# Sequential screening with unknown mean and spread

Daniel Danau, Annalisa Vinella

► To cite this version:

Daniel Danau, Annalisa Vinella. Sequential screening with unknown mean and spread. Economic Theory, 2025, pp.57. 10.1007/s00199-025-01652-6. hal-05045144

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Submitted on 11 Oct 2025

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# Sequential screening with unknown mean and spread

Daniel Danau<fnref n="1" /> ID · Annalisa Vinella<fnref n="2" /> ID

Received: 22 June 2024 / Accepted: 18 March 2025

© The Author(s) 2025

## Abstract

We study a sequential screening problem in which the information structure is characterized by neither first-order stochastic dominance (FOSD) nor mean-preserving spread (MPS). Specifically, we refer to a procurement contract for an activity entailing a cost of $γ = θ + εσ$. There is private information on the mean $θ$ and the spread $σ$, prior to contracting; and private observation of the zero-mean shock $ε$, after contracting. As in the literature on sequential screening, the screening instrument is the probability of the transaction taking place. At optimum, types are divided into groups, and the same allocation is assigned to all types within a group, as a way to discourage both overstatement of the mean and understatement of the spread within that same group. The distortion induced in the allocation depends on within-group average efficiency and cost of information. Relative to a scenario with FOSD, all types are recommended execution of the project more often. Relative to a scenario with MPS, performance is recommended more often from low-spread types and less often from high-spread types.

**Keywords** Sequential screening · Multidimensional screening · Mean and spread

**JEL Classification** D82

We are grateful to an Associate Editor and two anonymous Reviewers for their helpful suggestions. We also thank participants in the 2022 Conference on Mechanism and Institution Design (online via Singapore) and the 21st Annual International Industrial Organization Conference (Washington DC, April 2023) for their comments. The usual disclaimers apply.

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<footnote>Annalisa Vinella
annalisa.vinella@uniba.it</footnote>

<footnote>Daniel Danau
daniel.danau@unicaen.fr</footnote>

<footnote n="1">¹ Université de Caen Normandie - Centre de Recherche en Economie et Management, Esplanade de la Paix, 14000 Caen, France</footnote>

<footnote n="2">² Università degli Studi di Bari “Aldo Moro” - Dipartimento di Economia e Finanza, Largo Abbazia S. Scolastica, 53, 70124 Bari, Italy</footnote>
# 1 Introduction

In sequential screening problems, the agent is privately informed about the distribution of some relevant variable at the time when the contract with the principal is signed, and learns privately the realization of the relevant variable thereafter. The notion of sequential screening is introduced by Courty and Li (2000), who consider situations in which the distributions of the relevant variables (or types), included in a commonly known set, satisfy either first-order stochastic dominance (FOSD) or single-crossing mean-preserving spread (MPS). Under either such assumption, the authors derive a very useful result, namely, identification of the binding incentive constraints follows a similar procedure to static one-dimensional screening problems. Also the studies developed thereafter typically adopt one of the two notions of stochastic dominance. In this paper, we investigate optimal sequential screening in a setting with no stochastic dominance.

Together with indisputable benefits, the usual approach of assuming a stochastic ordering has also a limit. It is not consistent with the possibility of the two parameters of a distribution (the mean and the spread) being *both* privately known, which is what one would naturally expect. This restricts the applicability of the contractual solution derived in the literature. We believe that a good example of this issue is found in transportation projects. Indeed, in these projects, cost estimates are typically inaccurate, in that real values lie outside confidence intervals. Among scholars it has long been acknowledged that estimates are so inaccurate that strategic misrepresentation cannot be excluded.<fnref n="1" /> Without considering that those estimates include both point and interval estimates, one cannot assess what is wrong in the design of contracts for public projects and, hence, how the problem should be addressed.<fnref n="2" />

In theory, scholars seem to agree on the importance of generalizing the usual approach by taking the mean and the spread of the distribution to be both private information. However, so far no study has investigated this possibility. This is probably due to the complexity of multidimensional screening problems and the restrictions under which analytical solutions are usually found.

Based on the definition of MPS introduced by Rothschild and Stiglitz (1970), according to which a random variable is characterized by its mean and a noise, and the riskiness depends on the properties of the noise, we take the variable to be the cost $γ = θ + εσ$ of a project, where $E[ε] = 0$ and the spread $σ > 0$ gives the measure of riskiness. Instead of imposing that the mean be equal across distributions, we generalize this approach and let distributions differ in terms of both the mean $θ$ and the

<footnote n="1">¹ There are many references to this claim. Flyvbjerg et al. (2002) mention comparative analyses of actual and estimated costs in support of it, and also report frequent and significant differences between estimated and real cost values in transportation projects worldwide.</footnote>

<footnote n="2">² For instance, suppose that, in a project, the cost of activity is drawn either from the range [10, 30] or from the range [0, 40], and distributed uniformly. If the agent (a firm) is screened only on the basis of the point estimate, along the usual practice, then an incentive compatible contract would induce truthful revelation of the expected cost of 20. Suppose that execution of the activity is convenient only if the cost is found to be, at most, equal to 25. One problem here is that, if the true cost range is [0, 40], but contractual terms are set with reference to the range [10, 30], then the firm is over-compensated, in expected terms. Indeed, if the cost is between 25 and 40, then the activity will be shut down and the firm will gain nothing anyway; instead, if the cost is in the range [0, 10), then the firm will be paid as if the cost were higher.</footnote>
spread $\sigma$. Our focus is on a problem of task delegation, in which the principal does not observe the values of these parameters prior to signing the contract. In line with previous works on sequential screening, the surplus of the principal is a lump sum and the screening instrument is the probability of requiring the agent to execute the task.<fnref n="3" />

The screening problem we deal with is both sequential and multidimensional. In the analysis, the effect that the second-stage value of the random variable has on contractual decisions is captured by the cumulative distribution function of the shock $\varepsilon$. If second-stage private observation of the shock adds no agency costs (which we find to be true), then a solution to the screening problem in the contracting stage, which is multidimensional, can be identified. Indeed, our first result is that the principal should use the percentile of the shock distribution as a decision variable for an optimal allocation to be attained. As a consequence, multidimensional screening in the contracting stage can be treated as a static problem.

The other findings are as follows. In the absence of any incentive schemes, the agent would be prone to claim a falsely high expected cost (or mean) *vis-à-vis* the principal, and also to pretend that he is little uncertain about his point estimate of the cost, namely, that the *ex-post* value of the cost would not diverge significantly from the (high) expected value. To prevent overstatement of the mean, the principal should seldom require performance from high-mean types. Instead, to prevent understatement of the spread, the principal should require performance from low-spread types very often. Overall, the choice of the allocation has opposite effects on the incentives to misrepresent the two pieces of information. As a consequence, types are optimally bunched into groups. *Within* each group, the same allocation is assigned to all types so as to reconcile the incentives to misrepresent mean and spread in opposite directions. Moreover, incentives to lie *across* groups are eliminated if and only if the project is executed less often as one switches from one group of types to another that is characterized by a higher value of the mean and/or the spread. Actually, cheating across groups entails overstating the mean to a great (little) extent and understating the spread to a little (great) extent. From our investigation, it emerges that these incentives are eliminated if the monotonicity condition aforementioned holds across groups. As a final step of analysis, we characterize the optimal contract and develop a method for comparing the solution with those arising under FOSD and MPS. Because the solution to the two-dimensional screening problem is parameterized, with different values of the parameter identifying different groups of types, the method consists in parameterizing the solution derived in each of the two one-dimensional screening contexts (FOSD and MPS) so as to allow for a direct comparison with the solution obtained in the two-dimensional screening problem. Relative to sequential screening under FOSD, all types are recommended execution of the project more often. Relative to sequential screening under MPS, performance is recommended more often from low-spread types and less often from high-spread types. This precisely reflects the

<footnote n="3">³ In Courty and Li (2000) and more recent studies, the principal is a monopolist who does not know the distribution of consumer preferences, and the counterpart for the lump sum here referred to is the cost incurred by the monopolist to offer the product (or service) to consumers. The decision variable is the likelihood of the good or service being delivered. Given the linearity of the payoff, which we also assume, that likelihood is either one or zero.</footnote>
opposite effects that the choice of the allocation, as a screening instrument, has on the incentives to overstate the mean relative to the incentives to understate the spread.

This paper is, first, related to Courty and Li (2000) and the literature on sequential screening and dynamic contracting developed thereafter. Our general contribution to that literature resides in the identification of a methodology for the derivation of an analytical solution when FOSD applies to one subset of types and MPS to another, so that no stochastic dominance property applies to the entire set of types. Our main results depend on the incentives to misrepresent information *between* the two subsets of types rather than within each subset. We further outline a method for showing that, under public observation of the second-stage signal, non-local incentive constraints are irrelevant. The method permits us to prove that the irrelevance of second-stage private information, as highlighted by Esö and Szentes (2007) under FOSD, also extends to the two-dimensional sequential screening we consider. It consists in making use of the compatibility conditions of the partial differential equation induced by local incentive constraints, to establish the general form of the solution, prior to solving the program. In particular, with this, one does not need to verify global incentive compatibility *ex post*, at the solution to a reduced problem, which is the usual approach in multidimensional screening problems.<fnref n="4" />

The paper is also related to the narrow literature on multidimensional screening problems with one-dimensional screening instrument. That literature typically focuses on monopoly pricing under two privately known characteristics of consumer preferences, whereas we concentrate on procurement contracts. Laffont et al. (1987) consider situations in which the monopolist knows neither the intercept nor the slope of the demand curve. Araujo et al. (2022) extend the analysis to the case of a convex demand, and use the method of characteristics to find the solution to the quasi-linear partial differential equation implied by the incentive constraints. In so doing, they pin down a necessary condition for the contractual allocation to be locally incentive compatible. In a technical report, Deneckere and Severinov (2017) outline a general method for solving such problems. The similarity between this literature and our study resides, essentially, in the use of the method of characteristics to derive the solution. In addition, as aforementioned, we develop a procedure for showing that local incentive constraints in the first stage are the only constraints to induce agency costs. In our results, the necessary condition for an optimal allocation depends on the average surplus of the activity for each group of types, as in multidimensional screening settings with multiple screening instruments (Rochet and Choné 1998). Besides, given the purpose of our study, unlike the authors aforementioned, we do not restrict the domains of the two pieces of private information to the interval [0, 1] and do not impose any initial conditions on the two-dimensional screening problem.

To the best of our knowledge, the focus on task delegation is a novelty in sequential screening analysis, and is motivated by consideration of economic problems pertaining to delegation of public services. Following Riordan and Sappington (1987), who are the first to investigate the optimal monopoly franchising contract under private information on the signals of the cost distribution, that literature has developed along two main

<footnote n="4">⁴ One example is found in Armstrong and Rochet (1999), who provide a user guide for solving multidimensional screening problems.</footnote>
strands. The first strand includes models in which the firm decides whether to take some action before learning the true cost or delay the decision until after the cost will be realized (Kjerstad and Vagstad 2000; Board 2007; Mougeot and Naegelen 2011). The second strand includes models on cost overruns in which the firm is privately informed about the expected cost in the delegation stage, whereas the true cost is observed publicly in a later stage (Spulber 1990; Chen and Smith 2001). The issue of cost overruns is also argued to be critical in recent studies on public-private partnerships, such as Laffont (2003), Guasch et al. (2003) and Guasch et al. (2006), Iossa and Martimort (2012), Danau and Vinella (2015). In these studies, the expected cost is commonly known, whereas its realization is privately observed by the firm. None of them considers the possibility of the agent holding private knowledge of both the distribution and the true cost in the delegation of public activities.

The remainder of the paper is organized as follows. In Sect. 2, we describe the model and formalize the program of the principal. In Sect. 3, we focus on conditions for local incentive compatibility. In particular, we determine the solution to a partial differential equation, which must hold for the allocation to be incentive compatible. In Sect. 4, we state a reduced problem accounting for the partial differential equation as a constraint, and characterize its solution. The omitted constraints are considered in Sect. 5. It is shown that it is sufficient to have a monotonicity condition satisfied for the solution previously characterized to be implementable. Indeed, non-local incentive constraints and private observation of the shock are irrelevant. In Sect. 6, we draw a comparison between our solution and the solution to one-dimensional sequential screening under FOSD and MPS. In Sect. 7, we briefly highlight an additional incentive problem that would arise in a discrete-type setting. Section 8 concludes. Mathematical derivations are relegated to an appendix.

## 2 The model

We consider a principal P who delegates an activity to an agent. Execution of the activity occasions a cost of $\hat{\gamma}(\theta, \sigma, \varepsilon) \equiv \theta + \varepsilon\sigma$ to the agent, where $\theta \in \Theta$, $\sigma \in \Sigma$, $\Theta \equiv [\underline{\theta}, \bar{\theta}]$ and $\Sigma \equiv [\underline{\sigma}, \bar{\sigma}]$ are continuous and closed sets of strictly positive values. Furthermore, $\varepsilon$ captures a zero-mean shock that is unknown when the parties meet at the contracting table. For simplicity, we take $\varepsilon \in [-1, 1]$; also, henceforth, we refer to $\eta \equiv \frac{1+\varepsilon}{2} \in [0, 1]$, instead of $\varepsilon$, as the shock, and to $\gamma(\theta, \sigma, \eta) \equiv \hat{\gamma}(\theta, \sigma, 2\eta - 1)$ as the cost function. We further take $\theta > \sigma$, $\forall \theta \in \Theta$, $\forall \sigma \in \Sigma$, and denote, respectively, $M(\theta)$, $N(\sigma)$ and $\Xi(\eta)$ the cumulative distribution functions of $\theta$, $\sigma$ and $\eta$, independent of each other; their frequencies $m(\theta)$, $n(\sigma)$ and $\chi(\eta)$ are strictly positive everywhere.<fnref n="5" /> Prior to contracting with P, the agent observes the pair $(\theta, \sigma)$, which is the two-dimensional type in the first stage of the relationship, when the contract is signed. In a later (second) stage, after the contract is signed and before the activity is executed,

<footnote n="5">⁵ Our assumption that $\theta$ and $\sigma$ are independently distributed is in line with the narrow literature on multidimensional screening problems with a one-dimensional screening instrument. Our analytical results would carry over under a generalization of the regularity assumptions made on the distributions of $\theta$ and $\sigma$ (see footnote 17).</footnote>
the agent observes the shock $\eta$ as well. The activity run by the agent yields a surplus of $S$ to $P$.

The relationship between $P$ and the agent takes place in the two stages aforementioned, under full commitment. The contract is signed in the first stage, in which the agent has private information about the mean $\theta$ and the spread $\sigma$ of the cost distribution, whereas the realization of $\eta$ is learned by the agent in the second stage. With this information structure, our model generalizes the hypothesis made by Courty and Li (2000) on the ordering of distributions. The authors assume that the set of distributions of second-stage types is such that one between FOSD and single-crossing MPS holds. Here, the two-dimensional first-stage type $(\theta, \sigma)$ is a signal about the distribution of $\gamma(\theta, \sigma, \eta)$. The conditional distribution $G(\gamma|\theta, \sigma)$ is such that $G(\gamma|\theta, \sigma) = \Xi(\frac{\gamma-\theta+\sigma}{2\sigma})$. If $\theta < \theta'$, then $G(\gamma|\theta, \sigma)$ dominates $G(\gamma|\theta', \sigma)$ in a first-order stochastic sense: $G(\gamma|\theta', \sigma) > G(\gamma|\theta, \sigma)$. If $\sigma > \sigma'$, then $G(\gamma|\theta, \sigma') > G(\gamma|\theta, \sigma)$ if and only if $\eta < \frac{1}{2}$, so that the distributions satisfy the single-crossing MPS property. Instead, nothing can be said about the ordering of the distributions $G(\gamma|\theta, \sigma)$ and $G(\gamma|\theta', \sigma')$, which differ in terms of both mean and spread. Remarkably, our model departs from Courty and Li (2000) also in a second respect, namely, the support of second-stage types is bounded and depends on first-stage types. By making the hypothesis of a bounded support, we account for the fact that the cost cannot take negative values.<fnref n="6" /> Relative to multidimensional screening problems with one screening instrument, the value function (here, the cost function $\gamma(\theta, \sigma, \eta)$) is monotonic in one information dimension (here, $\theta$) and not in the other.<fnref n="7" />

## 2.1 The program of the principal

The Revelation Principle applies and the contractual offer is a pair $(t(\theta, \sigma, \eta); y(\theta, \sigma, \eta))$, for any given triplet $(\theta, \sigma, \eta)$, where $t(\theta, \sigma, \eta)$ is the transfer $P$ makes to compensate an agent of type $(\theta, \sigma)$ when the shock $\eta$ is realized, and $y(\theta, \sigma, \eta)$ is the probability of an agent of that type being required to execute the project. The ex-post utility of the agent is given by

$$u(\theta, \sigma, \eta) = t(\theta, \sigma, \eta) - y(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta). \quad (1)$$

<footnote n="6">⁶ It is, perhaps, worth clarifying that Courty and Li (2000) do not work with a specific form of the payoff function to derive the main results of their study. We assume a specific cost function as that helps us move from a setting with a single-crossing MPS distribution (as defined in the spirit of Rothschild and Stiglitz 1970) to a setting in which mean values differ across distributions, for a subset of distributions of the cost. It is also worth mentioning that, in more recent studies, Esö and Szentes (2007, 2017) proceed to an orthogonalization of any second-stage (or later) information. This makes it easier to solve the screening problem, in that one can make use of the probability integral transform theorem. In our setting, that is unnecessary because the distribution of the shock $\eta$ is independent of the type $(\theta, \sigma)$. Zik (2023) extends the setting of Courty and Li (2000) to situations in which the second-stage valuation depends on both first and second-stage private information, as the cost in our study. However, in line with most of the literature, the author assumes that the conditional distribution of the valuation satisfies FOSD, which is not the case of $G(\gamma|\theta, \sigma)$ here instead.</footnote>

<footnote n="7">⁷ See, for instance, Laffont et al. (1987) and Araujo et al. (2022).</footnote>
In expected terms, $U(\theta, \sigma) \equiv \int_{0}^{1} u(\theta, \sigma, \eta) \chi(\eta) d\eta$. We further denote

$$ \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \tilde{\eta}) \equiv t(\theta', \sigma', \tilde{\eta}) - \gamma(\theta, \sigma, \eta) y(\theta', \sigma', \tilde{\eta}) \quad (2) $$

the utility an agent of type $(\theta, \sigma)$ obtains when the shock $\eta$ is realized, if he reports $(\theta', \sigma')$ in the first stage and $\tilde{\eta}$ in the second stage. The program of P, denoted $\Gamma$, is written as follows:

$$ \max_{\{t(\theta, \sigma, \eta), y(\theta, \sigma, \eta)\}} \int_{\Theta} \int_{\Omega} \left( \int_{0}^{1} (y(\theta, \sigma, \eta) S - t(\theta, \sigma, \eta)) \chi(\eta) d\eta \right) n(\sigma) m(\theta) d\sigma d\theta $$

subject to

$$ U(\theta, \sigma) \geq \int_{0}^{1} \max_{\tilde{\eta}} \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \tilde{\eta}) \chi(\eta) d\eta, \quad \forall (\theta, \sigma, \eta) \quad (3) $$

$$ u(\theta, \sigma, \eta) \geq u(\theta, \sigma, \tilde{\eta}), \quad \forall (\theta, \sigma), \forall (\eta, \tilde{\eta}) \quad (4) $$

$$ U(\theta, \sigma) \geq 0, \quad \forall (\theta, \sigma) \quad (5) $$

$$ y(\theta, \sigma, \eta) \in [0, 1], \quad \forall (\theta, \sigma, \eta) \quad (6) $$

The incentive constraints (3) are those whereby the agent is unwilling to lie in the first stage, anticipating the report he would make in the second stage, for any given realization of $\eta$. The incentive constraints (4) are those whereby the agent is unwilling to misrepresent $\eta$ in the second stage, for any given combination of mean and spread. As is standard, the latter constraints imply that $y(\theta, \sigma, \eta)$ is continuous and differentiable almost everywhere with respect to $\eta$ such that $y_{\eta}(\theta, \sigma, \eta) \leq 0$. The outside opportunity of the agent is normalized to zero, which explains the formulation of the participation constraint (5). Lastly, (6) is a feasibility constraint on the probability of performance.

**Assumption 1** $S > \theta, \forall \theta, \text{ and } \sigma - \frac{1-N(\sigma)}{n(\sigma)} > 0, \forall \sigma$.

**Assumption 2** $\frac{M(\theta)}{m(\theta)}$ is non-decreasing $\forall \theta; \frac{1-N(\sigma)}{n(\sigma)}$ is non-increasing $\forall \sigma$.

The first assumption is made to rule out uninteresting cases. Obviously, $S > \theta$ is a necessary condition for the contractual relationship to be appealing to P, when the distribution of the agent's cost has a mean of $\theta$. As will become apparent, the second part of Assumption 1 ensures that there is enough uncertainty for P to optimally impose some restriction on the execution of the activity for some bad shock realization.<fnref n="8" /> The two parts of Assumption 2, both standard in mechanism design, require monotonicity of the hazard rates of the distributions $M(\theta)$ and $N(\sigma)$.

Henceforth, we use subscripts to denote partial derivatives of a function. In particular, $\frac{\partial U(\theta, \sigma)}{\partial \theta} \equiv U_{\theta}(\theta, \sigma)$ and $\frac{\partial U(\theta, \sigma)}{\partial \sigma} \equiv U_{\sigma}(\theta, \sigma)$.

<footnote n="8">⁸ For a similar assumption in multidimensional screening problems, see Armstrong (1996) for instance.</footnote>
For identification of the binding incentive constraints, it is useful to recall that in multidimensional screening problems the contract designer is to handle two essential issues. The first issue is to figure out the direction of incentives, which is a local problem, essentially. The second issue is to ascertain that global incentive constraints are satisfied, which is usually the case under very restrictive conditions in settings with a continuum of types. In addition, here it is also necessary to ensure that second-stage incentive constraints are satisfied.

## 3 Local incentive compatibility

Let us begin by considering the local first- and second-order conditions that are necessary for the type $(\theta, \sigma)$ to be revealed in the first stage, provided $\eta$ is truthfully reported in the second stage. The incentive constraints (3) are rewritten as

$$U(\theta, \sigma) \geq \int_{0}^{1} \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) d\eta, \quad \forall (\theta, \sigma, \eta), \quad (7)$$

namely, the utility type $(\theta, \sigma)$ obtains in the first stage when telling the truth in both stages is at least as great as the utility it would obtain by reporting $(\theta', \sigma')$ in the first stage and telling the truth in the second stage.

**Lemma 1** Suppose $y(\theta, \sigma, \eta)$ is non-increasing in $\theta$ and $(2\eta - 1) y(\theta, \sigma, \eta)$ non-increasing in $\sigma$. Then, the incentive constraints (7) are satisfied locally only if

1. $U_{\theta}(\theta, \sigma) = -\int_{0}^{1} y(\theta, \sigma, \eta) \chi(\eta) d\eta$ and
   $U_{\sigma}(\theta, \sigma) = -\int_{0}^{1} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta$;

2. $\frac{\partial}{\partial\theta} \int_{0}^{1} y(\theta, \sigma, \eta) \chi(\eta) d\eta \leq 0$ and $\frac{\partial}{\partial\sigma} \int_{0}^{1} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta \leq 0$;

3. $\left(\frac{\partial}{\partial\theta} \int_{0}^{1} y(\theta, \sigma, \eta) \chi(\eta) d\eta\right) \left(\frac{\partial}{\partial\sigma} \int_{0}^{1} y(\theta, \sigma, \eta) (2\eta - 1) \chi(\eta) d\eta\right) \geq \left(\frac{\partial}{\partial\theta} \int_{0}^{1} y(\theta, \sigma, \eta) (2\eta - 1) \chi(\eta) d\eta\right)^{2}$.

