Making Decisions under Model Misspecification
Simone Cerreia-Vioglio, Lars Peter Hansen, Fabio Maccheroni, Massimo Marinacci
aa r X i v : . [ ec on . T H ] A ug Making Decisions under Model Misspecification ∗ Simone Cerreia–Vioglio a , Lars Peter Hansen b , Fabio Maccheroni a and Massimo Marinacci aa Universit`a Bocconi and Igier, b University of Chicago
August 28, 2020
Abstract
We use decision theory to confront uncertainty that is sufficiently broad to incorporate“models as approximations.” We presume the existence of a featured collection of what wecall “structured models” that have explicit substantive motivations. The decision makerconfronts uncertainty through the lens of these models, but also views these models assimplifications, and hence, as misspecified. We extend min-max analysis under modelambiguity to incorporate the uncertainty induced by acknowledging that the models usedin decision-making are simplified approximations. Formally, we provide an axiomaticrationale for a decision criterion that incorporates model misspecification concerns. ∗ We thank the audiences at Advances in Decision Analysis 2019, the Blue Collar Working Group 2.0,SAET 2019 and Caltech for their very useful comments. We thank for the financial support the Alfred P.Sloan Foundation (grant G-2018-11113), the ERC (grants SDDM-TEA and INDIMACRO) and a PRIN grant(2017CY2NCA). ome l’araba fenice:che vi sia, ciascun lo dice;dove sia, nessun lo sa. The consequences of a decision may depend on exogenous contingencies and uncertain out-comes that are outside the control of a decision maker. This uncertainty takes on many forms.Economic applications typically feature risk , where the decision maker knows probabilitiesbut not necessarily outcomes. Statisticians and econometricians have long wrestled with howto confront ambiguity over models or unknown parameters within a model. Each model isitself a simplification or an approximation designed to guide or enhance our understandingof some underlying phenomenon of interest. Thus, the model, by its very nature, is mis-specified , but in typically uncertain ways. How should a decision maker acknowledge modelmisspecification in a way that guides the use of purposefully simplified models sensibly? Thisconcern has certainly been on the radar screen of statisticians and control theorists, but ithas been largely absent in formal approaches to decision theory. Indeed, the statisticiansBox and Cox have both stated the challenge succinctly in complementary ways:Since all models are wrong, the scientist must be alert to what is importantlywrong. It is inappropriate to be concerned about mice when there are tigersabroad. Box (1976).... it does not seem helpful just to say that all models are wrong. The veryword “model” implies simplification and idealization. The idea that complexphysical, biological or sociological systems can be exactly described by a fewformulae is patently absurd. The construction of idealized representations thatcapture important stable aspects of such systems is, however, a vital part ofgeneral scientific analysis and statistical models, especially substantive ones ...Cox (1995).While there are formulations of decision and control problems that intend to confront modelmisspecification, the aim of this paper is: (i) to develop an axiomatic approach that willprovide a rigorous guide for applications and (ii) to enrich formal decision theory whenapplied to environments with uncertainty through the guise of models. “Like the Arabian phoenix: that it exists, everyone says; where it is, nobody knows.” A passage from alibretto of Pietro Metastasio. In Hansen (2014) and Hansen and Marinacci (2016) three kinds of uncertainty are distinguished based onthe knowledge of the decision maker, the most challenging being model misspecification viewed as uncertaintyinduced by the approximate nature of the models under consideration.
1n this paper, we explore formally decision making with multiple models, each of whichis allowed to be misspecified. We follow Hansen and Sargent (2020) by referring to thesemultiple models as “structured models.” These structured models are ones that are explicitlymotivated or featured, such as ones with substantive motivation or scientific underpinnings,consistent with the use of the term “models” by Box and Cox. They may be based onscientific knowledge relying on empirical evidence and theoretical arguments or on reveal-ing parameterizations of probability models with parameters that are interpretable to thedecision maker. In posing decision problems formally, it is often assumed, following Wald(1950), that the correct model belongs to the set of models that decision makers posit. Thepresumption that a decision maker identifies, among their hypotheses, the correct modelis often questionable – recalling the initial quotation, the correct model is often a decisionmaker phoenix. We embrace, rather than push aside, the “models are approximations” per-spective of many applied researchers, as articulated by Box, Cox and others. To exploremisspecification formally, we introduce a potentially rich collection of probability distribu-tions that depict possible representations of the data without formal substantive motivation.We refer to these as “unstructured models.” We use such alternative models as a way tocapture how models could be misspecified. This distinction between structured and unstructured is central to the analysis in thispaper and is used to distinguish aversion to ambiguity over models and aversion to potentialmodel misspecification. At a decision-theoretic level, a proper analysis of misspecificationconcerns has remained elusive so far. Indeed, some of the few studies dealing with economicagents confronting model misspecification still assume that they are conventional expectedutility decision makers who treat model misspecification as if it were model ambiguity, despitebeing aware of a misspecification issue. We extend the analysis of Hansen and Sargent (2020)by providing an axiomatic underpinning for a corresponding decision theory along with arepresentation of the implied preferences that can guide applications. In so doing, we showan important connection with the analysis of subjective and objective rationality of Gilboaet al. (2010).
Criterion
This paper proposes a first decision-theoretic analysis of decision making undermodel misspecification. We consider a classic setup in the spirit of Wald (1950), but relativeto his seminal work we explicitly remove the assumption that the correct model belongs tothe set of posited models and we allow for nonneutrality toward this feature. More formally,we assume that decision makers posit a set Q of structured (probabilistic) models q onstates, motivated by their information, but they are afraid that none of them is correct andso face model misspecification. For this reason, decision makers contemplate what we call Such a distinction is also present in earlier work by Hansen and Sargent (2007) and Hansen and Miao(2018) but without specific reference to the terms “structured” and “unstructured.” See, e.g., Esponda and Pouzo (2016) and Fudenberg et al. (2017). nstructured models in ranking acts f , according to a conservative decision criterion V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q c ( p, q ) (cid:27) (1)To interpret this problem, let c Q ( p ) = min q ∈ Q c ( p, q )where we presume that c Q ( q ) = 0 when q ∈ Q . In this construction, c Q ( p ) is a (Hausdorff)distance between a model p and the posited compact set Q of structured models. Thisdistance is nonzero if and only if p is unstructured, that is, p / ∈ Q . More generally, p ’s thatare closer to the set of structured models Q have a less adverse impact on the preferences,as evident by rewriting (1) as: V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + c Q ( p ) (cid:27) This representation is a special case of the variational representation axiomatized by Mac-cheroni et al. (2006). The unstructured models are statistical artifacts that allow the decisionmaker to assess formally the potential consequences of misspecification as captured by theconstruction of c Q . In this paper we provide a formal interpretation of c Q as an index ofmisspecification fear: the lower the index, the higher the fear. A protective belt
When c takes the entropic form λR ( p || q ), with λ >
0, criterion (1)takes the form min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:27) (2)proposed by Hansen and Sargent (2020). It is the most tractable version of criterion (1),which for a singleton Q further reduces to a standard multiplier criterion a la Hansen andSargent (2001, 2008). By exchanging orders of minimization, we preserve this tractabilityand provide a revealing link to this earlier research,min q ∈ Q (cid:26) min p ∈ ∆ (cid:26)Z u ( f ) dp + λR ( p || q ) (cid:27)(cid:27) (3)The inner minimization problem gives rise to the minimization problem featured by Hansenand Sargent (2001, 2008) to confront the potential misspecification of a given probabilitymodel q . Unstructured models lack the substantive motivation of structured models, yetin (1) they act as a protective belt against model misspecification. The importance of Throughout the paper ∆ denotes the set of all probabilities (Section 2.1). To ease terminology, we often refer to “misspecification” rather than “model misspecification.” The Hansen and Sargent (2001, 2008) formulation of preferences builds on extensive literature in controltheory starting with Jacobson (1973)’s deterministic robustness criterion and a stochastic extension given byPetersen et al. (2000), among several others. λ ) to their proximity to the set Q , a measure oftheir plausibility in view of the decision maker information. The outer minimization overstructured models is the counterpart to the Wald (1950) and the more general Gilboa andSchmeidler (1989) max-min criterion.Our analysis provides a decision-theoretic underpinning for incorporating misspecificationconcerns in a distinct way from ambiguity aversion. Observe that misspecification fear isabsent when the index min q ∈ Q c ( p, q ) equals the indicator function δ Q of the set of structuredmodels Q , that is, min q ∈ Q c ( p, q ) = ( p ∈ Q + ∞ elseIn this case, which corresponds to λ = + ∞ in (2), criterion (1) takes a max-min form: V ( f ) = min q ∈ Q Z u ( f ) dq (4)This max-min criterion thus characterizes decision makers who confront model misspecifi-cation but are not concerned by it, so are misspecification neutral (see Section 4.1). Thecriterion in (1) may thus be viewed as representing decision makers who use a more pruden-tial variational criterion (1) than if they were to max-minimize over the set of structuredmodels which they posited. In particular, the farther away an unstructured model is fromthe set Q (so the less plausible it is), the less it is weighted in the minimization. Axiomatics
We use the entropic case (2) to outline our axiomatic approach. Start witha singleton Q = { q } . Decision makers, being afraid that the reference model q might notbe correct, contemplate also unstructured models p ∈ ∆ and rank acts f according to themultiplier criterion V λ,q ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + λR ( p || q ) (cid:27) (5)Here the positive scalar λ is interpreted as an index of misspecification fear. When decisionmakers posit a nonsingleton set Q of structured models, but are concerned that none of themis correct, then the multiplier criterion (5) gives only an incomplete dominance relation : f % ∗ g ⇐⇒ V λ,q ( f ) ≥ V λ,q ( g ) ∀ q ∈ Q (6)With (6), decision makers can safely regard f better than g . This type of ranking has,however, little traction because of the incomplete nature of % ∗ . Nonetheless, the burdenof choice will have decision makers to select between alternatives, be they rankable by % ∗ or not. A cautious way to complete the binary relation % ∗ is given by the preference % represented by (2), or equivalently by (3), that is, V λ,Q ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:27) (7)4his criterion thus emerges in our analysis as a cautious completion of a multiplier dominancerelation % ∗ . Suitably extended to a general preference pair ( % ∗ , % ), this approach permitsto axiomatize criterion (1) as the representation of the behavioral preference % and theunanimity criterion f % ∗ g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q as the representation of the incomplete dominance relation % ∗ . Basic notions
We consider a non-trivial event σ -algebra Σ of subsets of a state space S .We denote by B (Σ) the space of Σ-measurable simple functions ϕ : S → R , endowed withthe supnorm k k ∞ . The dual of B (Σ) can be identified with the space ba (Σ) of all boundedfinitely additive measures on ( S, Σ).We denote by ∆ the set of probabilities in ba (Σ) and endow ∆ and any of its subsetswith the weak* topology. In particular, ∆ σ denotes the subset of ∆ formed by the countablyadditive probability measures. Given a subset Q in ∆, we denote by ∆ ( Q ) the collectionof all probabilities p which are absolutely continuous with respect to Q , that is, if A ∈ Σand q ( A ) = 0 for all q ∈ Q , then p ( A ) = 0. Moreover, ∆ σ ( q ) denotes the set of elementsof ∆ σ which are absolutely continuous with respect to a single q ∈ ∆ σ , i.e., ∆ σ ( q ) = { p ∈ ∆ σ : p ≪ q } . Unless otherwise specified, throughout all the subsets of ∆ are to beintended non-empty.The (convex analysis) indicator function δ C : ∆ → [0 , ∞ ] of a convex subset C of ∆ isdefined by δ C ( p ) = ( p ∈ C + ∞ elseThroughout we adopt the convention 0 · ±∞ = 0.The effective domain of f : C → ( −∞ , ∞ ], denoted by dom f , is the set { p ∈ C : f ( p ) < ∞} where f takes on a finite value. The function f is:(i) grounded if the infimum of its image is 0, i.e., inf p ∈ C f ( p ) = 0;(ii) strictly convex if, given any distinct p, q ∈ C , we have f ( αp + (1 − α ) q ) < αf ( p ) +(1 − α ) f ( q ) for all α ∈ (0 ,
1) such that αp + (1 − α ) q ∈ dom f . Divergences and statistical distances
Given a non-empty subset Q of ∆, a function c : ∆ × Q → [0 , ∞ ] is a divergence ( for the set Q ) if5i) the sections c q : ∆ → [0 , ∞ ] are grounded, lower semicontinuous and convex for each q ∈ Q ;(ii) the function c Q : ∆ → [0 , ∞ ] defined by c Q ( · ) = min q ∈ Q c ( · , q ) is well defined,grounded, lower semicontinuous and convex;(iii) c − Q (0) = Q , that is, c Q ( p ) = 0 if and only if p ∈ Q .A divergence c that satisfies the distance property c ( p, q ) = 0 ⇐⇒ p = q (8)is called statistical distance ( for the set Q ). In particular, c Q ( p ) is now an Hausdorffstatistical distance between p and Q .The next lemma provides a simple condition for a function c : ∆ × Q → [0 , ∞ ] to be astatistical distance. Lemma 1
Let Q be a compact and convex subset of ∆ . A jointly lower semicontinuous andconvex function c : ∆ × Q → [0 , ∞ ] is a statistical distance if and only if it satisfies thedistance property (8). Given a continuous strictly convex function φ : [0 , ∞ ) → [0 , ∞ ) such that φ (1) = 0 andlim t →∞ φ ( t ) /t = ∞ , define a φ -divergence D φ : ∆ × ∆ σ → [0 , ∞ ] by D φ ( p || q ) = R φ (cid:18) dpdq (cid:19) dq if p ∈ ∆ σ ( q ) ∞ otherwiseHere we adopt the conventions 0 / −∞ . The most important example ofa divergence is the relative entropy given by φ ( t ) = t ln t − t + 1 and denoted by R ( p || q ). Another important example is the
Gini relative index given by the quadratic function φ ( t ) =( t − / χ ( p || q ).A φ -divergence D φ : ∆ × ∆ σ → [0 , ∞ ] is jointly lower semicontinuous and convex. Nextwe show that, when suitably restricted, it is a statistical distance, an important property forour purposes.
