october 29, 2019 arxiv:1910.12516v1 [math.oc] 28 oct 2019

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Robust Contracting in General Contract Spaces

Julio Backhoff-Veraguas ·

Patrick Beissner · Ulrich Horst

March 18, 2021

Abstract We consider a general framework of optimal mechanism design underadverse selection and ambiguity about the type distribution of agents.We prove theexistence of optimal mechanisms under minimal assumptions on the contract spaceand prove that centralized contracting implemented via mechanisms is equivalentto delegated contracting implemented via a contract menu under these assump-tions. Our abstract existence results are applied to a series of applications thatinclude models of optimal risk sharing and of optimal portfolio delegation.

Keywords robust contracts · nonmetrizable contract spaces · ambiguity ·financial markets

1 Introduction

We consider a model of optimal mechanism design under adverse selection underminimal assumptions on the contract space. We allow the agent’s preferences to betype-dependent (private information of the agent) and the principal to hold am-biguous beliefs about the distribution of these types. The principal’s preferencesfor ambiguity are captured by the maxmin expected utility model of [23] or, more

Horst and Beissner gratefully acknowledge financial support by the CRC TRR 190 Rationality

and competition - the economic performance of individuals and firms

Julio Backhoff-VeraguasUniversity of Vienna, Faculty of Mathematics. Austria.E-mail: [email protected]

Patrick BeissnerThe Australian National University, Research School of Economics. Australia.E-mail: [email protected]

Ulrich HorstHumboldt-Universitat zu Berlin, Department of Mathematics and School of Business andEconomics. Germany.E-mail: [email protected]

http://arxiv.org/abs/1910.12516v2

2 Julio Backhoff-Veraguas, Patrick Beissner, Ulrich Horst

generally, by variational preferences a la [26]. Our main result establishes the exis-tence of optimal contracts under minimal assumptions on the contract space andthe set of beliefs. Importantly, we do not require the contract space to be metriz-able. Nonmetrizable contract spaces arise naturally in principal-agent models whenthe utility levels of the agent are the relevant contract variables. The contractingproblem over the original contracting space, and the one over the space of theresulting utility levels, are equivalent under mild conditions on the agent’s utilityfunction. Quantifying the decision variable of the principal accounted in utilityunits of the agent is a well-known approach; see [36], [34] and more recently [32].In our case, the transformation has a different purpose. After the transformation,the agent’s utility, as a function of type and contract, reduces to a bilinear formon the contract/type space pair. This significantly simplifies the incentive compat-ibility constraint. Moreover, conditions on the continuity of the agent’s utility boildown to the joint continuity of this bilinear form. As a result, a solvable modelof optimal contracts with large type (contract) space typically only allows for asmall contract (type) space. This dichotomy is well known for the commodity-priceduality in general equilibrium analysis. We show in a series of applications thatthis seemingly mathematical generalization is of economic relevance. Our appli-cations include models of optimal reinsurance and market optimized risk sharing,and models of optimal portfolio delegation. The latter models combine adverseselection and moral hazard effects.

Our abstract existence result extends [30]1 by dropping the assumption of ametrizable contract space. The central idea of this line of reasoning is to considercontract menus (i.e. closed subsets of contracts) rather than contract mechanisms(i.e. functions). However, without a metric on the contract space, we must sub-stitute the usual Hausdorff metric topology on the set of contract menus by theFell topology (the topology of closed convergence) under which the hyperspace ofall closed subsets is compact, as long as the contract space is compact.2 Workingwith the space of closed subsets under the Fell topology, we can show continuityof the utility functions defined on sets and hence the existence of an optimal con-tract menu. Subsequently, we prove that delegated contracting implemented viacontract menus is equivalent to centralized contracting implemented via contractmechanisms. This equivalence is not an obvious result, because the usual measur-able selection theorems on which the proof of the revelation principle is based, donot apply in nonmetrizable spaces.

Our framework is flexible enough to allow the principal to hold uncertain beliefsabout the type of the agent. As such this paper also contributes to the growing lit-erature on robust design of incentives. Following [17] this literature can be roughlydecomposed as follows. First, “robustness to technology or preferences” where aprincipal does not exactly know the agent’s possible actions and/or the distri-bution over relevant output variables; see [15], [21], [22] for more details. Second,“robustness to beliefs and strategic behavior”, where agents hold ambiguous beliefsabout each other. For instance, [16] considers a planner designing a mechanism fortrade who is unsure about the beliefs that other players have about each other,and whose objective is worst-case expected welfare. [14], following up on [19] con-

1 In this study, a characterization of incentive compatibility is established in a Polish typespace combined with a compact metric contract space.

2 Appendix A provides a detailed account of all involved topological concepts.

Robust Contracting in General Contract Spaces 3

sider robustness to beliefs in common-value auctions where buyers observe noisyinformation about the value of the good and the auctioneer is uncertain aboutwhat the buyers know. Third, “robustness to distribution” where a principal hasonly partial information about the distribution over agents’ preference types.3 Forinstance, [11] and [2] consider buyer-seller models with private, respectively com-mon values where the buyer (principal) is confronted with ambiguity about thequality distribution of a good in question.

The present paper focuses on robustness to preference types with a specialemphasis on applications to settings in which types represent the agent’s beliefabout the states of the world. The main goal of the paper is to identify minimalassumptions on the contract space that still guarantee the existence of optimalcontracts and to understand the relation between contract and type space in suchsettings. This level of generality clearly prevents closed form solutions for optimalcontracts; the impact of robustness on the specific structure of optimal contractsis much better analyzed on a case-by-case basis and is well beyond the scope ofthis paper.

The rest of the paper is organized as follows. Section 2 provides several exam-ples that account for robustness and a contract space that cannot be captured byexisting results. Section 3 provides the results on the existence of an optimal con-tract. Section 4 provides a series of applications. Section 5 concludes by comparingthe results of the present work with those in the literature. Appendix A recalls aseries of abstract topological results; Appendix B presents the proofs.

2 Motivating Examples

In this section, we present three motivating examples that illustrate the way inwhich nonmetrizable compact contract spaces over utility units arise naturally inmodels of reinsurance and optimal risk sharing with non-compact metric contractspaces. We analyze each example in greater detail in Section 4.

In what follows we denote by L0(Q), L1(Q), and L∞(Q) the class of all al-most surely finite, integrable, respectively essentially bounded random variablesdefined on some probability space (Ω,F , Q). On L0(Q) we consider the topol-ogy of convergence in Q-probability; the set Lp(Q), 1 ≤ p ≤ ∞, is equippedwith the usual Lp-norm ‖ · ‖p. A set of random variables X is called uniformlynorm bounded if supx∈X ‖x‖∞ < ∞. It is called bounded in Q-probability iflimRր∞ supx∈X Q(|x| ≥ R) = 0.

2.1 A reinsurance model with utilities on the positive half line

Let us consider a reinsurance model between an uninformed principal and aninformed agent. Risk exchange can occur over a set X of random variables definedon a probability space (Ω,F , Q). The principal’s endowment is given by a randomvariable ep ∈ L1(Q). She may hedge her risk by exchanging payoffs x ∈ X withthe agent. Her utility from a risk transfer x is given by

EP ′ [v(ep − x)]

3 For completeness, other forms of robustness are related to collusion, renegotiation, androbustness of standard mechanisms.


for some bounded, concave utility function v : R+ → R on the positive half line,and some belief P ′ about the distribution of the states of the world. We assumethat the belief is equivalent to Q with bounded density.