Conditions 1 and 2 are standard. The first condition in 1 tells that to a marginal increase in $\theta$, for given $\sigma$, should correspond a reduction in the information rent of the agent equal to the expected cost of production of the marginal unit. With this, it is possible to prevent misrepresentation in a neighborhood of $\theta$ from both below and above. The second condition in 1 suggests that, for analogous reason, to a marginal increase in $\sigma$, for given $\theta$, should correspond an increase in the rent equal to the decrease in per-unit expected total cost.<fnref n="9" /> Taken together, conditions 1 highlight that the direction of incentives at the local level is similar to that identified in one-dimensional

<footnote n="9">⁹ To see why $U_{\sigma}(\theta, \sigma) \geq 0$, recall that constraints (35) imply that $y(\theta, \sigma, \eta)$ is non-increasing in $\eta$. Also, $\int_{0}^{1} (2\eta - 1) \chi(\eta) d\eta = 0$ by assumption. Accordingly, $\int_{0}^{1} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta = \int_{0}^{\frac{1}{2}} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta + \int_{\frac{1}{2}}^{1} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta \leq \int_{0}^{1} (2\eta - 1) y(\theta, \sigma, \frac{1}{2}) \chi(\eta) d\eta + \int_{\frac{1}{2}}^{1} (2\eta - 1) y(\theta, \sigma, \frac{1}{2}) \chi(\eta) d\eta = y(\theta, \sigma, \frac{1}{2}) \int_{0}^{1} (2\eta - 1) \chi(\eta) d\eta = 0$. It follows that $\int_{0}^{1} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta \leq 0$.</footnote>
sequential screening problems. Those in 2 are monotonicity conditions with respect to $\theta$ and to $\sigma$, as implied by the second-order local incentive compatibility constraints. When $\theta < \theta'$, the expected likelihood of performance being required decreases with $\theta$, if local incentive constraints are satisfied. When $\sigma > \sigma'$, the likelihood of performance being required is higher, on average, if $\eta < \frac{1}{2}$ than is in the converse case, because the cost is then below its mean value: $\gamma(\theta, \sigma, \eta) = \theta + \sigma(2\eta - 1) < \theta$. Condition 3 refers to the two-dimensional nature of the first-stage types. It ensures that the determinant of the Hessian matrix of $\int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) d\eta$, as evaluated in a critical point, is non-negative. If the condition were violated, then the critical point would be a saddle point. Importantly, condition 3 is not sufficient when it holds as an equality.

To characterize the solution to $\Gamma$, we need to focus on conditions 1 and identify the information rent that should be assigned to any type $(\theta, \sigma)$ for the contract to be locally incentive compatible. Besides, we also need to consider that, in the contract, the agent should not display path-dependent incentives to lie. This requires that the Schwarz integrability condition hold, i.e., $U_{\theta\sigma}(\theta, \sigma) = U_{\sigma\theta}(\theta, \sigma)$, where $U_{\theta\sigma}(\theta, \sigma) \equiv \frac{\partial^2 U(\theta, \sigma)}{\partial\theta\partial\sigma}$ and $U_{\sigma\theta}(\theta, \sigma) \equiv \frac{\partial^2 U(\theta, \sigma)}{\partial\sigma\partial\theta}$.

## 3.1 Information rent

**Lemma 2** Suppose $y(\theta, \sigma, \eta)$ is non-increasing in $\theta$ and $(2\eta - 1)y(\theta, \sigma, \eta)$ non-increasing in $\sigma$. In the incentive compatible contract, the information rent is given by

$$ U(\theta, \sigma) = U(\bar{\theta}, \underline{\sigma}) + \int_{\theta}^{\bar{\theta}} \int_{0}^{1} y(x, \underline{\sigma}, \eta) \chi(\eta) d\eta dx - \int_{\underline{\sigma}}^{\sigma} \int_{0}^{1} (2\eta - 1) y(\theta, z, \eta) \chi(\eta) d\eta dz, \forall (\theta, \sigma). \quad (8) $$

According to (8), the incremental rent $U(\theta, \sigma) - U(\bar{\theta}, \underline{\sigma})$, over that obtained by the worst type $(\bar{\theta}, \underline{\sigma})$, includes two components, which are such that P should be ready to grant a higher compensation to the agent the lower his value of $\theta$ is and the higher his value of $\sigma$ is. First, along the frontier $\sigma = \underline{\sigma}$, to a reduction in the mean value from $\bar{\theta}$ to $\theta$ corresponds an additional compensation, which is tantamount to the benefit the agent could obtain by overstating the mean. Next, with $\theta$ being fixed, to an increase in the spread value from $\underline{\sigma}$ to $\sigma$ corresponds, again, an additional compensation, which is tantamount to the benefit the agent could obtain by understating the spread.<fnref n="10" /> Taking

<footnote n="10">¹⁰ Based on Lemma 1, in the expression of $U(\theta, \sigma)$ in (8), up to the constant $U(\bar{\theta}, \underline{\sigma})$, one identifies the differences $U(\theta, \underline{\sigma}) - U(\bar{\theta}, \underline{\sigma}) = \int_{\bar{\theta}}^{\bar{\theta}} \int_{0}^{1} y(x, \underline{\sigma}, \eta) \chi(\eta) d\eta dx$ and $U(\theta, \sigma) - U(\bar{\theta}, \underline{\sigma}) = -\int_{\underline{\sigma}}^{\sigma} \int_{0}^{1} (2\eta - 1) y(\theta, z, \eta) \chi(\eta) d\eta dz$. Alternatively, up to $U(\bar{\theta}, \underline{\sigma})$, one could express $U(\theta, \sigma)$ in terms of the differences $U(\bar{\theta}, \sigma) - U(\bar{\theta}, \underline{\sigma}) = -\int_{\underline{\sigma}}^{\sigma} \int_{0}^{1} (2\eta - 1) y(\bar{\theta}, z, \eta) \chi(\eta) d\eta dz$ and $U(\theta, \sigma) - U(\bar{\theta}, \sigma) = \int_{\bar{\theta}}^{\bar{\theta}} \int_{0}^{1} y(x, \sigma, \eta) \chi(\eta) d\eta dx$, $U(\bar{\theta}, \underline{\sigma})$. Although the expression of $U(\theta, \sigma)$ would change, the rent would be equal with the two methods.</footnote>
(8) in expectation, integration by parts over $\Theta$ and $\Sigma$ yields

$$ \int_{\Theta} \int_{\Sigma} U(\theta, \sigma) n(\sigma) m(\theta) d\sigma d\theta \\ = \int_{\Theta} \int_{\Sigma} \left[ \int_{0}^{1} \left( y(\theta, \underline{\sigma}, \eta) \frac{M(\theta)}{m(\theta)} - y(\theta, \sigma, \eta) (2\eta - 1) \frac{1 - N(\sigma)}{n(\sigma)} \right) \chi(\eta) d\eta \right] \\ \times n(\sigma) m(\theta) d\sigma d\theta + U(\bar{\theta}, \underline{\sigma}). \quad (9) $$

Considering the information rent to be conceded in order to attain local incentive compatibility, together with the expected transfer

$$ \int_{\Theta} \int_{\Sigma} \int_{\eta} t(\theta, \sigma, \eta) \chi(\eta) n(\sigma) m(\theta) d\eta d\sigma d\theta \\ = \int_{\Theta} \int_{\Sigma} U(\theta, \sigma) n(\sigma) m(\theta) d\sigma d\theta \\ + \int_{\Theta} \int_{\Sigma} \int_{\eta} y(\theta, \sigma, \eta) [\theta + (2\eta - 1)\sigma] \chi(\eta) n(\sigma) m(\theta) d\eta d\sigma d\theta, $$

the objective function in $\Gamma$ is reformulated as

$$ \int_{\Theta} \int_{\Sigma} \left[ \int_{0}^{1} \left( y(\theta, \sigma, \eta) F(\theta, \sigma, \eta) - y(\theta, \underline{\sigma}, \eta) \frac{M(\theta)}{m(\theta)} \right) \chi(\eta) d\eta \right] \\ \times n(\sigma) m(\theta) d\sigma d\theta - U(\bar{\theta}, \underline{\sigma}), \quad (10) $$

where

$$ F(\theta, \sigma, \eta) \equiv S - \theta - (2\eta - 1) \left( \sigma - \frac{1 - N(\sigma)}{n(\sigma)} \right). \quad (11) $$

The following result is straightforward.<fnref n="11" />

**Lemma 3** Suppose $y(\theta, \sigma, \eta)$ is non-increasing in $\theta$ and $(2\eta - 1) y(\theta, \sigma, \eta)$ non-increasing in $\sigma$. At optimum: $U(\bar{\theta}, \underline{\sigma}) = 0$; $U(\theta, \sigma) > 0, \forall (\theta, \sigma) \neq (\bar{\theta}, \underline{\sigma})$; and $y(\theta, \underline{\sigma}, \eta) = 0, \forall \theta, \forall \eta$.

## 3.2 Partial differential equation

We mentioned that eliminating path-dependent incentives to lie requires $U_{\theta\sigma}(\theta, \sigma) = U_{\sigma\theta}(\theta, \sigma)$. This yields the partial differential equation (henceforth, PDE) stated in the next lemma.

<footnote n="11">¹¹ One can easily show that if the multivariate density of $(\theta, \sigma)$ were some $\rho(\theta, \sigma) > 0$, with the two variables being possibly correlated, then the hazard rates $\frac{M(\theta)}{m(\theta)}$ and $\frac{1 - N(\sigma)}{n(\sigma)}$ would be replaced, respectively, by the conditional hazard rates $\frac{\int_{\theta}^{\theta} \rho(x, \sigma) dx}{\rho(\theta, \sigma)}$ and $\frac{1 - \int_{\sigma}^{\sigma} \rho(\theta, z) dz}{\rho(\theta, \sigma)}$. For instance, the former of these rates, that of $\theta$, is found by considering the likelihood of its value falling into the interval $[\underline{\theta}, \theta]$ as well as the occurrence of the spread value $\sigma$. Our results would carry over if the conditional hazard rates would satisfy similar conditions to those stated in Assumption 1 and 2.</footnote>
**Lemma 4** *Conditions 1 in Lemma 1 imply*

$$ \int_{0}^{1} [(2\eta - 1) y_{\theta}(\theta, \sigma, \eta) - y_{\sigma}(\theta, \sigma, \eta)] \chi(\eta) d\eta = 0, \forall (\theta, \sigma). \quad (12) $$

In general, there exist no analytical solutions to optimization problems including PDE constraints, unless one can first solve the PDE to identify its general solution, which is parameterized, and then use this solution in the unconstrained objective function, as is done in static multidimensional screening problems. Relative to those problems, the one here considered displays the peculiarity that the PDE refers to variations expressed in expected terms. One needs to characterize the expected allocation $\int_0^1 y(\theta, \sigma, \eta) \chi(\eta) d\eta$ such that the equation is satisfied. To that end, the PDE can be written as the following family of PDEs:

$$ (2\eta - 1) y_{\theta}(\theta, \sigma, \eta) - y_{\sigma}(\theta, \sigma, \eta) = \frac{f(\eta)}{\chi(\eta)}, \quad \forall \eta \in [0, 1], \quad (13) $$

where $f(\cdot)$ is any function such that $\int_0^1 f(\eta) d\eta = 0$. The solution to the original PDE is given by the value of $y(\theta, \sigma, \eta)$ determined at the solution to (13) for all shocks $\eta$. Indeed, the original PDE is recovered by integrating both sides of (13) over the range of $\eta$, namely $[0, 1]$, and considering that $\int_0^1 f(\eta) d\eta = 0$.

**Lemma 5** *Suppose $y(\theta, \sigma, \eta)$ solves (13). Then, there exists a function $y^0$ such that*

$$ y(\theta, \sigma, \eta) = y^0(s, \eta) - (\sigma - \underline{\sigma}) \frac{f(\eta)}{\chi(\eta)}, \quad (14) $$

for any $\theta, \sigma, \eta$ and $s$ satisfying

$$ \theta = s - (\sigma - \underline{\sigma})(2\eta - 1). \quad (15) $$

From now on, we denote $\hat{\theta}(s, \sigma, \eta)$ the solution to (15). Once $y^0(s, \eta)$ is known, the value of $y(\theta, \sigma, \eta)$ satisfying (14) is obtained as follows. For any given $\eta$, $s$ varies within the domain of $\theta$. For any given $s$ in the domain, one identifies the curve (15) passing through the point with coordinates $\theta = s$ and $\sigma = \underline{\sigma}$, and having a slope of $-(2\eta - 1)$. For any given $\eta$, along each such line (hence, for each $s$), $y^0(s, \eta)$ is constant. The solution is as in (14) for all the $(\theta, \sigma)$-pairs belonging to the curve.

To explain the two parts of (14), it is useful to rewrite it as

$$ y(\theta, \sigma, \eta) = y^h(\theta, \sigma, \eta) + y^p(\theta, \sigma, \eta), \quad (16) $$

where $y^h(\theta, \sigma, \eta) \equiv y^0(s, \eta)$ is the general solution to the homogeneous equation

$$ (2\eta - 1) y_{\theta}(\theta, \sigma, \eta) - y_{\sigma}(\theta, \sigma, \eta) = 0, \quad (17) $$
and $y^p(\theta, \sigma, \eta) = -(\sigma - \bar{\sigma}) \frac{f(\eta)}{\chi(\eta)}$ is a particular solution to (13). With this expression, $y^p(\theta, \sigma, \eta)$ is independent of $\theta$, and one can see it as an initial condition of (13) for some given $\theta$, say $\theta = \bar{\theta}$.<fnref n="12" /> However, as $y^p(\theta, \sigma, \eta)$, taken alone, is a solution to (13), it should also be a solution in the incentive compatible contract for its value to be non-zero. In fact, it is impossible to identify some $y^p(\theta, \sigma, \eta)$ in the feasible range [0, 1] such that conditions 2 in Lemma 1 are satisfied, unless $f(\eta) = 0$ and so $y^p(\theta, \sigma, \eta) = 0$. For instance, a function $f(\cdot)$ such that $\int_0^1 f(\eta) d\eta = 0$ is given by $f(\eta) = c \frac{2\eta-1}{\chi(\eta)}$, $\forall c$, in which case $y^p(\theta, \sigma, \eta) = -(\sigma - \bar{\sigma}) c \frac{2\eta-1}{\chi(\eta)}$. Then, at the solution to the program of P, it must be the case that $c = 0$.

As a consequence, (14) reduces to $y(\theta, \sigma, \eta) = y^0(s, \eta)$, for all $\theta, \sigma, \eta$ and $s$ satisfying (15). Basically, for the allocation to be incentive compatible, types are divided into groups, and those belonging to the same group - hence, to the same curve (15), for some given $s$ - are all assigned the same allocation. Furthermore, within any given group identified at the solution to (12), cheating is unattractive: no type can obtain more than the rent (8) it receives by mimicking some other type included in that same group. Therefore, an allocation satisfying the partial differential equation (12) is not only necessary but also sufficient to prevent types from mimicking each other within the same group.

**Lemma 6** *Take $(\theta', \sigma') \equiv (\theta + ra, \sigma + rb)$, for some $a$ and $b$ constant, together with $Z(r) \equiv \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) d\eta$. Then, $Z'(r) = 0$ if the vector $(a, b)$ is such that $\theta_r = a$ and $\sigma_r' = b$ are derived as ordinary equations at the solution to (12).*

In substance, this result suggests that the incentive constraints whereby no type is attracted by any other type bunched in the same allocation, are all binding. Whereas it clarifies that the incentive constraints are all satisfied within groups, it involves that the second-order test of the agent's maximization problem is inconclusive. That is, the necessary condition 3 in Lemma 1 does not hold strictly, hence it is not sufficient for local incentive compatibility. To account for this, in the reduced problem to be presented in a moment, not only the global incentive constraints, but also the local incentive constraints are taken to be satisfied, as long as they refer to types located in a neighborhood of any given group.<fnref n="13" />

<footnote n="12">¹² Noticeably, in the proof of Lemma 16, the functions $y^0(s, \eta)$ and $y^p(\theta, \sigma, \eta)$ are derived by setting the initial value of $\sigma$ to $\bar{\sigma}$, and letting the initial values of $\theta$ and $y(\cdot, \cdot, \cdot)$ vary freely, in line with the expression of the rent provided in (8). Alternatively, one could start from $\theta = \bar{\theta}$ and let the initial values of $\sigma$ and $y(\cdot, \cdot, \cdot)$ vary freely. The general solution to the homogeneous equation and the particular solution would both be different. However, they would still sum up to $y(\theta, \sigma, \eta)$, as in the previous case. Indeed, if any other particular solution $\hat{y}^p(\theta, \sigma, \eta)$ of (13) is considered, together with the associated general solution $\hat{y}^h(\theta, \sigma, \eta)$ of (17), then the general solution $y(\theta, \sigma, \eta)$ to (13) is such that the difference $y(\theta, \sigma, \eta) - \hat{y}^p(\theta, \sigma, \eta)$ solves (17), so that $y(\theta, \sigma, \eta) = \hat{y}^h(\theta, \sigma, \eta) + \hat{y}^p(\theta, \sigma, \eta)$.</footnote>

<footnote n="13">¹³ Related to this result, Dodds (2024) points out that in multidimensional screening problems in general (hence, not only those with one-dimensional screening instruments), the first-order approach fails when types are bunched because the second-order conditions are only weakly satisfied.</footnote>
## 4 A reduced problem and its solution

We now incorporate the result stated in Lemma 3 and the solution to the PDE in the objective function (10), and focus on a reduced problem, denoted $\Gamma_1$, in which all remaining constraints are omitted:

$$ \left\{ \begin{array}{l} \max_{y(\theta, \sigma, \eta), \forall (\theta, \sigma, \eta)} \int_{\Theta} \int_{\Omega} \left[ \int_{0}^{1} y(\theta, \sigma, \eta) F(\theta, \sigma, \eta) \chi(\eta) d\eta \right] n(\sigma) m(\theta) d\sigma d\theta \\ \text{subject to} \quad y(\theta, \sigma, \eta) = y^0(s, \eta) \text{ and } \theta = s - (\sigma - \bar{\sigma}) (2\eta - 1). \end{array} \right. $$

The following hypotheses are made in $\Gamma_1$:

1. the monotonicity conditions in Lemma 1 are satisfied;

2. those conditions are sufficient for local incentive compatibility in the first stage;

3. they are also sufficient for non-local incentive compatibility in the first stage;

4. $\eta$ is public observation.

As $y^h(\theta, \sigma, \eta) = y^0(s, \eta)$ for all $\theta, \sigma, \eta$ and $s$ satisfying (15), the objective function must be reformulated so as to refer to the $(s, \sigma, \eta)$-space. This yields

$$ \int_0^1 \int_{\underline{\theta}}^{\max\{\bar{\theta}; \bar{\theta} + (2\eta-1)(\bar{\sigma}-\bar{\sigma}_0)\}} y^0(s, \eta) \int_{\sigma_1(s, \eta)}^{\sigma_2(s, \eta)} F(\hat{\theta}(s, \sigma, \eta), \sigma, \eta) \\ \times m(\hat{\theta}(s, \sigma, \eta)) n(\sigma) \chi(\eta) d\sigma ds d\eta, \quad (18) $$

where

$$ \sigma_1(s, \eta) \equiv \begin{cases} \underline{\sigma} & \text{if } \eta \le \frac{1}{2} \\ \max\{\underline{\sigma}; \underline{\sigma} + \frac{s-\bar{\theta}}{2\eta-1}\} & \text{if } \eta > \frac{1}{2} \end{cases} $$

and

$$ \sigma_2(s, \eta) \equiv \begin{cases} \min\{\bar{\sigma}; \underline{\sigma} + \frac{s-\bar{\theta}}{2\eta-1}\}, & \text{if } \eta \le \frac{1}{2} \\ \min\{\bar{\sigma}; \underline{\sigma} + \frac{s-\theta}{2\eta-1}\}, & \text{if } \eta > \frac{1}{2}. \end{cases} $$

To interpret the ranges of integration over the values of $s$ and $\sigma$ in (18), it must be considered that the $(\hat{\theta}(s, \sigma, \eta), \sigma)$-pairs must all fall within the domain $\Theta \times \Sigma$. In the graphs in Fig. 1, the curve $\hat{\theta}(s, \sigma, \eta)$ is represented as a function of $\sigma$, and one can visualize those ranges for different values of $\eta$. In particular, the graph on the left refers to a case where the value of $s$ is below $\bar{\theta}$, that to the right to a case where it is above $\bar{\theta}$. First consider shocks that are either favorable or neutral, i.e. $\eta \le \frac{1}{2}$, so that the function $\hat{\theta}(s, \sigma, \eta)$ is either constant or upward sloping with respect to $\sigma$. With the initial value being $\hat{\theta}(s, \underline{\sigma}, \eta) = s$, and the function being non-decreasing in $\sigma$, by taking $s \ge \underline{\theta}$ the relevant range of $s$ turns out to be $\Theta$. Indeed, in (18), the domain of $s$ is identified as being $[\underline{\theta}, \max\{\bar{\theta}; \bar{\theta} + (\bar{\sigma} - \underline{\sigma})(2\eta - 1)\}] = \Theta$. Next, for high values of at least one between $s$ and $\sigma$, the function $\hat{\theta}(s, \sigma, \eta)$ takes values outside $\Theta$.
<figure><img src="image_4.png" /><figcaption>Fig. 1 Examples of curves $\theta(s, \sigma, \eta) = s - (\sigma - \bar{\sigma}) (2\eta - 1)$ for $\eta_1 < \eta_2 < \frac{1}{2} < \eta_3 < \eta_4$, together with $s \in (\underline{\theta}, \bar{\theta})$ (graph on the left) and $s > \bar{\theta}$ (graph on the right)</figcaption></figure>

In the graph on the left, this is the case of $\eta = \eta_1$, which explains why, in that case, the upper bound of $\sigma$ in (18) is $\sigma_2(s, \eta_1) = \underline{\sigma} + \frac{s-\bar{\theta}}{2\eta_1-1} < \bar{\sigma}$. The converse is true for $\eta = \eta_2$. Now consider unfavorable shocks ($\eta > \frac{1}{2}$). One can see that there are values of $s$ exceeding $\bar{\theta}$. With this, it is possible to account for values of $\hat{\theta}(s, \sigma, \eta)$ close to $\bar{\theta}$ that would otherwise be excluded, given that the curve $\hat{\theta}(s, \sigma, \eta)$ is downward sloping with respect to $\sigma$. In the graph on the right, this is the case of the shocks $\eta_3$ and $\eta_4$. For such shock realizations and high values of $s$, the curve $\hat{\theta}(s, \sigma, \eta)$ may fall outside the range when $\sigma$ takes low values. This is why the spread is bounded from below at $\sigma_1(s, \eta) > \underline{\sigma}$. In addition, if the shock is as high as $\eta_4$ in the graph on the right, then the curve $\hat{\theta}(s, \sigma, \eta)$ falls outside the range of $\sigma$ on both sides.

