2

Taking expectations, we get: 1 U (θ) = max − ˆ 2 θ∈Θ

2 X ˆ − θ)2 (¯ xi (θ) i=1

!

2 ˆ ˆ ˆ = max −(¯ ˆ − θ)2 − y¯ (θ) − z(θ), − z(θ) x(θ) ˆ 4 θ∈Θ

29

ˆ = where z(θ)

1 2

2 ˆ i=1 σi (θ)

P2

≥ 0. From this, it immediately follows that U (·) is absolutely continu-

ous with a derivative defined almost everywhere as:

ˆ − θ), U˙ (θ) = 2(¯ x(θ)

(A.31)

with

U˙ 2 (θ) y¯2 (θ) − . 4 4 The principal’s expected loss with such a stochastic mechanism can be written as: ! ! Z 1 Z 2 X 1 (xi − θ − δi )2 dµ(x1 , x2 |θ) f (θ)dθ 2 K×K 0 i=1  Z 1 ∆ ∆2 2 ˙ =− U (θ) + δU (θ) + y¯(θ) − δ − f (θ)dθ. 2 4 0 0 ≤ z(θ) = −U (θ) −

(A.32)

The principal problem when stochastic mechanisms are allowed can be written as: s (P∆ )

:

min

Z

{U ∈W 1,1 (Θ),z≥0} 0

1

Ls∆ (U (θ), U˙ (θ), z(θ))f (θ)dθ,

where Ls∆ (U (θ), U˙ (θ), z(θ)) = −U (θ) − δU˙ (θ) − ∆

s

−U (θ) −

U˙ 2 (θ) ∆2 − z(θ) + δ2 + . 4 4

Clearly, the pointwise solution to this problem when ∆ > 0 is achieved for z(θ) = 0, i.e., for deterministic mechanisms. Proof of Proposition 8. Suppose that the optimal solution U ∗ is such that U ∗ (θ) = 0 on an interval I with non-empty interior, i.e., x∗1 (θ) = x∗2 (θ) = θ (or equivalently x∗ (θ) = θ and t∗ (θ) = 0) on I. Let [a, b] be any connected interval including I having this property (the proof extends easily to the case of several such intervals). Note in particular that incentive compatibility implies continuity of U ∗ and thus U ∗ (a) = U ∗ (b) = 0 with x∗ (a+ ) = limx→a+ x∗ (θ) = a and x∗ (b− ) = limx→b− x∗ (θ) = b. Note also that U ∗ is increasing in the left-neighborhood of a and decreasing in the right-neighborhood of b. First assume that 0 < a < b < 1. Fix some ǫ > 0 small enough and consider a new (continuous) ˜ such that: utility profile U ˜ (θ) = U

( U ∗ (θ) if θ ∈ [0, 1]\[a − u(ǫ), b + v(ǫ)] −ǫ2 if θ ∈ [a − u(ǫ), b + v(ǫ)],

(A.33)

where u(ǫ) and v(ǫ) are such that 2

ǫ =

Z

a

a−u(ǫ)

U˙ ∗ (θ)dθ = − 30

Z

b+v(ǫ) b

U˙ ∗ (θ)dθ.

(A.34)

Observe that u(ǫ) and v(ǫ) are positive when ǫ is small enough, and converge to zero with ǫ because U˙ ∗ (θ) = 2(x∗ (θ) − θ) is bounded. The corresponding decisions (˜ x(θ), t˜(θ)) are such that ( ( t∗ (θ) if θ ∈ [0, 1]\[a − u(ǫ), b + v(ǫ)] x∗ (θ) if θ ∈ [0, 1]\[a − u(ǫ), b + v(ǫ)] and t˜(θ) = 2 x ˜(θ) = ǫ if θ ∈ [a − u(ǫ), b + v(ǫ)]. θ if θ ∈ [a − u(ǫ), b + v(ǫ)]

Although (˜ x(θ), t˜(θ)) may not necessarily be continuous at a − u(ǫ) or b + v(ǫ), it is incentive

compatible. Indeed, x ˜(θ) is monotonically increasing both on the interiors of [a − u(ǫ), b + v(ǫ)] by

construction and on [0, 1]\[a − u(ǫ), b + v(ǫ)] because it is equal to x∗ (θ) there and this function is

increasing. Moreover, the monotonicity of x∗ also implies that x ˜(a − u(ǫ)) = a = x∗ (a+ ) ≥ x∗ (θ)

for θ ≤ a − u(ǫ) and similarly x ˜(b + v(ǫ)) = θ = x∗ (b− ) ≤ x∗ (θ) for θ ≥ b + u(ǫ) which proves

monotonicity everywhere.