The agent is endowed with a claim ea ∈ L1(Q). His utility from a risk transferx ∈ X is given by

EP [u(ea + x)]

for some strictly increasing, concave, continuous utility function u : R+ → R onthe positive half line and some belief P that is also equivalent to Q with boundeddensity. We refer to P as the agent’s type. Types are private knowledge to theagent; the principal only knows the set of possible types Q. We identify Q with theset of densities dP

dQ . Incorporating the non-negativity constraint on both parties’

payoff, the set of admissible transfers (the contract space in payoff units) is givenby

X := x ∈ X : −ea ≤ x ≤ ep .

2.1.1 The contracting problem over contingent claims

The principal faces the problem of designing an optimal risk-sharing rule x : Q →X that assigns a transfer to any agent type P ∈ Q and that maximizes her utility,subject to the usual individual rationality and incentive compatibility constraint.We assume that the principal’s utility from a risk-sharing mechanism x : Q → Xis given by a variational utility function (see [26]) of the form

infκ∈K

∫

Q

EP ′ [v (ep − xP )] dκ(P ) + α(κ)

(1)

where K⊂ ∆(Q) is a set of probability measures on Q, and α : K → R is a convexpenalty function. Our choice of utility function allows the principal to be uncertainabout the distribution of the agent type. The principal’s optimization problem isthus given by: find a mechanism x : Q → X that maximizes

infκ∈K

∫

Q

EP ′ [v (ep−xP )] dκ(P ) + α(κ)

subject to P 7→ EP ′ [v (ep−xP )] is measurable,

EP [u(xP+ea)− u(ea)] ≥ 0, P ∈ Q

EP [u(xP+ea)− u(xP+ea)] ≥ 0, P, P ∈ Q,

(2)

where the last two constraints denote the individual rationality (i.e. participation)and truth-telling (i.e. incentive compatibility) constraints, respectively. The firstof these guarantees that the mechanism is better than “offering nothing” and thesecond one makes sure that the agent is better-off if he reports his true type ratherthan lie. The set X is σ(L1(Q), L∞(Q))-compact.4 However, it is usually not normcompact.5 At the same time, the agent’s utility function is continuous w.r.t. the

4 This follows from the Dunford-Pettis theorem, see [20, Theorem A.45].5 For the lack of norm compactness, observe as an illustration that when −ea < ep are

both constants, Ω = [0, 1], and Q is Lebesgue measure on [0, 1], then X contains a sequenceof −ea, ep-valued functions that oscillate increasingly, thus having no accumulation pointw.r.t. Q-a.s. convergence and hence neither in L1(Q)-norm. By the Riesz lemma, the unitball x : ‖x‖ ≤ 1 in a normed vector space (X, ‖ · ‖) is compact if and only if X is finitedimensional.


norm topology under mild technical conditions but usually fails to be continuousw.r.t. the σ(L1(Q), L∞(Q))-topology (unless the agent is risk neutral).6

Remark 1 We emphasize that the choice of contract space X requires minimal as-sumptions and that the weak compactness requires no further restrictions on theset of possible contracts. By contrast, by the Kolmogorov-Riesz compactness the-orem norm compactness of the contract space would require the set of admissibletransfers to be uniformly equicontinuous on average.

2.1.2 The contracting problem over utility units

Without continuity assumptions on the agent’s utility function and compactnessconditions on the contract space, it is difficult to establish the existence of asolution for the contracting problem over contingent claims. To overcome thisproblem, we follow an approach that goes back at least to [36] and [34] and thathas more recently been used by [32] and many others, and consider instead thefollowing set

C := u(ea + x) : x ∈ X

of the agent’s utility levels as the new contract space. Since u is continuous andconcave, C is σ(L1(Q), L∞(Q))-compact. This leads to the following equivalentproblem over a compact contract space: find a mechanism c : Q → C that maxi-mizes

infκ∈K

∫

Q

EP ′

[

v(

ep + ea − u−1( cP ))]

dκ(P ) + α(κ)

subject to P 7→ EP ′

[

v(

ep + ea − u−1( cP ))]

is measurable,

EP [cP − u(ea)] ≥ 0, P ∈ Q

EP [cP − cP ] ≥ 0, P, P ∈ Q.

(3)

Although the change of variables helps to overcome the continuity problem (atthe level of utility units, the agent’s utility functional is linear), a new problememerges: space L∞(Q) is essentially never separable, and hence, set C cannot beexpected to be metrizable when equipped with the weak topology7. We show inSection 4 how our general existence result from Section 3 can be used to overcomethis problem, and to establish the existence of an optimal risk-sharing rule underthe preceding assumptions on utility functions u and v if all the densities in Q areuniformly L∞(Q) bounded, that is, uniformly norm bounded.

Remark 2 If the initial endowments must belong to L2(Q), then X and henceC would be σ(L2(Q), L2(Q))-compact. For instance, this excludes any α-Paretodistribution of wealth with α ≤ 2. Unlike the σ(L1(Q), L∞(Q))-topology theσ(L2(Q), L2(Q))-topology is metrizable on bounded sets if L2(Q) is separable.

6 Although (2) is a static problem, an extension to continuous time, as in [27], is possible.7 The standard way to metrize the weak topology on C is by considering a sequence (φn) ⊂

L∞(Q) and defining e.g. C×C ∋ (f, g) 7→ d(f, g) :=∑

n|∫φn(f−g)dQ|2n‖φn‖∞

. However, unless (φn)

is dense, this is a pseudo-metric which is not a metric (d(f, g) = 0 6⇒ f = g). If Q is not purelyatomic, then no dense sequence in L∞(Q) exists.


This illustrates again the economic rational for working with non-metrizable con-tract spaces: Imposing minimal assumptions on the initial endowments naturallyleads to L1(Q) - and hence nonmetrizable - contract spaces. The contract space ismetrizable only under additional restrictions on the agent’s initial endowments orutility function.

2.2 A reinsurance model with utilities on the whole real line

Let us now consider a modification of the preceding reinsurance model. We assumeagain that risk transfer occurs over a set of random variables X . The agent isendowed with a bounded initial claim ea, his utility function u : R+ → R is stilldefined on the positive half-line but the principal’s utility function v : R → R

is now defined on the whole real line, such as the exponential utility function.Thus, in contrast to Subsection 2.1, the principal’s payoff is not subject to a non-negativity constraint. In this case, we choose as our set of admissible transfers aset

X ⊂ x ∈ X : −ea ≤ x (4)

that is closed in L0(Q), convex and bounded in Q-probability. For instance, ifX is convex we may consider all transfers that are bounded below by −ea andbounded above by some positive random variable y ∈ L0(Q). Unlike in the pre-vious subsection, set X is not necessarily weakly compact. Nonetheless, we provein Subsection 4.1.2 that set C := u(ea + x) : x ∈ X of the agent’s utilitylevels is again σ(L1(Q), L∞(Q))-compact under the above assumptions on X ifu satisfies a certain asymptotic elasticity condition. Moreover, we prove that thecontracting problem (3) has a solution if the agent’s maximal attainable utilityfrom reinsurance is finite and the type space Q is L∞(Q)-norm compact.

Remark 3 If the principal’s utility function v is bounded as it was assumed in thepreceding subsection, then the type space only needs to be norm bounded. In thepresent setup, we can no longer assume that v is bounded because it is now definedon the whole real line.

2.3 A hedging problem with financial markets

We close this section with a financial market framework for the general settingof Section 2.2. Specifically, we consider an incomplete continuous-time financialmarket for which the asset price dynamics is described by a d-dimensional semi-martingale (St) defined on a filtered probability space (Ω, (Ft),F , Q). We considera contracting problem between an investor (the principal) and a broker (the agent).The broker’s assessment of the financial market dynamics is private knowledge.We assume that the principal cannot (or does not want to) trade in the financialmarket, either because she has no access to the market or because she finds it toocostly. Instead, she may hedge her risk by exchanging payoffs with the agent. Theagent is endowed with a bounded claim ea and accepts all transfers that he couldsuper-replicate at zero initial investment by trading in the financial market over thetime interval [0, T ]. Here, a contingent claim e ∈ L0(Q) is called super-replicableat zero initial investment if an admissible trading strategy π exists such that the


resulting gains from trading satisfy e ≤ (π·S)T :=∫ T

0πtdSt. An admissible trading

strategy is an (Ft) adapted d-dimensional stochastic process (πs) such that (π ·S)Tis uniformly bounded from below. Let X be the set of super-replicable claims. Weprove in Subsection 4.1.3 that the resulting set X in (4) is indeed closed in L0(Q),convex and bounded in Q-probability.