Lemma 2
Let Q be a compact and convex subset of ∆ σ . A restricted φ -divergence D φ :∆ × Q → [0 , ∞ ] is a statistical distance. By a “statistical distance” we do not restrict ourselves to a metric and in particular, given p, q ∈ Q , c ( p, q ) is not necessarily equal to c ( q, p ). The function dp/dq is any version of the Radon-Nikodym derivative of p with respect to q . Given the conventions 0 / · ±∞ = 0, it holds φ (0) = 0 ln 0 − · −∞ + 1 = 1. See Chapter 1 of Liese and Vajda (1987). We refer to this book for properties of φ -divergences. φ -divergence is an instance of a (universal) statistical distance c : ∆ × ∆ σ → [0 , ∞ ]whose restriction to each compact and convex subset Q of ∆ σ is a statistical distance for Q .Finally, given a coefficient λ ∈ (0 , ∞ ], the function λD φ : ∆ × Q → [0 , ∞ ]is also a statistical distance. Indeed, when λ = ∞ we have( ∞ ) D φ ( p || q ) = δ { q } ( p ) = ( p = q ∞ elsebecause of the convention 0 · ∞ = 0. Setup
We consider a generalized Anscombe and Aumann (1963) setup where a decisionmaker chooses among uncertain alternatives described by (simple) acts f : S → X , whichare Σ-measurable simple (i.e., finite valued) functions from a state space S to a consequencespace X . This latter set is assumed to be a non-empty convex subset of a vector space (forinstance, X is the set of all simple lotteries defined on a prize space). The triple( S, Σ , X ) (9)forms an (Anscombe-Aumann) decision framework .Let us denote by F the set of all acts. Given any consequence x ∈ X , we denote by x ∈ F also the constant act that takes value x . Thus, with a standard abuse of notation, weidentify X with the subset of constant acts in F . Given a function u : X → R , we denoteby Im u its image. Observe that u ◦ f ∈ B (Σ) when f ∈ F .A preference % is a binary relation on F that satisfies the so-called basic conditions (cf.Gilboa et al., 2010), i.e., it is:(i) reflexive and transitive ;(ii) monotone : if f, g ∈ F and f ( s ) % g ( s ) for all s ∈ S , then f % g ;(iii) continuous : if f, g, h ∈ F , the sets { α ∈ [0 ,
1] : αf + (1 − α ) g % h } and { α ∈ [0 ,
1] : h % αf + (1 − α ) g } are closed;(iv) non-trivial : there exist f, g ∈ F such that f ≻ g .Moreover, a preference % is unbounded if, for each x, y ∈ X with x ≻ y , there exist z, z ′ ∈ X such that 12 z + 12 y % x ≻ y % x + 12 z ′ x ≻ y , a bet on an event A is the act xAy defined by xAy ( s ) = ( x if s ∈ Ay elseIn words, a bet on event A is a binary act that yields a more preferred consequence if A obtains. Comparative uncertainty aversion
As in Ghirardato and Marinacci (2002), given twopreferences % and % on F , we say that % is more uncertainty averse than % if, for eachconsequence x ∈ X and act f ∈ F , f % x = ⇒ f % x In words, a preference is more uncertainty averse than another one if, whenever this prefer-ence is “bold enough” to prefer an uncertain alternative over a sure one, so does the otherone.
Decision criteria
A complete preference % on F is variational if it is represented by adecision criterion V : F → R given by V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p ) (cid:27) (10)where the affine utility function u is non-constant and the index of uncertainty aversion c : ∆ → [0 , ∞ ] is grounded, lower semicontinuous and convex. In particular, given twounbounded variational preferences % and % on F that share the same u , but differentindexes c and c , we have that % is more uncertainty averse than % if and only if c ≤ c (see Maccheroni et al., 2006, Propositions 6 and 8).When the function c has the entropic form c ( p, q ) = λR ( p || q ) with respect to a referenceprobability q ∈ ∆ σ , criterion (10) takes the multiplier form V λ,q ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + λR ( p || q ) (cid:27) analyzed by Hansen and Sargent (2001, 2008). If, instead, the function c has the indicatorform δ C , with C compact and convex, criterion (10) takes the max-min form V ( f ) = min p ∈ C Z u ( f ) dp axiomatized by Gilboa and Schmeidler (1989).All these criteria are here considered in their classical interpretation, so Waldean for themax-min criterion, in which the elements of ∆ are interpreted as models. Strzalecki (2011) provides the behavioral assumptions that characterize multiplier preferences amongvariational preferences. Models and preferences
The consequences of the acts among which decision makers have to choose depend on ex-ogenous states that are outside their control. They know that states obtain according toa probabilistic model described by a probability measure in ∆, the so-called true or cor-rect model . If decision makers knew the true model, they would confront only risk, whichis the randomness inherent to the probabilistic nature of the model. Our decision makers,unfortunately, may not know the true model. Yet, they are able to posit a set of structured probabilistic models Q , based on their information (which might well include existing scien-tific theories, say economic or physical), that form a set of alternative hypotheses regardingthe true model. It is a classical assumption, in the spirit of Wald (1950), in which Q is a setof posited hypotheses about the probabilistic behavior of a, natural or social, phenomenonof interest.A classical decision framework is described by a quartet:( S, Σ , X, Q ) (11)in which a set Q of models is added to a standard decision framework (9). The true modelmight not be in Q , that is, the decision makers information may be unable to pin it down.Throughout the paper we assume that decision makers know this limitation of their informa-tion and so confront model misspecification. This is in contrast with Wald (1950) and mostof the subsequent decision-theoretic literature, which assumes that decision makers eitherknow the true model and so face risk or, at least, know that the true model belongs to Q and so face model ambiguity. In what follows we assume that Q is a compact and convex subset of ∆ σ . As usual,convexity significantly simplifies the analysis. Yet, conceptually it is not an innocuous prop-erty: a hybrid model that mixes two structured models can only have a less motivationthan either of them. Decision criterion (1), however, accounts for the lower appeal of hybridmodels when c ( p, q ) is also convex in q (as, for instance, when c is a φ -divergence). To seewhy, observe that min p ∈ ∆ (cid:8)R u ( f ) dp + c ( p, q ) (cid:9) is, for each act f , convex in q . In turn, thisimplies that hybrid models negatively affect criterion criterion (1). This negative impact ofmixing thus features an “aversion to model hybridization” attitude, behaviorally captured byaxiom A.7. Remarkably, (2) the relative entropy criterion turns out to be neutral to modelhybridization. In this important special case, the assumption of convexity of Q is actuallywithout any loss of generality (as Appendix A.1.3 clarifies). Aydogan et al. (2018) propose an experimental setting that reveals the relevance of model misspecificationfor decision making. The model ambiguity (or uncertainty) literature is reviewed in Marinacci (2015). Q can be also justified in a robust Bayesian interpretation of our analysisthat regards Q as the set of the so-called predictive distributions, which are combinations of“primitive” models (typically extreme points of Q ) weighted according to alternative priorsover them. For instance, if the primitive models describe states through i.i.d. processes, theelements of Q describe them via exchangeable processes that combine primitive models andpriors (as in the Hewitt and Savage, 1955, version of the de Finetti Representation Theorem).Under this interpretation, the p ’s are introduced to provide a protective shield for each ofthe predictive distributions constructed from the alternative priors that are considered. We consider a two-preference setup, as in Gilboa et al. (2010), with a mental preference % ∗ and a behavioral preference % . Definition 1
A preference % is ( subjectively ) rational if it is:a. complete;b. risk independent: if x, y, z ∈ X and α ∈ (0 , , then x ∼ y implies αx + (1 − α ) z ∼ αy + (1 − α ) z . The behavioral preference % governs the decision maker choice behavior and so it is natu-ral to require it to be complete because, eventually, the decision maker has to choose betweenalternatives (burden of choice). It is subjectively rational because, in an “argumentative”perspective, the decision maker cannot be convinced that it leads to incorrect choices. Riskindependence ensures that % is represented on the space of consequences X by an affineutility function u : X → R , for instance an expected utility functional when X is the set ofsimple lotteries. So, risk is addressed in a standard way and we abstract from non-expectedutility issues.The mental preference % ∗ on F represents the decision maker “genuine” preference overacts, so it has the nature of a dominance relation for the decision maker. As such, it mightwell not be complete because of the decision maker inability to compare some pairs of acts. Definition 2
A preference % ∗ is a dominance relation (or is objectively rational ) if it is:a. c-complete: if x, y ∈ X , then x % ∗ y or y % ∗ x ;b. weak c-independent: if f, g ∈ F , x, y ∈ X and α ∈ (0 , , αf + (1 − α ) x % ∗ αg + (1 − α ) x = ⇒ αf + (1 − α ) y % ∗ αg + (1 − α ) y c. convex: if f, g, h ∈ F and α ∈ (0 , , f % ∗ h and g % ∗ h = ⇒ αf + (1 − α ) g % ∗ h f % ∗ g we say that f dominates g ( strictly if f ≻ ∗ g ). The dominance relation is,axiomatically, a variational preference which is not required to be complete. It is objectivelyrational because the decision maker can convince others of its reasonableness, for instancethrough arguments based on scientific theories (a case especially relevant for our purposes).Momentarily, axiom A.3 will further clarify its nature.Along with the classical decision framework (11), the preferences % ∗ and % form a two-preference classical decision environment ( S, Σ , X, Q, % ∗ , % ) (12)The next two assumptions, which we take from Gilboa et al. (2010), connect the twopreferences % ∗ and % .A.1 Consistency . For all f, g ∈ F , f % ∗ g = ⇒ f % g Consistency asserts that, whenever possible, the mental ranking informs the behavioral one.The next condition says that the decision maker opts, by default, for a sure alternative x over an uncertain one f , unless the dominance relation says otherwise.A.2 Caution.
For all x ∈ X and all f ∈ F , f % ∗ x = ⇒ x % f Unlike the previous assumptions, the next two are peculiar to our analysis. They bothlink Q to the two preferences % ∗ and % of the decision maker. We begin with the dominancerelation % ∗ . Here we write f Q = g when q ( f = g ) = 1 for all q ∈ Q , i.e., f and g are equalalmost everywhere according to each structured model.A.3 Objective Q -coherence. For all f, g ∈ F , f Q = g = ⇒ f ∼ ∗ g and % ∗ is complete when Q is a singleton.This axiom first provides a preferential translation of the special status of structured modelsover unstructured ones: if they all regard two acts to be almost surely identical, the decisionmaker “genuine” preference % ∗ follows suit and ranks them indifferent. Convexity is stronger than uncertainty aversion a la Schmeidler (1989), which merely requires that f ∼ ∗ g implies αf + (1 − α ) g % ∗ g . Yet, under completeness of % ∗ convexity and uncertainty aversion coincide (see,e.g., Lemma 56 of Cerreia-Vioglio et al., 2011). % ∗ by requiring that model ambiguity,i.e., a nonsingleton Q , is what underlies it. When Q is a singleton, the dominance relation iscomplete and yet, because of model misspecification, satisfies only a weak form of indepen-dence. In other words, in our approach model misspecification may cause violations of theindependence axiom for the dominance relation. Later in the paper, Section 4.2 will furtherdiscuss this important feature of our analysis.To introduce the second assumption, recall that a rational preference % admits an affineutility function u : X → R because it satisfies risk independence. This permits to define,given a model p ∈ ∆, a consequence x pf ∈ X for each act f via the equality u ( x pf ) = Z u ( f ) dp We can interpret x pf as the certainty equivalent of act f if p were the correct model. This no-tion of certainty equivalent permits to relate the posited set of models Q with the behavioralpreference % , here assumed to be rational.A.4 Subjective Q -coherence . For all f ∈ F and all x ∈ X , we have x ≻ ∗ x pf = ⇒ x ≻ f if and only if p ∈ Q .In words, p ∈ ∆ is a structured model, so belongs to Q , if and only if decision makers takeit seriously, that is, they never choose an act f that would be strictly dominated if p werethe correct model. Such a salience of p for the decision makers’ preference is the preferentialfootprint of a structured model, which decision makers take seriously under considerationbecause of its informational, possibly scientific, status (as opposed to an unstructured model,which decision makers regard as a statistical artifact). We now show how the assumptions on the mental and behavioral preferences permit tocharacterize criterion (1) for a given set Q , that is, for a DM’s given structured information.To this end, we say that a divergence c : ∆ × Q → [0 , ∞ ] is uniquely null if, for all ( p, q ) ∈ ∆ × Q , the sets c − p (0) and c − q (0) are at most singletons. For instance, statistical distancesare easily seen to be uniquely null because of the distance property (8).We are now ready to state our first representation result. Theorem 1
Let ( S, Σ , X, Q, % ∗ , % ) be a two-preference classical decision environment, where ( S, Σ) is a standard Borel space. The following statements are equivalent: i) % ∗ is an unbounded dominance relation and % is a rational preference that are both Q -coherent and jointly satisfy consistency and caution;(ii) there exist an onto affine function u : X → R and a divergence c : ∆ × Q → [0 , ∞ ] ,with dom c Q ⊆ ∆ ( Q ) , such that, for all acts f, g ∈ F , f % ∗ g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q (13) and f % g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + min q ∈ Q c ( p, q ) (cid:27) (14) If, in addition, c is uniquely null, then c : ∆ × Q → [0 , ∞ ] can be chosen to be a statisticaldistance. This result identifies, in particular, the main preferential assumptions underlying a rep-resentation of the type V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q c ( p, q ) (cid:27) (15)for the preference % . While this representation is of interest for a general divergence withrespect to a set Q , it is of particular interest when c : ∆ × Q → [0 , ∞ ] is a statistical distance.In this case, the partial ordering % ∗ is more easily interpreted. Though a technical conditionof “unique nullity” is imposed to pin down statistical distances, our representation arguablyhas more general applicability and captures the preferential underpinning of criterion (15).The Hausdorff statistical distance min q ∈ Q c ( p, q ) between p and Q is strictly positive ifand only if p is an unstructured model, i.e., p / ∈ Q . In particular, the more distant from Q is an unstructured model, the more it is penalized as reflected in the minimization problemthat criterion (15) features. A misspecification index
A behavioral preference % represented by (15) is variationalwith index min q ∈ Q c ( p, q ). So, if two unbounded preferences % and % represented by (15)share the same u but feature different statistical distances min q ∈ Q c ( p, q ) and min q ∈ Q c ( p, q ),then % is more uncertainty averse than % if and only ifmin q ∈ Q c ( p, q ) ≤ min q ∈ Q c ( p, q )In the present “classical” setting we interpret this comparative result as saying that the loweris min q ∈ Q c ( p, q ), the higher is the fear of misspecification. We thus regard the function p min q ∈ Q c ( p, q ) (16)as an index of aversion to model misspecification and we call it, for short, a misspecificationindex . The lower is this index, the higher is the fear of misspecification.13 pecifications and computability Two specifications of our representation are notewor-thy. First, when c is the entropic statistical distance λR ( p || q ), with λ ∈ (0 , ∞ ], we have thefollowing important special case of our representation V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:27) (17)which gives tractability to our decision criterion under model misspecification. Specifically,for λ ∈ (0 , ∞ ), min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:27) = min q ∈ Q − λ log Z e − u ( f ) .λ dq (18)This result is well known when Q is a singleton, that is, when (17) is a standard multipliercriterion. A second noteworthy special case of our representation is the Gini criterion V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q χ ( p || q ) (cid:27) . Remarkably, we havemin p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q χ ( p || q ) (cid:27) = min q ∈ Q (cid:26)Z u ( f ) dq − λ Var q ( u ( f )) (cid:27) for all acts f for which the mean-variance (in utils) criteria on the r.h.s. are monotone. So,the Gini criterion is a monotone version of the max-min mean-variance criterion.As to computability, in the important case when criterion (1) features a φ -divergence, likethe specifications just discussed, we need only to know the set Q to compute it, no integralwith respect to unstructured models is needed. This is proved in the next result which is aconsequence of a duality formula of Ben-Tal and Teboulle (2007). Proposition 1
Given Q ⊆ ∆ σ and λ > , for each act f ∈ F it holds V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q D φ ( p || q ) (cid:27) = λ inf q ∈ Q sup η ∈ R (cid:26) η − Z φ ∗ (cid:18) η − u ( f ) λ (cid:19) dq (cid:27) for all u : X → R . The r.h.s. formula computes criterion (1) for φ -divergences by using only integrals withrespect to structured models. This formula substantially simplifies computations and thusconfirms the analytical tractability of the previous specifications. When λ = ∞ , we have min p ∈ ∆ (cid:8)R u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:9) = min q ∈ Q R u ( f ) dq . See Appendix A.1.3 for the simple proof of (18). Here φ ∗ denotes the Fenchel conjugate of φ . As usual, φ is extended to R by setting φ ( t ) = + ∞ if t < φ ∗ is increasing. .1 Interpretation of the decision criterion In the Introduction we outlined a “protective belt” interpretation of decision criterion (15),i.e., V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q c ( p, q ) (cid:27) To elaborate, we begin by observing that the misspecification index (16) has the followingbounds 0 ≤ min q ∈ Q c ( p, q ) ≤ δ Q ( p ) ∀ p ∈ ∆ (19)where δ Q is the indicator function of the set Q of structured models. So, fear of misspeci-fication is absent when the misspecification index is δ Q – e.g., when λ = + ∞ in (17) – inwhich case criterion (15) takes a Wald (1950) max-min form V ( f ) = min q ∈ Q Z u ( f ) dq (20)This max-min criterion characterizes a decision maker who confronts model misspecificationbut is not concerned by it. In other words, this Waldean decision maker is a natural candidateto be (model) misspecification neutral. The next limit result further corroborates this insightby showing that, when the fear of misspecification vanishes, the decision maker becomesWaldean. Proposition 2
For each act f ∈ F , we have lim λ ↑∞ min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:27) = min q ∈ Q Z u ( f ) dq These observations, via bounds and limits, call for a proper decision-theoretic analysis ofmisspecification neutrality. To this end, note that structured models may be incorrect, yetuseful as Box (1976) famously remarked. This motivates the next notion. Recall that act xAy , with x ≻ y , represents a bet on event A . Definition 3
A preference % is bet-consistent if, given any x ≻ y , q ( A ) ≥ q ( B ) ∀ q ∈ Q = ⇒ xAy % xBy for all events A, B ∈ Σ . Under bet-consistency, a decision maker may fear model misspecification yet regardsstructured models as good enough to choose to bet on events that they unanimously rankas more likely. Preferences that are bet-consistent can be classified as exhibiting a relativelymild form of fear of model misspecification. The following result shows that an importantclass of preferences, which includes the ones represented by criterion (17), are bet-consistent. To ease matters, we state the result in terms of criterion (17). A general version can be easily establishedvia an increasing sequence of misspecification indexes, with c nQ ≤ c n +1 Q for each n and lim c nQ ( p ) = ∞ for each p Q . For example, c nQ ( p ) = λ n min q ∈ Q D φ ( p || q ) where λ n ↑ ∞ . roposition 3 If λ ∈ (0 , ∞ ) and c = λD φ , then a preference % represented by (15) isbet-consistent. Next we substantially strengthen bet-consistency by considering all acts, not just bets.