The function (18) is to be maximized point-wise with respect to $y^0(s, \eta)$, for any given $(s, \eta)$-pair and associated range of values $[\sigma_1(s, \eta), \sigma_2(s, \eta)]$. This leads to a cut-off solution $h(s)$, such that

$$y^0(s, \eta) = \begin{cases} 1 & \text{if } \eta \le h(s) \\ 0 & \text{otherwise.} \end{cases}$$

For any pair $(s, \sigma)$ and associated value $\theta(s, \sigma) \equiv \hat{\theta}(s, \sigma, h(s))$, at optimum, one has

$$h(s) = \frac{1}{2} \left( 1 + \frac{\int_{\sigma_1(s,h(s)))}^{\sigma_2(s,h(s)))} (S - \theta(s, \sigma)) m(\theta(s, \sigma)) n(\sigma) d\sigma}{\int_{\sigma_1(s,h(s)))}^{\sigma_2(s,h(s)))} \left(\sigma - \frac{1-N(\sigma)}{n(\sigma)}\right) m(\theta(s, \sigma)) n(\sigma) d\sigma} \right), \quad (19)$$

when this is below one, and $h(s) = 1$ otherwise. Recall that, under Assumption 1, the right-hand side of (19) is finite for any $s$-group. To rule out uninteresting cases, henceforth we only focus on interior solutions of $h(s)$, $\forall s$. Whereas (19) is an implicit

<figure><img src="image_5.png" /></figure>
equation defining $h(s)$, its specific value depends on the distribution functions $M(\theta)$ and $N(\sigma)$. However, from this implicit expression of $h(s)$, we already see that $h(s) > \frac{1}{2}$, regardless of the specific properties of the two distribution functions. This condition has a simple interpretation: if it is worth requiring some type $(\theta(s), \sigma(s), \sigma)$ to perform when unfavorable shocks $(\eta > \frac{1}{2})$ are realized, then it is clearly worth requiring that same type to perform when any favorable shock $(\eta < \frac{1}{2})$ is realized. Given $h(s)$, one obtains the three results stated hereafter.

**Proposition 1** For any type $(\theta, \sigma)$, if $y(\theta, \sigma, \eta)$ solves $\Gamma_1$, then the following statements are true:

1. The optimal expected allocation is some percentile $\phi(\theta, \sigma)$ of the distribution $\Xi(\cdot)$:

$$ \int_{0}^{1} y(\theta, \sigma, \eta) \chi(\eta) d\eta = \int_{0}^{\phi(\theta, \sigma)} \chi(\eta) d\eta = \Xi(\phi(\theta, \sigma)), \quad (20) $$

where $\phi(\theta, \sigma)$ is chosen by $P$.

2. Types are divided into groups, with $\phi(\theta, \sigma)$ being constant within each group: $\phi(\theta, \sigma) = h(s) > \frac{1}{2}$ for any $(\theta, \sigma, s)$ such that

$$ \theta = s - \left(\sigma - \underline{\sigma}\right) (2h(s) - 1). \quad (21) $$

3. Within each group identified by the value of $s$, $h(s)$ is directly proportional to the ratio between the average value of the benefit $S - \theta(s, \sigma)$ and the average virtual spread $\sigma - \frac{1-N(\sigma)}{n(\sigma)}$, along the curve $\theta(s, \sigma)$.

Let us first consider result 1 in the proposition. It suggests that one could refer directly to the percentile $\phi(\theta, \sigma)$ as a decision variable, and pin down its optimal value, as in a context of static multidimensional screening. This is potentially very useful in applications. Noticeably, if the screening problem is solved by referring to $\phi(\theta, \sigma)$ as the decision variable (along the lines of the proof of the next remark), then the solution is identical to that previously derived, while the methodology appears to be more in line with that usually followed in the literature. Being based on this result, henceforth we refer to $(t(\theta, \sigma, \eta); \phi(\theta, \sigma))$, instead of $(t(\theta, \sigma, \eta); y(\theta, \sigma, \eta))$, as contracting variables.

**Remark 1** If the contractual allocation is given by the percentile $\phi(\theta, \sigma)$ of the distribution $\Xi(\cdot), \forall (\theta, \sigma)$, then the PDE (12) becomes

$$ (2\phi(\theta, \sigma) - 1)\phi_{\theta}(\theta, \sigma) = \phi_{\sigma}(\theta, \sigma). \quad (22) $$

By solving $\Gamma_1$, it is found that, at optimum, $\phi(\theta, \sigma) = h(s)$ for any $(\theta, \sigma, s)$ such that (21) is satisfied.<fnref n="14" />

<footnote n="14">¹⁴ The procedure followed in the proof to solve the reduced problem is reminiscent of that used in static multidimensional screening problems, such as Laffont et al. (1987) and Araujo et al. (2022). Unlike in these latter studies, here we do not impose any initial conditions to get rid of terms like $\phi(\theta, \sigma)$.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 2 Direction of incentives of type $(\theta, \sigma)$ (graph on the left). Family of curves $\theta(s, \sigma)$ with $h(s)$ decreasing in $s$ (graph on the right)</figcaption></figure>

To explain result 2 in Proposition 1, we refer to the graph on the left in Fig. 2, and focus on the incentives of type $(\theta, \sigma)$ as represented by point A. If types were all assigned the same allocation, then type $(\theta, \sigma)$ would have something to gain from either overstating the mean or understating the spread, namely, reporting either $(\theta', \sigma)$ or $(\theta, \sigma')$, where $\theta' > \theta$ and $\sigma' < \sigma$. Basically, if one takes each of these incentives separately, then this conclusion replicates what Courty and Li (2000) find in a context of one-dimensional sequential screening. Here, there is obviously a mix of the two incentives. If the allocation is not differentiated across types, then type $(\theta, \sigma)$ has something to gain from both overstating the mean and understating the spread. However, a distortion in the allocation, which allows to discourage misrepresentation of one information dimension, ends up encouraging misrepresentation of the other. Thereby, the only possibility is to bunch low-mean-and-high-spread types with high-mean-and-low-spread types, and assign them all the same allocation. This explains why the agent is indifferent between revealing information and mimicking any other type included in the same group as his own type (Lemma 6). In the graph on the left in Fig. 2, this is true for types $(\theta, \sigma)$ and $(\theta', \sigma')$ as long as they belong to the same $\theta(s, \sigma)$-curve, and the result extends to all types located on that curve. In the graph on the right, an entire family of $\theta(s, \sigma)$-curves are represented, with $h(s)$ being lower for types located on upper curves (we explain in a later stage why $h(s)$ is taken to be monotonic in $s$). Overall, at the solution to the reduced problem, in which only the solution to the PDE is considered, bunching is the tool that allows P to eliminate the incentives to mimic types along each curve $\theta(s, \sigma)$.

Interpretation of result 3 in Proposition 1 follows from bunching of types. The allocation corresponding to any $\theta(s, \sigma)$-curve depends on the expected virtual surplus associated with the types located on it. To see this, it is useful to rewrite (19) as $\Phi(s, h(s)) = 0$, where

$$ \Phi(s, \eta) = \int_{\sigma_1(s, \eta)}^{\sigma_2(s, \eta)} (S - \hat{\theta}(s, \sigma, \eta)) m(\hat{\theta}(s, \sigma, \eta)) n(\sigma) d\sigma $$

<figure><img src="image_7.png" /></figure>
$$ -\int_{\sigma_1(s, \eta)}^{\sigma_2(s, \eta)} (2\eta - 1) \left( \sigma - \frac{1 - N(\sigma)}{n(\sigma)} \right) m(\hat{\theta}(s, \sigma, \eta)) n(\sigma) d\sigma \quad (23) $$

is the expected virtual surplus generated by the group of types identified by $s$ in program $\Gamma_1$. It is given by the difference between the expected (net) benefit (the first line of (23)) and the expected agency cost (the second line) associated with the $s$-group. The level of $h(s)$ is the greatest value of the shock such that the expected virtual surplus is non-negative. Presented in this way, the solution can be regarded as a generalization of that obtained by Courty and Li (2000) under MPS. In Sect. 6 we show that, as transposed in our setting, the solution obtained by Courty and Li (2000) for a case where $\theta$ (not $\sigma$) is common knowledge (i.e., distributions $G(\gamma | \theta, \sigma)$ respect MPS) boils down to $\Psi(\sigma, \phi(\theta, \sigma)) = 0$, where

$$ \Psi(\theta, \sigma, \eta) = (S - \theta) - \left( \sigma - \frac{1 - N(\sigma)}{n(\sigma)} \right) (2\eta - 1) \quad (24) $$

is the virtual surplus associated with type $\sigma$ and shock $\eta$. Here, $\Phi(s, \eta)$ is the virtual surplus as parameterized by the group-identifier $s$ and taken in expectation over the mean and spread values corresponding to the types falling into that group.<fnref n="15" /> In Sect. 6, we provide a more detailed comparison with the one-dimensional sequential screening solution. Before turning to that, in the next section, we complete the analysis and verify the constraints omitted in $\Gamma_1$.

# 5 Incentive constraints in the general program

## 5.1 Monotonicity across groups of type

Recall that the monotonicity conditions omitted in the reduced problem $\Gamma_1$ concern (i) the monotonicity of both $y(\theta, \sigma, \eta)$ and its expected value computed over $\theta$, and (ii) the monotonicity of both $(2\eta - 1)y(\theta, \sigma, \eta)$ and its expected value computed over $\sigma$ (Lemma 1). Intuitively, with the decision variable being the percentile $\phi(\cdot, \cdot)$, those conditions are satisfied, indeed, if and only if $\phi(\cdot, \cdot)$ is non-increasing in both $\theta$ and $\sigma$. As types within groups are bunched, monotonicity of $\phi(\cdot, \cdot)$ with respect to $\theta$ and to $\sigma$ translates into monotonicity of $h(s)$ across groups of types.

**Proposition 2** At the solution to $\Gamma_1$, monotonicity of $y(\theta, \sigma, \eta)$ and condition 2. in Lemma 1 boil down to $\phi_\theta(\theta, \sigma) \le 0$ and $\phi_\sigma(\theta, \sigma) \le 0$, where $\phi(\theta(s, \sigma), \sigma) = h(s)$, for all $h(s) > \frac{1}{2}$. Those conditions are both satisfied if $h'(s) \le 0$.

In one-dimensional sequential screening problems, the optimal allocation is monotone if the virtual surplus is monotonic both in the first-stage privately known variable and in the choice variable. This condition is satisfied given the monotonicity of the

<footnote n="15">¹⁵ Noticeably, our solution would be a generalization of the one-dimensional solution with FOSD, had we used the alternative formulation of the rent mentioned in footnote 10.</footnote>
hazard rate of the type distribution (see Krahmer and Strausz (2017) for a discussion). In our setting, $h(s)$ is monotonic if the expected virtual surplus $\Phi(s, h(s))$ is monotonic in both $s$ and $h(s)$. To verify this, observe that the virtual surplus

$$S - \theta(s, \sigma) - (2h(s) - 1) \left( \sigma - \frac{1 - N(\sigma)}{n(\sigma)} \right) \quad (25)$$

is monotonic both within and across groups of types. On the one hand, it is monotonic within groups as it decreases with the spread value $\sigma$ all over the feasible range $[\sigma_1(s, h(s)), \sigma_2(s, h(s))]$ delimiting each $s$-group of types. As in one-dimensional sequential screening, this is true as the hazard rate of the distribution of $\sigma$ is monotonic (Assumption 2). On the other hand, the virtual surplus is monotonic across groups, for two reasons. First, (25) decreases with $s$, for any given $\sigma$, as the group identified by the higher value of $s$ includes higher-mean types, which are associated with a lower virtual surplus. Second, a raise in $s$ induces a rightward shift in the range $[\sigma_1(s, h(s)), \sigma_2(s, h(s))]$, when strictly included in $\Sigma$, since the domain of mean values $\Theta$ is bounded. Therefore, the newly concerned group includes not only higher-mean types but also higher-spread types, which are associated with a lower virtual surplus too, as we said. If the allocation is kept unchanged for this group, then it is still the case that incentives to overstate the mean and to understate the spread compensate within the group, provided (21) holds. However, $h(s)$ is not at its optimal level. For instance, if type frequencies do not vary significantly across groups, then, the monotonicity of the virtual surplus just mentioned results in a lower expected virtual surplus for higher-$s$ groups. To restore optimal $(\Phi(s, h(s)) = 0)$, performance of higher-$s$ groups is to be shut down for some bad shock realizations, i.e., a higher $s$ should correspond to a lower $h(s)$ so as to bring the virtual surplus back to a higher level for all types included in the group. If instead type frequency does vary significantly, as one switches from one group to another, then the condition $h'(s) \le 0$ holds as long as the frequency of types yielding low virtual surplus increases more (decreases less) relative to that of other types within the same group. In this case, the expected virtual surplus is still decreasing in $s$, and $h(s)$ should be downsized at optimum. Noticeably, any higher-$s$ group can be treated either as a group with equal $\sigma$ but higher $\theta$ for nearly all types, or as a group with higher $\sigma$ for nearly all types but equal $\theta$, or with both higher $\theta$ and higher $\sigma$ for all types. For this reason, it suffices to look at the distribution of one between $\theta$ and $\sigma$ to learn how the allocation varies across groups. Being based on this conclusion, we hereafter content ourselves with providing a few examples of distributions of $\theta$, and abstain from considering distributions of $\sigma$ to avoid redundancy.<fnref n="16" />

**Uniform distribution of $\theta$** First take $m(\theta) = \frac{1}{\theta-\theta}$, $\forall \theta \in \Theta$. As there is no change in the frequency of mean values when switching from one group of types to another, the effects previously explained, following a change in $s$ and $h(s)$, are fully at work. As a

<footnote n="16">¹⁶ The analytical development of the examples here provided is found in Appendix G. Consideration of specific distributions of $\sigma$ rather than $\theta$ would lead to analogous conclusions; the analytical development is available with the authors.</footnote>
result, monotonicity holds across groups, for any distribution function of $\sigma$ satisfying
the regularity conditions in Assumption 1 and 2.

**Truncated exponential distribution of $\theta$** Take $m(\theta) = \frac{\lambda e^{-\lambda(\theta-\theta)}}{1-e^{-\lambda(\bar{\theta}-\theta)}}, \forall \theta \in \Theta$, for some constant $\lambda > 0$. In this case, a shift from one group of types to another does involve a change in frequencies from the original to the newly concerned group. That is, for any given $\sigma$ and $s_2 > s_1$, type $(\theta(s_2, \sigma), \sigma)$ has a lower frequency than type $(\theta(s_1, \sigma), \sigma)$. If $h(s_2)$ is set below $h(s_1)$, then the $\theta(s_2, \sigma)$-curve is flatter than the $\theta(s_1, \sigma)$-curve. As a result, a shift from the $s_1$-group including type $(\theta_1, \sigma)$ to the $s_2$-group including type $(\theta_2, \sigma)$ corresponds to a greater reduction in frequencies to the right than to the left of $(\theta_2, \sigma)$, namely, to a greater reduction in the frequency of higher-spread types within the group. Remarkably, these types are associated with lower levels of virtual surplus. Thus, relative to the case of the uniform distribution, there is here an additional reason for setting $h(s_2)$ below $h(s_1)$ to restore optimality $(\Phi(s_2, h(s_2)) = 0)$.

**Truncated normal distribution of $\theta$** Take $m(\theta) = \frac{e^{-\frac{(\theta-E[\theta])^2}{2\nu^2}}}{\int_{\bar{\theta}}^{\bar{\theta}} e^{-\frac{(x-E[\theta])^2}{2\nu^2}} dx}, \forall \theta \in \Theta$, where $\mathbb{E}[\theta]$ and $\nu > 0$ are the mean and the standard deviation of the parental normal distribution. Frequency is now increasing for $\theta < \mathbb{E}[\theta]$ and decreasing for $\theta > \mathbb{E}[\theta]$. Therefore, unlike in the case of the exponential distribution, there are now asymmetric changes in frequencies when shifting from low-$s$ to high-$s$ group, even if $h(s_2)$ is set equal to $h(s_1)$. With such changes, types generating a lower virtual surplus are more frequent. Furthermore, if $h(s_2)$ is set below $h(s_1)$, then the range of types with mean values above $\mathbb{E}[\theta]$ (hence, with lower frequencies) is widened. It follows that the lower $h(s_2)$ is set, the less frequent are the types located to the left of some given $(\theta(s_2, \sigma), \sigma)$, relative to the types located to the right. This effect works in an opposite direction to that of the exponential distribution. For this reason, the monotonicity condition does not hold for all values of the parameters of the distribution. A sufficient condition is to have $\frac{\bar{\theta}-\theta}{\nu^2} \le \rho$, for some $\rho > 0$, entailing that frequencies do not differ significantly across types. In that case, prevailing effects are those already described with reference to the uniform distribution.<fnref n="17" />

## 5.2 Sufficiency for local incentive compatibility

The need to verify local incentive compatibility *ex post* rests with the circumstance that
the usual second derivative test on critical points is inconclusive in our environment.

<footnote n="17">¹⁷ Remarkably, the sufficient condition for monotonicity mentioned in the text is satisfied in the limit case where $\bar{\theta} - \theta = 0$ and the sequential screening problem boils down to one with MPS. Specifically, the range $[\sigma_1(s, h(s)), \sigma_2(s, h(s))]$ reduces to some $\sigma(s)$ solving (21) for given $\theta$, $\forall s$, and the expected virtual surplus $\Phi(s, \eta)$ reduces to the virtual surplus $\Psi(\theta, \sigma(s), \eta)$. As $\sigma(s)$ increases with $s, h(s)$ is found to be monotonic simply because the virtual surplus $\Psi(\theta, \sigma(s), \eta)$ is monotonic in $\sigma$.</footnote>
The reason why the test fails here is that, due to the necessity to eliminate path-dependent incentives to lie, condition 3 in Lemma 1 must hold as an equality. As already explained, any type located on a given curve (15) obtains the same rent, regardless of whether he tells the truth or mimic some other type located on the same curve (Lemma 6). The rent decreases in a neighborhood of the true type instead, if the mimicked type does not belong to that curve. This result follows a similar logic to global incentive compatibility, as will be discussed in a moment.

**Proposition 3** Assume that $h'(s) \le 0, \forall s$. The conditions in Proposition 2 are sufficient for the solution to $\Gamma_1$ to be locally incentive compatible.

## 5.3 Sufficiency for global incentive compatibility

In one-dimensional screening problems, when the single-crossing property holds, only local first- and second-order conditions may be binding. The local second-order condition is equivalent to the monotonicity condition. In sequential screening problems, the stochastic ordering of the distributions plays the same role as the single-crossing condition.<fnref n="18" /> Taken together with the monotonicity of the allocation with respect to both first- and second-stage type, it is sufficient for global incentive compatibility. Unlike in those problems, here no stochastic ordering is identified. Therefore, we are faced with a common issue in multidimensional screening problems, namely, the lack of a property that would act as the counterpart of the single-crossing property (see Rochet and Stole 2003 for further explanations).

In the preliminary result that follows, we identify the necessary and sufficient condition that the allocation $\phi(\theta, \sigma)$ is to satisfy for type $(\theta, \sigma)$ - for which is it designed - to be unwilling to mimic some other type $(\theta', \sigma')$.

**Lemma 7** Define

$$ \psi(\theta, \sigma, \theta', \sigma') = \int_{\theta}^{\theta'} \int_{\phi(\theta', \sigma')}^{\phi(x, \sigma')} \chi(\eta) \, d\eta \, dx + \int_{\sigma'}^{\sigma} \int_{\phi(\theta, z)}^{\phi(\theta', \sigma')} (2\eta - 1) \, \chi(\eta) \, d\eta \, dz. \quad (26) $$

Incentive constraints are satisfied if and only if

$$ \psi(\theta, \sigma, \theta', \sigma') \ge 0, \forall (\theta, \sigma), (\theta', \sigma'). \quad (27) $$

The expression of $\psi(\theta, \sigma, \theta', \sigma')$ represents the surface of incremental increase in the expected cost $\int_0^{\phi(\theta, \sigma)} \gamma(\theta, \sigma, \eta) \chi(\eta) \, d\eta = \int_0^{\phi(\theta, \sigma)} [\theta + (2\eta - 1)\sigma] \chi(\eta) \, d\eta$, when shifting from type $(\theta, \sigma)$ to type $(\theta', \sigma')$. For type $(\theta, \sigma)$ to be discouraged from claiming $(\theta', \sigma')$, it must receive a rent that is at least as great as $\psi(\theta, \sigma, \theta', \sigma')$. Here, a complication rests with the circumstance that the two terms in $\psi(\theta, \sigma, \theta', \sigma')$ are not directly comparable. To circumvent this complication, we adopt the method

<footnote n="18">¹⁸ Courty and Li (2000) show that, under both FOSD and single-crossing MPS, identification of the binding incentive constraints follows a similar procedure to static one-dimensional screening problems. Krahmer and Strausz (2017) show that FOSD in a sequential screening problem is equivalent to the single-crossing property in a static problem in which the contract designed by the principal is stochastic.</footnote>
hereafter described, which could also suit other environments without a clear stochastic
dominance among distributions. Provided local incentive compatibility requires that
the allocation be group-specific, we identify the surface covering all groups falling
into the range of types from $(\theta, \sigma)$ to $(\theta', \sigma')$ and use the allocations designed for
those types to evaluate $\psi(\theta, \sigma, \theta', \sigma')$. With this, also taking into account that the
allocation is monotonic across groups, we derive the following result.

**Proposition 4** For all $h(s) > \frac{1}{2}$ such that $h'(s) \le 0$, and $\forall (\theta, \sigma)$ such that $(\theta, \sigma) = (\theta(s, \sigma), \sigma)$ for any given $s$, (27) holds. Consequently, no type $(\theta, \sigma) = (\theta(s, \sigma), \sigma)$ has incentives to mimic some other type $(\theta', \sigma') = (\theta(s', \sigma'), \sigma')$, for $s' \ne s$.

To see why monotonicity of the allocation across groups is sufficient, first take the
two pieces of private information separately and recall that FOSD holds for $\theta$, MPS
for $\sigma$. Monotonicity of the allocation with respect to $\theta$ ($\phi_{\theta}(\theta, \sigma) \le 0$) is sufficient to
prevent cheating on that dimension, provided that the contract is incentive compatible
at a local level. Indeed, (27) holds for $\sigma = \sigma'$. The same is true as regards monotonicity
with respect to $\sigma$ ($\phi_{\sigma}(\theta, \sigma) \le 0$). Because of this, (27) holds for $\theta = \theta'$. For similar
reasons, incentives to lie on both pieces of information in the same direction (namely,
to either overstate or understate both of the two) are eliminated. Therefore, the only
issue with global incentive compatibility may arise if the agent has an interest in
misrepresenting both $\theta$ and $\sigma$ in opposite directions. Accordingly, one needs to verify
the cases where $\theta' > (<) \theta$ and $\sigma' < (> \sigma) \sigma$. Furthermore, in light of Lemma 6, the
concern is limited to types belonging to different groups.

To prove that cheating is prevented from such types, we first define $s(\theta, \sigma)$ as the
inverse function of (21). With this, we can rewrite the two terms in $\psi(\theta, \sigma, \theta', \sigma')$
by considering the allocations $h(s(\cdot, \cdot))$ identified over the ranges $[\theta, \theta']$ and $[\sigma', \sigma]$
when $\theta' > \theta$ and $\sigma' < \sigma$, and over the ranges $[\theta', \theta]$ and $[\sigma, \sigma']$ when $\theta' < \theta$
and $\sigma' > \sigma$. This allows us to make the two expressions of $\psi(\theta, \sigma, \theta', \sigma')$ directly
comparable. For instance, when $\theta' > \theta$ and $\sigma' < \sigma$, one identifies either (i) the set of
allocations that corresponds to the groups of types with mean values falling into the
range $[\theta, \theta']$ and spread values into a subset of $[\sigma', \sigma]$; or (ii) the set that corresponds
to the groups of types with spread values falling into the range $[\sigma', \sigma]$ and mean values
into a subset of $[\theta, \theta']$. Defining $\xi(\theta')$ the spread value such that types $(\theta', \sigma')$ and
$(\theta, \xi(\theta'))$ are included in the same group, case (i) arises when type $(\theta, \sigma)$ lies to
the right of the curve passing through points $(\theta', \sigma')$ and $(\theta, \xi(\theta'))$; case (ii) when
type $(\theta, \sigma)$ lies to the left of that curve instead. In either case, we can reformulate the
second term in the expression of $\psi(\theta, \sigma, \theta', \sigma')$ as a surface comprised between two
mean values, which is then comparable with the surface expressed by the first term.
This all allows us to show that (27) holds.