3 An Abstract Adverse Selection Problem

Let (Θ, τΘ) and (C, τC) be topological spaces. We endow each of them with theirBorel sigma-algebras, denoted B(Θ) and B(C) respectively. Space C is the spaceof contracts that the principal may offer the agent. Space Θ is the space of possibletypes that the agent may have.

3.1 Menus

If the agent is of type θ ∈ Θ and contract c ∈ C is chosen, the agent’s utility isdenoted by U(θ, c) and the principal’s utility is denoted by V (θ, c). Let

K 6= ∅

be a set of regular Borel probability measures on Θ. Set K comprises the principal’sbeliefs about the agent’s type distribution.

Assumption 1 Throughout, we make the following standing assumptions:

(a) (C, τC) and (Θ, τΘ) are compact Hausdorff spaces.(b) U : Θ × C → R is jointly continuous on Θ × C.(c) V : Θ × C → R is jointly upper semicontinuous on Θ × C.

Remark 4 The standard situation we focus on is one in which C and Θ are subsetsof Banach spaces E and E′ that form a dual pair 〈E,E′〉, where τC is the weaktopology on C and τΘ is a norm topology on Θ, and where U(θ, c) is a continuousfunction of the duality product 〈c, θ〉. Since the dual of a Banach space is largeif the space itself is small and vice versa, the duality of E and E′ implies that asolvable adverse-selection model with large type (contract) space typically requiresa small contract (type) space. In Section 4 we consider the case E = L1(Q) andE′ = L∞(Q) and consider a weakly compact subset C ⊂ E as our contract space.As illustrated in the previous section this choice of contract space is natural whenmaking the agent’s utility the principal’s decision variables.

The family of all τC-closed subsets of C (excluding ∅) is denoted by CL(C).

Definition 1 A menu is a non-empty closed subset of contracts. In other words,menus are the elements of CL(C).

A menu D ∈ CL(C) is automatically compact. Whenever C is not metrizable,we cannot introduce the familiar Hausdorff distance on CL(C). Instead, we equipCL(C) with the Fell topology τF . We recall the Fell topology and the related


concept of Kuratowski-Painleve convergence8 in Appendix A. The main benefit ofworking with the Fell topology is that the family of closed subsets of C is compactas shown by the following lemma. The result is well-known; we give a proof inAppendix B for the reader’s convenience.

Lemma 1 Space (CL(C), τF ) is compact Hausdorff.

The use of the Fell topology is convenient if the contract space is non-metrizable,for instance if C is a weakly compact subset of some L1-space. Indeed, we provethat the agent’s indirect utility function when presented a menu D ∈ CL(C) iscontinuous and that the corresponding argmax correspondence is upper semicon-tinuous. From this we conclude that the principal’s utility function as a functionof types and menus is upper semicontinuous. Since the set of menus is compactin the Fell topology, this implies that an optimal menu exists. From this, we canthen conclude the existence of an optimal contract mechanism.

3.2 Agent

Let us consider the agent’s utility over menus. Given a menu D ∈ CL(C), anagent of type θ derives as utility the maximum U∗(θ,D) he can attain out of thecontracts in the menu:

U∗ : Θ × CL(C) → R, U∗(θ,D) := supd∈D

U(θ, d).

Proposition 1 U∗ is continuous on (Θ × CL(C), τΘ × τF ).

For the agent let Φ be the optimal contracts, depending on types and menu:

Φ : Θ × CL(C) ։ C, Φ(θ,D) := argmaxd∈D

U(θ, d). (5)

Set Φ(θ,D) contains all contracts within menu D, which are optimal for the agentof type θ.

Proposition 2 Φ is a jointly upper hemicontinuous correspondence with compactnon-empty values.

To formulate the individual rationality constraint, we assume that the agentagrees with a proposed menu only if he is left better off than if he had just retainedhis given (type-dependent) reservation utility u, that is, if

U∗(θ,D) ≥ u(θ), for all θ ∈ Θ. (6)

Equation (6) imposes a constraint on the menu D, by requiring for each agent typethe existence of a contract in D which is better than a given outside option foran agent of that type. We assume that function u : Θ → R is common knowledgeand the following standing assumption holds:9

8 Limits in this sense are where the sets accumulate. Note that we have to work with netsinstead of sequences unless the contract set is metrizable.

9 The assumption states that at least one individually rational contract exists. For instance,one formally expects that the action “c∗ =do nothing” (often c∗ = 0 is a concrete model)belongs to C and u(θ) := U(θ, c∗) holds for all θ ∈ Θ.


Assumption 2 There exists c∗ ∈ C such that U(θ, c∗) ≥ u(θ), for all θ ∈ Θ.

To describe the set of individually rational menus, let T : Θ ։ CL(C) be thecorrespondence defined by

T (θ) :=

D ∈ CL(C) : U∗(θ,D) ≥ u(θ)

= U∗(θ, ·)−1[u(θ),∞),

which is closed-valued (by Proposition 1) and hence compact-valued. By Assump-tion 2, c∗ ∈ C exists such that c∗ ∈ T (θ) for all θ ∈ Θ. As a result, we mightdefine, with a slight abuse of notation,

T :=⋂

θ∈Θ

T (θ) ⊂ CL(C), (7)

which is compact and non-empty. We call T the set of individually rational menus.

3.3 Principal

Given an agent’s type θ ∈ Θ and a menu D, the principal will only consider thosecontracts in D that are optimal w.r.t. her utility for this given type. This definesa type-dependent utility for the principal, namely:

V ∗ : Θ × CL(C) → R, V ∗(θ,D) := supd∈Φ(θ,D)

V (θ, d).

The counterpart to Proposition 1 is the following:

Proposition 3 V ∗ is upper semicontinuous on (Θ × CL(C), τΘ × τF ).

To specify the principal’s utility, we recall that the (non-empty) set K consistsof regular Borel probability measures on the type space and thus it collects theprincipal’s possible priors over the type of the agent. We fix a penalty function

α : K → R

and define the principal’s utility of a menu D in variational form as follows:

infκ∈K

∫

θ∈Θ

V ∗(θ,D)dκ(θ) + α(κ)

. (8)

We stress that these integrals are well-defined if we assume that the type space iscompact, as we have assumed so far, or if merely the principal’s utility function isbounded. The principal’s optimization problem over menus reads:

supD∈T

infκ∈K

∫

θ∈Θ

V ∗(θ,D) dκ(θ) + α(κ)

(9)

The following result states a general existence of optimal contracts.

Theorem 3 Under the standing Assumptions 1 and 2, an optimal solution D∗ ∈T to problem (9) exists.


We now present two extensions of the preceding theorem. The first deals withnon-compact type spaces but assumes that the principal’s utility is bounded.

Proposition 4 Let V be uniformly bounded. Further suppose that Assumption 2holds, as well as Assumptions 1 where the compactness of the type space (Θ, τΘ)is dropped. Then, an optimal solution D∗ ∈ T to the problem (9) exists.

The second extension applies to a situation in which the principal evaluatesher performance according to a worst-case approach. In this case, the existence ofan optimal menu can be shown under considerably weaker conditions. The proofis a straightforward modification of the proof of Theorem 3.