Definition 4
A rational preference % on F is ( model ) misspecification neutral if Z u ( f ) dq ≥ Z u ( g ) dq ∀ q ∈ Q = ⇒ f % g for all acts f, g ∈ F . In this case, a decision maker trusts models enough so to follow them when, if correct,they would unanimously rank pairs of acts. Fear of misspecification thus plays no role inthe decision maker preference, so it is decision-theoretically irrelevant. For this reason, thedecision maker attitude toward model misspecification can be classified as neutral. The nextresult shows that this may happen if and only if the decision maker adopts the max-mincriterion (20).
Proposition 4
A preference % represented by criterion (15) is misspecification neutral ifand only if it is represented by the max-min criterion (20). This result provides the sought-after decision-theoretic argument for the interpretation ofthe max-min criterion as the special case of decision criterion (15) that corresponds to aver-sion to model ambiguity, with no fear of misspecification. As remarked in the Introduction, itsuggests that a decision maker using such a criterion may be viewed as a decision maker who,under model ambiguity, would max-minimize over the set of structured models which sheposited but that, for fear of misspecification, ends up using the more prudential variationalcriterion (15). Unstructured models lack the informational status of structured models, yetin the criterion (15) they act as a “protective belt” against model misspecification.Note that under this interpretation of criterion (15), the special multiplier case of a sin-gleton Q = { q } corresponds to a decision maker who, with no fear of misspecification, wouldadopt the expected utility criterion R u ( f ) dq . In other words, a singleton Q corresponds toan expected utility decision maker who fears misspecification.Summing up, in our analysis decision makers adopt the max-min criterion (20) if theyeither confront only model ambiguity (an information trait) or are averse to model ambiguity(a taste trait) with no fear of model misspecification. As just argued, the singleton Q = { q } special casemin p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) (21)16f decision criterion (15) is an expected utility criterion under fear of misspecification (of theunique posited q ). Via the relation f % ∗ g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q (22)the representation theorem thus clarifies the interpretation of % ∗ as a dominance relationunder model misspecification by showing that it amounts to uniform dominance across allstructured models with respect to criterion (21).It is easy to see that strict dominance amounts to (22), with strict inequality for some q ∈ Q . This observation raises a question: is there a notion of dominance that correspondsto strict inequality for all q ∈ Q ? To address this question, we introduce a strong dominance relation by writing f ≻≻ ∗ g if, for all acts h, l ∈ F ,(1 − δ ) f + δh ≻ ∗ (1 − δ ) g + δl for all small enough δ ∈ [0 , By taking h = f and l = g , we have the basic implication f ≻≻ ∗ g = ⇒ f ≻ ∗ g Strong dominance is a strengthening of strict dominance in which the decision maker canconvince others “beyond reasonable doubt.” The next characterization corroborates thisinterpretation and, at the same time, answers the previous question in the positive. Proposition 5
Let c : ∆ × Q → [0 , ∞ ] be a divergence, u : X → R an onto and affinefunction and % ∗ an unbounded dominance relation represented by (22). For all acts f, g ∈ F ,we have f ≻≻ ∗ g if and only if there exists ε > such that min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) + ε ∀ q ∈ Q This characterization shows that ≻ ∗ and ≻≻ ∗ agree on consequences and, more impor-tantly, that f ≻≻ ∗ g = ⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) > min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q In turn, this easily implies f ≻≻ ∗ g = ⇒ f ≻ g (23)We can diagram the relationships among the different dominance notions as follows: ≻≻ ∗ = ⇒ ≻ ∗ = ⇒ ≻⇓ ⇓≻ = ⇒ % Strong dominance has been introduced by Cerreia-Vioglio et al. (2020). Up to an ε that ensures a needed uniformity of the strict inequality across structured models.
17n instance when f ≻ ∗ g = ⇒ f ≻ g (24)may fail is the max-min criterion (20).We close by discussing misspecification neutrality, which in view of Proposition 4 ischaracterized by the misspecification index min q ∈ Q c ( p, q ) = δ Q ( p ). Lemma 3
Let c be a statistical distance c : ∆ × Q → [0 , ∞ ] . We have min q ∈ Q c ( p, q ) = δ Q ( p ) if and only if, for each q ∈ Q , c ( p, q ) = ∞ for all p / ∈ Q . In words, misspecification neutrality is characterized by a statistical distance that max-imally penalizes unstructured models, which end up playing no role. From a statisticaldistance angle, this confirms that misspecification neutrality is the attitude of a decisionmaker who confronts model misspecification, but does not care about it (and so has no usefor unstructured models).This angle becomes relevant here because it shows that, under misspecification neutrality,the representation (22) of the dominance relation becomes f % ∗ g ⇐⇒ min q ′ ∈ Q (cid:26)Z u ( f ) dq ′ + c (cid:0) q ′ , q (cid:1)(cid:27) ≥ min q ′ ∈ Q (cid:26)Z u ( g ) dq ′ + c (cid:0) q ′ , q (cid:1)(cid:27) ∀ q ∈ Q Unstructured models play no role here. Next we show that also statistical distances play norole, so representation (22) further reduces to f % ∗ g ⇐⇒ Z u ( f ) dq ≥ Z u ( g ) dq ∀ q ∈ Q (25)when the dominance relation satisfies the independence axiom. This means, inter alia , thatfear of model misspecification may cause violations of the independence axiom for such arelation, thus providing a new rationale for violations of this classic axiom.All this is shown by the next result, which is the version for our setting of the main resultof Gilboa et al. (2010). Proposition 6
Let ( S, Σ , X, Q, % ∗ , % ) be a two-preference classical decision environment.The following statements are equivalent:(i) % ∗ is an unbounded dominance relation that satisfies independence and % is a rationalpreference that are both Q -coherent and jointly satisfy consistency and caution;(ii) there exist an onto affine function u : X → R and a statistical distance c : ∆ × Q → [0 , ∞ ] , with c ( p, q ) = δ { q } ( p ) for all ( p, q ) ∈ ∆ × Q , such that (13) and (14) hold, i.e., f % ∗ g ⇐⇒ Z u ( f ) dq ≥ Z u ( g ) dq ∀ q ∈ Q and f % g ⇐⇒ min q ∈ Q Z u ( f ) dq ≥ min q ∈ Q Z u ( g ) dq % ∗ thus takes a misspecification neutralform, while the preference % is represented by the max-min criterion. So far, we carried out our analysis for a given set Q of structured models. Indeed, a two-preference classical decision environment (12) should be more properly written as (cid:0) S, Σ , X, Q, % ∗ Q , % Q (cid:1) with the dependence of preferences on Q highlighted. Decision environments, however, mayshare common state and consequence spaces, but differ on the posited sets of structuredmodels because of different information that decision makers may have. It then becomesimportant to ensure that decision makers use decision criteria that, across such environments,are consistent.To address this issue, in this section we consider a family (cid:8)(cid:0) S, Σ , X, Q, % ∗ Q , % Q (cid:1)(cid:9) Q ∈Q of classical decision environments that differ in the set Q of posited models and we introduceaxioms on the family n % ∗ Q o Q ∈Q that connect these environments. In keeping with whatassumed so far, Q is the collection of compact and convex subsets of ∆ σ .A.5 Monotonicity ( in model ambiguity ). If Q ′ ⊆ Q then, for all f, g ∈ F , f % ∗ Q g = ⇒ f % ∗ Q ′ g According to this axiom, if the “structured” information underlying a set Q is good enoughfor the decision maker to establish that an act dominates another one, a better informationwhich decreases model ambiguity can only confirm such judgement. Its reversal would be,indeed, at odds with the objective rationality spirit of the dominance relation.Next we consider a separability assumption.A.6 Q -separability . For all f, g ∈ F , f % ∗ q g ∀ q ∈ Q = ⇒ f % ∗ Q g In words, an act dominates another one when it does, separately, through the lenses ofeach structured model. In this axiom the incompleteness of % ∗ Q arises as that of a Paretianorder over the, complete but possibly misspecification averse, preferences % ∗ q determined bythe elements of Q . 19hese two assumptions, paired with the ones of Theorem 1, guarantee that all dominancerelations % ∗ Q agree on X . We can thus just write % ∗ , dropping the subscript Q . To statethe next axiom, we need a last piece of notation: we denote by x f,q the certainty equivalentof act f for preference % ∗ q .A.7 Model hybridization aversion . Given any q, q ′ ∈ ∆ σ , λx f,q + (1 − λ ) x f,q ′ % ∗ x f,λq +(1 − λ ) q ′ for all λ ∈ (0 ,
1) and all f ∈ F .According to this axiom, the decision maker dislikes, ceteris paribus , facing a hybridstructured model λq + (1 − λ ) q ′ that, by mixing two structured models q and q ′ , could onlyhave a less substantive motivation (cf. Section 3.1).We close with a continuity axiom.A.8 Lower semicontinuity . For all x ∈ X and all f ∈ F , the set { q ∈ ∆ σ : x % ∗ x f,q } isclosed.We can now state the extension of Theorem 1 to families of decision environments. Theorem 2
Let (cid:8)(cid:0) S, Σ , X, Q, % ∗ Q , % Q (cid:1)(cid:9) Q ∈Q be a family of two-preference classical decision environments. The following statements areequivalent:(i) n % ∗ Q o Q ∈Q is monotone, Q -separable, lower semicontinuous, averse to model hybridiza-tion and, for each Q ∈ Q , the preferences % ∗ Q and % Q satisfy the hypotheses of Theorem1;(ii) there exist an onto affine function u : X → R and a jointly lower semicontinuous andconvex statistical distance c : ∆ × ∆ σ → [0 , ∞ ] , with dom c Q ⊆ ∆ ( Q ) for all Q ∈ Q ,such that, for all acts f, g ∈ F , f % ∗ Q g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q (26) and f % Q g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + min q ∈ Q c ( p, q ) (cid:27) (27) Moreover, u is unique up to a positive affine transformation and, given u , c is unique. Q is permitted to vary across the collection Q of compact andconvex subsets of ∆ σ . A two-preference classical decision problem is a septet (cid:0) F, S, Σ , X, Q, % ∗ Q , % Q (cid:1) (28)where F ⊆ F is a non-empty choice set formed by the acts among which a decision makerhas actually to choose, % ∗ Q and % Q are preferences represented by (26) and (27).Given a compact and convex set Q in ∆ σ , the decision maker chooses the best act in F according to % Q . In particular, the value function v : Q → ( −∞ , ∞ ] is given by v ( Q ) = sup f ∈ F min p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q c ( p, q ) (cid:27) (29)Yet, it is the dominance relation % ∗ Q that permits to introduce admissibility. Definition 5
An act f ∈ F is ( weakly ) admissible if there is no act g ∈ F that (strongly)strictly dominates f . To relate this notion to the usual notion of admissibility, observe that g ≻ ∗ Q f amountsto min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q with strict inequality for some q ∈ Q . We are thus purposefully defining admissibility interms of the structured models Q , not the larger class of models ∆, with a model-by-modeladjustment for misspecification that makes our notion different from the usual one.The next result relates optimality and admissibility. Proposition 7
Consider a decision problem (28).(i) Optimal acts are weakly admissible. They are admissible provided (24) holds. See, e.g., Ferguson (1967) p. 54. ii) Unique optimal acts are admissible. Optimal acts (if exist) might not be admissible because the max-min nature of decisioncriterion (15) may lead to violations of (24). Yet, the last result ensures that they belong tothe collection of weakly admissible acts F ∗ Q = (cid:8) f ∈ F : ∄ g ∈ F, g ≻≻ ∗ Q f (cid:9) Next we build on this property to establish a comparative statics exercise across decisionproblems (28) that differ on the posited set Q of structured models. Proposition 8
We have Q ⊆ Q ′ = ⇒ v ( Q ) ≥ v (cid:0) Q ′ (cid:1) and v ( Q ) = max f ∈ F ∗ Q min p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q c ( p, q ) (cid:27) provided the sup in (29) is achieved. Smaller sets of structured models are, thus, more valuable. Indeed, in decision problemsthat feature a larger set of structured models – so, a more discordant information – thedecision maker exhibits, ceteris paribus , a higher fear of misspecification: Q ⊆ Q ′ = ⇒ min q ∈ Q c ( p, q ) ≥ min q ∈ Q ′ c ( p, q )In turn, this easily implies v ( Q ) ≥ v ( Q ′ ).The decision maker thus dislikes information discordance. In a finite state space, infor-mation discordance is maximal, so information is inconclusive, when Q = ∆. Indeed, by thedistance property (8) we have min q ∈ Q c ( p, q ) = 0 for all p ∈ ∆ if and only if Q = ∆. So, thesimplex case represents maximal misspecification fear, given any c (so, any attitude towardmodel misspecification). Criterion (15) then takes an extreme statewise max-minimizationform V ( f ) = min s ∈ S u ( f ( s ))which embodies a form of the precautionary principle that, here, thus emerges out of inconclu-sive information (e.g., based on inconclusive scientific knowledge). In contrast, informationdiscordance is absent when Q is a singleton. Infinite state spaces require some technicalities. A divergence twist
In our analysis a notion of set divergence naturally arises. Specifically, denoting by Q thecollection of all compact and convex subsets of ∆ σ , say that a function C : ∆ × Q → [0 , ∞ ]is a set divergence if(i) C ( · , Q ) : ∆ → [0 , ∞ ] is grounded, lower semicontinuous and convex for each Q ∈ Q ;(ii) C ( p, Q ) = 0 if and only if p ∈ Q .If we consider a jointly lower semicontinuous and convex function c : ∆ × ∆ σ → [0 , ∞ ] suchthat c ( p, q ) = 0 if and only if p = q , by Lemma 1, we can define a set divergence by setting C ( p, Q ) = min q ∈ Q c ( p, q ). In particular, C ( p, { q } ) = c ( p, q ). This is the Hausdorff-type setdivergence that characterizes our decision criterion (1). Yet, for a generic set divergence C ,not necessarily pinned down by an underlying statistical distance c , our criterion generalizesto V Q ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + C ( p, Q ) (cid:27) Since C ( p, Q ) ≤ δ Q ( p ) for all p ∈ ∆, this variational criterion still represents a preferencethat is more uncertainty averse than the max-min one in (4). Though the analysis of thisgeneral criterion is beyond the scope of this paper, this brief discussion should help to putour exercise in a better perspective. Quantitative researchers use models to enhance their understanding of economic phenomenaand to make policy assessments. In essence, each model tells its own quantitative story. Werefer to such models as “structured models.” Typically, there are more than just one suchtype of model, with each giving rise to a different quantitative story. Statistical and eco-nomic decision theories have addressed how best to confront the ambiguity among structuredmodels. Such structured models are, by their very nature, misspecified. Nevertheless, thedecision maker seeks to use such models in sensible ways. This problem is well recognized byapplied researchers, but it is typically not part of formal decision theory. In this paper, weextend decision theory to confront model misspecification concerns. In so doing, we recovera variational representation of preferences that includes penalization based on discrepancymeasures between “unstructured alternatives” and the set of structured probability models.