To interpret the result in Proposition 4, it is useful to make reference to the graph in
Fig. 3, which represents the direction of incentives, taking $\theta < \theta'$ and $\sigma > \sigma' > \sigma''$.
If type $(\theta, \sigma)$ declares $(\theta', \sigma')$, where $\sigma'$ is slightly below $\sigma$, then this type attaches a
high importance to the benefit associated with overstatement of the mean, relative to
that associated with understatement of the spread. However, such a benefit is withheld
because an upper curve $\theta(s, \sigma)$ is associated with a higher $s$, hence with a slightly
lower likelihood of performance. If type $(\theta, \sigma)$ declares $(\theta', \sigma'')$, where $\sigma''$ is largely
<figure><img src="image_8.png" /><figcaption>**Fig. 3** Curves $\theta(s, \sigma)$ along which types $(\theta, \sigma)$, $(\theta', \sigma')$ and $(\theta', \sigma'')$ are identified (respectively, in point A, B and C)</figcaption></figure>

below $\sigma$, then it attaches a high importance to the benefit associated with understate-
ment of the spread, relative to that associated with overstatement of the mean. This
benefit disappears as well because now the mimicked type belongs to a lower curve,
corresponding to a higher likelihood of performance, and this is not convenient to type
$(\theta, \sigma)$. Indeed, this type would be exposed to a higher risk of being due to perform in
a bad state, in which the realized cost is higher than that of type $(\theta', \sigma'')$.

**5.4 Private observation of η is irrelevant**

There are two more sets of incentive constraints that must hold. These are the incentive
constraints whereby the agent is unwilling to cheat on the shock in the second stage,
and those whereby the agent is unwilling to deliver coordinated lies in the first and
second stage of the relationship. As is well known, in sequential screening problems,
second-stage incentive constraints are not an issue, if the agent reports truthfully in
the first stage, given that the contract is signed *before* the agent learns the shock. The
very issue rests with the first-stage incentive constraints, which may be tighter relative
to a case where the shock is public observation, due to the possibility for the agent of
coordinating lies between the two stages.

**Proposition 5** *There exists a transfer rule t such that the contract (y, t) is incentive compatible when η is privately observed.*

This result extends the finding of Esö and Szentes (2007) to a two-dimensional
setting. By considering a one-dimensional sequential screening problem, the authors
use the probability integral transform theorem to show that a type that claims to be
less efficient in the first stage, will want to “correct” that lie in the second stage and

<figure><img src="image_9.png" /></figure>
pretend to be experiencing a more favorable shock than the true one. The reason is that, after claiming to be less efficient in the first stage, the agent is less likely to be required to perform. Following an unfavorable shock, the agent would like to execute the activity, as the actual cost is lower than P believes based on the first-stage report. It follows that private information on the shock is not costly to P.<fnref n="19" /> It is worth mentioning that Liu et al. (2020) already ascertain that the result of Esö and Szentes (2007) extends to settings in which the support of second-stage types shifts with the first-stage signal. Similar arguments are found to apply also in the two-dimensional framework, in which the support can extend in both directions as the spread increases. Noticeably, as Proposition 5 suggests, here the result ensues directly from the local incentive constraints rather than from the solution to a reduced program.

# 6 Comparison with one-dimensional sequential screening

We now compare the solution presented in Proposition 1 with the findings of Courty and Li (2000). To that end, it is useful to recall the solution that is obtained in our setting, if private information concerns either only the mean or only the spread. When $\sigma$ is public observation, the rent of the agent is given by

$$U^{\theta}(\theta, \sigma) \equiv U^{\theta}(\bar{\theta}, \sigma) + \int_{\theta}^{\bar{\theta}} \int_{0}^{1} y(x, \sigma, \eta) \chi(\eta) d\eta dx.$$

When $\theta$ is public observation, it is given by

$$U^{\sigma}(\theta, \sigma) \equiv U^{\sigma}(\theta, \underline{\sigma}) - \int_{\underline{\sigma}}^{\sigma} \int_{0}^{1} (2\eta - 1) y(\theta, z, \eta) \chi(\eta) d\eta dz.$$

Standard computations lead to the following result.<fnref n="20" />

**Lemma 8** If the spread of the cost $\sigma$ is public observation, then the optimal allocation is such that

$$\phi^{\theta}(\theta, \sigma) = \frac{1}{2} \left[ 1 + \frac{1}{\sigma} \left( S - \theta - \frac{M(\theta)}{m(\theta)} \right) \right]. \qquad (28)$$

If the mean of the cost $\theta$ is public observation, then the optimal allocation is such that

$$\phi^{\sigma}(\theta, \sigma) = \frac{1}{2} \left( 1 + \frac{S - \theta}{\sigma - \frac{1 - N(\sigma)}{n(\sigma)}} \right). \qquad (29)$$

<footnote n="19">¹⁹ Krahmer and Strausz (2015) show that if the first-stage type (say, the mean) is drawn from a discrete set, then the second-stage private observation is costly to the principal.</footnote>

<footnote n="20">²⁰ If transposed in the Courty and Li setting, the solution under FOSD, for instance, would be $\phi^{\theta}(\theta, \sigma) = \frac{1}{2} \left[ 1 + \frac{1}{\sigma} \left( S - \theta + \frac{\partial G(\gamma|\theta, \sigma)}{\partial \theta} \frac{M(\theta)}{m(\theta)} \right) \right]$. As $\frac{\partial G(\gamma|\theta, \sigma)}{\partial \theta} = \frac{\partial}{\partial \theta} \Xi \left( \frac{\gamma - \theta + \sigma}{2\sigma} \right) = -\frac{\chi(\eta)}{2\sigma}$ and $\frac{\partial G(\gamma|\theta, \sigma)}{\partial \gamma} = \frac{\partial}{\partial \gamma} \Xi \left( \frac{\gamma - \theta + \sigma}{2\sigma} \right) = \chi(\eta)$, the solution boils down to (28).</footnote>
<figure><img src="image_10.png" /><figcaption>Fig. 4 Comparison between two-dimensional solution and uni-dimensional solution with publicly known $\sigma$ when distributions are uniform and $\hat{s} < \bar{\theta}$, where $\sigma_2(\hat{s}, h(\hat{s})) = \bar{\sigma}$</figcaption></figure>

In line with the FOSD hypothesis made by Courty and Li (2000), a distribution with a higher mean stochastically dominates a distribution with a lower mean, provided the two distributions have an equal spread, which is commonly known. No distortion is induced in the allocation designed for the most efficient type (here, the one with the lowest mean); for all other types, the allocation is distorted below the first-best level, in consideration of the incentives to overstate the mean. As the hazard rate $\frac{M(\theta)}{m(\theta)}$ is monotonically increasing in $\theta$, the downward distortion is higher for less efficient types (the counterpart of Proposition 3.5 in Courty and Li (2000)). A distribution with a higher spread is a MPS of a distribution with a lower spread, provided the two distributions have equal mean, which is commonly known. Any distribution but that with the highest spread is associated with an allocation that is distorted upwards so as to contain the benefit that higher-spread types could obtain by understating the spread. As the hazard rate $\frac{1-N(\sigma)}{n(\sigma)}$ is monotonically decreasing in $\sigma$ (by assumption), and given that $\phi^\sigma(\theta, \sigma) > \frac{1}{2}$, the upward distortion is more pronounced for types that are less uncertain about the cost (the counterpart of Proposition 3.6 in Courty and Li (2000)).

In the two-dimensional setting, a downward distortion should be induced in the allocation designed for high-mean types, whereas an upward distortion should be induced in that designed for low-spread types. Therefore, the optimal likelihood is the outcome of two opposite effects, and is likely to diverge significantly when the entire distribution is private information.

**Proposition 6 At optimum:**

1. $\forall (\theta, \sigma), \phi(\theta, \sigma) > \phi^\theta(\theta, \sigma)$, As $\frac{M(\theta)}{m(\theta)}$ is monotonically increasing in $\theta$, the difference $\phi(\theta, \sigma) - \phi^\theta(\theta, \sigma)$ is higher the higher the mean is and/or the lower the spread is.

2. $\forall \sigma, \phi(\theta, \sigma) \ge \phi^\sigma(\theta, \sigma)$ for low values of $\frac{1-N(\sigma)}{n(\sigma)}$; the converse is true for high values.

<figure><img src="image_11.png" /></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. 5** Comparison between two-dimensional solution and one-dimensional solution with publicly known $\theta$ when distributions are uniform and $\hat{s} < \bar{\theta}$, where $\sigma_2(\hat{s}, h(\hat{s})) = \bar{\sigma}$</figcaption></figure>

If private information on the level of uncertainty were neglected, then the likelihood of performance would be set too low, especially for high-mean-and-low-spread types. Whereas a low likelihood is considered to be optimal in light of the incentives to overstate the mean of the cost, a high likelihood is optimal when uncertainty is low, in that it discourages more uncertain types from pretending to be less uncertain about the expected cost. Consequently, if the latter incentive were disregarded, then not only would efficiency be reduced, but also a higher rent would be conceded to more uncertain types. This explains why $\phi(\theta, \sigma) > \phi^\theta(\theta, \sigma)$. On the other hand, if one were to neglect private information on the mean, then most types would be assigned a likelihood that is either too high or too low. In particular, types whose cost is highly uncertain would be required to execute the activity too rarely, in spite of not representing attractive reports for other types. Formally, considering that the hazard rate $\frac{1-N(\sigma)}{\eta(\sigma)}$ decreases with the spread, it is for high values of the spread that $\phi(\theta, \sigma) > \phi^\theta(\theta, \sigma)$.

The case that lends itself better to a graphical representation of the comparison previously outlined is one where the mean and the spread of the cost are both uniformly distributed. This is considered in Figs. 4 and 5, which highlight the two results stated in Proposition 6 for the case of uniform distributions (details are relegated to Appendix M). In the two figures, we first identify $\phi(\theta(s, \sigma), \sigma)$, which takes three distinct functional forms over three sub-intervals of $s$, according to which the boundaries $\sigma_1(\cdot, \cdot)$ and $\sigma_2(\cdot, \cdot)$ are determined. Next, in Fig. 4, $\phi(\theta(s, \sigma), \sigma)$ is compared with $\phi^\theta(\theta(s, \sigma), \sigma)$, which is the optimal allocation obtained when the spread is public information, for three distinct values of the spread. One sees that, as previously explained, to a higher value of $s$, hence to a greater mean, corresponds a wider discrepancy between the two solutions, for any given value of the spread. One further sees that the divergence between one- and two-dimensional solution is more pronounced for lower values of $\sigma$. This is because the lower the spread is, the more the likelihood is distorted upwards. In Fig. 5, $\phi(\theta(s, \sigma), \sigma)$ is compared with $\phi^\sigma(\theta(s, \sigma), \sigma)$, which is the optimal allocation when the mean is public observation. For any interior value of the spread, distinct regions are identified, over which the likelihood is either higher or
<figure><img src="image_13.png" /><figcaption>**Fig. 6** The direction of relevant incentives for type $(\theta, \sigma')$ under a linear approximation of the rent</figcaption></figure>

lower in a context of two-dimensional screening. For a higher spread, it is more likely that the agent be required to perform less often under two-dimensional screening than in a setting with MPS. To the extremes, crossing occurs at the beginning of the curve, where $s = \underline{\theta}$, if it is also $\sigma = \underline{\sigma}$; it occurs at the end of the curve, where $s = \bar{\sigma}$, if it is also $\sigma = \bar{\sigma}$. The former and the latter crossing points respectively correspond to type $(\underline{\theta}, \underline{\sigma})$ and type $(\bar{\theta}, \bar{\sigma})$, for which the two-dimensional solution replicates that obtained under MPS.

## 7 Additional incentives when types are discrete

To conclude our analysis, we show that the results derived so far hold as long as types are drawn from continuous sets, not otherwise. To that end, it is useful to consider a linear approximation of the information rent in (8). Actually, with this, the expression of the rent is similar to the formulation it would have, if the distributions of mean and spread were discrete.

**Proposition 7** In a neighborhood of $(\theta, \sigma)$, (8) is approximated linearly as follows:

$$ 
\begin{aligned}
U(\theta, \sigma) = U(\theta', \sigma') &+ (\theta' - \theta) \Xi(\phi(\theta', \sigma')) \\
& - (\sigma - \sigma') \left[ (2\phi(\theta', \sigma') - 1) \Xi(\phi(\theta', \sigma')) - 2 \int_0^{\phi(\theta', \sigma')} \Xi(\eta) d\eta \right], \\
\forall (\theta, \sigma).
\end{aligned} 
\quad (30) $$

Take $\theta < \theta'$ and $\sigma > \sigma'$. A contract granting the information rent $U(\theta, \sigma)$ to type $(\theta, \sigma)$ is not incentive compatible if $\phi(\theta, \sigma) > \phi(\theta', \sigma')$.

This situation is represented in Fig. 6, in which types $(\theta, \sigma)$, $(\theta', \sigma')$ and $(\theta, \sigma')$ are all drawn from discrete sets, and each two of them are adjacent in at least one
dimension. Thus, the rent is linear, as in (30). If $\phi(\theta, \sigma) > \phi(\theta', \sigma')$, then type $(\theta, \sigma')$ may find $(\theta, \sigma)$ more attractive than $(\theta', \sigma')$. With a continuum of types, the triangle ABC collapses onto the curve BC, if the three types are all in a neighborhood, so that the unusual incentive of type $(\theta, \sigma')$ is eliminated. This is no longer true with discrete types because bunching might not be desirable in that environment.

## 8 Conclusion

Whereas among scholars it has long been acknowledged that private information on a distribution should be meant as private information on both the parameters characterizing it, hitherto the literature has not investigated a screening problem with an information structure of this kind. The difficulties with solving multidimensional screening problems are, perhaps, one of the main reasons for the lack of analysis. The fact that, in this study, an analytical solution was obtained without imposing any important restrictions on the primitives of the model, particularly, on the distributions of the variables that the distribution of the cost depends upon, shows that optimal contracts with privately known distributions can be characterized and considered in applications. Documented examples of mismatching between reported estimates and real data, mirroring incentive problems of the kind we explored, are recorded in transportation projects. Further examples of a similar nature are found in central/local government hierarchies dealing with environmental issues. In these examples, initial multidimensional information and later private observation following a shock are typically intertwined.

We believe that the results of our study also provide a tool for making progress with empirical analysis in the field. Focusing on auctions, recent empirical studies investigate whether or not observed bids are consistent with the assumed information structure and the resulting theoretical predictions. As far as procurement and other contractual relationships are concerned, the finding that private information about an entire distribution (rather than single parameters, as in environments with FOSD or MPS) results in a different incentive pattern, suggests that there would be scope for exploring, in empirical terms, what exact information structure is at the root of errors that are too distant from forecast errors for not conjecturing on incentive reasons.

We highlighted that if types are discrete, then the agent displays unusual incentives to pretend to be more efficient, as a way of raising the likelihood of being required to produce for the principal (Sect. 7). Under FOSD, Krahmer and Strausz (2015) find that, with discrete types, incentives of this kind arise due to the possibility for the agent of coordinating lies between the two stages of the relationship. That is, in the second stage, the agent pretends to be more efficient than actually is in order to “correct” for the fake report delivered in the first stage, so that the principal will require performance more often. This raises agency costs *ex ante*. The simple result in Proposition 7 suggests that incentives to mimic more efficient types already appear in the first stage.

One last remark is worth making. As mentioned in Introduction, the screening problem we investigated is tractable under the usual assumption that the objective function of the principal is linear in the decision variable. Absent this assumption, which is common in sequential screening models, it would become more complex in
two respects. First, as highlighted in our analysis, the PDE to be solved would depend on a set, rather than on a single value. Second, the properties of the marginal surplus function would be critical to the combination of binding incentive constraints in the optimal contract, with the mean and the spread being both privately known. Indeed, the efficiency loss incurred to contain a rent is not the same when the marginal surplus is convex in the size of the activity and when it is concave. A specific investigation would be necessary to gain a better understanding of the optimal screening in that context. This is on our research agenda.

A Proof of Lemma 1

Recall that $U(\theta, \sigma)$ is found from (1) as follows:

$$ 
\begin{aligned}
U(\theta, \sigma) &= \int_{0}^{1} u(\theta, \sigma, \eta) \chi(\eta) d\eta \\
&= \int_{0}^{1} (t(\theta, \sigma, \eta) - y(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta.
\end{aligned} 
\quad (31) $$

Then, for any $(\theta, \sigma)$ and $(\theta', \sigma')$, the incentive constraints are:

$$ 
\begin{aligned}
U(\theta, \sigma) &\ge U(\theta', \sigma') + \int_{0}^{1} \gamma(\theta', \sigma', \eta) y(\theta', \sigma', \eta) \chi(\eta) d\eta \\
&\quad - \int_{0}^{1} \gamma(\theta, \sigma, \eta) y(\theta', \sigma', \eta) \chi(\eta) d\eta \\
U(\theta', \sigma') &\ge U(\theta, \sigma) - \left( \int_{0}^{1} \gamma(\theta', \sigma', \eta) y(\theta, \sigma, \eta) \chi(\eta) d\eta \right. \\
&\quad \left. - \int_{0}^{1} \gamma(\theta, \sigma, \eta) y(\theta, \sigma, \eta) \chi(\eta) d\eta \right).
\end{aligned} 
$$

Under the same hypothesis, $y(\theta, \sigma, \eta)$ is differentiable almost everywhere with respect to both $\theta$ and $\sigma$, hence so is $t(\theta, \sigma, \eta)$ as well. The local first-order conditions are given by

$$ \int_{0}^{1} (t_{\theta}(\theta, \sigma, \eta) - y_{\theta}(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta = 0 \quad (32) $$

and

$$ \int_{0}^{1} (t_{\sigma}(\theta, \sigma, \eta) - y_{\sigma}(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta = 0. \quad (33) $$

Differentiating (31) with respect to $\theta$ and to $\sigma$ and using (32) and (33), conditions 1 in the lemma are found. The local second-order conditions are given by

$$ \int_{0}^{1} (t_{\theta\theta}(\theta, \sigma, \eta) - y_{\theta\theta}(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta \le 0, \quad (34) $$
$$ \int_{0}^{1} (t_{\sigma\sigma}(\theta, \sigma, \eta) - y_{\sigma\sigma}(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta \le 0 \quad (35) $$

and

$$ \begin{aligned} & \int_{0}^{1} (t_{\theta\theta}(\theta, \sigma, \eta) - y_{\theta\theta}(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta \\ & \quad \times \int_{0}^{1} (t_{\sigma\sigma}(\theta, \sigma, \eta) - y_{\sigma\sigma}(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta \\ & \ge \int_{0}^{1} (t_{\theta\sigma}(\theta, \sigma, \eta) - y_{\theta\sigma}(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta. \end{aligned} \quad (36) $$

Differentiating both sides of (32) and (33) with respect to $\theta$ and to $\sigma$ respectively, (34) and (35) are rewritten as conditions 2 in the lemma. From (32) one has

$$ \int_{0}^{1} (t_{\theta\sigma}(\theta, \sigma, \eta) - y_{\theta\sigma}(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta = \int_{0}^{1} y_{\theta}(\theta, \sigma, \eta) \gamma_{\sigma}(\theta, \sigma, \eta) \chi(\eta) d\eta. $$

Using this and $\gamma_{\sigma}(\theta, \sigma, \eta) = 2\eta - 1$, (36) is rewritten as condition 3 in the lemma.

## B Proof of Lemma 2

Integrating over $\theta$ the first condition in point 1 of Lemma 1 yields

$$ U(\theta, \sigma) = U(\bar{\theta}, \sigma) + \int_{\theta}^{\bar{\theta}} \int_{0}^{1} y(x, \sigma, \eta) \chi(\eta) d\eta dx. \quad (37) $$

Differentiating both sides of (37) with respect to $\sigma$ and rearranging, one obtains

$$ U_{\sigma}(\bar{\theta}, \sigma) = U_{\sigma}(\theta, \sigma) - \int_{\theta}^{\bar{\theta}} \int_{0}^{1} y_{\sigma}(x, \sigma, \eta) \chi(\eta) d\eta dx. \quad (38) $$

Recalling $U_{\sigma}(\theta, \sigma) = -\int_{0}^{1}(2\eta - 1)y(\theta, \sigma, \eta)\chi(\eta)d\eta$ from Lemma 1, and using it in the right-hand side of (38) yields

$$ U_{\sigma}(\bar{\theta}, \sigma) = -\int_{0}^{1} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta - \int_{\theta}^{\bar{\theta}} \int_{0}^{1} y_{\sigma}(x, \sigma, \eta) \chi(\eta) d\eta dx. $$

Taking integrals over $\sigma$ further yields

$$ \begin{aligned} U(\bar{\theta}, \sigma) = U(\bar{\theta}, \underline{\sigma}) & - \int_{\underline{\sigma}}^{\bar{\sigma}} \int_{0}^{1} (2\eta - 1) y(\theta, z, \eta) \chi(\eta) d\eta dz \\ & - \int_{\theta}^{\bar{\theta}} \int_{0}^{1} (y(x, \sigma, \eta) - y(x, \underline{\sigma}, \eta)) \chi(\eta) d\eta dx. \end{aligned} $$
Replacing in (37), (8) is found.