Proposition 5 Assume that Assumption 2 and the following conditions hold:

(a’) (C, τC) is a compact Hausdorff space.(b’) U : Θ × C → R is continuous on C for any fixed type θ.(c’) V : Θ × C → R is upper semicontinuous on C for any fixed type θ.

Then, an optimal solution D∗ ∈ T to the problem of maximizing over T the func-tional D 7→ infθ∈Θ V ∗(θ,D) exists.

3.4 On optimal contract mechanisms

A contract mechanism is a mapping ϕ : Θ → C from the type space into thecontract space. A contract mechanism is individually rational if

U(θ, ϕ(θ)) ≥ u(θ), for all θ ∈ Θ. (10)

and incentive compatible if

U(θ, ϕ(θ)) ≥ U(θ, ϕ(θ′)), for all θ, θ′ ∈ Θ. (11)

We denote by M the set of all contract mechanisms that are individually ratio-nal and incentive compatible. In defining a principal-agent problem over optimalmechanisms, rather than over contract menus, the first difficulty is that the integral

∫

Θ

V (θ, ϕ(θ))dκ(θ),

is not defined unless θ 7→ V (θ, ϕ(θ)) is measurable. Thus, we introduce set

M := ϕ ∈ M : V (·, ϕ(·)) is measurable

and define the principal-agent problem over contract mechanisms as follows:

supϕ∈M

infκ∈K

∫

Θ

V (θ, ϕ(θ))dκ(θ) + α(κ)

. (12)

Theorem 4 Under the assumptions of either Theorem 3 or Proposition 4, anoptimal solution ϕ∗ ∈ M for problem (12) exists.

The proof of Theorem 4 relies on Theorem 3/Proposition 4. In fact, this proofreveals that delegated contracting implemented via contract menus is equivalent tocentralized contracting implemented via contract mechanisms. That is, the princi-pal achieves the same optimal utility level by either proposing a suitable contractto each type of agent, or by proposing a suitable set of contracts (a menu) andletting the agent decide for himself.


4 Applications

In this section, we consider several examples for which we can establish the ex-istence of an optimal contract by using the results of Section 3. We are mostlyinterested in the case in which the type space Θ is a subset of L∞(Q) and thecontract space C in utility units is a σ(L1(Q), L∞(Q))-compact subset of L1(Q).The agent’s utility function then depends on the duality product

〈·, ·〉 : L1(Q)× L∞(Q) → R. (13)

The duality product is sequentially continuous when L∞(Q) is equipped with theσ(L∞(Q), L1(Q))-topology (see [1, Theorem 9.37]) but may fail to be continuousin general. Therefore, to guarantee continuity of the duality product we need toconsider the norm topology on L∞(Q) and assume that Θ is subset of L∞(Q).

First, we revisit the motivating examples presented in Section 2. Subsequently,we consider two examples in which the agent can trade in a financial market aftertransacting with the principal.

4.1 The motivating examples revisited

4.1.1 The reinsurance model of Section 2.1

Since the agent’s utility function is strictly increasing, the contracting problems (2)and (3) are equivalent. We apply our abstract existence results to the contractingproblem over utility levels. In the notation of Section 3, τC is the σ(L1(Q), L∞(Q))-topology on the contract space

C = u(ea + x) : x ∈ [−ea, ep]

of utility levels, τΘ is the L∞(Q) norm-topology on the type space Θ = Q, whereas

U(P, c) = EP [c] and V (P ′, c) = EP ′ [v(ep + ea − u−1(c))].

The utility function U is jointly continuous for our choice of topologies. To establishthe upper semicontinuity of the utility function V , we recall the following result;see Proposition 2.10 in [8].

Lemma 2 Let L be a normed vector space and let f : L → R be concave. Then,norm upper-semicontinuity of f is equivalent to the weak upper-semicontinuity off .

Since u−1 is convex by definition, −u−1(·) is concave, and hence V is concavebecause v is concave. Since V is also L1(Q)-norm upper semicontinuous by Fatou’slemma, it follows that V is (jointly) weakly upper semicontinuous. Moreover, ifQ ⊂ L∞(Q) is norm-bounded then V is uniformly bounded because v is bounded.Thus, if an individually rational contract exists, then the existence of a solutionto the contracting problem follows from Theorem 4. By reverting the change ofvariables from X to C, we obtain an optimal mechanism for problem (2).


4.1.2 The reinsurance model of Section 2.2

This section analyzes the model of Section 2.2 in a broader context. Our analysis isinspired by [25, Section 2]. We assume that the agent’s utility function u : R+ → R

satisfies the following conditions.

Assumption 5 The utility function u : R+ → R is lower bounded, strictly in-creasing, strictly concave, continuously differentiable, satisfies the Inada conditionand the following asymptotic elasticity (AE) condition:

lim supz→∞

zu′(z)

u(z)< 1.

The (AE) condition was first introduced by [24] to show that optimal solutionsexist for a wide class of utility maximization problems. For our purposes, this yieldssuitable compactness of the set of utility levels. In economic terms, it is a conditionon the ratio of the marginal utility u′(z) and the average utility u(z)/z for z large.We refer to [33, Definition 3.1] and thereafter, for an exhaustive discussion on thiscondition and its interpretation.

Example 1 The CRRA class uγ(z) = zγ

γfor γ ∈ (0, 1) satisfies Assumption 5.

To handle utilities v on the whole real line, let X ⊂ L0(Q) be the contractspace in payoff units that satisfies the following condition:

Assumption 6 Set X is convex and closed in L0(Q). Further, for any λ ∈ R thefollowing set is bounded in Q-probability:

x ∈ X : x ≥ λ.

We assume that the agent is endowed with a bounded random variable ea ∈L∞(Q) and introduce the set of feasible risk transfers

X (ea) := x ∈ L0(Q) : 0 ≤ x ≤ ea + x for some x ∈ X (14)

= (ea + X − L0+(Q)) ∩ L0

+(Q).

This set represents the set of risk transfers that are feasible to the agent. It followsfrom Assumption 6 and the Komlos lemma that X (ea) is convex, solid,10 andclosed in L0(Q). Next, we introduce the polar set of X (ea), namely,

X (ea)0 := x0 ∈ L0

+(Q) : EQ[xx0] ≤ 1 for all x ∈ X (ea)

along with the convex conjugate u∗ of u:

u∗(y) := supz∈R

u(z)− zy.

Lemma 3 Under Assumptions 5 and 6, assume that some y > 0 and a randomvariable x0 ∈ X (ea)

0 exist such that

EQ[u∗(x0y)] < ∞. (15)

Then, the contract space in utility units C := u(X (ea)) = u(x) : x ∈ X (ea) iscontained in L1(Q) and is σ(L1(Q), L∞(Q))-compact.

10 This means that if x ∈ X (ea) and y ≤ x, then y ∈ X (ea).


The following lemma gives a sufficient condition that guarantees condition (15).It essentially states that the agent’s maximal attainable utility from risk transferis finite. The proofs of Lemmas 3 and 4 are presented in Appendix B.

Lemma 4 Let ‖ea‖ be considered as a constant function on Ω. If

supx∈X (‖ea‖)

EQ[u(x)] < ∞, (16)

where X (‖ea‖) defined in (14), then condition (15) holds.

With Lemma 3 and 4 at hand, we can now use the same arguments as in thepreceding subsection along with Theorem 4 to establish the following result. Itshows that the contracting problem introduced in Section 2.2 has a solution if theagent’s utility from risk sharing is finite.