A Proofs and related analysis
In the appendix, we provide the proofs of our main results plus some ancillary results.Appendix A.1 contains all the results pertaining statistical φ -divergences and distances.23ppendix A.3 contains the proofs of our representation results (Theorems 1 and 2). AppendixA.4 contains the proofs of the remaining results. A.1 Preamble
A.1.1 Proof of Lemma 1
We substantially need to prove that the function c Q : ∆ → [0 , ∞ ], defined by c Q ( p ) =min q ∈ Q c ( p, q ), is well defined, grounded, lower semicontinuous and convex. This fact followsfrom the following version of a well known result (see, e.g., Fiacco and Kyparisis, 1986). Lemma 4
Let Q be a compact and convex subset of ∆ . If c : ∆ × Q → [0 , ∞ ] is a jointlylower semicontinuous and convex function such that there exist ¯ p ∈ ∆ and ¯ q ∈ Q such that c ( ¯ p, ¯ q ) = 0 , then c Q : ∆ → [0 , ∞ ] defined by c Q ( p ) = min q ∈ Q c ( p, q ) ∀ p ∈ ∆ is well defined, grounded, lower semicontinuous and convex. Proof
Since c is lower semicontinuous and Q is non-empty and compact, c Q is well defined.Moreover, we have that 0 ≥ c ( ¯ p, ¯ q ) ≥ c Q (¯ p ) ≥
0, proving that c Q is grounded. We nextshow that c Q is lower semicontinuous. Consider ˜ U = { p ∈ ∆ : c Q ( p ) > α } where α ∈ R . If˜ U is empty, then it is open. Otherwise, consider ¯ p ∈ ˜ U . It follows that(¯ p, q ) ∈ (cid:8)(cid:0) p ′ , q ′ (cid:1) ∈ ∆ × Q : c (cid:0) p ′ , q ′ (cid:1) > α (cid:9) = ¯ U ∀ q ∈ Q Since c is jointly lower semicontinuous, then ¯ U is open in the product topology. Thus, foreach q ∈ Q there exist two neighborhoods U q and V q such that(¯ p, q ) ∈ U q × V q ⊆ ¯ U Since q ∈ V q for all q ∈ Q , we have that { V q } q ∈ Q is an open cover of Q . Since Q is compact,it admits a finite subcover { V q i } ni =1 . Define the open set U = ∩ ni =1 U q i . Since ¯ p ∈ U q forall q ∈ Q , note that ¯ p ∈ U . Consider p ∈ U and q ′ ∈ Q . It follows that q ′ ∈ V q i for some i ∈ { , ..., n } . This implies that ( p, q ′ ) ∈ U q i × V q i ⊆ ¯ U . We can conclude that c ( p, q ′ ) > α .Since p and q ′ were arbitrarily chosen in U and Q , we have that c Q ( p ) = min q ′ ∈ Q c ( p, q ′ ) > α for all p ∈ U , proving that ¯ p ∈ U ⊆ ˜ U and so lower semicontinuity of c Q .If p , p ∈ ∆, then define q , q ∈ Q to be such that c ( p , q ) = min q ∈ Q c ( p , q ) = c Q ( p ) and c ( p , q ) = min q ∈ Q c ( p , q ) = c Q ( p )24onsider λ ∈ (0 , p λ = λp + (1 − λ ) p and q λ = λq + (1 − λ ) q ∈ Q . Since c isjointly convex, it follows that c Q ( p λ ) = min q ∈ Q c ( p λ , q ) ≤ c ( p λ , q λ ) ≤ λc ( p , q ) + (1 − λ ) c ( p , q )= λc Q ( p ) + (1 − λ ) c Q ( p )proving convexity. (cid:4) Proof of Lemma 1
We first prove the “If” part. We need to prove that c is a divergence thatsatisfies (8). In particular, we need to show that c Q and c q are well defined, grounded, lowersemicontinuous and convex for all q ∈ Q . As for c q , since c is jointly lower semicontinuousand convex, so is c q and we only need to prove that c q is grounded. Since c ≥ c q ( q ) = c ( q, q ) = 0, proving that c q ≥ Q is compact and convex and c is jointly lower semicontinuous and convex andsuch that c ( q, q ) = 0 for all q ∈ Q , then c Q : ∆ → [0 , ∞ ] is well defined, grounded, lowersemicontinuous and convex. Finally, since c satisfies (8), note that c Q ( p ) = 0 if and only if c ( p, q ) = 0 for some q ∈ Q if and only if p = q for some q ∈ Q if and only if p ∈ Q .As for the “Only if” part, it is trivial since a statistical distance function, by definition,satisfies (8). (cid:4) A.1.2 Proof of Lemma 2
We actually prove a more complete result. A piece of notation: we write p ∼ Q if thereexists a control measure q ∈ Q such that p ∼ q . Lemma 5
Let Q be a compact and convex subset of ∆ σ . A restricted φ -divergence D φ :∆ × Q → [0 , ∞ ] is a statistical distance. Moreover,(i) if q ∈ Q , then D φ ( ·|| q ) : ∆ → [0 , ∞ ] is strictly convex;(ii) if p ∈ ∆ σ and p ∼ Q , then D φ ( p ||· ) : Q → [0 , ∞ ] is strictly convex. Proof
It is well known that on ∆ × ∆ σ the function D φ is jointly lower semicontinuous andconvex and satisfies the property D φ ( p || q ) = 0 ⇐⇒ p = q Though a routine result, for the sake of completeness, we provide a proof since we did not find oneallowing S to be infinite (see Topsoe, 2001, p. 178 for the finite case). A probability q ∈ Q is a control measure of Q if q ′ ≪ q for all q ′ ∈ Q . When Q is a compact and convexsubset of ∆ σ , Q has a control measure (see, e.g., Maccheroni and Marinacci, 2001). Such a measure mightnot be unique, yet any two control measures of Q are equivalent. So, the notion p ∼ Q is well defined andindependent of the chosen control measure.
25y Lemma 1, it follows that D φ : ∆ × Q → [0 , ∞ ] is a statistical distance. We next provepoints (i) and (ii).(i). Consider q ∈ Q . Let p ′ , p ′′ ∈ ∆ and α ∈ (0 ,
1) be such that p ′ = p ′′ and D φ ( αp ′ + (1 − α ) p ′′ || q ) < ∞ . If either D φ ( p ′ || q ) or D φ ( p ′′ || q ) are not finite, we trivially conclude that D φ ( αp ′ + (1 − α ) p ′′ || q ) < ∞ = αD φ ( p ′ || q ) + (1 − α ) D φ ( p ′′ || q ). Let us then assume that both D φ ( p ′ || q ) and D φ ( p ′′ || q )are finite. By definition of D φ and since ∆ σ ( q ) is convex, this implies that p ′ , p ′′ ∈ ∆ σ ( q )as well as αp ′ + (1 − α ) p ′′ ∈ ∆ σ ( q ). Since p ′ and p ′′ are distinct, we have that dp ′ /dq and dp ′′ /dq take different values on a set of strictly positive q -measure: call it ˜ S . Since φ isstrictly convex, it follows that φ (cid:18) α dp ′ dq ( s ) + (1 − α ) dp ′′ dq ( s ) (cid:19) < αφ (cid:18) dp ′ dq ( s ) (cid:19) + (1 − α ) φ (cid:18) dp ′′ dq ( s ) (cid:19) ∀ s ∈ ˜ S By definition of D φ , this implies that D φ (cid:0) αp ′ + (1 − α ) p ′′ || q (cid:1) = Z S φ (cid:18) d [ αp ′ + (1 − α ) p ′′ ] dq ( s ) (cid:19) dq = Z S φ (cid:18) α dp ′ dq ( s ) + (1 − α ) dp ′′ dq ( s ) (cid:19) dq = Z ˜ S φ (cid:18) α dp ′ dq ( s ) + (1 − α ) dp ′′ dq ( s ) (cid:19) dq + Z S \ ˜ S φ (cid:18) α dp ′ dq ( s ) + (1 − α ) dp ′′ dq ( s ) (cid:19) dq< α Z S φ (cid:18) dp ′ dq ( s ) (cid:19) dq + (1 − α ) Z S φ (cid:18) dp ′′ dq ( s ) (cid:19) dq = αD φ (cid:0) p ′ || q (cid:1) + (1 − α ) D φ (cid:0) p ′′ || q (cid:1) We conclude that D φ ( ·|| q ) : ∆ → [0 , ∞ ] is strictly convex.(ii). Before starting, we make three observations.a. Since Q is a non-empty, compact and convex subset of ∆ σ , note that there exists¯ q ∈ Q such that q ≪ ¯ q for all q ∈ Q . Since p ∼ Q , we have that p ∼ ¯ q . This implies alsothat q ≪ p for all q ∈ Q .b. If q ∼ p , then ( dp/dq ) − is well defined almost everywhere (with respect to either p or q ) and can be chosen (after defining arbitrarily the function over a set of zero measure)to be the Radon-Nikodym derivative dq/dp .c. Since φ is strictly convex, if we define φ ⋆ : (0 , ∞ ) → [0 , ∞ ) by φ ⋆ ( x ) = xφ (1 /x ) for all x >
0, then also φ ⋆ is strictly convex. By point b, if p ∈ ∆ σ and q ∈ Q are such that p ∼ q p = dp/dq , then q ( { ˙ p = 0 } ) = 0 = p ( { ˙ p = 0 } ) and D φ ( p || q ) = Z S φ (cid:18) dpdq (cid:19) dq = Z { ˙ p =0 } φ (cid:18) dpdq (cid:19) dq + Z { ˙ p> } φ (cid:18) dpdq (cid:19) dq = Z { ˙ p> } φ (cid:16) dpdq (cid:17) − dq = Z { ˙ p> } φ ⋆ (cid:18) dqdp (cid:19) dpdq dq = Z { ˙ p> } φ ⋆ (cid:18) dqdp (cid:19) dp We can now prove the statement. Let q ′ , q ′′ ∈ Q and α ∈ (0 ,
1) be such that q ′ = q ′′ and D φ ( p || αq ′ + (1 − α ) q ′′ ) < ∞ . If either D φ ( p || q ′ ) or D φ ( p || q ′′ ) are not finite, we triviallyconclude that D φ ( p || αq ′ + (1 − α ) q ′′ ) < ∞ = αD φ ( p || q ′ ) + (1 − α ) D φ ( p || q ′′ ). Let us thenassume that both D φ ( p || q ′ ) and D φ ( p || q ′′ ) are finite. By definition of D φ , we can concludethat p ≪ q ′ and p ≪ q ′′ . By point a, this yields that q ′ ∼ p ∼ q ′′ and p ∼ αq ′ + (1 − α ) q ′′ .Since q ′ and q ′′ are distinct, we have that dq ′ /dp and dq ′′ /dp take different values on a set ofstrictly positive p -measure: call it ˜ S . By point c, we have that p (cid:18)(cid:26) dpd [ αq ′ + (1 − α ) q ′′ ] = 0 (cid:27)(cid:19) = p (cid:18)(cid:26) dpdq ′ = 0 (cid:27)(cid:19) = p (cid:18)(cid:26) dpdq ′′ = 0 (cid:27)(cid:19) = 0Thus, by point c and since dq ′ /dp and dq ′′ /dp take different values on a set of strictly positive p -measure, there exists a p -measure 1 set ˜ S such that D φ (cid:0) p || αq ′ + (1 − α ) q ′′ (cid:1) = Z ˜ S φ ⋆ (cid:18) d [ αq ′ + (1 − α ) q ′′ ] dp (cid:19) dp< α Z ˜ S φ ⋆ (cid:18) dq ′ dp (cid:19) dp + (1 − α ) Z ˜ S φ ⋆ (cid:18) dq ′′ dp (cid:19) dp = αD φ (cid:0) p || q ′ (cid:1) + (1 − α ) D φ (cid:0) p || q ′′ (cid:1) proving point (ii). A.1.3 Non-convex set of structured models
Let us consider two decision makers who adopt criterion (17), the first one posits a, possiblynon-convex, set of structured models Q and the second one posits its closed convex hull co Q .So, the second decision maker considers also all the mixtures of structured models positedby the first decision maker. Next we show that their preferences over acts actually agree.It is thus without loss of generality to assume that the set of posited structured models isconvex, as it was assumed in the main text. Before doing so we prove formula (18). Observethat given a compact subset Q ⊆ ∆ σ , be that convex or not, we have27in p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:27) = min p ∈ ∆ min q ∈ Q (cid:26)Z u ( f ) dp + λR ( p || q ) (cid:27) = min q ∈ Q min p ∈ ∆ (cid:26)Z u ( f ) dp + λR ( p || q ) (cid:27) = min q ∈ Q φ − λ (cid:18)Z φ λ ( u ( f )) dq (cid:19) where φ λ ( t ) = − e − λ t for all t ∈ R where λ > Proposition 9 If Q ⊆ ∆ σ is compact, then for each f ∈ F min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:27) = min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ co Q R ( p || q ) (cid:27) Proof