ˆ t˜(θ)) ˆ ˆ and (x∗ (θ), ˆ t∗ (θ)) ˆ ˆ We now compute the loss difference between mechanisms (˜ x(θ), θ∈Θ θ∈Θ as: Z 1 ˜ (θ), U ˜˙ (θ))f (θ)dθ − L∆ (U ∗ (θ), U˙ ∗ (θ))f (θ)dθ L∆ (U 0 0 Z a ˙ ˜ (θ), U ˜ (θ)) − L∆ (U ∗ (θ), U˙ ∗ (θ)))f (θ)dθ (L∆ (U =

Z

+ +

1

Z

Z

a−u(ǫ) b

a

b+v(ǫ) b

(A.35)

˜ (θ), U ˜˙ (θ)) − L∆ (U ∗ (θ), U˙ ∗ (θ)))f (θ)dθ (L∆ (U ˜ (θ), U ˜˙ (θ)) − L∆ (U ∗ (θ), U˙ ∗ (θ)))f (θ)dθ. (L∆ (U

Computing the second of those integrals, we get: A(ǫ) ≡

Z

b a

˜ (θ), U ˜˙ (θ)) − L∆ (U ∗ (θ), U˙ ∗ (θ)))f (θ)dθ = (ǫ2 − ǫ∆)(F (b) − F (a)). (L∆ (U

Turning now to the first of those integrals, observe that on the interval [a − u(ǫ), a]: s ˙∗ 2 ˜ (θ), U ˜˙ (θ)) − L∆ (U ∗ (θ), U˙ ∗ (θ)) = ǫ2 − ǫ∆ + U ∗ (θ) + δU˙ ∗ (θ) + ∆ −U ∗ (θ) − (U ) (θ) L∆ (U 4 Because 0 ≥ U ∗ (θ) ≥ −ǫ2 on this interval, we get: ˜ (θ), U ˜˙ (θ)) − L∆ (U ∗ (θ), U˙ ∗ (θ)) ≤ ǫ2 − ǫ∆ + δU˙ ∗ (θ) + ∆ L∆ (U

−U ∗ (θ) −

(U˙ ∗ )2 (θ) 4

p ≤ ǫ2 − ǫ∆ + δU˙ ∗ (θ) + ∆ −U ∗ (θ) ≤ ǫ2 + δU˙ ∗ (θ).

Hence, |B(ǫ)| = |

s

Z

a a−u(ǫ)

˜ (θ), U ˜˙ (θ)) − L∆ (U ∗ (θ), U˙ ∗ (θ)))f (θ)dθ| ≤ (L∆ (U 31

Z

a a−u(ǫ)

(ǫ2 + δ|U˙ ∗ (θ)|)f (θ)dθ.

From the definition of u(ǫ) and the fact that U ∗ is increasing in the interval [a − u(ǫ), a] for ǫ small enough: Z

a a−u(ǫ)

(ǫ2 + δ|U˙ ∗ (θ)|)f (θ)dθ ≤ (max f (θ)) ǫ2 u(ǫ) + δ θ

Z

a

a−u(ǫ)

U˙ ∗ (θ)dθ

!

= (max f (θ))ǫ2 (u(ǫ) + δ). θ

Hence, B(ǫ) is of order 2 in ǫ. A similar upper bound applies to the third integral C(ǫ) = R b+v(ǫ) ˜ (θ), U ˜˙ (θ)) − L∆ (U ∗ (θ), U˙ ∗ (θ)))f (θ)dθ. B(ǫ) and C(ǫ) are thus negligible compared (L∆ (U b

to A(ǫ). The sign of the left-hand side of (A.35) is thus given by the sign of A(ǫ), i.e., it is negative when ǫ is small enough.

ˆ t˜(θ)) ˆ ˆ is incentive compatible, we get a contradiction with the Since the mechanism (˜ x(θ), θ∈Θ ∗ ∗ ˆ ˆ posited optimality of (x (θ), t (θ)) ˆ and the corresponding utility profile U ∗ . θ∈Θ

The construction is similar when 0 < a < b = 1, by letting v(ǫ) = 0 and replacing EquaRa tion (A.34) by ǫ2 = a−u(ǫ) U˙ ∗ (θ)dθ, in which case the integral C(ǫ) is equal to zero; similarly, when R b+v(ǫ) ∗ 0 = a < b < 1, we let u(ǫ) = 0 and replace Equation (A.34) by ǫ2 = − U˙ (θ)dθ, in which b

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

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