Theorem 7 Under Assumptions 5 and 6, if (16) holds and if Q ⊂ L∞(Q) is normcompact,11 then the contracting problem of finding a mechanism x : Q → X (ea)that maximizes

infκ∈K

∫

Q

EP ′ [v (ep − ea + xP ))] dκ(P ) + α(κ)

subject to P 7→ EP ′ [v (ep − ea + xP ))] , is measurable

EP [u(xP )− u(ea)] ≥ 0, P ∈ Q

EP [u(xP )− u(xP )] ≥ 0, P, P ∈ Q,

(17)

has a solution as soon as an individually rational contract exists (this is the caseif, for instance, 0 ∈ X ).

Owing to Section 3, the principal achieves in fact the same optimal utilitylevel by either proposing a suitable contract to each type of agent (as in (17)),or by proposing a suitable set of contracts (a menu) and letting agents decide forthemselves. The same applies to the next application.

4.1.3 The hedging problem of Section 2.3

The analysis of the hedging problem is related to the analysis in [18]. A tradingstrategy π is called admissible, if the gain from trade, modelled as a stochasticintegral (π.S)T =

∫ T

0πtdSt, is well–defined and lower bounded. We put

X0 := x : x ≤ (π.S)T , π is admissible.

For the considered financial market, we require a form of no-arbitrage:

Assumption 8 A probability measure Q′ ≈ Q exists such that the process t 7→(π.S)t is a Q′-local martingale for every admissible strategy π. Moreover,

umax := supx∈X0

EQ[u(x+ ea)] < ∞.

11 Special cases of norm-compact subsets of L∞(Q) can be constructed using Arzela andAscoli’s theorem when Ω is a topological space.


The key observation, made in [25, Lemma 1] for the case without randomendowments, but easily obtainable in our setting as well, is the following:

Lemma 5 Assumptions 5 and 8 imply σ(L1(Q), L∞(Q))-compactness of C =u(x) : x ∈ (X0 + ea) ∩ L0

+(Q).

Again, in the notation of Section 3, τC is the σ(L1(Q), L∞(Q))-topology on Cof utility levels, τΘ is the L∞(Q) norm-topology on set Θ = Q, and

U(P, c) = EP [c] and V (P ′, c) = EP ′ [v(e+ ea − u−1(c))].

The agent’s utility U is jointly continuous and the principal’s utility V is uppersemicontinuous. As a result, the hedging problem introduced in Section 2.3 has asolution if the set of densities in Q is L∞(Q)-norm-compact. This result is formallycovered by Theorem 7, but not quite, since hypothesis umax < ∞ is weaker than(16).

4.2 Optimized risk sharing under incomplete markets

In the previous examples, the agent had to decide whether to accept (or reject) acontract based on his portfolio after the risk transfer. In this section we considertwo examples in which he decides to accept (or reject) a contract based on the“indirect utility” that arises after investing in a financial market about whichhe has private information. Specifically, we consider a continuous time financialmarket model with one risk-free and one risky asset. The price of the risk-free assetis normalized to one. The discounted price of the risky asset follows a geometricBrownian motion with drift:

dSt

St= dWt + f ′(Wt)dt, S0 = s, 0 ≤ t ≤ T

where W is a one-dimensional standard Brownian motion defined on the Wienerspace (Ω,F , (Ft)0≤t≤T , Q) and f ∈ F where F is a class of real-valued twice con-tinuously differentiable, uniformly bounded, uniformly equicontinuous functionsf : R → R with common compact support that satisfy the normalization con-straint f(0) = 012. The closure F of F w.r.t. the topology of uniform convergenceis a compact subset of the set of continuous functions on R w.r.t. the supremumnorm by the theorem of Arzela and Ascoli. By Girsanov’s theorem, for any f ∈ F

there exists a unique equivalent martingale measure Pf whose density is given by

dPf

dQ= ef(WT ).

We assume that the pricing kernel is private knowledge to the agent and chooseas our type set the function space

Θ := F

equipped with the topology of uniform convergence. In particular, the financialmarket is complete from the agent’s point of view. The agent is endowed with

12 We think of f ′ as being a constant outside a large compact set. In this case, the pricedynamics essentially reduces to geometric Brownian motion.


a bounded claim ea ∈ L∞(Q). His preferences over payoffs are defined by anexpected utility functional of the form EQ[u(·)] for some Bernoulli utility functionu : R → R that we choose to be the exponential (Subsection 4.2.1), respectivelythe logarithmic (Subsection 4.2.2) one. The principal is endowed with a boundedclaim ep ∈ L∞(Q); her preferences over payoffs are defined by an expected utilityfunctional of the form EQ[v(·)] for some concave Bernoulli utility function v :R → R. An admissible trading strategy for the agent is an adapted real-valuedstochastic process vector ξ that satisfies EQ[

∫ T

0ξ2t dt] < ∞.

4.2.1 Risk-sharing with hedging

Let us first consider a risk-sharing problem in which the agent can hedge his riskby trading in the financial market after transacting with the principal. We assumethat

u(y) = 1− e−αy

with CARA parameter α > 0.The agent as an expert has better information about the financial market and

has a single pricing measure Pf . Given his initial endowment ea ∈ L∞(Q), thebudget set of agent f ∈ F is given by

B(f) :=

x ∈ L1(Q) : Ef [x− ea] ≤ 0

(18)

where Ef denotes the expectation under the pricing measure Pf . We notice thatEf [ea] is finite because Pf has a bounded density w.r.t. Q. The following resultcan be inferred from [20, Chapter 3, Example 3.37]. It yields the optimal claimand the optimal utility from trading for any agent type.

Lemma 6 The optimal attainable payoff over all admissible trading strategies foran agent of type f in the budget set (18) is

x∗ = −1

αln

(

dPf

dQ

)

+ Ef [ea] +1

αH(Pf |Q),

and the optimal utility from trading in the market is

EQ[u(x∗)] = 1− e−αEf [ea]−H(Pf |Q).

Here, H(Pf |Q) := Ef [ln(dPf

dQ )] = EQ

[

ef(WT )f(WT )]

< ∞ denotes the relative

entropy of Pf with respect to the Wiener measure Q.

The densitydPf

dQcorresponds to the private state price density of agent f . Thus,

Lemma 6 clarifies how the optimal payoff depends on the distance between Pf andQ.

Let X ⊂ L1(Q) be a closed set of uniformly integrable financial positions x :Ω → R. By the Dunford-Pettis theorem, this is equivalent to X being σ(L1(Q), L∞(Q))-compact. We take X , equipped with the σ(L1(Q), L∞(Q)) topology as the contractspace:

C = X .


When offered a payoff x ∈ X the agent’s income before trading is ea+x. It followsfrom the above lemma that his optimal optimal utility after trading in the financialmarket, that is, his indirect utility function is given by

U (f, x) = 1− e−αEf [ea+x]−H(Pf |Q).

Since all functions f ∈ F are uniformly bounded it follows from the dominatedconvergence theorem that the mapping f 7→ H(Pf |Q) is continuous. As a result,the mapping

(f, x) 7→ U (f, x)

on Θ × X is jointly continuous. A direct computation shows that a mechanismf 7→ xf is incentive compatible if and only if

Ef [xf − xf ′ ] ≥ 0, for all f, f ′ ∈ F.

We assume that at least one individually rational incentive compatible mechanismexists. The principal’s utility from transacting with agent f is

V (xf ) = EQ[v(eP − xf )].

Since v is concave, V is upper semicontinuous, and thus, it follows from Theorem4 that the principal’s optimization problem to find a mechanism x : F → C thatmaximizes

infκ∈K

∫

F

V (xf ) dκ(f) + α(κ)

subject to the usual incentive compatibility, individual rationality, and measur-ability condition, has a solution. Furthermore, owing to Section 3, the principalachieves in fact the same optimal utility level by either proposing a suitable con-tract to each type of agent (where indirect utilities are specified), or by proposinga suitable set of contracts (a menu) and letting agents decide for themselves ac-cording to their indirect utilities. The same applies to the next application.