First observe that co Q ⊆ ∆ σ . Indeed, since Q is a compact subset of ∆ σ , theset function ν : Σ → [0 , ν ( E ) = min q ∈ Q q ( E ) for all E ∈ Σ is an exactcapacity which is continuous at S . This implies that Q ⊆ core ν ⊆ ∆ σ , yielding thatco Q ⊆ core ν ⊆ ∆ σ . Given what we have shown before we can conclude thatmin p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:27) = min q ∈ Q φ − λ (cid:18)Z φ λ ( u ( f )) dq (cid:19) = φ − λ (cid:18) min q ∈ Q (cid:18)Z φ λ ( u ( f )) dq (cid:19)(cid:19) = φ − λ (cid:18) min q ∈ co Q (cid:18)Z φ λ ( u ( f )) dq (cid:19)(cid:19) = min q ∈ co Q φ − λ (cid:18)Z φ λ ( u ( f )) dq (cid:19) = min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ co Q R ( p || q ) (cid:27) proving the statement. (cid:4) A.2 Proof of Proposition 1
The result follows from the following lemma. Here, as usual, φ is extended to R by setting φ ( t ) = + ∞ if t / ∈ [0 , + ∞ ). In particular, φ ∗ is non-decreasing. Lemma 6
For each Q ⊆ ∆ σ and each λ > , inf p ∈ ∆ (cid:26)Z u ( f ) dp + λ inf q ∈ Q D φ ( p || q ) (cid:27) = λ inf q ∈ Q sup η ∈ R (cid:26) η − Z φ ∗ (cid:18) η − u ( f ) λ (cid:19) dq (cid:27) for all u : X → R and all f : S → X such that u ◦ f is bounded and measurable. roof By Theorem 4.2 of Ben-Tal and Teboulle (2007), for each q ∈ ∆ σ it holdsinf p ∈ ∆ (cid:26)Z ξdp + D φ ( p || q ) (cid:27) = sup η ∈ R (cid:26) η − Z φ ∗ ( η − ξ ) dq (cid:27) for all ξ ∈ L ∞ ( q ). Then, if u ◦ f is bounded and measurable, from u ◦ f ∈ L ∞ ( q ) for all q ∈ ∆ σ , it follows that, for all λ > p ∈ ∆ (cid:26)Z u ( f ) dp + λD φ ( p || q ) (cid:27) = λ inf p ∈ ∆ (cid:26)Z u ( f ) λ dp + D φ ( p || q ) (cid:27) = λ sup η ∈ R (cid:26) η − Z φ ∗ (cid:18) η − u ( f ) λ (cid:19) dq (cid:27) for all λ >
0, as desired. (cid:4)
Proof of Proposition 1
In view of the last lemma, it is enough to observe that, if f : S → X is simple and measurable, then u ◦ f is simple and measurable for all u : X → R . (cid:4) A.3 Representation results
The proof of Theorem 1 is based on four key steps. We first provide a representation for anunbounded and objectively Q -coherent dominance relation % ∗ (Appendix A.3.1). Second,we provide a representation for a pair of binary relations ( % ∗ , % ) which satisfy all of theassumptions of Theorem 1 with the exception of subjective Q -coherence (Appendix A.3.2).Third, we provide two results regarding variational preferences which will help isolate theset of structured models Q in the main representation (Appendix A.3.3). Finally, we mergethese three steps to prove our first representation result (Appendix A.3.4). The proof ofTheorem 2 instead is presented as one result and it relies on some of the aforementionedresults. In what follows, given a function c : ∆ × Q → [0 , ∞ ], where Q is a compact andconvex subset of ∆ σ , we say that c is a weak divergence ( for the set Q ) if it satisfies the firsttwo properties defining a divergence. A.3.1 A Bewley-type representation
The next result is a multi-utility (variational) representation for unbounded dominance re-lations.
Lemma 7
Let % ∗ be a binary relation on F , where ( S, Σ) is a standard Borel space. Thefollowing statements are equivalent:(i) % ∗ is an unbounded dominance relation which satisfies objective Q -coherence; ii) there exist an onto affine function u : X → R and a weak divergence c : ∆ × Q → [0 , ∞ ] such that dom c ( · , q ) ⊆ ∆ ( Q ) for all q ∈ Q and f % ∗ g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q (30)To prove this result, we need to introduce one mathematical object. Let (cid:23) ∗ be a binaryrelation on B (Σ). We say that (cid:23) ∗ is convex niveloidal if and only if (cid:23) ∗ is a preorder thatsatisfies the following five properties:1. For each ϕ, ψ ∈ B (Σ) and for each k ∈ R ϕ (cid:23) ∗ ψ = ⇒ ϕ + k (cid:23) ∗ ψ + k
2. If ϕ, ψ ∈ B (Σ) and { k n } n ∈ N ⊆ R are such that k n ↑ k and ϕ − k n (cid:23) ∗ ψ for all n ∈ N ,then ϕ − k (cid:23) ∗ ψ ;3. For each ϕ, ψ ∈ B (Σ) ϕ ≥ ψ = ⇒ ϕ (cid:23) ∗ ψ
4. For each k, h ∈ R and for each ϕ ∈ B (Σ) k > h = ⇒ ϕ + k ≻ ∗ ϕ + h
5. For each ϕ, ψ, ξ ∈ B (Σ) and for each λ ∈ (0 , ϕ (cid:23) ∗ ξ and ψ (cid:23) ∗ ξ = ⇒ λϕ + (1 − λ ) ψ (cid:23) ∗ ξ Lemma 8 If % ∗ is an unbounded dominance relation, then there exists an onto affine func-tion u : X → R such that x % ∗ y ⇐⇒ u ( x ) ≥ u ( y ) (31) Proof
Since % ∗ is a non-trivial preorder on F that satisfies c-completeness, continuity andweak c-independence, it is immediate to conclude that % ∗ restricted to X satisfies weakorder, continuity and risk independence. By Herstein and Milnor (1953), it follows thatthere exists an affine function u : X → R that satisfies (31). Since % ∗ is a non-trivial preorder To prove that % ∗ satisfies risk independence, it suffices to deploy the same technique of Lemma 28 ofMaccheroni et al. (2006) and observe that % ∗ is a complete preorder on X . This yields that x ∼ ∗ y = ⇒ x + 12 z ∼ ∗ y + 12 z ∀ z ∈ X By Theorem 2 of Herstein and Milnor (1953) and since % ∗ satisfies continuity, we can conclude that % ∗ satisfies risk independence. F that satisfies monotonicity, we have that % ∗ is non-trivial on X . By Lemma 59 ofCerreia-Vioglio et al. (2011) and since % ∗ is non-trivial on X and satisfies unboundedness,we can conclude that u is onto. (cid:4) Since u is affine and onto, note that { u ( f ) : f ∈ F } = B (Σ). In light of this observation,we can define a binary relation (cid:23) ∗ on B (Σ) by ϕ (cid:23) ∗ ψ ⇐⇒ f % ∗ g where u ( f ) = ϕ and u ( g ) = ψ (32) Lemma 9 If % ∗ is an unbounded dominance relation, then (cid:23) ∗ , defined as in (32), is a welldefined convex niveloidal binary relation. Moreover, if % ∗ is objectively Q -coherent, then ϕ Q = ψ implies ϕ ∼ ∗ ψ . Proof
We begin by showing that (cid:23) ∗ is well defined. Assume that f , f , g , g ∈ F aresuch that u ( f i ) = ϕ and u ( g i ) = ψ for all i ∈ { , } . It follows that u ( f ( s )) = u ( f ( s ))and u ( g ( s )) = u ( g ( s )) for all s ∈ S . By Lemma 8, this implies that f ( s ) ∼ ∗ f ( s ) and g ( s ) ∼ ∗ g ( s ) for all s ∈ S . Since % ∗ is a preorder that satisfies monotonicity, this impliesthat f ∼ ∗ f and g ∼ ∗ g . Since % ∗ is a preorder, if f % ∗ g , then f % ∗ f % ∗ g % ∗ g = ⇒ f % ∗ g that is, f % ∗ g implies f % ∗ g . Similarly, we can prove that f % ∗ g implies f % ∗ g . Inother words, f % ∗ g if and only if f % ∗ g , proving that (cid:23) ∗ is well defined. It is immediateto prove that (cid:23) ∗ is a preorder. We next prove properties 1–5.1. Consider ϕ, ψ ∈ B (Σ) and k ∈ R . Assume that ϕ (cid:23) ∗ ψ . Let f, g ∈ F and x, y ∈ X besuch that u ( f ) = 2 ϕ , u ( g ) = 2 ψ , u ( x ) = 0 and u ( y ) = 2 k . Since u is affine, it followsthat u (cid:18) f + 12 x (cid:19) = 12 u ( f ) + 12 u ( x ) = ϕ (cid:23) ∗ ψ = 12 u ( g ) + 12 u ( x ) = u (cid:18) g + 12 x (cid:19) proving that f + x % ∗ g + x . Since % ∗ satisfies weak c-independence and u isaffine, we have that f + y % ∗ g + y , yielding that ϕ + k = 12 u ( f ) + 12 u ( y ) = u (cid:18) f + 12 y (cid:19) (cid:23) ∗ u (cid:18) g + 12 y (cid:19) = 12 u ( g ) + 12 u ( y ) = ψ + k
2. Consider ϕ, ψ ∈ B (Σ) and { k n } n ∈ N ⊆ R such that k n ↑ k and ϕ − k n (cid:23) ∗ ψ for all n ∈ N . We have two cases: 31a) k >
0. Consider f, g, h ∈ F such that u ( f ) = ϕ , u ( g ) = ϕ − k and u ( h ) = ψ Since k > k n ↑ k , there exists ¯ n ∈ N such that k n > n ≥ ¯ n . Define λ n = 1 − k n /k for all n ∈ N . It follows that λ n ∈ [0 ,
1] for all n ≥ ¯ n . Since u isaffine, for each n ≥ ¯ nu ( λ n f + (1 − λ n ) g ) = λ n u ( f ) + (1 − λ n ) u ( g ) = ϕ − k n (cid:23) ∗ ψ = u ( h )yielding that λ n f + (1 − λ n ) g % ∗ h for all n ≥ ¯ n . Since % ∗ satisfies continuityand λ n →
0, we have that g % ∗ h , that is, ϕ − k = u ( g ) (cid:23) ∗ u ( h ) = ψ (b) k ≤
0. Since { k n } n ∈ N is convergent, { k n } n ∈ N is bounded. Thus, there exists h > k n + h > n ∈ N . Moreover, k n + h ↑ k + h >
0. By point 1, wealso have that ϕ − ( k n + h ) = ( ϕ − k n ) − h (cid:23) ∗ ψ − h for all n ∈ N . By subpoint a,we can conclude that ( ϕ − k ) − h = ϕ − ( k + h ) (cid:23) ∗ ψ − h . By point 1, we obtainthat ϕ − k (cid:23) ∗ ψ .3. Consider ϕ, ψ ∈ B (Σ) such that ϕ ≥ ψ . Let f, g ∈ F be such that u ( f ) = ϕ and u ( g ) = ψ . It follows that u ( f ( s )) ≥ u ( g ( s )) for all s ∈ S . By Lemma 8, this impliesthat f ( s ) % ∗ g ( s ) for all s ∈ S . Since % ∗ satisfies monotonicity, this implies that f % ∗ g , yielding that ϕ = u ( f ) (cid:23) ∗ u ( g ) = ψ .4. Consider k, h ∈ R and ϕ ∈ B (Σ). We first assume that k > h and k = 0. By point3, we have that ϕ = ϕ + k (cid:23) ∗ ϕ + h . By contradiction, assume that ϕ ∗ ϕ + h . Itfollows that ϕ ∼ ∗ ϕ + h , yielding that I = { w ∈ R : ϕ ∼ ∗ ϕ + w } is a non-empty setwhich contains 0 and h . We next prove that I is an unbounded interval, that is, I = R .First, consider w , w ∈ I . Without loss of generality, assume that w ≥ w . By point3 and since w , w ∈ I , we have that for each λ ∈ (0 , ϕ (cid:23) ∗ ϕ + w (cid:23) ∗ ϕ + ( λw + (1 − λ ) w ) (cid:23) ∗ ϕ + w (cid:23) ∗ ϕ proving that ϕ ∼ ∗ ϕ + ( λw + (1 − λ ) w ), that is, λw + (1 − λ ) w ∈ I . Next, weobserve that I ∩ ( −∞ , = ∅ 6 = I ∩ (0 , ∞ ). Since h ∈ I and h <
0, we have that I ∩ ( −∞ , = ∅ . Since I is an interval and 0 , h ∈ I , we have that h/ ∈ I . Bypoint 1 and since ϕ ∼ ∗ ϕ + h/
2, we have that ϕ − h/ ∼ ∗ ( ϕ + h/ − h/ ϕ ,proving that 0 < − h/ ∈ I ∩ (0 , ∞ ). By definition of I , note that if w ∈ I \ { } ,then ϕ + w ∼ ∗ ϕ . By point 1 and since w/ ∈ I and (cid:23) ∗ is a preorder, we have that( ϕ + w ) + w/ ∼ ∗ ϕ + w/ ∼ ∗ ϕ , that is, w, w ∈ I . Since I is an interval, we have32hat either (cid:0) w, w (cid:1) ⊆ I if w < (cid:0) w, w (cid:1) ⊆ I if w >
0. This will help us inproving that I is unbounded from below and above. By contradiction, assume that I is bounded from below and define m = inf I . Since I ∩ ( −∞ , = ∅ , we have that m <
0. Consider { w n } n ∈ N ⊆ I ∩ ( −∞ ,
0) such that w n ↓ m . Since (cid:0) w n , w n (cid:1) ⊆ I forall n ∈ N , it follows that m ≤ w n for all n ∈ N . By passing to the limit, we obtainthat m ≤ m <
0, a contradiction. By contradiction, assume that I is bounded fromabove and define M = sup I . Since I ∩ (0 , ∞ ) = ∅ , we have that M >