C Proof of Lemma 5

As (13) is quasi-linear, it can be solved by using the method of characteristics. With this, (13) is transformed into a sequence of ordinary differential equations. Take $\mathbb{S} = \{\theta, \sigma, y(\theta, \sigma, \eta)\}$ to be the solution surface. Using the total derivative formula, any solution belonging to $\mathbb{S}$ is found to be such that

$$((2\eta - 1), -1, 0) (y_{\theta}, y_{\sigma}, -1) = 0.$$

Indeed, $(y_{\theta}, y_{\sigma}, -1)$ is the normal to the solution surface $\mathbb{S}$. In each point of $\mathbb{S}$, the vector $((2\eta - 1), -1, 0)$ lies on the plane tangent to $\mathbb{S}$. For any given $\eta$, one can thus define a curve $C = \{\tilde{\theta}(r), \tilde{\sigma}(r), \tilde{y}(r)\}$, as parameterized by $r$, such that the vector $((2\eta - 1), -1, 0)$ is tangent to the curve in each point belonging to it, so that the following ordinary differential equations are satisfied:

$$\tilde{\theta}_r(r) = 2\eta - 1 \qquad (39)$$

$$\tilde{\sigma}_r(r) = -1 \qquad (40)$$

$$\tilde{y}_r(r) = \frac{f(\eta)}{\chi(\eta)}. \qquad (41)$$

As the vector $((2\eta - 1), -1, 0)$ is tangent to curve $C$, the curve lies on the solution surface. (39) to (41) are the characteristic equations of the PDE. The solution to the PDE is equivalent to the solution to these three ordinary differential equations. One obtains:

$$\tilde{\theta}(r) - \tilde{\theta}(0) = (2\eta - 1)r, \qquad (42)$$

$$\tilde{\sigma}(r) - \tilde{\sigma}(0) = -r, \qquad (43)$$

$$\tilde{y}(r) - \tilde{y}(0) = \frac{f(\eta)}{\chi(\eta)}. \qquad (44)$$

Take $\tilde{\sigma}(0) = \underline{\sigma}$. As $\sigma = \tilde{\sigma}(r)$, (43) becomes $r = -(\sigma - \underline{\sigma})$. Using it in (42) and (44) yields

$$\tilde{\theta}(r) = \tilde{\theta}(0) - (\sigma - \underline{\sigma})(2\eta - 1) \qquad (45)$$

$$\tilde{y}(r) = \tilde{y}(0) - (\sigma - \underline{\sigma}) \frac{f(\eta)}{\chi(\eta)}. \qquad (46)$$

$\tilde{\theta}(0)$ and $\tilde{y}(0)$ can both vary freely. Take $\tilde{\theta}(0) = s$, for some parameter $s$. Then, also $\tilde{y}(0)$ is a function of $s$, for any given $\eta$. Denote $y_0(s, \eta) = \tilde{y}(0)$. As $\tilde{\theta}(r) = \theta$, (45) boils down to (15). As $\tilde{y}(r) = y(\theta, \sigma, \eta)$, (46) is rewritten as (16).
D Proof of Lemma 6

Take Z (r) ≡ ∫ 1 0 ˜ u ( θ, σ, η, θ $^{′}$, σ$^{′}$, η ) χ ( η ) dη, where θ $^{′}$= θ + ar and σ ′ = σ + br, for some a and b constant. Letting Z $^{′}$( r ) be the directional derivative of Z ( r ) in the direction of the vector ( a, b ), we can write it as

$$
\begin{align*}
Z'(r) ={}& a \frac{\partial}{\partial \theta'} \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) \, d\eta \\
& + b \frac{\partial}{\partial \sigma'} \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) \, d\eta.
\end{align*}
$$

Recall from the proof of Lemma 1 that the information rent is written as

$$
\begin{align*}
& \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) \, d\eta \\
&= U(\theta', \sigma') + \int_0^1 (\gamma(\theta', \sigma', \eta) - \gamma(\theta, \sigma, \eta)) y(\theta', \sigma', \eta) \chi(\eta) \, d\eta.
\end{align*}
$$

Thus,

$$
\begin{align*}
Z'(r) ={}& a U_{\theta'}(\theta', \sigma') + b U_{\sigma'}(\theta', \sigma') \\
& + \int_0^1 (\gamma(\theta', \sigma', \eta) - \gamma(\theta, \sigma, \eta)) (ay_{\theta'}(\theta', \sigma', \eta) \\
& \qquad + by_{\sigma'}(\theta', \sigma', \eta)) \chi(\eta) \, d\eta \\
& + \int_0^1 [a + b(2\eta - 1)] y(\theta', \sigma', \eta) \chi(\eta) \, d\eta.
\end{align*}
$$

Using conditions 1 in Lemma 1 and rearranging, one further obtains

$$
Z'(r) = \int_{0}^{1} \left( \gamma(\theta', \sigma', \eta) - \gamma(\theta, \sigma, \eta) \right) \left( ay_{\theta'}(\theta', \sigma', \eta) + by_{\sigma'}(\theta', \sigma', \eta) \right) \chi(\eta) \, d\eta.
$$

As a = θ$_{r}$ and b = σ$_{r}$ , one can further write

$$
Z'(r) = \int_0^1 \left( \gamma(\theta', \sigma', \eta) - \gamma(\theta, \sigma, \eta) \right) \left( \theta'_r y_{\theta'}(\theta', \sigma', \eta) + \sigma'_r y_{\sigma'}(\theta', \sigma', \eta) \right) \chi(\eta) \, d\eta.
$$

Take ( θ, σ ) and ( θ $^{′}$, σ$^{′}$) to belong to the same characteristic curve at the solution to (13). Then, the directional derivative Z $^{′}$( r ) in the direction of the vector ( a, b ) is identified along the curve. In particular, θ ′ r and σ ′ r respectively satisfy (39) and (40) so that θ ′ r = 2 η − 1 and σ ′ r = − 1. Moreover, as y ( θ, σ, η ) = y $^{0}$( r, η ) for any characteristic curve parameterized by r , and y $^{0}$( r, η ) is found at the solution to the homogeneous equation (17), one has

$$
\theta'_{r} y_{\theta'}(\theta', \sigma', \eta) + \sigma'_{r} y_{\sigma'}(\theta', \sigma', \eta) = (2\eta - 1) y_{\theta'}(\theta', \sigma', \eta) - y_{\sigma'}(\theta', \sigma', \eta) = 0.
$$
Therefore, $Z'(r) = 0$.

## E Proof of Remark 1

With $\phi(\theta, \sigma)$ being a decision variable, $\Gamma_1$ is reformulated as

$$ \Gamma_1' \left\{ \begin{array}{l} \max_{\phi(\theta, \sigma), \forall(\theta, \sigma)} \int_{\Theta} \int_{\Sigma} \left\{ \int_{0}^{\phi(\theta, \sigma)} \left( F(\theta, \sigma, \eta) - y(\theta, \underline{\sigma}, \eta) \frac{M(\theta)}{m(\theta)} \right) \chi(\eta) d\eta \right\} n(\sigma) m(\theta) d\sigma d\theta \\ \text{subject to (22)} \end{array} \right. $$

Given the definition of $F(\theta, \sigma, \eta)$ in (11), the integrand in the objective function in $\Gamma_1'$ is developed as follows:

$$ \begin{aligned} & \int_{0}^{\phi(\theta, \sigma)} \left( F(\theta, \sigma, \eta) - y(\theta, \underline{\sigma}, \eta) \frac{M(\theta)}{m(\theta)} \right) \chi(\eta) d\eta \\ & = (S - \theta) \Xi(\phi(\theta, \sigma)) - \Xi(\phi(\theta, \underline{\sigma})) \frac{M(\theta)}{m(\theta)} \\ & \quad - \left( \sigma - \frac{1 - N(\sigma)}{n(\sigma)} \right) \left[ (2\phi(\theta, \sigma) - 1) \Xi(\phi(\theta, \sigma)) - 2 \int_{0}^{\phi(\theta, \sigma)} \Xi(\eta) d\eta \right] \\ & \equiv \hat{F}(\theta, \sigma). \end{aligned} \quad (47) $$

Hence, $\Gamma_1'$ is rewritten as

$$ \Gamma_1'' \left\{ \begin{array}{l} \max_{\phi(\theta, \sigma), \forall(\theta, \sigma)} \int_{\Theta} \int_{\Sigma} \hat{F}(\theta, \sigma) n(\sigma) m(\theta) d\sigma d\theta \\ \text{subject to (22)} \end{array} \right. $$

Proceeding as in the proof of Lemma 5, the following ordinary differential equations are satisfied at the solution to (22):

$$ \tilde{\theta}_r(r) = 2\tilde{h}(r) - 1 \quad (48) $$

$$ \tilde{\sigma}_r(r) = -1 \quad (49) $$

$$ \chi(\tilde{h}(r)) \tilde{h}_r(r) = 0. \quad (50) $$

From these, (21) is found at the solution, which is such that $\phi(\theta(s, \sigma), \sigma) = h(s)$. Using this result to rewrite the objective function in $\Gamma_1''$ in the $(s, \sigma)$-space, one has

$$ \int_{\Theta} \int_{\Sigma} \hat{F}(\theta, \sigma) n(\sigma) m(\theta) d\sigma d\theta = \int_{S} E(s, h(s)) ds, $$

where

$$ E(s, h(s)) = \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \hat{F}(\theta(s, \sigma), \sigma) \frac{d\theta(s, \sigma)}{ds} m(\theta(s, \sigma)) n(\sigma) d\sigma $$
$$= \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \widehat{F}(\theta(s, \sigma), \sigma) (1 + \theta_h(s, \sigma) h'(s)) m(\theta(s, \sigma)) n(\sigma) d\sigma$$

With this, $\Gamma_1''$ becomes

$$\max_{h(s)} \int_s E(s, h(s)) ds.$$

By continuity of $h(s)$, the solution is obtained through point-wise maximization with respect to $h(s)$. To that end, one needs to compute the Euler equation $E_h = \frac{\partial E_h'}{\partial s}$.

For the remainder of the proof, we drop arguments from the functions $F(\cdot, \cdot)$ and $\theta(\cdot, \cdot)$, for simplicity. We compute

$$E_h = \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} (F_h + F_{\theta} \theta_h + F_{\Xi} \chi(h)) (1 + \theta_h h'(s)) m(\theta) n(\sigma) d\sigma \\ + \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} F \theta_h h'(s) m(\theta) n(\sigma) d\sigma \\ + \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} F (1 + \theta_h h'(s)) m'(\theta) \theta_h n(\sigma) d\sigma + \mathcal{L},$$

where

$$\mathcal{L} = \frac{\partial \sigma_2(s, h(s))}{\partial h} \widehat{F}(\theta(s, \sigma_2), \sigma_2) \frac{d\theta(s, \sigma_2)}{ds} m(\theta(s, \sigma_2)) n(\sigma_2) \\ - \frac{\partial \sigma_1(s, h(s))}{\partial h} \widehat{F}(\theta(s, \sigma_1), \sigma_1) \frac{d\theta(s, \sigma_1)}{ds} m(\theta(s, \sigma_1)) n(\sigma_1).$$

By the definition of $\sigma_i(s, h(s))$, $\forall i \in \{1, 2\}$, one between $\sigma_i(s, h(s))$ and $\theta(s, \sigma_i)$ lies on an extreme point of the $\theta(s, \sigma)$-curve, which is a constant. Hence, $\forall i$, either $\frac{\partial \sigma_2(s, h(s))}{\partial h} = 0$ or $\frac{\partial \theta(s, \sigma_2)}{\partial s} = 0$, so that $\mathcal{L} = 0$. As it is also $\theta_{hh} = 0$, one has

$$E_h = \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} (F_h + F_{\theta} \theta_h + F_{\Xi} \chi(h)) (1 + \theta_h h'(s)) m(\theta) n(\sigma) d\sigma \\ + \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} F (1 + \theta_h h'(s)) \theta_h m'(\theta) n(\sigma) d\sigma.$$

Furthermore,

$$E_{h'} = \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} F \theta_h m(\theta) n(\sigma) d\sigma$$

and

$$\frac{\partial E_{h'}}{\partial s} = \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} [F_{\theta}(1 + \theta_h h'(s)) + F_h h'(s) + F_{\Xi} \chi(h) h'(s)] \theta_h m(\theta) n(\sigma) d\sigma$$
$$ 
\begin{aligned}
&+ \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} F(\theta_{hh}h'(s) + \theta_{hs}) m(\theta) n(\sigma) d\sigma \\
&+ \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} F\theta_h(1 + \theta_h h'(s)) m'(\theta) n(\sigma) d\sigma + 0,
\end{aligned}
$$

where, again, the zero term appears because, $\forall i$, one between $\sigma_i(s, h(s))$ and $\theta(s, \sigma_i)$ is a constant. Moreover, $\theta_{hh} = \theta_{hs} = 0$, so that

$$ \frac{\partial E_{h'}}{\partial s} = \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} [F_{\theta}(1 + \theta_h h'(s)) + F_h h'(s) + F_{\Xi} \chi(h) h'(s)] \theta_h m(\theta) n(\sigma) d\sigma \\ + \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} F(1 + \theta_h h'(s)) \theta_h m'(\theta) n(\sigma) d\sigma. $$

Thus, the Euler equation $E_h = \frac{\partial E_{h'}}{\partial s}$ amounts to

$$ \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} (F_h + F_{\Xi} \chi(h)) m(\theta) n(\sigma) d\sigma = 0. $$

Given that $\phi(\theta(s, \sigma), \sigma) = h(s)$ and $\phi(\theta(s, \sigma), \underline{\sigma}) = h(\theta(s, \sigma))$, one can further write

$$ \hat{F}(\theta(s, \sigma), \sigma) = (S - \theta(s, \sigma)) \Xi(h(s)) - \Xi(h(\theta(s, \sigma))) \frac{M(\theta(s, \sigma))}{m(\theta(s, \sigma))} \quad (51) \\ - \left(\sigma - \frac{1-N(\sigma)}{n(\sigma)}\right) \left[(2h(s)-1)\Xi(h(s)) - 2 \int_{0}^{h(s)} \Xi(\eta) d\eta\right]. $$

Computing $F_h + F_{\Xi} \chi(h)$ from (51), the Euler equation is further specified as

$$ \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} \left[ S - \theta(s, \sigma) - \left( \sigma - \frac{1-N(\sigma)}{n(\sigma)} \right) (2h(s) - 1) \right] m(\theta(s, \sigma)) n(\sigma) d\sigma = 0. \quad (52) $$

By rearranging, one obtains (21). Lastly, the function $\hat{F}(\theta, \sigma)$, as defined in (47), is concave in $\phi(\theta, \sigma)$, given that $\sigma - \frac{1-N(\sigma)}{n(\sigma)} > 0$ (second part of Assumption 1). Hence, the solution to each of the above Euler equations is also sufficient for optimality.

## F Proof of Proposition 2

Monotonicity of $y(\theta, \sigma, \eta)$ with respect to $\theta$ follows from $\phi_{\theta} \le 0$; monotonicity of $(2\eta - 1) y(\theta, \sigma, \eta)$ with respect to $\sigma$ follows from $\phi(\theta, \sigma) > \frac{1}{2}$ and $\phi_{\sigma} \le 0$. From Proposition 1:

$$ \int_{0}^{1} y(\theta, \sigma, \eta) \chi(\eta) d\eta = \int_{0}^{\phi(\theta, \sigma)} \chi(\eta) d\eta $$

<figure><img src="image_14.png" /></figure>
$$ \int_{0}^{1} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta = \int_{0}^{\phi(\theta, \sigma)} (2\eta - 1) \chi(\eta) d\eta. $$

Accordingly, conditions 2 in Lemma 1 are rewritten as

$$ \begin{aligned} \frac{\partial}{\partial \theta} \int_{0}^{1} y(\theta, \sigma, \eta) \chi(\eta) d\eta \le 0 &\Leftrightarrow \phi_{\theta}(\theta, \sigma) \le 0 \\ \frac{\partial}{\partial \sigma} \int_{0}^{1} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta \le 0 &\Leftrightarrow (2\phi(\theta, \sigma) - 1) \phi_{\sigma}(\theta, \sigma) \le 0. \end{aligned} $$

As $\phi(\theta(s, \sigma), \sigma) = h(s) > \frac{1}{2}, \forall s$ (Proposition 1), $(2\phi(\theta, \sigma) - 1)\phi_{\sigma}(\theta, \sigma) \le 0$ is equivalent to $\phi_{\sigma}(\theta, \sigma) \le 0$. First differentiate both sides of $\phi(\theta(s, \sigma), \sigma) = h(s)$ with respect to $s$. It yields $\phi_{\theta}(\theta(s, \sigma), \sigma) \frac{d\theta(s, \sigma)}{ds} = h'(s)$. As $h'(s) \le 0$, it is $\phi_{\theta}(\theta(s, \sigma), \sigma) \le 0$ if and only if $\frac{d\theta(s, \sigma)}{ds} \ge 0$. From (21), $\frac{d\theta(s, \sigma)}{ds} = 1 - 2h'(s)(\sigma - \underline{\sigma})$. As $h'(s) \le 0$, it is also $\frac{d\theta(s, \sigma)}{ds} > 0$, hence $\phi_{\theta}(\theta(s, \sigma), \sigma) \le 0$. Next differentiate both sides of $\phi(\theta(s, \sigma), \sigma) = h(s)$ with respect to $\sigma$. It yields $\phi_{\theta}(\theta(s, \sigma), \sigma)\theta_{\sigma}(s, \sigma) + \phi_{\sigma}(\theta(s, \sigma), \sigma) = 0$, and so $\phi_{\sigma}(\theta(s, \sigma), \sigma) = -\phi_{\theta}(\theta(s, \sigma), \sigma)\theta_{\sigma}(s, \sigma) = (2h(s) - 1)\phi_{\theta}(\theta(s, \sigma), \sigma)$. Because $\phi_{\theta}(\theta(s, \sigma), \sigma) \le 0$ and $h(s) > \frac{1}{2}$, one has $(2h(s) - 1)\phi_{\theta}(\theta(s, \sigma), \sigma) \le 0$. Hence, $\phi_{\sigma}(\theta(s, \sigma), \sigma) \le 0$.

## G Proof of $h'(s) \le 0$ for specific distributions of $\theta$

Using (21) in the definition of $h(s)$ and rearranging, we obtain

$$ h(s) = \frac{1}{2} \left( 1 + \frac{(S-s) \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))) m(\theta(s, \sigma)) n(\sigma) d\sigma}{\int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))) (\underline{\sigma} - \frac{1-N(\sigma)}{n(\sigma)}) m(\theta(s, \sigma)) n(\sigma) d\sigma} \right). \quad (53) $$

Based on this, we can write

$$ \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} J(s, \sigma) m(\theta(s, \sigma)) n(\sigma) d\sigma = 0, \quad (54) $$

where

$$ J(s, \sigma) \equiv S - s - (2h(s) - 1) \left( \underline{\sigma} - \frac{1 - N(\sigma)}{n(\sigma)} \right). \quad (55) $$

As (54) holds for any $s$, we differentiate both sides with respect to $s$ and rearrange to obtain

$$ h'(s) = \frac{1}{2} \frac{A(s) + B(s)}{\int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} \left( \underline{\sigma} - \frac{1-N(\sigma)}{n(\sigma)} \right) m(\theta(s, \sigma)) n(\sigma) d\sigma + V(s)}, \quad (56) $$
where

$$A(s) \equiv \frac{d\sigma_2(s, h(s))}{\partial s} J(s, \sigma_2(s, h(s))) - \frac{d\sigma_1(s, h(s))}{\partial s} J(s, \sigma_1(s, h(s))),$$

$$B(s) \equiv \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \left( J(s, \sigma) \frac{m'(\theta(s, \sigma))}{m(\theta(s, \sigma))} - 1 \right) m(\theta(s, \sigma)) n(\sigma) d\sigma,$$

$$V(s) \equiv \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} J(s, \sigma) (\sigma - \bar{\sigma}) \frac{m'(\theta(s, \sigma))}{m(\theta(s, \sigma))} m(\theta(s, \sigma)) n(\sigma) d\sigma.$$

Using (55) together with 2, one finds $\frac{\partial J(s, \sigma)}{\partial \sigma} \le 0$, with strict inequality at least for some $\sigma$, when $h(s)$ is an interior solution. It follows from (54) that $J(s, \sigma_2(s, h(s))) \le 0 \le J(s, \sigma_1(s, h(s)))$, with at least one of these inequalities being strictly satisfied. From the definitions of $\sigma_1(s, \eta)$ and $\sigma_2(s, \eta)$, one also has $\frac{d\sigma_2(s, h(s))}{ds} \ge 0$ together with $\frac{d\sigma_1(s, h(s))}{ds} \ge 0$, if $h'(s) \le 0$. Therefore, if $h'(s) \le 0$, then $A'(s) \le 0$. Furthermore, by Assumption 1 and 2, $\int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} (\bar{\sigma} - \frac{1-N(\sigma)}{n(\sigma)}) m(\theta(s, \sigma)) n(\sigma) d\sigma \ge 0$. We are left with checking the sign of $B(s)$ and $V(s)$ for the distributions of $\theta$ presented in the main text.

## Uniform and truncated exponential distribution of $\theta$

Take $m(\theta) = \frac{\lambda e^{-\lambda(\theta-\bar{\theta})}}{1-e^{-\lambda(\bar{\theta}-\theta)}}$, $\forall \theta \in [\bar{\theta}, \bar{\theta}]$, where $\lambda \ge 0$ is a constant. For $\lambda > 0$ this represents an exponential distribution, for $\lambda = 0$ a uniform distribution. One finds $\frac{m'(\theta)}{m(\bar{\theta})} = -\lambda$, $\forall \theta \in [\bar{\theta}, \bar{\theta}]$. First, using this expression in $B(s)$ and recalling (54), we see that $B(s) = 0$. Next, denoting $\hat{\sigma}$ such that $J(s, \hat{\sigma}) = 0$ and recalling again (54), we can write

$$\begin{aligned} V(s) &= \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} J(s, \sigma) (\sigma - \hat{\sigma}) m(\theta(s, \sigma)) n(\sigma) d\sigma \\ &= \int_{\sigma_1(s, h(s))}^{\hat{\sigma}} J(s, \sigma) (\sigma - \hat{\sigma}) m(\theta(s, \sigma)) n(\sigma) d\sigma \\ &\quad + \int_{\hat{\sigma}}^{\sigma_2(s, h(s))} J(s, \sigma) (\sigma - \hat{\sigma}) m(\theta(s, \sigma)) n(\sigma) d\sigma. \end{aligned}$$

This is strictly negative because $J(s, \sigma) \ge 0$ if $\sigma \in [\sigma_1(s, h(s)), \hat{\sigma}]$, and $J(s, \sigma) \le 0$ if $\sigma \in [\hat{\sigma}, \sigma_2(s, h(s))]$, where at least one of the two inequalities holds strictly when $h(s)$ is an interior solution. Therefore, $h'(s) \le 0$, $\forall \lambda \ge 0$.

## Truncated normal distribution of $\theta$

Take $m(\theta) = \frac{e^{-\frac{(\theta-\bar{\theta})^2}{2\hat{\nu}^2}}}{\int_{\bar{\theta}}^{\bar{\theta}} e^{-\frac{(x-\bar{\theta})^2}{2\hat{\nu}^2}} dx}$, where $\mathbb{E}[\theta]$ and $\hat{\nu} > 0$ are as defined in the main text.

One has $\frac{m'(\theta)}{m(\bar{\theta})} = \frac{\mathbb{E}[\theta] - \bar{\theta}}{\hat{\nu}^2}$. Therefore, $\frac{m'(\theta)}{m(\bar{\theta})} > 0$ if $\theta < \mathbb{E}[\theta]$ and $\frac{m'(\theta)}{m(\bar{\theta})} < 0$ if $\theta > \mathbb{E}[\theta]$.
Accordingly,

$$V(s) = \int_{\sigma_1(s,h(s))}^{\hat{\sigma}} J(s, \sigma) (\sigma - \bar{\sigma}) \frac{\mathbb{E}[\theta] - \theta(s, \sigma)}{v^2} m(\theta(s, \sigma)) n(\sigma) d\sigma \\ + \int_{\hat{\sigma}}^{\sigma_2(s,h(s))} J(s, \sigma) (\sigma - \bar{\sigma}) \frac{\mathbb{E}[\theta] - \theta(s, \sigma)}{v^2} m(\theta(s, \sigma)) n(\sigma) d\sigma$$

which is strictly positive. Thus, the denominator to the right-hand side of (56) is strictly positive, so that $h'(s) \le 0$, if and only if $A(s) + B(s) \le 0$. Taking again $\hat{\sigma}$ such that $J(s, \hat{\sigma}) = 0$ and using again (54), we can write

$$B(s) = \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} J(s, \sigma) \frac{\mathbb{E}[\theta] - \theta(s, \sigma)}{v^2} m(\theta(s, \sigma)) n(\sigma) d\sigma \\ < \int_{\sigma_1(s,h(s))}^{\hat{\sigma}} J(s, \sigma) \frac{\mathbb{E}[\theta] - \bar{\theta}}{v^2} m(\theta(s, \sigma)) n(\sigma) d\sigma \\ + \int_{\hat{\sigma}}^{\sigma_2(s,h(s))} J(s, \sigma) \frac{\mathbb{E}[\theta] - \bar{\theta}}{v^2} m(\theta(s, \sigma)) n(\sigma) d\sigma \\ = \frac{\bar{\theta} - \bar{\theta}}{v^2} \int_{\sigma_1(s,h(s))}^{\hat{\sigma}} J(s, \sigma) m(\theta(s, \sigma)) n(\sigma) d\sigma.$$

We see that $A(s) + B(s) \le 0$ if and only if

$$\frac{\bar{\theta} - \theta}{v^2} \le \varrho \equiv \frac{\int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} m(\theta(s, \sigma)) n(\sigma) d\sigma - A(s)}{\int_{\sigma_1(s,h(s))}^{\hat{\sigma}} J(s, \sigma) m(\theta(s, \sigma)) n(\sigma) d\sigma}.$$

The ratio to the right-hand side of this condition is strictly positive when $h'(s) \le 0$ since $A(s) < 0$ in that case. Hence, $\varrho > 0$.