4.2.2 A model of optimal portfolio delegation

In the previous applications, the principal’s utility function was independent of theagent type f ∈ F. In this section, we consider a simple model of optimal portfoliodelegation in which the principal’s utility function depends on the agent type. Wenow assume that the agent’s Bernoulli utility function is

u(y) = ln(y).

We retain the assumption that the drift of the stock price process is private infor-mation to the agent. The budget set of an agent of type f is the same as in (18).The following result can again be inferred from [20, Chapter 3, Example 3.43]:

Lemma 7 The optimal attainable payoff over all admissible trading strategies foran agent of type f in the budget set (18) is

x∗ = Ef [ea]dQ

dPf

,

and the optimal utility from trading in the market is

EQ[u(x∗)] = lnEf [ea] +H(Q|Pf ) = lnEf [ea]− EQ[f(WT )].


The principal, considered an investor, outsources her portfolio selection to amanager (the agent) who has private information about the financial market dy-namics and whose investment decisions in the above-specified financial marketcannot be monitored. Following [28] and [3], we restrict the class of admissiblecontracts to be of an affine linear form: the principal can offer the agent somestate dependent payoff x plus a β-fraction of the gains or losses from trading. Inparticular, the agent is liable for possible losses. Formally, a contract consists ofsome F-measurable random variable x, that we again assume to belong to someσ(L1(Q), L∞(Q))-compact set X and some number β ∈ [0, 1], the proportion ofwealth the agent can keep. The contract space

C = X × [0, 1],

is equipped with the product topology. When offered a contract (x, β) ∈ C, theagent’s income from an admissible trading strategy ξ is

ea + x+ β

∫ T

0

ξtdSt.

By Lemma 7, since asset prices are martingales under measure Pf the agent’soptimal utility after trading in the market can be described by the continuousindirect utility function

U(f, x) = lnEf [ea + x] +H(Q|Pf ).

In particular, the agent’s optimal utility is independent of the performancepart

∫

ξdS of his contract13. Hence, and as in Subsection 4.2.1, a mechanismf 7→ ϕ(f) = (xf , βf ) is incentive compatible if and only if Ef [xf − xf ′ ] ≥ 0 for allf, f ′ ∈ F.

Although the agent’s optimal utility is independent of the performance partof his contract, the optimal trading strategy ξ∗ and thus the income w∗ to theprincipal depend on f . Since the optimal claim for the agent can be decomposedas x∗ = x+ ea + β

∫ T

0ξ∗t dSt the optimal gains from trading are given by

∫ T

0

ξ∗t dSt =1

β

(

Ef [ea + x]dQ

dPf− x− ea

)

and hence, the principal’s income from outsourcing the portfolio selection is givenby

w∗ =1− β

β

(

Ef [ea + x]dQ

dPf− (ea + x)

)

.

Hence, unlike in the previous section, the principal’s utility function now dependson the type of the agent with whom she interacts. Her utility when interactingwith an agent of type f and offering a contract (x, β) ∈ C is given by

V (f, (x, β)) = EQ

[

v

(

ep − x+1− β

β

(

Ef [ea + x]dQ

dPf− (ea + x)

))]

.

13 This has previously been observed in [28] and [3].


This function is jointly upper semicontinuous.We can now use the same argumentsas in Subsection 4.2.1 to conclude that the principal’s optimization problem offinding a mechanism (x, β) : F → C that maximizes

infκ∈K

∫

F

V (f, (xf , βf )) dκ(f) + α(κ)

subject to the incentive compatibility, individual rationality, and measurabilitycondition, has a solution (if a feasible mechanism exists) by Theorem 4.

5 Concluding Remarks

We considered a very general setting for a mechanism design problem in the pres-ence of adverse selection. Under mild hypotheses we proved that optimal contractsexist and that centralized contracting implemented via contract mechanisms isequivalent to delegated contracting implemented via contract menus. The guidingprinciple of our work was to use the utility levels of the agent as the relevant con-tract variables. When doing so, the agent’s utility function is given by a bilinearform. This does not only significantly simplify the incentive compatibility condi-tion but also yields a natural duality between the type and the contract type.Simultaneously, it naturally results in a framework in which the relevant topo-logical spaces for the contracting problem may lack metrizability (and possiblyseparability too), as we illustrated with various examples. It would be interestingto derive further insights into the structure of optimal contracts and to quantifythe impact of robustness or ambiguity on optimal contracts as in [2] or [35]. Inview of the many facets ambiguity adds to optimal contracting problems, a formalanalysis of the characteristics of equilibrium contracts in our general environmentdoes not seem appropriate. Conditions for pooling or separating equilibrium arerelevant for many applications but these questions are best analyzed in a lessgeneral framework or even on a case-by-case basis.

We conclude this paper with a couple of technical comments. Our study wasgreatly influenced by the works [30,31], and in particular, by Page’s idea of consid-ering contract menus rather than mechanisms. However, the lack of metrizabilityprecludes us from applying the results therein, because it rules out the use ofmeasurable selection results. For instance, in [30], the contract space is given bysequentially compact subsets of L∞. Under an additional separability assumptionthat would not be needed in our setting, the contract space is metrizable. Com-pared with [30], we reverse the role of the contract and type space: in all ourexamples, the contract spaces are subsets of L1, which is very natural when theagent is an expected utility maximizer, whereas the type spaces are subsets ofL∞. The choice of norm-bounded, respectively compact, type spaces in L∞ is animmediate consequence of the fact that the agent’s utility function is given by abilinear form after the transformation of variables.

To the best of our knowledge, the existence of mechanisms does not followfrom existing Komlos-type results used or established in other studies, such as[29,6,5,4]. Our setting does not seem to fulfill the hypotheses needed for sucharguments. Under various topological assumptions on the contract space, it hasbeen shown that if the principal knows the distribution µ of the agent types and


if the set of incentive compatible mechanisms is convex,14 then a sequence ϕkof strongly measurable incentive compatible mechanisms admits an almost surelyCesaro convergent subsequence, that is, a subsequence nk exists such that

1

nk

nk∑

i=1

ϕi(θ) → ϕ∗(θ) µ-a.s. as k → ∞.

Our setting does not seem to fulfill the hypotheses needed for such arguments. Morestrikingly, our framework allows to consider very general sets of beliefs about thetype distribution. By contrast, Komlos-type arguments always need a referencedistribution w.r.t. which the principal’s beliefs are absolutely continuous. Such areference distribution does not exist if the principal evaluates her performance ac-cording to a worst-case approach w.r.t. the agent’s type. A further advantage ofthe approach used here is that we show that suitably-measurable contract mech-anisms, and contract menus, are largely equivalent even though we cannot usemeasurable selection arguments. In other words, we do not require mechanismsto be measurable functions; only the principal’s utility from implementing thesemechanisms needs to be measurable.

A Abstract Results

A.1 The Fell Topology

We denote by C a compact Hausdorff topological space and recall that CL(C) stands for thefamily of all non-empty closed subsets of C.

Definition 2 The Fell topology on CL(C), which we denote τF , is the topology generated bythe subbase consisting of all sets of the form

V − := A ∈ CL(C) : A ∩ V 6= ∅,

andW+ := A ∈ CL(C) : A ⊂ W,

where V and W are non-empty open subsets of C.

The definition of Fell topology when C is only Hausdorff can be found in [9, Definition5.1.1.], and it reduces to Definition 2 in the present case of C being compact.