0. Consider { w n } n ∈ N ⊆ I ∩ (0 , ∞ ) such that w n ↑ M . Since (cid:0) w n , w n (cid:1) ⊆ I for all n ∈ N , it followsthat M ≥ w n for all n ∈ N . By passing to the limit, we obtain that M ≥ M > I is a non-empty unbounded interval, that is, I = R .This implies that ϕ ∼ ∗ ϕ + w for all w ∈ R . In particular, select w = k ϕ k ∞ + 1 and w = − k ϕ k ∞ −
1. Since (cid:23) ∗ is a preorder, we have that ϕ + w ∼ ∗ ϕ + w . Moreover, ϕ + w ≥ > − ≥ ϕ + w . By point 3, this implies that ϕ + w (cid:23) ∗ (cid:23) ∗ − (cid:23) ∗ ϕ + w .Since (cid:23) ∗ is a preorder and ϕ + w ∼ ∗ ϕ + w , we can conclude that 1 ∼ ∗ −
1. Note alsothat there exist x, y ∈ X such that u ( x ) = 1 and u ( y ) = −
1. By Lemma 8, this impliesthat x ≻ ∗ y . By definition of (cid:23) ∗ and since u ( x ) = 1 ∼ ∗ − u ( y ), we also have that y % ∗ x , a contradiction. Thus, we proved that if k > h and k = 0, then ϕ + k ≻ ∗ ϕ + h .Assume simply that k > h . This implies that 0 > h − k and ϕ ≻ ∗ ϕ + ( h − k ). Bypoint 1, we can conclude that ϕ + k ≻ ∗ ϕ + ( h − k ) + k = ϕ + h .5. Consider ϕ, ψ, ξ ∈ B (Σ) and λ ∈ (0 , ϕ (cid:23) ∗ ξ and ψ (cid:23) ∗ ξ . Let f, g, h ∈ F be such that u ( f ) = ϕ , u ( g ) = ψ and u ( h ) = ξ . By assumption anddefinition of (cid:23) ∗ , we have that f % ∗ h and g % ∗ h . Since % ∗ satisfies convexity and u is affine, this implies that λf + (1 − λ ) g % ∗ h , yielding that λϕ + (1 − λ ) ψ = λu ( f ) + (1 − λ ) u ( g ) = u ( λf + (1 − λ ) g ) (cid:23) ∗ u ( h ) = ξ .Points 1–5 prove the first part of the statement. Finally, consider ϕ, ψ ∈ B (Σ). Notethat there exist a partition { A i } ni =1 of S and { α i } ni =1 and { β i } ni =1 in R such that ϕ = n X i =1 α i A i and ψ = n X i =1 β i A i Note that { s ∈ S : ϕ ( s ) = ψ ( s ) } = ∪ i ∈{ ,...,n } : α i = β i A i . Since ϕ Q = ψ , we have that q ( A i ) = 0for all q ∈ Q and for all i ∈ { , ..., n } such that α i = β i . Since u is unbounded, define { x i } ni =1 ⊆ X to be such that u ( x i ) = α i for all i ∈ { , ..., n } . Since u is unbounded, define { y i } ni =1 ⊆ X to be such that y i = x i for all i ∈ { , ..., n } such that α i = β i and u ( y i ) = β i otherwise. Define f, g : S → X by f ( s ) = x i and g ( s ) = y i for all s ∈ A i and for all i ∈ { , ..., n } . It is immediate to see that f Q = g as well as u ( f ) = ϕ and u ( g ) = ψ . Since % ∗ is objectively Q -coherent, we have that f ∼ ∗ g , yielding that ϕ ∼ ∗ ψ and proving the secondpart of the statement. (cid:4) (cid:23) ∗ . This paired with Lemma 8 and Proposition 11 will yield the proof of Lemma7. Lemma 10
Let (cid:23) ∗ be a convex niveloidal binary relation. If ψ ∈ B (Σ) , then U ( ψ ) = { ϕ ∈ B (Σ) : ϕ (cid:23) ∗ ψ } is a non-empty convex set such that:1. ψ ∈ U ( ψ ) ;2. if ϕ ∈ B (Σ) and { k n } n ∈ N ⊆ R are such that k n ↑ k and ϕ − k n ∈ U ( ψ ) for all n ∈ N ,then ϕ − k ∈ U ( ψ ) ;3. if k > , then ψ − k U ( ψ ) ;4. if ϕ ≥ ϕ and ϕ ∈ U ( ψ ) , then ϕ ∈ U ( ψ ) ;5. if k ≥ and ϕ ∈ U ( ψ ) , then ϕ + k ∈ U ( ψ ) . Proof
Since (cid:23) ∗ is reflexive, we have that ψ ∈ U ( ψ ), proving that U ( ψ ) is non-empty andpoint 1. Consider ϕ , ϕ ∈ U ( ψ ) and λ ∈ (0 , ϕ (cid:23) ∗ ψ and ϕ (cid:23) ∗ ψ . Since (cid:23) ∗ satisfies convexity, we have that λϕ + (1 − λ ) ϕ (cid:23) ∗ ψ , proving convexityof U ( ψ ). Consider ϕ ∈ B (Σ) and { k n } n ∈ N ⊆ R such that k n ↑ k and ϕ − k n ∈ U ( ψ ) for all n ∈ N . It follows that ϕ − k n (cid:23) ∗ ψ for all n ∈ N , then ϕ − k (cid:23) ∗ ψ , that is, ϕ − k ∈ U ( ψ ),proving point 2. If k >
0, then 0 > − k and ψ = ψ + 0 ≻ ∗ ψ − k , that is, ψ − k U ( ψ ),proving point 3. Consider ϕ ≥ ϕ such that ϕ ∈ U ( ψ ), then ϕ (cid:23) ∗ ϕ and ϕ (cid:23) ∗ ψ ,yielding that ϕ (cid:23) ∗ ψ and, in particular, ϕ ∈ U ( ψ ), proving point 4. Finally, to provepoint 5, it is enough to set ϕ = ϕ + k in point 4. (cid:4) Before stating the next result, we define few properties that will turn out to be usefullater on. A functional I : B (Σ) → R is:1. a niveloid if I ( ϕ ) − I ( ψ ) ≤ sup s ∈ S ( ϕ ( s ) − ψ ( s )) for all ϕ, ψ ∈ B (Σ);2. normalized if I ( k ) = k for all k ∈ R ;
3. monotone if for each ϕ, ψ ∈ B (Σ) ϕ ≥ ψ = ⇒ I ( ϕ ) ≥ I ( ψ )4. (cid:23) ∗ consistent if for each ϕ, ψ ∈ B (Σ) ϕ (cid:23) ∗ ψ = ⇒ I ( ϕ ) ≥ I ( ψ ) With the usual abuse of notation, we denote by k both the real number and the constant function takingvalue k .
34. concave if for each ϕ, ψ ∈ B (Σ) and λ ∈ (0 , I ( λϕ + (1 − λ ) ψ ) ≥ λI ( ϕ ) + (1 − λ ) I ( ψ )6. translation invariant if for each ϕ ∈ B (Σ) and k ∈ R I ( ϕ + k ) = I ( ϕ ) + k Lemma 11
Let (cid:23) ∗ be a convex niveloidal binary relation. If ψ ∈ B (Σ) , then the functional I ψ : B (Σ) → R , defined by I ψ ( ϕ ) = max { k ∈ R : ϕ − k ∈ U ( ψ ) } ∀ ϕ ∈ B (Σ) is a concave niveloid which is (cid:23) ∗ consistent and such that I ψ ( ψ ) = 0 . Moreover, we havethat:1. The functional ¯ I ψ = I ψ − I ψ (0) is a normalized concave niveloid which is (cid:23) ∗ consistent.2. If (cid:23) ∗ satisfies ψ Q = ψ ′ = ⇒ ψ ∼ ∗ ψ ′ then ψ Q = ψ ′ = ⇒ I ψ = I ψ ′ and ¯ I ψ = ¯ I ψ ′ Proof
Consider ϕ ∈ B (Σ). Define C ϕ = { k ∈ R : ϕ − k ∈ U ( ψ ) } . Note that C ϕ is non-empty. Indeed, if we set k = − k ϕ k ∞ −k ψ k ∞ , then we obtain that ϕ − k = ϕ + k ϕ k ∞ + k ψ k ∞ ≥ k ψ k ∞ ≥ ψ ∈ U ( ψ ). By property 4 of Lemma 10, we can conclude that ϕ − k ∈ U ( ψ ),that is, k ∈ C ϕ . Since U ( ψ ) is convex, it follows that C ϕ is an interval. Since ϕ ∈ B (Σ),note that there exists ˆ k ∈ R such that ψ ≥ ϕ − ˆ k . It follows that ψ (cid:23) ∗ ϕ − ˆ k . In particular,we can conclude that ψ ≻ ∗ ϕ − (cid:16) ˆ k + ε (cid:17) for all ε >
0. This yields that C ϕ is bounded fromabove. Finally, assume that { k n } n ∈ N ⊆ C ϕ and k n ↑ k . By property 2 of Lemma 10, we canconclude that k ∈ C ϕ . To sum up, C ϕ is a non-empty bounded from above interval of R thatsatisfies the property { k n } n ∈ N ⊆ C ϕ and k n ↑ k = ⇒ k ∈ C ϕ (33)The first part yields that sup { k ∈ R : ϕ − k ∈ U ( ψ ) } = sup C ϕ ∈ R is well defined. By(33), we also have that sup C ϕ ∈ C ϕ , that is, sup C ϕ = max C ϕ , proving that I ψ is welldefined. Next, we prove that I ψ is a concave niveloid. We first show that I ψ is monotoneand translation invariant. By Proposition 2 of Cerreia-Vioglio et al. (2014), this impliesthat I ψ is a niveloid. Rather than proving monotonicity, we prove that I ψ is (cid:23) ∗ consistent. Since if ϕ ≥ ϕ , then ϕ (cid:23) ∗ ϕ , it follows that (cid:23) ∗ consistency implies monotonicity. ϕ , ϕ ∈ B (Σ) such that ϕ (cid:23) ∗ ϕ . By the properties of (cid:23) ∗ and definition of I ψ ,we have that ϕ − I ψ ( ϕ ) (cid:23) ∗ ϕ − I ψ ( ϕ ) and ϕ − I ψ ( ϕ ) ∈ U ( ψ )and, in particular, ϕ − I ψ ( ϕ ) (cid:23) ∗ ψ . Since (cid:23) ∗ is a preorder, this implies that ϕ − I ψ ( ϕ ) (cid:23) ∗ ψ , that is, ϕ − I ψ ( ϕ ) ∈ U ( ψ ) and I ψ ( ϕ ) ∈ C ϕ , proving that I ψ ( ϕ ) ≥ I ψ ( ϕ ). We nextprove translation invariance. Consider ϕ ∈ B (Σ) and k ∈ R . By definition of I ψ , we canconclude that ( ϕ + k ) − ( I ψ ( ϕ ) + k ) = ϕ − I ψ ( ϕ ) ∈ U ( ψ )This implies that I ψ ( ϕ ) + k ∈ C ϕ + k and, in particular, I ψ ( ϕ + k ) ≥ I ψ ( ϕ ) + k . Since k and ϕ were arbitrarily chosen, we have that I ψ ( ϕ + k ) ≥ I ψ ( ϕ ) + k ∀ ϕ ∈ B (Σ) , ∀ k ∈ R This yields that I ψ ( ϕ + k ) = I ψ ( ϕ ) + k for all ϕ ∈ B (Σ) and for all k ∈ R . We move to prove that I ψ is concave. Consider ϕ , ϕ ∈ B (Σ) and λ ∈ (0 , I ψ , we have that ϕ − I ψ ( ϕ ) ∈ U ( ψ ) and ϕ − I ψ ( ϕ ) ∈ U ( ψ )Since U ( ψ ) is convex, we have that( λϕ + (1 − λ ) ϕ ) − ( λI ψ ( ϕ ) + (1 − λ ) I ψ ( ϕ ))= λ ( ϕ − I ψ ( ϕ )) + (1 − λ ) ( ϕ − I ψ ( ϕ )) ∈ U ( ψ )yielding that λI ψ ( ϕ )+(1 − λ ) I ψ ( ϕ ) ∈ C λϕ +(1 − λ ) ϕ and, in particular, I ψ ( λϕ + (1 − λ ) ϕ ) ≥ λI ψ ( ϕ ) + (1 − λ ) I ψ ( ϕ ).Finally, since ψ ∈ U ( ψ ), note that 0 ∈ C ψ and I ψ ( ψ ) ≥
0. By definition of I ψ , if I ψ ( ψ ) >
0, then ψ − I ψ ( ψ ) ∈ U ( ψ ), a contradiction with property 3 of Lemma 10.1. It is routine to check that ¯ I ψ is a normalized concave niveloid which is (cid:23) ∗ consistent.2. Clearly, we have that if ψ ∼ ∗ ψ ′ , then U ( ψ ) = U ( ψ ′ ), yielding that I ψ = I ψ ′ and, inparticular, I ψ (0) = I ψ ′ (0) as well as ¯ I ψ = ¯ I ψ ′ . The point trivially follows. (cid:4) Proposition 10
Let (cid:23) ∗ be a binary relation on B (Σ) . The following statements are equiv-alent:(i) (cid:23) ∗ is convex niveloidal; Observe that if ϕ ∈ B (Σ) and k ∈ R , then − k ∈ R and I ψ ( ϕ ) = I ψ (( ϕ + k ) − k ) ≥ I ψ ( ϕ + k ) − k yielding that I ψ ( ϕ + k ) ≤ I ψ ( ϕ ) + k . ii) there exists a family of concave niveloids { I α } α ∈ A on B (Σ) such that ϕ (cid:23) ∗ ψ ⇐⇒ I α ( ϕ ) ≥ I α ( ψ ) ∀ α ∈ A (34) (iii) there exists a family of normalized concave niveloids (cid:8) ¯ I α (cid:9) α ∈ A on B (Σ) such that ϕ (cid:23) ∗ ψ ⇐⇒ ¯ I α ( ϕ ) ≥ ¯ I α ( ψ ) ∀ α ∈ A (35) Proof (iii) implies (i). It is trivial.(i) implies (ii). Let A = B (Σ). We next show that ϕ (cid:23) ∗ ϕ ⇐⇒ I ψ ( ϕ ) ≥ I ψ ( ϕ ) ∀ ψ ∈ B (Σ)where I ψ is defined as in Lemma 11 for all ψ ∈ B (Σ). By Lemma 11, we have that I ψ is (cid:23) ∗ consistent for all ψ ∈ B (Σ). This implies that ϕ (cid:23) ∗ ϕ = ⇒ I ψ ( ϕ ) ≥ I ψ ( ϕ ) ∀ ψ ∈ B (Σ)Vice versa, consider ϕ , ϕ ∈ B (Σ). Assume that I ψ ( ϕ ) ≥ I ψ ( ϕ ) for all ψ ∈ B (Σ). Let ψ = ϕ . By Lemma 11, we have that I ϕ ( ϕ ) ≥ I ϕ ( ϕ ) = 0yielding that ϕ ≥ ϕ − I ϕ ( ϕ ) ∈ U ( ϕ ). By point 4 of Lemma 10, this implies that ϕ ∈ U ( ϕ ), that is, ϕ (cid:23) ∗ ϕ .(ii) implies (iii). Given a family of concave niveloids { I α } α ∈ A , define ¯ I α = I α − I α (0) forall α ∈ A . It is immediate to verify that ¯ I α is a normalized concave niveloid for all α ∈ A .It is also immediate to observe that I α ( ϕ ) ≥ I α ( ϕ ) ∀ α ∈ A ⇐⇒ ¯ I α ( ϕ ) ≥ ¯ I α ( ϕ ) ∀ α ∈ A proving the implication. (cid:4) Remark 1