## H Proof of Proposition 3

Recall $Z(r) = \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) d\eta$, where $\theta' = \theta + ar$ and $\sigma' = \sigma + br$, for some $a$ and $b$ constant. We found that the directional vector of $Z(r)$ in the direction of the vector $(a, b)$ is

$$Z'(r) = a \frac{\partial}{\partial \theta'} \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) d\eta \\ + b \frac{\partial}{\partial \sigma'} \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) d\eta.$$
We can further compute

$$
\begin{align}
Z''(r) ={}& a^2 \frac{\partial^2}{\partial \theta'^2} \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) \, d\eta \nonumber \\
& + 2ab \frac{\partial^2}{\partial \theta' \partial \sigma'} \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) \, d\eta \nonumber \\
& + b^2 \frac{\partial^2}{\partial \sigma'^2} \int_0^1 \tilde{u}(\theta, \sigma, \eta, \theta', \sigma', \eta) \chi(\eta) \, d\eta. \tag{57}
\end{align}
$$

At $r = 0$, i.e. for $(\theta', \sigma') = (\theta, \sigma)$, we have

$$
\begin{align*}
Z''(0) ={}& a^2 \frac{\partial}{\partial \theta} \int_0^1 y(\theta, \sigma, \eta) \chi(\eta) \, d\eta + 2ab \frac{\partial}{\partial \theta} \int_0^1 (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) \, d\eta \\
& + b^2 \frac{\partial}{\partial \sigma} \int_0^1 y(\theta, \sigma, \eta) (2\eta - 1) \chi(\eta) \, d\eta.
\end{align*}
$$

Being based on the definition of $\phi(\theta, \sigma)$, this further becomes

$$
Z''(0) = \Xi(\phi(\theta, \sigma)) \left[ a^2 \phi_{\theta}(\theta, \sigma) + 2ab(2\phi(\theta, \sigma) - 1)\phi_{\theta}(\theta, \sigma) \right. \\
\left. \qquad + b^2(2\phi(\theta, \sigma) - 1)\phi_{\sigma}(\theta, \sigma) \right].
$$

We see that the sign of $Z''(0)$ is the same as that of the quadratic equation

$$
Q(a) \equiv a^2 \phi_{\theta}(\theta, \sigma) + 2ab(2\phi(\theta, \sigma) - 1)\phi_{\theta}(\theta, \sigma) + b^2(2\phi(\theta, \sigma) - 1)\phi_{\sigma}(\theta, \sigma).
$$

The discriminant of the quadratic is given by

$$
4b^2 (2\phi(\theta, \sigma) - 1) \phi_{\theta}(\theta, \sigma) [(2\phi(\theta, \sigma) - 1) \phi_{\theta}(\theta, \sigma) - \phi_{\sigma}(\theta, \sigma)].
$$

Using (22), we see that this discriminant equals zero. Therefore, $Q(a)$ has only one root, given by $a = -b(2\phi(\theta, \sigma) - 1)$. Moreover, computing $Q'(a) = 2a\phi_{\theta}(\theta, \sigma) + 2b(2\phi(\theta, \sigma) - 1)\phi_{\theta}(\theta, \sigma)$ and $Q''(a) = 2\phi_{\theta}(\theta, \sigma)$, and considering that $\phi_{\theta}(\theta, \sigma) < 0$, $Q(a)$ is found to be concave down, so that $Q(a) < (=) 0 \Leftrightarrow Z''(0) < (=) 0$ for any $a \neq (=) -b(2\phi(\theta, \sigma) - 1)$. Next, because $a = \theta_r$ and $b = \sigma_r$, the condition $Q(a) = 0 \Leftrightarrow Z''(0) = 0$ is tantamount to

$$
\theta_r = -\sigma_r (2\phi(\theta, \sigma) - 1).
$$

Therefore, for some constant value $\phi(\theta, \sigma)$, any variation away from $(\theta, \sigma)$ must
satisfy the previous equation, which was found at the solution to (22). Indeed, taking
$\underline{\sigma}$ to be the initial value of $\sigma$ and $s$ the initial value of $\theta$, and denoting $h(s) = \phi(\theta, \sigma)$,
one finds (21). Therefore, $Z''(0) = 0$ if and only if any type $(\tilde{\theta}, \tilde{\sigma})$ that type $(\theta, \sigma)$
may want to mimic lies on the same curve as the mimicker. $Z''(r) < 0$ otherwise.

<figure><img src="image_15.png" /></figure>
I Proof of Lemma 7

Given the allocation $(t(\theta, \sigma, \eta); y(\theta, \sigma, \eta))$, type $(\theta, \sigma)$ is unwilling to report $(\theta', \sigma')$ if and only if

$$ \int_0^1 t(\theta, \sigma, \eta) \chi(\eta) d\eta - \int_0^1 y(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta) \chi(\eta) d\eta \\ \geq \int_0^1 t(\theta', \sigma', \eta) \chi(\eta) d\eta - \int_0^1 y(\theta', \sigma', \eta) \gamma(\theta, \sigma, \eta) \chi(\eta) d\eta. $$

This is rewritten as

$$ U(\theta, \sigma) \geq U(\theta', \sigma') + \int_0^1 y(\theta', \sigma', \eta) (\gamma(\theta', \sigma', \eta) - \gamma(\theta, \sigma, \eta)) \chi(\eta) d\eta. \quad (58) $$

One can write

$$ \begin{aligned} U(\theta, \sigma) &= \int_0^1 t(\theta, \sigma, \eta) \chi(\eta) d\eta - \int_0^1 y(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta) \chi(\eta) d\eta \\ &= U(\theta', \sigma') + \int_0^1 (t(\theta, \sigma, \eta) - t(\theta', \sigma', \eta)) \chi(\eta) d\eta \\ &\quad - \left( \int_0^1 \gamma(\theta, \sigma, \eta) y(\theta, \sigma, \eta) \chi(\eta) d\eta - \int_0^1 \gamma(\theta', \sigma', \eta) y(\theta', \sigma', \eta) \chi(\eta) d\eta \right). \end{aligned} $$

Replacing this expression of $U(\theta, \sigma)$, (58) boils down to $\psi(\theta, \sigma, \theta', \sigma') \geq 0$, where

$$ \begin{aligned} \psi(\theta, \sigma, \theta', \sigma') &\equiv \int_0^1 (t(\theta, \sigma, \eta) - t(\theta', \sigma', \eta)) \chi(\eta) d\eta \\ &\quad - \int_0^1 (y(\theta, \sigma, \eta) - y(\theta', \sigma', \eta)) \gamma(\theta, \sigma, \eta) \chi(\eta) d\eta. \end{aligned} $$

Denote $\Delta t \equiv \int_0^1 (t(\theta, \sigma, \eta) - t(\theta', \sigma', \eta)) \chi(\eta) d\eta$, so that

$$ \psi(\theta, \sigma, \theta', \sigma') = \Delta t - \int_0^1 (y(\theta, \sigma, \eta) - y(\theta', \sigma', \eta)) \gamma(\theta, \sigma, \eta) \chi(\eta) d\eta. \quad (59) $$

Using (32) and (33), one can compute

$$ \begin{aligned} \Delta t &= \int_0^1 \int_{\sigma'}^{\sigma} t_z(\theta, z, \eta) \chi(\eta) dz d\eta + \int_0^1 \int_{\theta'}^{\theta} t_x(\chi, \sigma', \eta) \chi(\eta) dxd\eta \\ &= \int_0^1 \int_{\sigma'}^{\sigma} y_z(\theta, z, \eta) \gamma(\theta, z, \eta) \chi(\eta) dz d\eta \end{aligned} $$
$$+ \int_{0}^{1} \int_{\theta'}^{\theta} y_x(x, \sigma', \eta) \gamma(x, \sigma', \eta) \chi(\eta) dxd\eta.$$

Taking part integrals of each term in this expression, one further obtains

$$\begin{aligned}
\Delta t = & \int_{0}^{1} y(\theta, \sigma, \eta) \gamma(\theta, \sigma, \eta) \chi(\eta) d\eta - \int_{0}^{1} \int_{\sigma'}^{\sigma} y(\theta, z, \eta) \gamma_z(\theta, z, \eta) \chi(\eta) dzd\eta \\
& - \int_{0}^{1} y(\theta', \sigma', \eta) \gamma(\theta', \sigma', \eta) \chi(\eta) d\eta \\
& - \int_{0}^{1} \int_{\theta'}^{\theta} y(x, \sigma', \eta) \gamma_x(x, \sigma', \eta) \chi(\eta) dxd\eta.
\end{aligned}$$

Using next in (59) and rearranging yields

$$\begin{aligned}
\psi(\theta, \sigma, \theta', \sigma') = & - \int_{0}^{1} \int_{\sigma'}^{\sigma} y(\theta, z, \eta) \gamma_z(\theta, z, \eta) \chi(\eta) dzd\eta \quad (60) \\
& - \int_{0}^{1} \int_{\theta'}^{\theta} y(x, \sigma', \eta) \gamma_x(x, \sigma', \eta) \chi(\eta) dxd\eta + \Delta\gamma,
\end{aligned}$$

where $\Delta\gamma \equiv \int_{0}^{1} y(\theta', \sigma', \eta) (\gamma(\theta, \sigma, \eta) - \gamma(\theta', \sigma', \eta)) \chi(\eta) d\eta$. This is rewritten as

$$\Delta\gamma = \int_{0}^{1} \left( \int_{\theta'}^{\theta} y(\theta', \sigma', \eta) \gamma_x(x, \sigma, \eta) dx + \int_{\sigma'}^{\sigma} y(\theta', \sigma', \eta) \gamma_z(\theta', z, \eta) dz \right) \chi(\eta) d\eta.$$

First replacing the above expression of $\Delta\gamma$ in (60), next replacing $\gamma_z(\cdot, z, \eta) = 2\eta - 1$ and $\gamma_x(x, \cdot, \eta) = 1$ and rearranging, yield

$$\begin{aligned}
\psi(\theta, \sigma, \theta', \sigma') = & \int_{\theta}^{\theta'} \int_{0}^{1} (y(x, \sigma', \eta) - y(\theta', \sigma', \eta)) \chi(\eta) d\eta dx \\
& - \int_{\sigma'}^{\sigma} \int_{0}^{1} (y(\theta, z, \eta) - y(\theta', \sigma', \eta)) (2\eta - 1) \chi(\eta) d\eta dz. \quad (61)
\end{aligned}$$

Using the definition of $\phi(\theta, \sigma)$ in (61), (26) is found.

## J Proof of Proposition 4

To develop the proof we consider the following three cases:

1. $\theta' \ge (\le) \theta$ and/or $\sigma' \ge (\le) \sigma$.

Recalling that $\phi_{\theta} \le 0$ and $\phi_{\sigma} \le 0$ (Proposition 2) and that $\phi(\theta, \sigma) > \frac{1}{2}, \forall (\theta, \sigma)$, it is clearly $\psi(\theta, \sigma, \theta', \sigma') \ge 0$.

<figure><img src="image_16.png" /></figure>
2. $\theta' > \theta$ and $\sigma' < \sigma$.

Rewrite (26) as $\psi(\theta, \sigma, \theta', \sigma') = \psi_1(\theta, \sigma, \theta', \sigma') + \psi_2(\theta, \sigma, \theta', \sigma')$, where

$$ \psi_1(\theta, \sigma, \theta', \sigma') = \int_{\theta}^{\theta'} \int_{\phi(\theta', \sigma')}^{\phi(x, \sigma')} \chi(\eta) \, d\eta \, dx $$

$$ \psi_2(\theta, \sigma, \theta', \sigma') = \int_{\sigma'}^{\sigma} \int_{\phi(\theta, z)}^{\phi(\theta', \sigma')} (2\eta - 1) \chi(\eta) \, d\eta \, dz. $$

Define $s(\theta, \sigma)$, $\forall (\theta, \sigma)$, the value of $s$ solving (21) for some given $(\theta, \sigma)$: $s(\theta, \sigma) = \theta + (\sigma - \underline{\sigma}) (2h(s(\theta, \sigma)) - 1)$. Then, $\phi(\theta, \sigma) = h(s(\theta, \sigma))$, $\forall (\theta, \sigma)$. Rewrite

$$ \psi_1(\theta, \sigma, \theta', \sigma') = \int_{\theta}^{\theta'} \int_{h(s(\theta', \sigma'))}^{h(s(x, \sigma'))} \chi(\eta) \, d\eta \, dx \quad (62) $$

$$ \psi_2(\theta, \sigma, \theta', \sigma') = - \int_{\sigma'}^{\sigma} \int_{h(s(\theta', \sigma'))}^{h(s(\theta, z))} (2\eta - 1) \chi(\eta) \, d\eta \, dz. \quad (63) $$

Further denote $\xi(\theta')$ the value of $\sigma$ such that $s(\theta', \sigma') = s(\theta, \xi(\theta'))$. Hence, $h(s(\theta', \sigma')) = h(s(\theta, \xi(\theta'))$. That is, $\xi(\theta')$ is such that type $(\theta, \xi(\theta'))$ lies on the same curve (21) as type $(\theta', \sigma')$, along which $h(s(\cdot, \cdot))$ is constant. There are two cases, according to the size of $\xi(\theta')$.

2.a $\xi(\theta') \in [\sigma', \sigma]:$ the curve $\theta(s, z)$ crosses the horizontal line $\theta$ at some $\xi(\theta') < \sigma$. Decompose $\psi_2(\theta, \sigma, \theta', \sigma')$ in (63) as

$$ \psi_2(\theta, \sigma, \theta', \sigma') = \psi_2^1(\theta, \sigma, \theta', \sigma') + \psi_2^2(\theta, \sigma, \theta', \sigma'), $$

where

$$ \psi_2^1(\theta, \sigma, \theta', \sigma') = - \int_{\sigma'}^{\xi(\theta')} \int_{h(s(\theta, \xi(\theta')))}^{h(s(\theta, z))} (2\eta - 1) \chi(\eta) \, d\eta \, dz \quad (64) $$

$$ \psi_2^2(\theta, \sigma, \theta', \sigma') = - \int_{\xi(\theta')}^{\sigma} \int_{h(s(\theta, \xi(\theta')))}^{h(s(\theta, z))} (2\eta - 1) \chi(\eta) \, d\eta \, dz. \quad (65) $$

We shall now rewrite $\psi_2^1(\theta, \sigma, \theta', \sigma')$ in (64) in a convenient manner. Using $s(\theta', \sigma') = s(\theta, \xi(\theta'))$, we can change the variable $z \in [\sigma', \xi(\theta')]$ with $s(\theta, z) \in [s(\theta, \sigma'), s(\theta, \xi(\theta'))]$ so that

$$ \psi_2^1(\theta, \sigma, \theta', \sigma') = - \int_{s(\theta, \sigma')}^{s(\theta, \xi(\theta'))} \left( \int_{h(s(\theta, \xi(\theta')))}^{h(s(\theta, z))} (2\eta - 1) \chi(\eta) \, d\eta \right) ds(\theta, z). \quad (66) $$
Denote $\zeta(z)$ the value of $\theta$ such that $s(\theta, z) = s(\zeta(z), \sigma')$, $\forall z$. Also, if $z = \xi(\theta')$, then $\zeta(z) = \zeta(\xi(\theta')) = \theta'$, so that $s(\theta, \xi(\theta')) = s(\theta', \sigma')$ (from the definition of $\xi(\theta')$). Replacing in (66) yields

$$ \psi_2^1(\theta, \sigma, \theta', \sigma') = - \int_{s(\theta, \sigma')}^{s(\theta', \sigma')} \left( \int_{h(s(\theta', \sigma'))}^{h(s(\zeta(z), \sigma'))} (2\eta - 1) \chi(\eta) d\eta \right) ds(\theta, z). $$

Next denote $x = \zeta(z)$. As $\zeta(\sigma') = \theta$ and $\zeta(\xi(\theta')) = \theta'$, $x \in [\theta, \theta']$. With a new change of variable, we can further rewrite

$$ \psi_2^1(\theta, \sigma, \theta', \sigma') = - \int_{\theta}^{\theta'} \int_{h(s(\theta', \sigma'))}^{h(s(x, \sigma'))} (2\eta - 1) \chi(\eta) d\eta dx, \quad (67) $$

where we have used $\frac{ds(x, \sigma')}{dx} = 1$. From (62), (65) and (67), we can compute

$$ \begin{aligned} \psi(\theta, \sigma, \theta', \sigma') &= \psi_1(\theta, \sigma, \theta', \sigma') + \psi_2^1(\theta, \sigma, \theta', \sigma') + \psi_2^2(\theta, \sigma, \theta', \sigma') \\ &= 2 \int_{\theta}^{\theta'} \int_{h(s(\theta', \sigma'))}^{h(s(x, \sigma'))} (1 - \eta) \chi(\eta) d\eta dx \\ &\quad + \int_{\xi(\theta')}^{\sigma} \int_{h(s(\theta, z))}^{h(s(\theta, \xi(\theta')))} (2\eta - 1) \chi(\eta) d\eta dz. \end{aligned} \quad (68) $$

As $s(\theta', \sigma') > s(x, \sigma')$ for any $x < \theta'$, $h'(s) \le 0$ and $h(s) \le 1$, $\forall s$, the first term here above is non-negative. As $\xi(\theta') < \sigma$, $s(\theta, z) > s(\theta, \xi(\theta'))$ for any $z > \xi(\theta')$, $h'(s) \le 0$, and $h(s) > \frac{1}{2}$, also the second term is non-negative. Thus, overall, $\psi(\theta, \sigma, \theta', \sigma') \ge 0$.

2.b $\xi(\theta') > \sigma$, which is equivalent to $\zeta(\sigma) < \theta'$. Using $s(\theta, z) = s(\zeta(z), \sigma')$, we can rewrite $\psi_2(\theta, \sigma, \theta', \sigma')$ in (63) as

$$ \psi_2(\theta, \sigma, \theta', \sigma') = - \int_{\sigma'}^{\sigma} \int_{h(s(\theta', \sigma'))}^{h(s(\zeta(z), \sigma'))} (2\eta - 1) \chi(\eta) d\eta dz $$

and, with a change of variable,

$$ \psi_2(\theta, \sigma, \theta', \sigma') = - \int_{s(\zeta(\sigma'), \sigma')}^{s(\zeta(\sigma), \sigma')} \left( \int_{h(s(\theta', \sigma'))}^{h(s(\zeta(z), \sigma'))} (2\eta - 1) \chi(\eta) d\eta \right) ds(\zeta(z), \sigma'). $$

Denote $x = \zeta(z)$. As $\zeta(\sigma') = \theta$, it is $x \in [\theta, \zeta(\sigma)]$. Then,

$$ \psi_2(\theta, \sigma, \theta', \sigma') = - \int_{s(\theta, \sigma')}^{s(\zeta(\sigma), \sigma')} \left( \int_{h(s(\theta', \sigma'))}^{h(s(x, \sigma'))} (2\eta - 1) \chi(\eta) d\eta \right) ds(x, \sigma') $$
and, with a new change of variable,

$$ \psi_2(\theta, \sigma, \theta', \sigma') = - \int_{\theta}^{\zeta(\sigma)} \int_{h(s(\theta', \sigma'))}^{h(s(x, \sigma'))} (2\eta - 1) \chi(\eta) d\eta dx, \quad (69) $$

where we have used $\frac{ds(x, \sigma')}{dx} = 1$. Because $\zeta(\sigma) < \theta'$, we can decompose $\psi_1(\theta, \sigma, \theta', \sigma')$ in (62) as

$$ \psi_1(\theta, \sigma, \theta', \sigma') = \psi_1^1(\theta, \sigma, \theta', \sigma') + \psi_1^2(\theta, \sigma, \theta', \sigma'), $$

where

$$ \psi_1^1(\theta, \sigma, \theta', \sigma') = \int_{\theta}^{\zeta(\sigma)} \int_{h(s(\theta', \sigma'))}^{h(s(x, \sigma'))} \chi(\eta) d\eta dx \quad (70) $$

$$ \psi_1^2(\theta, \sigma, \theta', \sigma') = \int_{\zeta(\sigma)}^{\theta'} \int_{h(s(\theta', \sigma'))}^{h(s(x, \sigma'))} \chi(\eta) d\eta dx. \quad (71) $$

From (69), (70) and (71), one has

$$ \begin{aligned} \psi(\theta, \sigma, \theta', \sigma') &= \psi_1^1(\theta, \sigma, \theta', \sigma') + \psi_2(\theta, \sigma, \theta', \sigma') + \psi_1^2(\theta, \sigma, \theta', \sigma') \\ &= 2 \int_{\theta}^{\zeta(\sigma)} \int_{h(s(\theta', \sigma'))}^{h(s(x, \sigma'))} (1 - \eta) \chi(\eta) d\eta dx \\ &\quad + \int_{\zeta(\sigma)}^{\theta'} \int_{h(s(\theta', \sigma'))}^{h(s(x, \sigma'))} \chi(\eta) d\eta dx. \end{aligned} \quad (72) $$

As $\zeta(\sigma) \in [\theta, \theta']$, $s(x, \sigma') < s(\theta', \sigma')$ for any $x < \theta'$, and $h'(s) \le 0$, the two terms in the expression here above are both non-negative. Overall, $\psi(\theta, \sigma, \theta', \sigma') \ge 0$.

3. $\theta' < \theta$ and $\sigma' > \sigma$.

The proof is analogous to that of case 2. Here, either $\xi(\theta') \in [\sigma, \sigma']$, in which case $\psi(\theta, \sigma, \theta', \sigma')$ is rewritten as in (68), or $\xi(\theta') < \sigma \Leftrightarrow \zeta(\sigma) \in [\theta', \theta]$, in which case $\psi(\theta, \sigma, \theta', \sigma')$ is rewritten as in (72). In both situations, $\psi(\theta, \sigma, \theta', \sigma') \ge 0$.

## K Proof of Proposition 5

From (2), the rent that type $(\theta, \sigma)$ obtains in the second stage, if $(\hat{\theta}, \hat{\sigma})$ is reported in the first stage and $\tilde{\eta}$ in the second stage, is given by

$$ \tilde{u}(\theta, \sigma, \eta, \hat{\theta}, \hat{\sigma}, \tilde{\eta}) = t(\hat{\theta}, \hat{\sigma}, \tilde{\eta}) - \gamma(\hat{\theta}, \hat{\sigma}, \eta) y(\hat{\theta}, \hat{\sigma}, \tilde{\eta}). $$
Further denote

$$ \tilde{U}(\theta, \sigma, \hat{\theta}, \hat{\sigma}) = \int_{0}^{1} \tilde{u}(\theta, \sigma, \eta, \hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}, \eta)) \chi(\eta) d\eta, $$

where $\hat{\eta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}, \eta)$ is the optimal report made by type $(\theta, \sigma)$ in the second stage, after reporting $(\hat{\theta}, \hat{\sigma})$ in the first stage and given that the shock $\eta$ is realized.

**Lemma 9** In an incentive compatible contract,

$$ \tilde{U}(\theta, \sigma, \hat{\theta}, \hat{\sigma}) = U(\hat{\theta}, \hat{\sigma}) + \int_{\theta}^{\hat{\theta}} \int_{0}^{1} y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(x, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) \chi(\eta) d\eta \\ + \int_{\sigma}^{\hat{\sigma}} \int_{0}^{1} (2\eta - 1) y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, z, \hat{\theta}, \hat{\sigma}, \eta)) \chi(\eta) d\eta. \quad (73) $$

**Proof** For shortness, in this proof, we drop arguments from the function $\hat{\eta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}, \eta)$.