Let us now recall the notions of lower and upper closed limits for nets of sets. For a netAλλ∈Λ in CL(C), let Li(Aλ) denote the set of all limit points of Aλλ∈Λ. These are thepoints c ∈ C such that each neighborhood of c intersects Aλ for all λ in some residual subset ofΛ. By contrast, Ls(Aλ) denotes the set of all cluster points of Aλλ∈Λ. These are the pointsc ∈ C such that each neighborhood of c intersects Aλ for all λ in some cofinal subset of Λ. Wealways have Li(Aλ) ⊂ Ls(Aλ).

Since C is a compact Hausdorff space, convergence w.r.t. the Fell topology and the notionof Kuratowski-Painleve convergence of nets coincide [9, Theorem 5.2.6]. Hence, we focus onthe latter convergence, which is easier to apply in the proofs.

Definition 3 LetAλλ∈Λ be a net in CL(C) and A ∈ CL(C). We say that Aλλ∈Λ con-verges in the Kuratowski-Painleve sense to A (denoted A = K limλ Aλ) if Li(Aλ) = A =Ls(Aλ).

14 This is the case if the contract space is convex and the agent’s utility function is affine.The latter is guaranteed if mixed contracts are considered.


A.2 Semicontinuity of an integral functional

We made use of the following result, which is nontrivial in the present nonmetrizable setting:

Lemma 8 Let X, Y be compact Hausdorff spaces, λ a regular probability measure on B(X),and g : X × Y → R a jointly upper semicontinuous function. Then,

Y ∋ y 7→

∫

X

g(x, y)dλ(x),

is well-defined and upper semicontinuous.

Proof Compact Hausdorff spaces are completely regular, and in a completely regular spaceevery finite upper semicontinuous function equals the pointwise infimum of its continuousmajorants:

g(x, y) = infc∈S

c(x, y), S := c : c ≥ g everywhere, c continuous;

see for example [12, Proposition 7, No 7, Chapter IX.10]. Let us assume for the moment15 thatfor c : X × Y → R continuous we have

yα → y implies supx

|c(x, yα)− c(x, y)| → 0. (19)

From the duality of continuous functions and finite measures, we find that y 7→∫

c(x, y)dλ(x),is continuous. Next, note that set S is downwards directed, and hence, in particular, it can beseen as a decreasing net. In line with [7, Proposition 2.13], where the regularity of λ is needed,applied to the (upper-bounded) upper semicontinuous function g, we easily deduce

∫

X

g(x, y)dλ(x) =

∫

X

(

infc∈S

c(x, y)

)

dλ(x) = infc∈S

∫

X

c(x, y)dλ(x).

Thus, the function on the l.h.s. is an infimum of continuous functions and therefore uppersemicontinuous.

Let us return to (19), which would be trivial in the metrizable setting. We fix y and letǫ > 0. For each x we have by continuity the existence of neighborhoods UX

x and V Yx of x and

y respectively, such that

|c(x, y)− c(x, y)| ≤ ǫ/2 for all x ∈ UXx ,∀y ∈ V Y

x .

We now cover X × y, a compact, with⋃

x∈X UXx × V Y

x . Therefore, we obtain x1, . . . , xn

such thatX × y ⊂

⋃

i≤n

UXxi

× V Yxi

,

and we introduce V := ∩i≤nVYxi

. This is an open set containing y, and thus,

X × y ⊂⋃

i≤n

UXxi

× V.

Therefore, given x arbitrary and y ∈ V , there is some i ≤ n such that (x, y) ∈ UXxi

× V andthus

|c(x, y)− c(x, y)| ≤ |c(x, y) − c(xi, y)|+ |c(xi, y) − c(x, y)|

≤ ǫ/2 + ǫ/2.

This shows that supx∈X |c(x, y)− c(x, y)| ≤ ǫ for all y ∈ V , as desired.

15 Actually, this follows immediately from the fact that the topologies on X and Y aregenerated by uniformities: we prefer to give self-contained arguments, perhaps paving the wayfor future extensions.


Corollary 1 Let X be a Hausdorff space, and let Y be a compact Hausdorff space. Let λa regular probability measure on B(X), and g : X × Y → R a jointly upper semicontinuous

bounded function. Then

Y ∋ y 7→

∫

X

g(x, y)dλ(x),

is well-defined and upper semicontinuous.

Proof Let (Xn) be an increasing sequence of compact subsets of X such that λ(Xn) ↑ 1, theexistence of which is guaranteed by the (inner) regularity of λ. By Lemma 8, the mappings

Gn(y) :=

∫

Xn

g(x, y)dλ(x)

are well-defined and upper semicontinuous for each n ∈ N. Subtracting from g its lower bound,we may assume that g is non-negative. For any y and λ a.e. x we have 1Xn (x)g(x, y) ր g(x, y),and hence, by sequential monotone convergence

limn→∞

Gn(y) = supn

Gn(y) =

∫

X

g(x, y)dλ(x) =: G(y).

Thus, G is well-defined, and its upper semicontinuity remains to be proved. Since g is bounded,we have ‖Gn−G‖∞ → 0 as n → ∞. Hence, if yα → y is a net, then for any ǫ > 0 and n = n(ǫ)big enough

lim supG(yα) ≤ ǫ+ limsupGn(yα) ≤ ǫ+Gn(y) ≤ ǫ +G(y),

which concludes the proof.

B Proofs

We now present the proof of Lemma 1, which actually follows by [9, Proposition 5.1.2 andExercise 5.1.4(a)]. For the reader’s convenience, we provide the complete argument:

Proof of Lemma 1 Since C is compact Haussdorf, for each A1, A2 ∈ CL(C) disjoint, we can find

U1, U2 ∈ τC disjoint such that Ai ⊂ Ui. Then, U+1 and U+

2 are disjoint open neighborhoodsin τF of A1 and A2 respectively. Thus τF is Hausdorff.

To prove that τF is compact, we follow [9, Theorem 5.1.3]. Let Vλ : λ ∈ Λ and Wσ :σ ∈ Σ be two families of non-empty open sets of C, such that

CL(C) = V −λ

: λ ∈ Λ ∪ W+σ : σ ∈ Σ.

By the Alexander subbase theorem, it suffices to check the existence of a finite subcoveringfor this kind of coverings, to obtain proof of compactness of τF . Note that if Λ is empty, thenC ∈ ∪σW

+σ , therefore, for some σ ∈ Σ we must have Wσ = C and consequently W+

σ = CL(C)is a finite subcovering. Hence, we now assume that Λ is non-empty. If Σ is empty, then ∀c ∈C : c ∈ ∪λV

−λ

, and thus, these Vλ’s form an open covering of C. Then, we find C = ∪ni=1Vλn

and getCL(C) = ∪n

i=1V−λn

is a finite subcovering.Finally, if both Λ and Σ are non-empty, some σ0 ∈ Σ must exist such that (Wσ0

)c ⊂ ∪λVλ.Indeed, if this was not the case, we may choose cσ ∈ (Wσ)c\ ∪λ Vλ for each σ, and then the

closed set cσ : σ ∈ Στintersects no Vλ and is not contained in any Wσ , contradicting the

covering assumption. Since (Wσ0)c is compact, it follows (Wσ0

)c ⊂ ∪mk=1Vλk

and then clearly

W+σ0

∪k V −λk

= CL(C).

Proof of Proposition 1: Let I : Θ × CL(C) ։ Θ × C be the identity correspondence

(θ,D) 7→ I(θ,D) := θ ×D.

It is easy to see that I is a continuous correspondence (see [1, Lemmata 17.4-17.5]) where thedomain is given the topology (τΘ×τF ) and the range is given (τΘ×τC). Since by Assumption1, function U is jointly continuous, it follows by [1, Berge Maximum Theorem 17.31] that U∗

is continuous too.


Proof of Proposition 2: Since D ∈ CL(C) is compact, we have Φ(θ,D) 6= ∅ by the continuityof U(θ, ·). Further, since set D ∩U(θ, ·)−1(U∗(θ,D)) ⊂ C is closed, it must be compact too,showing that Φ(θ,D) is compact.