Given a convex niveloidal binary relation (cid:23) ∗ on B (Σ), we call canonical (resp., canonical normalized ) the representation { I ψ } ψ ∈ B (Σ) (resp., (cid:8) ¯ I ψ (cid:9) ψ ∈ B (Σ) ) obtained fromLemma 11 and the proof of Proposition 10. By the previous proof, clearly, { I ψ } ψ ∈ B (Σ) and (cid:8) ¯ I ψ (cid:9) ψ ∈ B (Σ) satisfy (34) and (35) respectively.The next result clarifies what the relation is between any representation of (cid:23) ∗ and thecanonical ones. This will be useful in establishing an extra property of (cid:8) ¯ I ψ (cid:9) ψ ∈ B (Σ) inCorollary 1. 37 emma 12 Let (cid:23) ∗ be a convex niveloidal binary relation. If B is an index set and { J β } β ∈ B is a family of normalized concave niveloids such that ϕ (cid:23) ∗ ψ ⇐⇒ J β ( ϕ ) ≥ J β ( ψ ) ∀ β ∈ B then for each ψ ∈ B (Σ) I ψ ( ϕ ) = inf β ∈ B ( J β ( ϕ ) − J β ( ψ )) ∀ ϕ ∈ B (Σ) (36) and ¯ I ψ ( ϕ ) = inf β ∈ B ( J β ( ϕ ) − J β ( ψ )) + sup β ∈ B J β ( ψ ) ∀ ϕ ∈ B (Σ) (37) Proof
Fix ϕ ∈ B (Σ) and ψ ∈ B (Σ). By definition, we have that I ψ ( ϕ ) = max { k ∈ R : ϕ − k ∈ U ( ψ ) } Since { J β } β ∈ B represents (cid:23) ∗ and each J β is translation invariant, note that for each k ∈ R ϕ − k ∈ U ( ψ ) ⇐⇒ ϕ − k (cid:23) ∗ ψ ⇐⇒ J β ( ϕ − k ) ≥ J β ( ψ ) ∀ β ∈ B ⇐⇒ J β ( ϕ ) − k ≥ J β ( ψ ) ∀ β ∈ B ⇐⇒ J β ( ϕ ) − J β ( ψ ) ≥ k ∀ β ∈ B ⇐⇒ inf β ∈ B ( J β ( ϕ ) − J β ( ψ )) ≥ k Since ϕ − I ψ ( ϕ ) ∈ U ( ψ ), this implies that I ψ ( ϕ ) = inf β ∈ B ( J β ( ϕ ) − J β ( ψ )). Since ϕ and ψ were arbitrarily chosen, (36) follows. Since ¯ I ψ = I ψ − I ψ (0), we only need to compute − I ψ (0). Since each J β is normalized, we have that − I ψ (0) = − inf β ∈ B ( J β (0) − J β ( ψ )) = − inf β ∈ B ( − J β ( ψ )) = sup β ∈ B J β ( ψ ), proving (37). (cid:4) Corollary 1 If (cid:23) ∗ is a convex niveloidal binary relation, then ¯ I ≤ ¯ I ψ for all ψ ∈ B (Σ) . Proof
By Lemma 12 and Remark 1 and since each ¯ I ψ ′ is a normalized concave niveloid, wehave that¯ I ( ϕ ) = inf ψ ′ ∈ B (Σ) (cid:0) ¯ I ψ ′ ( ϕ ) − ¯ I ψ ′ (0) (cid:1) + sup ψ ′ ∈ B (Σ) ¯ I ψ ′ (0) = inf ψ ′ ∈ B (Σ) ¯ I ψ ′ ( ϕ ) ≤ ¯ I ψ ( ϕ ) ∀ ϕ ∈ B (Σ)for all ψ ∈ B (Σ), proving the statement. (cid:4) The next result will be instrumental in providing a niveloidal multi-representation of % ∗ when | Q | ≥
2. In order to discuss it, we need a piece of terminology. We denote by V thequotient space B (Σ) /M where M is the vector subspace n ϕ ∈ B (Σ) : ϕ Q = 0 o . Recall thatthe elements of V are equivalence classes [ ψ ] with ψ ∈ B (Σ) where ψ ′ , ψ ′′ ∈ [ ψ ] if and onlyif ψ Q = ψ ′ Q = ψ ′′ . Recall that Q is convex. 38 roposition 11 If ( S, Σ) is a standard Borel space and | Q | ≥ , then there exists a bijection f : V → Q . Proof
We begin by observing that: | ca (Σ) | ≤ | ca + (Σ) × ca + (Σ) | = | ca + (Σ) | = | (0 , ∞ ) × ∆ σ | = | ∆ σ | The first inequality holds because the map g : ca (Σ) → ca + (Σ) × ca + (Σ), defined by µ ( µ + , µ − ), is injective. Since Σ is non-trivial, ca + (Σ) is infinite and a bijection justifyingthe first equality exists by Theorem 1.4.5 of Srivastava (1998). As to the second equality,the map g : ca + (Σ) \ { } → (0 , ∞ ) × ∆ σ , defined by µ ( µ ( S ) , µ/µ ( S )), is a bijectionand so | ca + (Σ) \ { }| = | (0 , ∞ ) × ∆ σ | ; by Theorem 1.3.1 of Srivastava (1998), | ca + (Σ) | = | ca + (Σ) \ { }| = | (0 , ∞ ) × ∆ σ | . As to the last equality, by Theorem 1.4.5 and Exercise1.5.1 of Srivastava (1998), being | (0 , ∞ ) | = | (0 , | ≤ | ∆ σ | , we have | ∆ σ | ≤ | (0 , ∞ ) × ∆ σ | = | (0 , × ∆ σ | ≤ | ∆ σ × ∆ σ | = | ∆ σ | , yielding that | (0 , ∞ ) × ∆ σ | = | ∆ σ | .We conclude that | ca (Σ) | ≤ | ∆ σ | , that is, there exists an injective map g : ca (Σ) → ∆ σ .Since Q is a compact and convex subset of ∆ σ , there exists ¯ q ∈ Q such that q ≪ ¯ q for all q ∈ Q . We define h : V → ca (Σ) by h ([ ψ ]) ( A ) = Z A ψd ¯ q ∀ A ∈ ΣNote that h is well defined. For, if ψ ′ ∈ [ ψ ], that is, ψ Q = ψ ′ , then ψ ¯ q = ψ ′ , yielding that R A ψd ¯ q = R A ψ ′ d ¯ q for all A ∈ Σ. Similarly, h ([ ψ ]) = h ([ ψ ′ ]) implies that ψ ¯ q = ψ ′ . Since q ≪ ¯ q for all q ∈ Q , this implies that ψ Q = ψ ′ and [ ψ ] = [ ψ ′ ], proving h is injective. Thisimplies that ˜ f = g ◦ h is a well defined injective function from V to ∆ σ . Clearly, we have that | ∆ σ | ≥ (cid:12)(cid:12)(cid:12) ˜ f ( V ) (cid:12)(cid:12)(cid:12) ≥ | [0 , | . Since ( S, Σ) is a standard Borel space and Q is convex and | Q | ≥ | [0 , | ≥ | ∆ σ | ≥ | Q | ≥ | [0 , | . This implies that | V | = (cid:12)(cid:12)(cid:12) ˜ f ( V ) (cid:12)(cid:12)(cid:12) = | Q | ,proving the statement. (cid:4) Proof of Lemma 7 (ii) implies (i). It is trivial.(i) implies (ii). Since % ∗ is objectively Q -coherent, if | Q | = 1, that is Q = { ¯ q } , then % ∗ is complete. By Maccheroni et al. (2006) and since % ∗ is unbounded, it follows thatthere exists an onto and affine u : X → R and a grounded, lower semicontinuous and convex c ¯ q : ∆ → [0 , ∞ ] such that V : F → R defined by V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + c ¯ q ( p ) (cid:27) ∀ f ∈ F represents % ∗ . If we define c : ∆ × Q → [0 , ∞ ] by c ( p, q ) = c ¯ q ( p ) for all ( p, q ) ∈ ∆ × Q , thenwe have that c is a weak divergence. By Lemma 15 and since % ∗ is objectively Q -coherent,it follows that c ( p, q ) = ∞ for all p ∈ ∆ \ ∆ ( Q ) and for all q ∈ Q , proving the implication.39ssume | Q | >
1. By Lemma 8, there exists an onto affine function u : X → R whichrepresents % ∗ on X . By Lemma 9, this implies that we can consider the convex niveloidalbinary relation (cid:23) ∗ defined as in (32). By definition of (cid:23) ∗ and Proposition 10 (and Remark1), we have that f % ∗ g ⇐⇒ u ( f ) (cid:23) ∗ u ( g ) ⇐⇒ ¯ I ψ ( u ( f )) ≥ ¯ I ψ ( u ( g )) ∀ ψ ∈ B (Σ)where each ¯ I ψ is a normalized concave niveloid. As before, consider V = B (Σ) /M where M is the vector subspace n ϕ ∈ B (Σ) : ϕ Q = 0 o . For each equivalence class [ ψ ], select exactlyone ψ ′ ∈ B (Σ) such that ψ ′ ∈ [ ψ ]. In particular, let ψ ′ = 0 when [ ψ ] = [0]. We denote thissubset of B (Σ) by ˜ V . Clearly, we have that¯ I ψ ( u ( f )) ≥ ¯ I ψ ( u ( g )) ∀ ψ ∈ B (Σ) = ⇒ ¯ I ψ ( u ( f )) ≥ ¯ I ψ ( u ( g )) ∀ ψ ∈ ˜ V Vice versa, assume that ¯ I ψ ( u ( f )) ≥ ¯ I ψ ( u ( g )) for all ψ ∈ ˜ V . Consider ˆ ψ ∈ B (Σ). Itfollows that there exists [ ψ ] in V such that ˆ ψ ∈ [ ψ ]. Similarly, consider ψ ′ ∈ ˜ V such that ψ ′ ∈ [ ψ ]. It follows that ˆ ψ Q = ψ ′ . By Lemmas 9 and 11 and since % ∗ is objectively Q -coherent, then ¯ I ˆ ψ = ¯ I ψ ′ , yielding that ¯ I ˆ ψ ( u ( f )) ≥ ¯ I ˆ ψ ( u ( g )). Since ˆ ψ was arbitrarily chosen¯ I ψ ( u ( f )) ≥ ¯ I ψ ( u ( g )) for all ψ ∈ B (Σ). By construction, observe that there exists abijection ˜ f : ˜ V → V . By Proposition 11, we have that there exists a bijection f : V → Q .Define ¯ f = f ◦ ˜ f . By Corollary 1, if we define ˆ I q = ¯ I ¯ f − ( q ) for all q ∈ Q , then we have thatˆ I ¯ f (0) ≤ ˆ I q ∀ q ∈ Q and f % ∗ g ⇐⇒ ¯ I ψ ( u ( f )) ≥ ¯ I ψ ( u ( g )) ∀ ψ ∈ B (Σ) ⇐⇒ ¯ I ψ ( u ( f )) ≥ ¯ I ψ ( u ( g )) ∀ ψ ∈ ˜ V ⇐⇒ ˆ I q ( u ( f )) ≥ ˆ I q ( u ( g )) ∀ q ∈ Q Since each ˆ I q is a normalized concave niveloid, we have that for each q ∈ Q there exists afunction c q : ∆ → [0 , ∞ ] which is grounded, lower semicontinuous, convex and such thatˆ I q ( ϕ ) = min p ∈ ∆ (cid:26)Z ϕdp + c q ( p ) (cid:27) ∀ ϕ ∈ B (Σ)If we define c : ∆ × Q → [0 , ∞ ] by c ( p, q ) = c q ( p ) for all ( p, q ) ∈ ∆ × Q , then c satisfies thefirst property defining a divergence and (30) holds. By Lemma 15 and (30) and since % ∗ isobjectively Q -coherent, it follows that c ( p, q ) = ∞ for all p ∈ ∆ \ ∆ ( Q ) and for all q ∈ Q .Finally, recall that c ( p, q ) = sup ϕ ∈ B (Σ) (cid:26) ˆ I q ( ϕ ) − Z ϕdp (cid:27) ∀ p ∈ ∆ , ∀ q ∈ Q I ¯ f (0) ≤ ˆ I q for all q ∈ Q , we have that for each q ∈ Qc (cid:0) p, ¯ f (0) (cid:1) = sup ϕ ∈ B (Σ) (cid:26) ˆ I ¯ f (0) ( ϕ ) − Z ϕdp (cid:27) ≤ sup ϕ ∈ B (Σ) (cid:26) ˆ I q ( ϕ ) − Z ϕdp (cid:27) = c ( p, q ) ∀ p ∈ ∆Since c (cid:0) · , ¯ f (0) (cid:1) is grounded, lower semicontinuous and convex and ¯ f (0) ∈ Q , this impliesthat c Q ( · ) = min q ∈ Q c ( · , q ) = c (cid:0) · , ¯ f (0) (cid:1) is well defined and shares the same properties,proving that c is a weak divergence. (cid:4) A.3.2 A parametric representationLemma 13
Let ( % ∗ , % ) be two binary relations on F , where ( S, Σ) is a standard Borel space.The following statements are equivalent:(i) % ∗ is an unbounded dominance relation satisfying objective Q -coherence and % is arational preference that jointly satisfy consistency and caution;(ii) there exist an onto affine function u : X → R and a weak divergence c : ∆ × Q → [0 , ∞ ] such that dom c ( · , q ) ⊆ ∆ ( Q ) for all q ∈ Q and f % ∗ g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q as well as f % g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c Q ( p ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c Q ( p ) (cid:27) Proof (i) implies (ii). We proceed by steps. Before starting, we make one observation. ByLemma 7 and since % ∗ is an unbounded dominance relation which is objectively Q -coherentthere exist an onto affine function u : X → R and a weak divergence c : ∆ × Q → [0 , ∞ ] suchthat dom c ( · , q ) ⊆ ∆ ( Q ) for all q ∈ Q and f % ∗ g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q We are left to show that c Q : ∆ → [0 , ∞ ] is such that f % g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c Q ( p ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c Q ( p ) (cid:27) (38)To prove this we consider c is as in the proof of (i) implies (ii) in Lemma 7. This covers bothcases | Q | = 1 and | Q | >
1. In particular, for each q ∈ Q define ˆ I q : B (Σ) → R byˆ I q ( ϕ ) = min p ∈ ∆ (cid:26)Z ϕdp + c ( p, q ) (cid:27) ∀ ϕ ∈ B (Σ)41nd recall that there exists ˆ q (= ¯ f (0) when | Q | >
1) such that c ( · , ˆ q ) ≤ c ( · , q ), thus ˆ I ˆ q ≤ ˆ I q ,for all q ∈ Q . Step 1. % agrees with % ∗ on X . In particular, u : X → R represents % ∗ and % . Proof of the Step