First take $\hat{\eta} \in (0, 1)$ (interior solution). Recall that (35) implies that $y(\theta, \sigma, \eta)$ is differentiable almost everywhere with respect to $\eta$ and $y_{\eta}(\theta, \sigma, \eta) \le 0$. One further has

$$ t_{\hat{\eta}}(\hat{\theta}, \hat{\sigma}, \hat{\eta}) - \gamma(\theta, \sigma, \eta) y_{\hat{\eta}}(\hat{\theta}, \hat{\sigma}, \hat{\eta}) = 0. $$

Using (35) together with the envelope theorem and $\gamma_{\theta}(\theta, \sigma, \eta) = 1$, we compute

$$ \begin{aligned} \tilde{U}_{\theta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}) &= \int_{0}^{1} (t_{\hat{\eta}}(\hat{\theta}, \hat{\sigma}, \hat{\eta}) \hat{\eta}_{\theta} - \gamma_{\theta}(\hat{\theta}, \hat{\sigma}, \eta)) y(\hat{\theta}, \hat{\sigma}, \hat{\eta}) - \gamma(\hat{\theta}, \hat{\sigma}, \eta) y_{\hat{\eta}}(\hat{\theta}, \hat{\sigma}, \hat{\eta}) \hat{\eta}_{\theta}) \chi(\eta) d\eta \\ &= \int_{0}^{1} [(t_{\hat{\eta}}(\hat{\theta}, \hat{\sigma}, \hat{\eta}) - \gamma(\theta, \sigma, \eta) y_{\hat{\eta}}(\hat{\theta}, \hat{\sigma}, \hat{\eta})) \hat{\eta}_{\theta} - \gamma_{\theta}(\theta, \sigma, \eta) y(\hat{\theta}, \hat{\sigma}, \hat{\eta})] \chi(\eta) d\eta \\ &= - \int_{0}^{1} \gamma_{\theta}(\theta, \sigma, \eta) y(\hat{\theta}, \hat{\sigma}, \hat{\eta}) \chi(\eta) d\eta. \end{aligned} $$

Using $\gamma_{\theta}(\theta, \sigma, \eta) = 1$, this is rewritten as

$$ \tilde{U}_{\theta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}) = - \int_{0}^{1} y(\hat{\theta}, \hat{\sigma}, \hat{\eta}) \chi(\eta) d\eta. \quad (74) $$

Similarly, using $\gamma_{\sigma}(\theta, \sigma, \eta) = 2\eta - 1$, one finds

$$ \tilde{U}_{\sigma}(\theta, \sigma, \hat{\theta}, \hat{\sigma}) = - \int_{0}^{1} (2\eta - 1) y(\hat{\theta}, \hat{\sigma}, \hat{\eta}) \chi(\eta) d\eta. \quad (75) $$

Next take $\hat{\eta} \in \{0, 1\}$ (corner solution). Then, $\hat{\eta}_{\theta} = \hat{\eta}_{\sigma} = 0$. Using this in (74) and (75), analogous results are derived.
Using (74), the change in utility from type $(\hat{\theta}, \hat{\sigma})$ to type $(\theta, \hat{\sigma})$ is found to be

$$ \tilde{U}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}) - \tilde{U}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}) = \int_{\theta}^{\hat{\theta}} \int_{0}^{1} y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(x, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) \chi(\eta) d\eta. \quad (76) $$

Further using (75), the change in utility from $(\theta, \hat{\sigma})$ to $(\theta, \sigma)$ is found to be

$$ \tilde{U}(\theta, \sigma, \hat{\theta}, \hat{\sigma}) - \tilde{U}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}) = \int_{\sigma}^{\hat{\sigma}} \int_{0}^{1} (2\eta - 1) y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, z, \hat{\theta}, \hat{\sigma}, \eta)) \chi(\eta) d\eta. \quad (77) $$

Summing (76) and (77) side by side and replacing $\tilde{U}(\hat{\theta}, \hat{\sigma}, \hat{\theta}, \hat{\sigma}) = U(\hat{\theta}, \hat{\sigma})$, (73) is obtained. □

**Lemma 10** The contract is incentive compatible if the allocation is such that, for any $(\theta, \sigma)$ and $(\hat{\theta}, \hat{\sigma})$ such that $\theta < \hat{\theta}$ and $\sigma < \hat{\sigma}$,

$$ \int_{0}^{1} y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) \chi(\eta) d\eta \le \int_{0}^{1} y(\theta, \hat{\sigma}, \eta) \chi(\eta) d\eta \quad (78) $$

and

$$ \int_{0}^{1} (2\eta - 1) y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}, \eta)) \chi(\eta) d\eta \le \int_{0}^{1} (2\eta - 1) y(\theta, \sigma, \eta) \chi(\eta) d\eta. \quad (79) $$

21

**Proof** Recall (74) and start from $(\theta, \sigma) = (\hat{\theta}, \hat{\sigma})$, namely, the report $(\hat{\theta}, \hat{\sigma})$ coincides with the type $(\theta, \sigma)$. The change in utility from type $(\hat{\theta}, \hat{\sigma})$ to type $(\theta, \hat{\sigma})$ is given by

$$ \tilde{U}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}) - \tilde{U}(\hat{\theta}, \hat{\sigma}, \hat{\theta}, \hat{\sigma}) = \int_{\theta}^{\hat{\theta}} \int_{0}^{1} y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(x, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) \chi(\eta) d\eta. $$

Next use (75) to write the change in utility from $(\theta, \hat{\sigma})$ to $(\theta, \sigma)$ as

$$ \tilde{U}(\theta, \sigma, \hat{\theta}, \hat{\sigma}) - \tilde{U}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}) = \int_{\sigma}^{\hat{\sigma}} \int_{0}^{1} (2\eta - 1) y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, z, \hat{\theta}, \hat{\sigma}, \eta)) \chi(\eta) d\eta. $$

Taking the two previous differences together yields

$$ \begin{aligned} & \tilde{U}(\theta, \sigma, \hat{\theta}, \hat{\sigma}) - \tilde{U}(\hat{\theta}, \hat{\sigma}, \hat{\theta}, \hat{\sigma}) \\ & = \int_{\theta}^{\hat{\theta}} \int_{0}^{1} y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(x, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) d\eta \end{aligned} $$

<footnote n="21">21 In a different context, Esö and Szentes (2007) identify a condition similar to (78) as being sufficient for implementation of the allocation pinned down under public observation of the signal. Lemma 10 is an extension of their result to the two-dimensional sequential screening problem.</footnote>
$$ + \int_{\sigma}^{\hat{\sigma}} \int_{0}^{1} (2\eta - 1) y (\hat{\theta}, \hat{\sigma}, \hat{\eta} (\theta, z, \hat{\theta}, \hat{\sigma}, \eta)) \chi (\eta) d\eta. \quad (80) $$

We next use the equivalence

$$ \begin{aligned} \tilde{U} (\theta, \sigma, \hat{\theta}, \hat{\sigma}) - \tilde{U} (\theta, \sigma, \theta, \sigma) &= \tilde{U} (\theta, \sigma, \hat{\theta}, \hat{\sigma}) - \tilde{U} (\hat{\theta}, \hat{\sigma}, \hat{\theta}, \hat{\sigma}) - (\tilde{U} (\theta, \sigma, \theta, \sigma) - \tilde{U} (\hat{\theta}, \hat{\sigma}, \hat{\theta}, \hat{\sigma})). \end{aligned} $$

With $\tilde{U} (\theta, \sigma, \theta, \sigma) = U (\theta, \sigma), \forall (\theta, \sigma)$, the incentive constraint $\tilde{U} (\theta, \sigma, \hat{\theta}, \hat{\sigma}) - U (\theta, \sigma) \le 0$ is equivalent to the last line here above being non-positive, namely,

$$ \tilde{U} (\theta, \sigma, \hat{\theta}, \hat{\sigma}) - U (\hat{\theta}, \hat{\sigma}) \le U (\theta, \sigma) - U (\hat{\theta}, \hat{\sigma}). \quad (81) $$

From (8), the right-hand side of (81) is written as

$$ U (\theta, \sigma) - U (\hat{\theta}, \hat{\sigma}) = \int_{\theta}^{\hat{\theta}} \int_{0}^{1} y (x, \hat{\sigma}, \eta) \chi (\eta) d\eta dx - \int_{\hat{\sigma}}^{\sigma} \int_{0}^{1} (2\eta - 1) y (\theta, z, \eta) \chi (\eta) d\eta. $$

The left-hand side of (81) is developed as in (80). Overall, (81) is rewritten as

$$ \begin{aligned} & \int_{\theta}^{\hat{\theta}} \int_{0}^{1} y (\hat{\theta}, \hat{\sigma}, \hat{\eta} (x, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) \chi (\eta) d\eta dx \\ & + \int_{\sigma}^{\hat{\sigma}} \int_{0}^{1} (2\eta - 1) y (\hat{\theta}, \hat{\sigma}, \hat{\eta} (\theta, z, \hat{\theta}, \hat{\sigma}, \eta)) \chi (\eta) d\eta dz \\ & \le \int_{\theta}^{\hat{\theta}} \int_{0}^{1} y (x, \hat{\sigma}, \eta) \chi (\eta) d\eta dx + \int_{\sigma}^{\hat{\sigma}} \int_{0}^{1} (2\eta - 1) y (\theta, z, \eta) \chi (\eta) d\eta dz, \end{aligned} $$

which is satisfied if the two conditions in the lemma both hold. □

**Lemma 11** In the incentive compatible contract, $\hat{\eta} (\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta) < \eta$ and $\phi (\theta, \sigma) \ge \phi (\hat{\theta}, \hat{\sigma})$.

**Proof of $\hat{\eta} (\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta) < \eta$**

By monotonicity of $\gamma (\cdot, \cdot, \eta)$ with respect to $\eta$, there exists a unique $\eta_1$ such that $\gamma (\theta, \tilde{\sigma}, \eta) = \gamma (\hat{\theta}, \hat{\sigma}, \eta_1)$.

First take $\eta_1 \in (0, 1)$, involving that the report made by the agent in the second stage, after observing $\eta$, is an interior point. We show that $\eta_1 = \hat{\eta} (\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)$. Under incentive compatibility, if the type is $(\hat{\theta}, \hat{\sigma})$ and the shock is $\eta_1$, then

$$ \eta_1 = \arg \max_{\tilde{\eta}} \{u (\hat{\theta}, \hat{\sigma}, \tilde{\eta})\} $$
$$ = \arg\max_{\tilde{\eta}} \{t(\hat{\theta}, \hat{\sigma}, \tilde{\eta}) - \gamma(\hat{\theta}, \hat{\sigma}, \eta_1) y(\hat{\theta}, \hat{\sigma}, \tilde{\eta})\}. $$

As $\gamma(\theta, \tilde{\sigma}, \eta) = \gamma(\hat{\theta}, \hat{\sigma}, \eta_1)$, we can rewrite

$$ \eta_1 = \arg\max_{\tilde{\eta}} \{t(\hat{\theta}, \hat{\sigma}, \tilde{\eta}) - \gamma(\theta, \tilde{\sigma}, \eta) y(\hat{\theta}, \hat{\sigma}, \tilde{\eta})\}. \quad (82) $$

Next consider the utility of type $(\theta, \tilde{\sigma})$ when $(\hat{\theta}, \hat{\sigma})$ is reported in the first stage and $\hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)$ in the second stage, after observing $\eta$. The optimized utility is

$$ \tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) = t(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) - \gamma(\theta, \tilde{\sigma}, \eta) y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) $$

and

$$ \begin{aligned} \hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta) &= \arg\max_{\tilde{\eta}} \{\tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \tilde{\eta})\} \\ &= \arg\max_{\tilde{\eta}} \{t(\hat{\theta}, \hat{\sigma}, \tilde{\eta}) - \gamma(\theta, \tilde{\sigma}, \eta) y(\hat{\theta}, \hat{\sigma}, \tilde{\eta})\}. \end{aligned} \quad (83) $$

From (82) and (83), $\eta_1 = \hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)$, so that $\gamma(\theta, \tilde{\sigma}, \eta) = \gamma(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta))$, developed as

$$ 2\hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta) - 1 = \frac{\tilde{\sigma}}{\hat{\sigma}}(2\eta - 1) - \frac{\hat{\theta} - \theta}{\hat{\sigma}}. $$

As $\tilde{\sigma} \in \{\hat{\sigma}, \sigma\}$, with $\sigma < \hat{\sigma}$ and $\hat{\theta} > \theta$, $\hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta) < \eta$.

Next take $\eta_1 \le 0$, involving that $\gamma(\theta, \tilde{\sigma}, \eta) \le \gamma(\hat{\theta}, \hat{\sigma}, 0)$, $\forall \eta$. Using the second-stage incentive constraints, one can write

$$ u(\theta, \tilde{\sigma}, \eta) \ge u(\theta, \tilde{\sigma}, \tilde{\eta}) + (\gamma(\theta, \tilde{\sigma}, \tilde{\eta}) - \gamma(\theta, \tilde{\sigma}, \eta)) y(\theta, \tilde{\sigma}, \tilde{\eta}), \forall \eta, \tilde{\eta}. $$

One can further write

$$ \begin{aligned} \tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \tilde{\eta}) &= t(\hat{\theta}, \hat{\sigma}, \tilde{\eta}) - \gamma(\theta, \tilde{\sigma}, \eta) y(\hat{\theta}, \hat{\sigma}, \tilde{\eta}) \\ &= u(\hat{\theta}, \hat{\sigma}, \tilde{\eta}) + (\gamma(\hat{\theta}, \hat{\sigma}, \tilde{\eta}) - \gamma(\theta, \tilde{\sigma}, \eta)) y(\hat{\theta}, \hat{\sigma}, \tilde{\eta}). \end{aligned} $$

We show that $\tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \tilde{\eta})$ is a decreasing function of $\tilde{\eta}$, so that it is optimal to report $\tilde{\eta} = 0$. To that end, take $\eta_2 < \eta_3$. We thus have

$$ \begin{aligned} \tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \eta_2) - \tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \eta_3) \\ = u(\hat{\theta}, \hat{\sigma}, \eta_2) + (\gamma(\hat{\theta}, \hat{\sigma}, \eta_2) - \gamma(\theta, \tilde{\sigma}, \eta)) y(\hat{\theta}, \hat{\sigma}, \eta_2) \\ - [u(\hat{\theta}, \hat{\sigma}, \eta_3) + (\gamma(\hat{\theta}, \hat{\sigma}, \eta_3) - \gamma(\theta, \tilde{\sigma}, \eta)) y(\hat{\theta}, \hat{\sigma}, \eta_3)]. \end{aligned} \quad (84) $$
As $\gamma(\theta, \tilde{\sigma}, \eta) \le \gamma(\hat{\theta}, \hat{\sigma}, 0)$, one has $\gamma(\hat{\theta}, \hat{\sigma}, \eta_2) - \gamma(\theta, \tilde{\sigma}, \eta) > 0, \forall \eta_2 > 0$. Using this inequality together with $y(\hat{\theta}, \hat{\sigma}, \eta_2) > y(\hat{\theta}, \hat{\sigma}, \eta_3)$, one finds that

$$ 
\begin{aligned}
& \tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \eta_2) - \tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \eta_3) \\
& > u(\hat{\theta}, \hat{\sigma}, \eta_2) + (\gamma(\hat{\theta}, \hat{\sigma}, \eta_2) - \gamma(\theta, \tilde{\sigma}, \eta)) y(\hat{\theta}, \hat{\sigma}, \eta_3) \\
& \quad - [u(\hat{\theta}, \hat{\sigma}, \eta_3) + (\gamma(\hat{\theta}, \hat{\sigma}, \eta_3) - \gamma(\theta, \tilde{\sigma}, \eta)) y(\hat{\theta}, \hat{\sigma}, \eta_3)].
\end{aligned}
$$

This is rewritten as

$$ 
\begin{aligned}
& \tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \eta_2) - \tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \eta_3) \\
& > u(\hat{\theta}, \hat{\sigma}, \eta_2) - u(\hat{\theta}, \hat{\sigma}, \eta_3) - (\gamma(\hat{\theta}, \hat{\sigma}, \eta_3) - \gamma(\hat{\theta}, \hat{\sigma}, \eta_2)) y(\hat{\theta}, \hat{\sigma}, \eta_3).
\end{aligned}
$$

The right-hand side of this inequality is non-negative under the second-stage incentive constraints. Therefore, $\tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \eta_2) > \tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \eta_3)$, showing that $\tilde{u}(\theta, \tilde{\sigma}, \eta, \hat{\theta}, \hat{\sigma}, \tilde{\eta})$ decreases in $\tilde{\eta}$, so that $\hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta) = 0$. Hence, $\hat{\eta}(\theta, \tilde{\sigma}, \hat{\theta}, \hat{\sigma}, \eta) < \eta, \forall \eta > 0$.

Lastly, suppose that $\eta_1 \ge 1$, so that $\gamma(\theta, \tilde{\sigma}, \eta) > \gamma(\hat{\theta}, \hat{\sigma}, 1), \forall \eta < 1$. As $\theta < \hat{\theta}$ and $\tilde{\sigma} \le \hat{\sigma}$, the inequality $\gamma(\theta, \tilde{\sigma}, \eta) > \gamma(\hat{\theta}, \hat{\sigma}, 1)$ is violated, and we rule out this case.

## Proof of $\phi(\theta, \sigma) \ge \phi(\hat{\theta}, \hat{\sigma})$

Rewrite

$$ \phi(\theta, \sigma) - \phi(\hat{\theta}, \hat{\sigma}) = \phi(\theta, \sigma) - \phi(\hat{\theta}, \sigma) + \phi(\hat{\theta}, \sigma) - \phi(\hat{\theta}, \hat{\sigma}). $$

As $\theta < \hat{\theta}$ and $\phi_{\theta}(\theta, \sigma) \le 0, \phi(\theta, \sigma) - \phi(\hat{\theta}, \sigma) \ge 0$. As $\sigma < \hat{\sigma}$ and $\phi_{\sigma}(\theta, \sigma) \le 0, \phi(\hat{\theta}, \sigma) - \phi(\hat{\theta}, \hat{\sigma}) \ge 0$. Hence, $\phi(\theta, \sigma) \ge \phi(\hat{\theta}, \hat{\sigma})$. $\square$

The proof of the next lemma makes reference to Lemma 11.

**Lemma 12** *Conditions (78) and (79) are satisfied at the solution to $\Gamma_1$.*

*Proof* We begin by showing that $y(\theta, \hat{\sigma}, \eta) \ge y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)), \forall \eta$, which implies (78). As $\theta < \hat{\theta}$, it follows from the incentive constraints that $\phi(\theta, \hat{\sigma}) \ge \phi(\hat{\theta}, \hat{\sigma})$. First take $\eta \le \phi(\hat{\theta}, \hat{\sigma})$. As $\eta \le \phi(\hat{\theta}, \hat{\sigma}) \le \phi(\theta, \hat{\sigma})$, one has $y(\theta, \hat{\sigma}, \eta) = 1$. As it is also $\hat{\eta}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta) < \eta$, one further has $\hat{\eta}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta) < \phi(\hat{\theta}, \hat{\sigma})$, so that $y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta)) = 1$ as well. Hence, $y(\theta, \hat{\sigma}, \eta) = y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta))$. Next take $\eta \in [\phi(\hat{\theta}, \hat{\sigma}), \phi(\theta, \hat{\sigma})]$. With analogous reasoning, $y(\theta, \hat{\sigma}, \eta) = 1$ and $y(\hat{\theta}, \hat{\sigma}, \hat{\eta}) \le 1$. If $\eta > \phi(\theta, \hat{\sigma})$, then $y(\theta, \hat{\sigma}, \eta) = y(\hat{\theta}, \hat{\sigma}, \eta) = 0$. Overall, $y(\theta, \hat{\sigma}, \eta) \ge y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \hat{\sigma}, \hat{\theta}, \hat{\sigma}, \eta))$.

We now turn to show that $(2\eta - 1) y(\theta, \sigma, \eta) \ge (2\eta - 1) y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}, \eta)), \forall \eta$, which implies (79). As above, the inequalities $\phi(\hat{\theta}, \hat{\sigma}) \le \phi(\theta, \sigma)$ and $\hat{\eta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}, \eta) < \eta$ imply $y(\theta, \sigma, \eta) = y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}, \eta)) = 1$ if $\eta \le
$\phi(\hat{\theta}, \hat{\sigma})$, so that the condition holds as an equality. If $\eta \ge \phi(\hat{\theta}, \hat{\sigma})$, then it follows from Proposition 1 that $\eta > \frac{1}{2}$. Hence, one has $(2\eta - 1)y(\theta, \sigma, \eta) \ge (2\eta - 1)y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}, \eta))$ if $y(\theta, \sigma, \eta) \ge y(\hat{\theta}, \hat{\sigma}, \hat{\eta}(\theta, \sigma, \hat{\theta}, \hat{\sigma}, \eta))$, and the remainder of the proof proceeds as before. $\square$

# L Proof of Proposition 6

## Case 1: $\sigma$ is public observation

With a change of variable, we can reformulate the solution stated in (28) in the $(s, \sigma)$-space. Define the function $s: \{\Theta, \Sigma\} \to [\theta, \bar{s}]$ such that $s(\theta, \sigma) \equiv \theta + (2\phi^\theta(\theta, \sigma) - 1)(\sigma - \bar{\sigma})$. The inverse of this function is $\theta^\theta(s, \sigma) = s - (2h^\theta(s, \sigma) - 1)(\sigma - \bar{\sigma})$, where $h^\theta(s, \sigma) = \phi^\theta(\theta^\theta(s, \sigma), \sigma)$. Replacing $\theta^\theta(s, \sigma)$ in (28) and rearranging, one obtains

$$2h^{\theta}(s, \sigma) - 1 = \frac{1}{\bar{\sigma}} \left( S - s - \frac{M(\theta^{\theta}(s, \sigma))}{m(\theta^{\theta}(s, \sigma))} \right). \quad (85)$$

Next using (21) in (19) and rearranging, one obtains

$$2h(s) - 1 = \frac{(S - s) \int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} m(\theta(s, z)) n(z) dz}{\int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} \left(\bar{\sigma} - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz}. \quad (86)$$

By continuity and monotonicity of $h(s)$, $\theta^\theta(s^\theta, \sigma) = \theta(s, \sigma)$ for some $s^\theta(s)$ that is unique for each $s$. This is equivalent to

$$s - (2h(s) - 1)(\sigma - \bar{\sigma}) = s^\theta(s, \sigma) - (2h^\theta(s^\theta(s, \sigma)) - 1)(\sigma - \bar{\sigma}). \quad (87)$$

Replacing $2h^\theta(s^\theta) - 1$ and $2h(s) - 1$ from (85) and (86) in (87) and rearranging for $s^\theta$, one obtains

$$s^\theta(s, \sigma) = s \frac{\sigma}{\sigma} \frac{\int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} \left(\sigma - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz}{\int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} \left(\bar{\sigma} - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz} \\ - \frac{\sigma - \bar{\sigma}}{\sigma} \left\{ S \frac{\int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} \left(\frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz}{\int_{\sigma_1(s,h(s))}^{\sigma_2(s,h(s))} \left(\bar{\sigma} - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz} + \frac{M(\theta(s, \sigma))}{m(\theta(s, \sigma))} \right\}. \quad (88)$$
We check whether $s^\theta \le s$. Replacing the expression of $s^\theta$ stated in (88) and rearranging for $s$, it is found that $s^\theta \le s$ if and only if

$$s \le S + \frac{M(\theta(s, \sigma)) \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \left(\frac{\sigma}{n(z)} - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz}{m(\theta(s, \sigma)) \int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \frac{1-N(z)}{n(z)} m(\theta(s, z)) n(z) dz}.$$

Because $2h(s) - 1 \ge 0$ (Proposition 1) and (86), one deduces that $s < S$. Hence, $s^\theta \le s$, $\forall s$. It follows from (87) that $h(s) \ge h^\theta(s^\theta)$, which is equivalent to $\phi(\theta(s, \sigma), \sigma) \ge \phi^\theta(\theta(s, \sigma), \sigma)$.