We now prove upper hemicontinuity. Since C is compact, by [1, 17.11 Closed Graph The-orem] we need only check that the graph of Φ is closed. Let (θλ, Dλ)λ∈Λ be a net with(θλ, Dλ) → (θ,D), and let fλ ∈ Φ(θλ,Dλ) with fλ → f . We must show that f ∈ Φ(θ,D).Since f is a limit point of fλ, with fλ ∈ Dλ and D = Li(Dλ), we obtain f ∈ D. On theother hand, by definition

U(θλ, fλ) = U∗(θλ,Dλ).

Since U and U∗ are continuous (Proposition 1), we can go into the limit, finding

U(θ, f) = U∗(θ,D).

This finding and f ∈ D conclude the proof.

Proof of Proposition 3: By Proposition 2, the correspondence (θ,D) 7→ Φ(θ,D) is upperhemicontinuous and has non-empty compact values. On the other hand, by Assumption 1,the function (θ, f) 7→ V (θ, f) is upper semicontinuous. It follows that the mapping V ∗(·, ·) isupper semicontinuous; see [10, Theorem 2 (p. 116)] or [1, Lemma 17.30].

Proof of Theorem 3: First, define Fκ : CL(C) → R by

Fκ(D) :=

∫

Θ

V ∗(θ,D)dκ(θ),

for κ ∈ K. We have V ∗(θ,D) ≤ supθ∈Θ, c∈C V (θ, c), the r.h.s. of which is finite by compactnessand u.s.c. of V . By Lemma 8, taking g := V ∗, we have that Fκ is well-defined and, in fact,upper semicontinuous in CL(C). It follows that infκ∈K Fκ(·) is also upper semicontinuous. Theset T is non-empty and compact, see (7). The result follows.

Proof of Proposition 4: Since V is assumed bounded, the proof follows from the same argu-ments as for Theorem 3], using Corollary 1 instead of Lemma 8.

Proof of Theorem 4: Let D∗ be the optimizer of (3). Since set Φ(θ,D∗) is non-empty andcompact for each θ, we deduce that set

argmaxd∈Φ(θ,D∗)

V (θ, d)

is non-empty. By axiom of choice, we select ϕ∗(θ) ∈ argmaxd∈Φ(θ,D∗) V (θ, d), thus, by defini-tion,

V (θ, ϕ∗(θ)) = V ∗(θ,D∗), for all θ.

Observe that V (·, ϕ∗(·)) is measurable, because V ∗(·,D∗) is measurable. In addition, sinceϕ∗(θ) ∈ D∗ for each θ, we deduce in particular

ϕ∗(θ) ∈ Φ(θ,D∗),

which yields thatU(θ, ϕ∗(θ)) ≥ U(θ, ϕ∗(θ′)), for all θ′ ∈ Θ.

Finally, since D∗ ∈ T and ϕ∗(θ) ∈ Φ(θ,D∗) we conclude that U(θ, ϕ∗(θ)) ≥ u(θ). Thus far, we

have established that ϕ∗ ∈ M . Now, let ϕ ∈ M arbitrary, and let D be the closure of ϕ(Θ).It follows that ϕ(θ) ∈ Φ(θ,D), since ϕ is incentive compatible and by continuity of U . ThusV (θ, ϕ(θ)) ≤ V ∗(θ,D) by definition. Since ϕ is individually rational, it follows D ∈ T . All inall,

∫

Θ

V (θ, ϕ(θ))dκ(θ) ≤

∫

Θ

V ∗(θ,D)dκ(θ)

≤

∫

Θ

V ∗(θ,D∗)dκ(θ) =

∫

Θ

V (θ, ϕ∗(θ))dκ(θ),

from which ϕ is optimal for Problem (12).


Proof of Lemma 3: Without loss of generality, we may assume that u ≥ 0. By (15) and thedefinitions of v and X (ea)0 we obtain that C is bounded in L1(Q). By Eberlein-Smulian’stheorem, it is enough to check sequential compactness. By contradiction suppose that C is notσ(L1(Q), L∞(Q))-relatively compact, and hence not uniformly integrable. As in the proof of[25, Lemma 1], we obtain the existence of a sequence gn ⊂ X (ea), a sequence of disjointmeasurable events An and an α > 0 such that

EQ[u(gn)1An ] ≥ α.

Accordingly, if we define Hk :=∑

n≤k u(gn)1An we find EQ[Hk] ≥ kα. Note that as in [24,

Lemma 6.3], u∗(y/2) ≤ au∗(y) + b, and thus, in particular,

u∗(y2−m) ≤ amu∗(y) + (am−1 + · · ·+ a1 + a0)b.

Now, clearly

Hk = u

∑

n≤k

gn1An

≤ u∗(Y ) + Y∑

n≤k

gn1An ,

for every non-negative random variable Y . Let us temporarily assume that we could chooseY ∗ independent of k and such that for some L < α:

EQ

Y ∗∑

n≤k

gn1An

≤ Lk. (20)

From the above computations we then have kα ≤ EQ[Hk] ≤ EQ[u∗(Y ∗)] + Lk, and hence, iffurther

EQ[u∗(Y ∗)] < ∞, (21)

this would yield a contradiction with L < α.We now show the existence of Y ∗ fulfilling (20) and (21). By Assumption (15), there exists

y > 0, Y ∈ X (ea)0 such that EQ[u∗(yY )] < ∞. However, then

EQ[u∗(2−myY )] < ∞,

for all m > 0. We now take m large so that L := y2−m < α and define Y ∗ := LY . Then

EQ

[

Y ∗∑

n≤k

gn1An

]

≤ EQ

[

Y ∗∑

n≤k

gn]

≤ Lk.

Hence Y ∗ fulfills (20)-(21) as desired.It remains to show that C is weakly closed. First we show that C is convex. Since u−1 is

increasing and u concave, for X, X ∈ X (ea) and β ∈ [0, 1] we have

u−1(

βu(X) + (1− β)u(X))

≤ u−1(

u(βX + (1− β)X))

= βX + (1 − β)X ∈ X (ea).

Because X (ea) is solid, then u−1(

βu(X) + (1− β)u(X))

∈ X (ea), and therefore βu(X)+(1−

β)u(X) ∈ C. Since C is convex, it is weakly closed if and only if it is strongly closed. Let usrecall the bipolar theorem of [13, Theorem 1.3], which applies here since X (ea) is convex, solid,and closed in L0(Q), that states

X ∈ X (ea) ⇐⇒ EQ[XY ] ≤ 1 for all Y ∈ X (ea)0.

Now let U(Xn)n ⊂ C converge in L1-norm to Z, and let Y ∈ X (ea)0 be arbitrary. Then,

EQ[Y u−1 u(Xn)] = EQ[Y Xn] ≤ 1,

and therefore, by Fatou’s lemma EQ[Y u−1(Z)] ≤ 1 too, and by the bipolar theorem u−1(Z) ∈X (ea). Consequently, Z ∈ C, finishing the proof.


Proof of Lemma 4: Observe that X (ea) ⊂ X (‖ea‖), and, as a consequence,

∀X ∈ X (‖ea‖) : EQ[XY ] ≤ 1 ⇒ ∀X ∈ X (ea) : EQ[XY ] ≤ 1.

Assumption (16) allows us to apply [24, Theorem 3.1] to the pairing between X (‖ea‖) and itspolar. As a particular consequence, there exists y ≥ 0 and Y in the polar of X (‖ea‖) such that

EQ[u∗(yY )] < ∞.

Since the polar of X (ea) is larger, we conclude the proof.

Acknowledgements The authors would like to thank the anonymous referees and editorsfor their valuable advice.

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