We now turn to compare the difference $\phi(\theta, \sigma) - \phi^\theta(\theta, \sigma)$ for different values of $\sigma$. Take any $\sigma_0$ and $\sigma_1$ in the domain of $\sigma$ such that $\sigma_0 < \sigma_1$. One has $\phi(\theta(s, \sigma_0), \sigma_0) = \phi(\theta(s, \sigma_1), \sigma_1)$ and $\phi^\theta(\theta^\theta(s^\theta, \sigma_0), \sigma_0) = \phi^\theta(\theta^\theta(s^\theta, \sigma_1), \sigma_1)$, so that

$$\phi(\theta(s, \sigma_0), \sigma_0) - \phi^\theta(\theta^\theta(s^\theta, \sigma_0), \sigma_0) = \phi(\theta(s, \sigma_1), \sigma_1) - \phi^\theta(\theta^\theta(s^\theta, \sigma_1), \sigma_1).$$

By definition of $s^\theta$, $(\theta(s, \sigma_1), \sigma_1) = (\theta^\theta(s^\theta, \sigma_1), \sigma_1)$ and we can write

$$\phi(\theta(s, \sigma_0), \sigma_0) - \phi^\theta(\theta^\theta(s^\theta, \sigma_0), \sigma_0) = \phi(\theta(s, \sigma_1), \sigma_1) - \phi^\theta(\theta^\theta(s, \sigma_1), \sigma_1). \quad (89)$$

Because

$$\frac{d\phi^\theta(\theta^\theta(s^\theta, \sigma_0), \sigma_0)}{ds^\theta} = \frac{dh^\theta(s^\theta)}{ds^\theta} < 0$$

and $s^\theta < s$ (as previously found), one further has $\phi^\theta(\theta^\theta(s^\theta, \sigma_0), \sigma_0) > \phi^\theta(\theta^\theta(s, \sigma_0), \sigma_0)$. Hence,

$$\phi(\theta(s, \sigma_0), \sigma_0) - \phi^\theta(\theta^\theta(s^\theta, \sigma_0), \sigma_0) < \phi(\theta(s, \sigma_0), \sigma_0) - \phi^\theta(\theta^\theta(s, \sigma_0), \sigma_0).$$

Using the left-hand side of (89), this is rewritten as

$$\phi(\theta(s, \sigma_1), \sigma_1) - \phi^\theta(\theta(s, \sigma_1), \sigma_1) < \phi(\theta(s, \sigma_0), \sigma_0) - \phi^\theta(\theta^\theta(s, \sigma_0), \sigma_0).$$

Hence, the difference $\phi(\theta, \sigma) - \phi^\theta(\theta, \sigma)$ is higher the lower $\sigma$ is and the higher $\theta$ is along any given characteristic curve.

Taking any $\sigma_2 > \sigma_1$, and proceeding analogously, one deduces that the difference $\phi(\theta, \sigma) - \phi^\theta(\theta, \sigma)$ is higher the higher $\sigma$ is and the lower $\theta$ is along any given characteristic curve.
## Case 2: $\theta$ is public observation

Proceeding in a similar manner to Case 1, replace $\theta$ with $\theta^{\sigma}$ ($s, \sigma$) in (29) and rearrange to rewrite $\phi^{\sigma}(\theta, \sigma)$ as $\phi^{\sigma}(\theta^{\sigma}(s, \sigma), \sigma) = h^{\sigma}(s)$, where

$$2h^{\sigma}(s) - 1 = \frac{S - s}{\underline{\sigma} - \frac{1-N(\sigma)}{n(\sigma)}}. \qquad (90)$$

Take $s^{\sigma}$ to be such that $\theta^{\sigma}(s^{\sigma}, \sigma) = \theta(s, \sigma)$:

$$s - (2h(s) - 1)(\sigma - \underline{\sigma}) = s^{\sigma} - (2h^{\sigma}(s^{\sigma}) - 1)(\sigma - \underline{\sigma}). \qquad (91)$$

Using (86) and (90) in (91), and rewriting to obtain an expression of $s^{\sigma}$, one finds

$$s^{\sigma} = S - (S - s) \frac{\int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \left(\sigma - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz \underline{\sigma} - \frac{1-N(\sigma)}{n(\sigma)}}{\int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \left(\underline{\sigma} - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz \sigma - \frac{1-N(\sigma)}{n(\sigma)}}. \qquad (92)$$

Defining

$$\delta(s, \sigma) \equiv \frac{\int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \left(\sigma - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz \underline{\sigma} - \frac{1-N(\sigma)}{n(\sigma)}}{\int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \left(\underline{\sigma} - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz \sigma - \frac{1-N(\sigma)}{n(\sigma)}} - 1. \qquad (93)$$

one can rewrite (92) as $s^{\sigma} = s - (S - s)\delta$. Because $s \le \bar{s} < S$, one has $s^{\sigma} < (=) s$ if and only if $\delta(s, \sigma) > (=) 0$. Further rewrite the expression of $\delta(s, \sigma)$ in (93) as

$$\delta(s, \sigma) = \frac{\sigma - \frac{1-N(\sigma)}{n(\sigma)}}{\sigma - \frac{1-N(\sigma)}{n(\sigma)}} \left( \frac{\int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \left(\sigma - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz \underline{\sigma} - \frac{1-N(\sigma)}{n(\sigma)}}{\int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} \left(\underline{\sigma} - \frac{1-N(z)}{n(z)}\right) m(\theta(s, z)) n(z) dz \sigma - \frac{1-N(\sigma)}{n(\sigma)}} \right).$$

Subtracting and adding 1 to the term in brackets and rearranging, we obtain

$$\delta(s, \sigma) = \frac{\sigma - \underline{\sigma}}{\sigma - \frac{1-N(\sigma)}{n(\sigma)}} \left( \frac{L(s) - \frac{1-N(\sigma)}{n(\sigma)}}{\underline{\sigma} - L(s)} \right),$$

where

$$L(s) \equiv \frac{\int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} (1 - N(z)) m(\theta(s, z)) dz \underline{\sigma} - \frac{1-N(\sigma)}{n(\sigma)}}{\int_{\sigma_1(s, h(s))}^{\sigma_2(s, h(s))} m(\theta(s, z)) n(z) dz \sigma - \frac{1-N(\sigma)}{n(\sigma)}}. \qquad (94)$$

With $\sigma > \frac{1-N(\sigma)}{n(\sigma)}$, $\forall \sigma$ (by assumption), one has $\underline{\sigma} > L(s)$, $\forall s$. Hence, $\delta(s, \sigma) > (=) 0$ if and only if $L(s) > (=) \frac{1-N(\sigma)}{n(\sigma)}$. Therefore, $s^{\sigma} < (=) s$ is equivalent to $L(s) > (=) \frac{1-N(\sigma)}{n(\sigma)}$. Lastly, as $s^{\sigma} < (=) s$ is equivalent to $h(s) > (=) h^{\sigma}(s^{\sigma})$ and $(\theta, \sigma) = (\theta(s, \sigma), \sigma) = (\theta^{\sigma}(s^{\sigma}, \sigma), \sigma)$, one deduces that $\phi(\theta, \sigma) > (=) \phi^{\sigma}(\theta, \sigma)$ if and only if $L(s) > (=) \frac{1-N(\sigma)}{n(\sigma)}$.
# M An example with uniform distributions

We proceed in the following steps:

1. Determine $h(s) = \phi(\theta(s, \sigma), \sigma)$ from (19), when both $\theta$ and $\sigma$ are uniformly distributed.

2. Determine $h^\theta(s, \sigma) \equiv \phi(\theta^\theta(s, \sigma), \sigma)$ and $h^\sigma(s, \theta) \equiv \phi(\theta^\sigma(s, \sigma), \sigma)$. The former is determined from the equation $\theta^\theta(s, \sigma) = s - (2h^\theta(s, \sigma) - 1)(\sigma - \underline{\sigma})$, the second from the equation $\theta^\sigma(s, \sigma) = s - (2h^\sigma(s, \sigma) - 1)(\sigma - \overline{\sigma})$.

3. Determine $s^\theta(s, \sigma)$ such that $\theta(s, \sigma) = \theta^\theta(s^\theta(s, \sigma), \sigma)$ and $s^\sigma(s, \sigma)$ such that $\theta(s, \sigma) = \theta^\sigma(s^\sigma(s, \sigma), \sigma)$.

4. Evaluate the differences $D^\theta(s, \sigma) \equiv h(s) - h(s^\theta(s, \sigma), \sigma)$ and $D^\sigma(s, \sigma) \equiv h(s) - h(s^\sigma(s, \sigma), \sigma)$ as well as their signs.

## Determination of $h(s)$

From (19), and with $\theta$ and $\sigma$ being both uniformly distributed, we have

$$h(s) = \frac{1}{2} \left( 1 + \frac{\int_{\sigma_1(s,h(s))) \frac{(S - \theta(s, \sigma)) d\sigma}{\int_{\sigma_1(s,h(s))) \left(\sigma - \frac{1-N(\sigma)}{n(\sigma)}\right) d\sigma} \right).$$

Replacing $\theta(s, \sigma)$ from (21) together with $\sigma - \frac{1-N(\sigma)}{n(\sigma)} = 2\sigma - \bar{\sigma}$, and rearranging yields

$$2h(s) - 1 = \frac{2(S - s)}{\sigma_1(s, h(s)) + \sigma_2(s, h(s)) + 2\underline{\sigma} - 2\overline{\sigma}} \quad (95)$$

Take $\sigma_1(s, h(s)) = \underline{\sigma}$ and $\sigma_2(s, h(s)) = \overline{\sigma} + \frac{s-\theta}{2h(s)-1}$, which is equivalent to $s < \min\{\hat{s}, \bar{\sigma}\}$, for some given $\hat{s}$. Replacing in (95) and rearranging, one further obtains

$$2h(s) - 1 = \frac{2S - 3s + \theta}{2(2\underline{\sigma} - \overline{\sigma})} \quad (96)$$

Take $\sigma_1(s, h(s)) = \underline{\sigma}$ and $\sigma_2(s, h(s)) = \overline{\sigma}$. One obtains

$$2h(s) - 1 = 2 \frac{S - s}{3\underline{\sigma} - \overline{\sigma}} \quad (97)$$

This second case exists when $\hat{s} < \bar{\theta}$ and $s \in [\hat{s}, \bar{\theta}]$, where $\hat{s}$ is found by equalizing the right-hand sides of (96) and (97):

$$\hat{s} \equiv \underline{\theta} + 2(S - \hat{s}) \frac{\underline{\sigma} - \underline{\sigma}}{3\underline{\sigma} - \overline{\sigma}} \quad (98)$$

Hence, the hypothesis that $\hat{s} < \bar{\theta}$ holds if and only if $\frac{\bar{\theta} - \theta}{\bar{\sigma} - \underline{\sigma}} > \frac{2(S - \theta)}{\underline{\sigma} + \bar{\sigma}}$. If this condition is violated, then $\hat{s} \ge \bar{\theta}$. For $s \in [\bar{\theta}, \hat{s}]$, one has $\sigma_1(s, h(s)) = \underline{\sigma} + \frac{s - \bar{\theta}}{2h(s) - 1}$ and
$\sigma_2(s, h(s)) = \underline{\sigma} + \frac{s-\underline{\theta}}{2h(s)-1}$. Replacing in (95) and rearranging yields

$$2h(s) - 1 = \frac{S - 2s + \frac{\underline{\theta} + \bar{\theta}}{2}}{2\underline{\sigma} - \bar{\sigma}}. \qquad (99)$$

Lastly, if $s > \max\{\bar{\theta}, \hat{s}\}$, then $\sigma_1(s, h(s)) = \underline{\sigma} + \frac{s-\bar{\theta}}{2h(s)-1}$ and $\sigma_2(s, h(s)) = \bar{\sigma}$. Replacing in (95) and rearranging, one obtains

$$2h(s) - 1 = \frac{2S - 3s + \bar{\theta}}{3\underline{\sigma} - \bar{\sigma}}. \qquad (100)$$

Finally take $\bar{s}$ such that $\bar{\theta} = \bar{s} - (2h(\bar{s}) - 1)(\bar{\sigma} - \underline{\sigma})$. Replacing $h(\bar{s})$ from (100) and rearranging, one finds

$$\bar{s} = \bar{\theta} + (S - \bar{\theta}) \frac{\bar{\sigma} - \underline{\sigma}}{\bar{\sigma}}. \qquad (101)$$

Considering (96), (97), (99) and (100), one finds that

$$h(s) = \begin{cases} \frac{1}{2} \left(1 + \frac{2S-3s+\underline{\theta}}{2(2\underline{\sigma}-\bar{\sigma})}\right) & \text{if } s \in [\underline{\theta}, \min\{\bar{\theta}, \hat{s}\}] \\ \frac{1}{2} \left(1 + \frac{2S-2s}{3\underline{\sigma}-\bar{\sigma}}\right) & \text{if } \frac{\bar{\theta}-\underline{\theta}}{\bar{\sigma}-\underline{\sigma}} \ge \frac{2(S-\underline{\theta})}{\underline{\sigma}+\bar{\sigma}} \text{ and } s \in [\hat{s}, \bar{\theta}] \\ \frac{1}{2} \left(1 + \frac{S-2s+\frac{\underline{\theta}+\bar{\theta}}{2}}{2\underline{\sigma}-\bar{\sigma}}\right) & \text{if } \frac{\bar{\theta}-\underline{\theta}}{\bar{\sigma}-\underline{\sigma}} < \frac{2(S-\underline{\theta})}{\underline{\sigma}+\bar{\sigma}} \text{ and } s \in [\bar{\theta}, \hat{s}] \\ \frac{1}{2} \left(1 + \frac{2S-3s+\bar{\theta}}{3\underline{\sigma}-\bar{\sigma}}\right) & \text{if } s \in [\max\{\bar{\theta}, \hat{s}\}, \bar{s}], \end{cases} \qquad (102)$$

where $s \in [\underline{\theta}, \bar{s}], \hat{s} = \underline{\theta} + 2(S - \hat{s}) \frac{\bar{\sigma} - \underline{\sigma}}{3\underline{\sigma} - \bar{\sigma}}$ and $\bar{s} = \bar{\theta} + (S - \bar{\theta}) \frac{\bar{\sigma} - \underline{\sigma}}{\bar{\sigma}}$.

## Determination of $h^\theta(s, \sigma)$ and $h^\sigma(s, \sigma)$

The expression of $2h^\theta(s, \sigma) - 1$ in (85) is specified as $2h^\theta(s, \sigma) - 1 = \frac{1}{\underline{\sigma}}(S - s + \underline{\theta} - \theta^\theta(s, \sigma))$. Replacing $\theta^\theta(s, \sigma) = s - (\sigma - \underline{\sigma})(2h^\theta(s, \sigma) - 1)$ and rearranging, one obtains

$$2h^{\theta}(s, \sigma) - 1 = \frac{S - 2s + \underline{\theta}}{2\underline{\sigma} - \sigma}. \qquad (103)$$

The expression of $2h^\sigma(s, \sigma) - 1$ in (90) is specified as

$$2h^{\sigma}(s, \sigma) - 1 = \frac{S - s}{\underline{\sigma} + \sigma - \bar{\sigma}}. \qquad (104)$$
Determination of $D^\theta(s, \sigma)$ and $D^\sigma(s, \sigma)$

First determine $D^\theta(s, \sigma)$. Evaluate (103) at $s = s^\theta(s, \sigma)$ and rewrite it for $s^\theta(s, \sigma)$ as follows:

$$s^{\theta}(s, \sigma) = \frac{S - (2h^{\theta}(s^{\theta}(s, \sigma), \sigma) - 1)(2\underline{\sigma} - \sigma) + \underline{\theta}}{2}.$$

Using it in (87), one finds

$$2s - 2(2h(s) - 1)(\sigma - \underline{\sigma}) = S - \sigma(2h^{\theta}(s^{\theta}(s, \sigma), \sigma) - 1) + \underline{\theta}.$$

Rewriting it for $D^\theta(s, \sigma) = h(s) - h(s^\theta(s, \sigma), \sigma)$, one finds

$$D^{\theta}(s, \sigma) = \frac{(2\underline{\sigma} - \sigma)(2h(s) - 1) - (S + \underline{\theta} - 2s)}{2\sigma} \quad (105)$$

Let the numerator of $D(s, \sigma)$ be

$$d(s) = (2\underline{\sigma} - \bar{\sigma})(2h(s) - 1) - (S + \underline{\theta} - 2s). \quad (106)$$

We see that if $d(s) > 0$, then $D^\theta(s, \bar{\sigma}) > 0$ and $D^\theta_\sigma(s, \sigma) < 0$. Consequently, also $D^\theta(s, \sigma) > 0$. We shall now verify the sufficient condition $d(s) > 0$. Replacing $2h(s) - 1$ from (95) in (106) and rearranging, one obtains

$$d(s) = \frac{2(S-s)(2\underline{\sigma}-\bar{\sigma}) - (S+\underline{\theta}-2s)(\sigma_1(s,h(s)) + \sigma_2(s,h(s)) + 2\underline{\sigma}-2\bar{\sigma})}{\sigma_1(s,h(s)) + \sigma_2(s,h(s)) + 2\underline{\sigma}-2\bar{\sigma}} \\ > \frac{2(S-s)(2\underline{\sigma}-\bar{\sigma}) - 2(S+\underline{\theta}-2s)(2\underline{\sigma}-\bar{\sigma})}{\sigma_1(s,h(s)) + \sigma_2(s,h(s)) + 2\underline{\sigma}-2\bar{\sigma}} \\ > 0.$$

From (105), one has

$$D_s^\theta(s, \sigma) = \frac{1 + (2\underline{\sigma} - \sigma)h'(s)}{\sigma},$$

so that $D_s^\theta(s, \sigma) > 0$ if and only if $h'(s) > -\frac{1}{2\underline{\sigma}-\sigma}$. We next show that this condition holds, $\forall s$. Take $s < \min\{\hat{s}, \bar{\theta}\}$. From (96), $h'(s) = \frac{-3}{4(2\underline{\sigma}-\bar{\sigma})} > -\frac{1}{2\underline{\sigma}-\sigma}$. Take $\hat{s} < \bar{\theta}$ and $s \in [\hat{s}, \bar{\theta}]$. From (97), $h'(s) = \frac{-1}{3\underline{\sigma}-\bar{\sigma}} > -\frac{1}{2\underline{\sigma}-\sigma}$. Take $\hat{s} > \bar{\theta}$ and $s \in [\bar{\theta}, \hat{s}]$. From (99), $h'(s) = \frac{-1}{2\underline{\sigma}-\sigma}$. Lastly, take $s > \max\{\hat{s}, \bar{\theta}\}$. From (100), $h'(s) = \frac{-3}{2(3\underline{\sigma}-\bar{\sigma})} > -\frac{1}{2\underline{\sigma}-\sigma}$.

Finally determine $D^\sigma(s, \sigma)$. Evaluate (104) at $s = s^\sigma(s, \sigma)$ and use it in (91). Rewriting the latter for $D^\sigma(s, \sigma) = h(s) - h(s^\sigma(s, \sigma), \sigma)$, one obtains

$$D^{\sigma}(s, \sigma) = \frac{(2h(s) - 1)(\underline{\sigma} + \sigma - \bar{\sigma}) - (S - s)}{2\sigma - \bar{\sigma}} \quad (107)$$
Replacing $(2h(s) - 1)$ from (95) and rearranging, one further obtains

$$D^{\sigma}(s, \sigma) = \frac{S - s}{2\sigma - \bar{\sigma}} \frac{\sigma - \frac{\sigma_1(s,h(s)) + \sigma_2(s,h(s))}{2}}{\sigma_1(s, h(s)) + \sigma_2(s, h(s)) + 2\bar{\sigma} - 2\bar{\sigma}}$$

We see that $D^{\sigma}(s, \sigma) > (=) 0$ if and only if $\sigma > (=) L(s)$, where $L(s)$ is defined in (94) and such that $L(s) = \frac{\sigma_1(s,h(s)) + \sigma_2(s,h(s))}{2}$ with linear distributions. Moreover, $D_{\sigma}^{\sigma}(s, \sigma) > 0$ because $\sigma_1(s, h(s)) + \sigma_2(s, h(s)) - \bar{\sigma} > 0$. We are left with establishing the sign of $D_s^{\sigma}(s, \sigma)$. From (107), $D_s^{\sigma}(s, \sigma) < 0$ if and only if $h'(s) < \frac{-1}{2(\bar{\sigma} + \sigma - \bar{\sigma})}$. We now verify whether this condition holds in each sub-interval of $s$. Take $s < \min\{\hat{s}, \bar{\theta}\}$. From (96), $h'(s) = \frac{-3}{4(2\bar{\sigma} - \bar{\sigma})} < \frac{-1}{2(\bar{\sigma} + \sigma - \bar{\sigma})}$, so that $D_s^{\sigma}(s, \sigma) < 0$. Take $\hat{s} < \bar{\theta}$ and $s \in [\hat{s}, \bar{\theta}]$. From (97), $h'(s) = \frac{-1}{3\bar{\sigma} - \bar{\sigma}}$, so that $D_s^{\sigma}(s, \sigma) < 0$, if and only if $\sigma > \frac{\bar{\sigma} + \bar{\sigma}}{2} = L(s, \bar{\sigma}, \bar{\sigma})$. Take $\hat{s} > \bar{\theta}$ and $s \in [\bar{\theta}, \hat{s}]$. From (99), $h'(s) = \frac{-1}{2\bar{\sigma} - \bar{\sigma}} < \frac{-1}{2(\bar{\sigma} + \sigma - \bar{\sigma})}$, so that $D_s^{\sigma}(s, \sigma) < 0$. Lastly, take $s > \max\{\hat{s}, \bar{\theta}\}$. From (100), $h'(s) = \frac{-3}{2(3\bar{\sigma} - \bar{\sigma})}$. One has $h'(s) < \frac{-1}{2(\bar{\sigma} + \sigma - \bar{\sigma})}$, so that $D_s^{\sigma}(s, \sigma) < 0$, if and only if $\sigma > \frac{2}{3}\bar{\sigma}$.

## N Proof of Proposition 7

Take $\theta < \theta'$ and $\sigma > \sigma'$. Using (30), rewrite $U(\theta, \sigma')$ as follows:

$$U(\theta, \sigma') = U(\theta', \sigma') + (\theta' - \theta) \Xi(\phi(\theta', \sigma')). \quad (108)$$

Further rewrite the incentive constraint whereby type $(\theta, \sigma')$ is unwilling to report $(\theta, \sigma)$ as

$$U(\theta, \sigma') \geq U(\theta, \sigma) - (\sigma' - \sigma) \left[ (2\phi(\theta, \sigma) - 1) \Xi(\phi(\theta, \sigma)) - 2 \int_{0}^{\phi(\theta, \sigma)} \Xi(\eta) d\eta \right].$$

Replacing $U(\theta, \sigma)$ from (30) in this constraint and rearranging yields

$$U(\theta, \sigma') \geq U(\theta', \sigma') + (\theta' - \theta) \Xi(\phi(\theta', \sigma')) + (\sigma - \sigma') \Delta(\phi(\theta, \sigma), \phi(\theta', \sigma')),$$

where

$$\begin{aligned} \Delta(\phi(\theta, \sigma), \phi(\theta', \sigma')) &\equiv (2\phi(\theta, \sigma) - 1) \Xi(\phi(\theta, \sigma)) \\ &\quad - (2\phi(\theta', \sigma') - 1) \Xi(\phi(\theta', \sigma')) \\ &\quad - 2 \int_{\phi(\theta', \sigma')}^{\phi(\theta, \sigma)} \Xi(\eta) d\eta, \quad \forall (\phi(\theta, \sigma), \phi(\theta', \sigma')) \end{aligned}$$

is strictly positive if $\phi(\theta, \sigma) > \phi(\theta', \sigma')$. Therefore, the right-hand side of the constraint above includes an extra term, relative to the right-hand side of (108), which
contradicts the hypothesis that $U(\theta, \sigma')$ is given by (30) in the incentive compatible contract.

**Funding** Open access funding provided by Università degli Studi di Bari Aldo Moro within the CRUI-CARE Agreement.

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**Conflict of interest** The authors have no conflict of interest to declare that are relevant to the content of this article.

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