Note that % ∗ and % restricted to X are continuous weak orders that satisfyrisk independence. Moreover, by the observation above, % ∗ is represented by u . By Hersteinand Milnor (1953) and since % is non-trivial, it follows that there exists a non-constant andaffine function v : X → R that represents % on X . Since ( % ∗ , % ) jointly satisfy consistency,it follows that for each x, y ∈ Xu ( x ) ≥ u ( y ) = ⇒ v ( x ) ≥ v ( y )By Corollary B.3 of Ghirardato et al. (2004), u and v are equal up to an affine and positivetransformation, hence the statement. We can set v = u . (cid:3) Step 2. There exists a normalized, monotone and continuous functional I : B (Σ) → R suchthat f % g ⇐⇒ I ( u ( f )) ≥ I ( u ( g )) Proof of the Step
By Cerreia-Vioglio et al. (2011) and since % is a rational preferencerelation, the statement follows. (cid:3) Step 3. I ( ϕ ) ≤ inf q ∈ Q ˆ I q ( ϕ ) for all ϕ ∈ B (Σ) .Proof of the Step Consider ϕ ∈ B (Σ). Since each ˆ I q is normalized and monotone and u is onto, we have that ˆ I q ( ϕ ) ∈ [inf s ∈ S ϕ ( s ) , sup s ∈ S ϕ ( s )] ⊆ Im u for all q ∈ Q . Since ϕ ∈ B (Σ), it follows that there exists f ∈ F such that ϕ = u ( f ) and x ∈ X such that u ( x ) = inf q ∈ Q ˆ I q ( ϕ ). For each ε > x ε ∈ X such that u ( x ε ) = u ( x ) + ε .Since inf q ∈ Q ˆ I q ( ϕ ) = u ( x ), it follows that for each ε > q ∈ Q such thatˆ I q ( u ( f )) = ˆ I q ( ϕ ) < u ( x ε ) = ˆ I q ( u ( x ε )), yielding that f % ∗ x ε . Since ( % ∗ , % ) jointly satisfycaution, we have that x ε % f for all ε >
0. By Step 2, this implies that u ( x ) + ε = u ( x ε ) = I ( u ( x ε )) ≥ I ( u ( f )) = I ( ϕ ) ∀ ε > q ∈ Q ˆ I q ( ϕ ) = u ( x ) ≥ I ( ϕ ), proving the step. (cid:3) Step 4. I ( ϕ ) ≥ inf q ∈ Q ˆ I q ( ϕ ) for all ϕ ∈ B (Σ) .Proof of the Step Consider ϕ ∈ B (Σ). We use the same objects and notation of Step 3.Note that for each q ′ ∈ Q ˆ I q ′ ( u ( f )) = ˆ I q ′ ( ϕ ) ≥ inf q ∈ Q ˆ I q ( ϕ ) = u ( x ) = ˆ I q ′ ( u ( x ))42hat is, f % ∗ x . Since ( % ∗ , % ) jointly satisfy consistency, we have that f % x . By Step 2,this implies that I ( ϕ ) = I ( u ( f )) ≥ I ( u ( x )) = u ( x ) = inf q ∈ Q ˆ I q ( ϕ )proving the step. (cid:3) Step 5. I ( ϕ ) = min p ∈ ∆ (cid:8)R ϕdp + c Q ( p ) (cid:9) for all ϕ ∈ B (Σ) .Proof of the Step By Steps 3 and 4 and since ˆ I ˆ q ≤ ˆ I q for all q ∈ Q , we have that I ( ϕ ) = min q ∈ Q ˆ I q ( ϕ ) = ˆ I ˆ q ( ϕ ) ∀ ϕ ∈ B (Σ)Since c ( · , ˆ q ) = c Q ( · ), it follows that for each ϕ ∈ B (Σ) I ( ϕ ) = ˆ I ˆ q ( ϕ ) = min p ∈ ∆ (cid:26)Z ϕdp + c ( p, ˆ q ) (cid:27) = min p ∈ ∆ (cid:26)Z ϕdp + c Q ( p ) (cid:27) proving the step. (cid:3) Thus, (38) follows from Steps 2 and 5, this completes the proof.(ii) implies (i). It is routine. (cid:4)
A.3.3 Two variational lemmas
The next two lemmas will be key in characterizing subjective and objective Q -coherence. Lemma 14
Let % be a variational preference represented by V : F → R defined by V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p ) (cid:27) and let ¯ p ∈ ∆ . If % is unbounded, then the following conditions are equivalent:(i) c ( ¯ p ) = 0 ;(ii) x ¯ pf % f for all f ∈ F ;(iii) for each f ∈ F and for each x ∈ Xx ≻ x ¯ pf = ⇒ x ≻ f. Proof
We actually prove that (i)= ⇒ (ii) ⇐⇒ (iii), with equivalence when % is unbounded.(i) implies (ii). Let f ∈ F . It is enough to observe that c ( ¯ p ) = 0 implies V (cid:16) x ¯ pf (cid:17) = u (cid:16) x ¯ pf (cid:17) = Z u ( f ) d ¯ p + c (¯ p ) ≥ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p ) (cid:27) = V ( f )43ielding that x ¯ pf % f .(ii) implies (iii). Assume that x ¯ pf % f for all f ∈ F . Since % is complete and transitive,it follows that if x ≻ x ¯ pf , then x ≻ f .(iii) implies (ii). By contradiction, suppose that there exists f ∈ F such that f ≻ x ¯ pf .Let x f ∈ X be such that x f ∼ f . This implies that x f ≻ x ¯ pf and so x f ≻ f , a contradiction.(ii) implies (i). Let % be unbounded. Assume that x ¯ pf % f for all f ∈ F , i.e., V ( f ) ≤ R u ( f ) d ¯ p for all f ∈ F . So, ¯ p corresponds to a SEU preference that is less ambiguity aversethan % . By Lemma 32 of Maccheroni et al. (2006), we can conclude that c ( ¯ p ) = 0. (cid:4) Lemma 15
Let % be a variational preference represented by V : F → R defined by V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p ) (cid:27) If Q is a compact and convex subset of ∆ σ and % is unbounded and such that f Q = g = ⇒ f ∼ g then dom c ⊆ ∆ ( Q ) . Proof
Let p ∈ ∆ \ ∆ ( Q ). It follows that there exists A ∈ Σ such that q ( A ) = 0 for all q ∈ Q as well as p ( A ) >
0. Define I : B (Σ) → R by I ( ϕ ) = min p ∈ ∆ (cid:8)R ϕdp + c ( p ) (cid:9) for all ϕ ∈ B (Σ). Since u is unbounded, for each λ ∈ R there exists x λ ∈ X such that u ( x λ ) = λ . Similarly, there exists y ∈ X such that u ( y ) = 0. For each λ ∈ R define f λ = x λ Ay . By construction, we have that f λ Q = y for all λ ∈ R . This implies that I ( λ A ) = V ( f λ ) = V ( y ) = I (0) = 0. By Maccheroni et al. (2006) and since u is unbounded,we have that c ( p ) = sup ϕ ∈ B (Σ) (cid:26) I ( ϕ ) − Z ϕdp (cid:27) ≥ sup λ ∈ R { I ( λ A ) − λp ( A ) } = ∞ Since p was arbitrarily chosen, it follows that dom c ⊆ ∆ ( Q ). (cid:4) A.3.4 Proof of Theorem 1
We only prove (i) implies (ii), the converse being routine. By Lemma 13, there exist anonto and affine function u : X → R and a weak divergence c : ∆ × Q → [0 , ∞ ] such thatdom c ( · , q ) ⊆ ∆ ( Q ) for all q ∈ Q and % ∗ is represented by f % ∗ g ⇔ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q (39)and % is represented by V : F → R defined by V ( f ) = min p ∈ ∆ (cid:26)Z u ( f ) dp + c Q ( p ) (cid:27) (40)44y Lemma 14 and since % is subjectively Q -coherent and % ∗ and % coincide on X , weconclude that c − Q (0) = Q , proving the implication.Next, assume that c is uniquely null. Define the correspondence Γ : Q ⇒ Q byΓ ( q ) = { p ∈ ∆ : c ( p, q ) = 0 } = arg min c q Since c Q ≤ c q for all q ∈ Q and c − Q (0) = Q , we have that Γ is well defined. Since c q isgrounded, it follows that Γ ( q ) = ∅ for all q ∈ Q . Since c is uniquely null and c q is grounded,we have that c − q (0) is a singleton, that is, c ( p, q ) = c (cid:0) p ′ , q (cid:1) = 0 = ⇒ p = p ′ This implies that Γ ( q ) is a singleton, therefore Γ is a function. Since c − Q (0) = Q , observethat ∪ q ∈ Q Γ ( q ) = ∪ q ∈ Q arg min c q = arg min c Q = Q that is, Γ is surjective. Since c is uniquely null, we have that c − p (0) is at most a singleton,that is, c ( p, q ) = c (cid:0) p, q ′ (cid:1) = 0 = ⇒ q = q ′ yielding that Γ is injective. To sum up, Γ is a bijection. Define ˜ c : ∆ × Q → [0 , ∞ ]by ˜ c ( p, q ) = c (cid:0) p, Γ − ( q ) (cid:1) for all ( p, q ) ∈ ∆ × Q . Note that ˜ c ( · , q ) is grounded, lowersemicontinuous, convex and dom ˜ c ( · , q ) ⊆ ∆ ( Q ) for all q ∈ Q . Next, we show that ˜ c Q = c Q .Since c Q is well defined, for each p ∈ ∆ there exists q p ∈ Q such that˜ c ( p, Γ ( q p )) = c ( p, q p ) = min q ∈ Q c ( p, q ) ≤ c (cid:0) p, q ′ (cid:1) = ˜ c (cid:0) p, Γ (cid:0) q ′ (cid:1)(cid:1) ∀ q ′ ∈ Q Since Γ is a bijection, we have that ˜ c ( p, Γ ( q p )) ≤ ˜ c ( p, q ) for all q ∈ Q . Since p was arbitrarilychosen, it follows that c Q ( p ) = min q ∈ Q c ( p, q ) = ˜ c ( p, Γ ( q p )) = min q ∈ Q ˜ c ( p, q ) = ˜ c Q ( p ) ∀ p ∈ ∆To sum up, ˜ c Q = c Q and ˜ c − Q (0) = c − Q (0) = Q . In turn, since c Q is grounded, lowersemicontinuous and convex, this implies that ˜ c Q is grounded, lower semicontinuous andconvex. Since Γ is a bijection, we can conclude that (39) holds with ˜ c in place of c and (40)holds with ˜ c Q in place of c Q .We are left to show that ˜ c ( p, q ) = 0 if and only if p = q . Since c − q (0) is a singletonfor all q ∈ Q and Γ is a bijection, if ˜ c ( p, q ) = 0, then c (cid:0) p, Γ − ( q ) (cid:1) = 0, yielding that p = Γ (cid:0) Γ − ( q ) (cid:1) = q . On the other hand, ˜ c ( q, q ) = c (cid:0) q, Γ − ( q ) (cid:1) = 0. We can conclude that˜ c ( p, q ) = 0 if and only if p = q , proving that ˜ c is a statistical distance. (cid:4) .3.5 Proof of Theorem 2 We only prove (i) implies (ii), the converse being routine. We proceed by steps.
Step 1. % ∗ Q agrees with % ∗ Q ′ on X for all Q, Q ′ ∈ Q . In particular, there exists an affineand onto function u : X → R representing % ∗ Q for all Q ∈ Q . Proof of the Step
Let
Q, Q ′ ∈ Q be such that Q ⊇ Q ′ . Note that % ∗ Q and % ∗ Q ′ , restrictedto X , satisfy weak order, continuity and risk independence. By Herstein and Milnor (1953)and since % ∗ Q and % ∗ Q ′ are non-trivial, there exist two non-constant affine functions u Q , u Q ′ : X → R which represent % ∗ Q and % ∗ Q ′ , respectively. Since n % ∗ Q o Q ∈Q is monotone, we havethat u Q ( x ) ≥ u Q ( y ) = ⇒ u Q ′ ( x ) ≥ u Q ′ ( y )By Corollary B.3 of Ghirardato et al. (2004), u Q and u Q ′ are equal up to an affine andpositive transformation. Next, fix ¯ q ∈ Q . Set u = u ¯ q . Given any other q ∈ ∆ σ , consider¯ Q = co { ¯ q, q } . By the previous part, it follows that u ¯ Q , u q and u ¯ q are equal up to an affineand positive transformation. Given that q was arbitrarily chosen, we can set u = u q for all q ∈ Q . Similarly, given a generic Q ∈ Q , select q ∈ Q . Since Q ⊇ { q } , it follows that we canset u = u Q . Since each % ∗ Q is unbounded for all Q ∈ Q , we have that u is onto. (cid:3) Step 2. For each q ∈ ∆ σ there exists a normalized, monotone, translation invariant andconcave functional I q : B (Σ) → R such that f % ∗ q g ⇐⇒ I q ( u ( f )) ≥ I q ( u ( g )) Moreover, there exists a unique grounded, lower semicontinuous and convex function c q :∆ → [0 , ∞ ] such that I q ( ϕ ) = min p ∈ ∆ (cid:26)Z ϕdp + c q ( p ) (cid:27) ∀ ϕ ∈ B (Σ) (41) Proof of the Step
Fix q ∈ ∆ σ . Since % ∗ q is an unbounded dominance relation which iscomplete, we have that % ∗ q is a variational preference. By the proof of Theorem 3 andProposition 6 of Maccheroni et al. (2006), there exists a normalized, monotone, translationinvariant and concave functional I q : B (Σ) → R such that f % ∗ q g ⇐⇒ I q ( u ( f )) ≥ I q ( u ( g ))Moreover, we have that there exists a unique grounded, lower semicontinuous and convexfunction c q : ∆ → [0 , ∞ ] satisfying (41). (cid:3) c : ∆ × ∆ σ → [0 , ∞ ] by c ( p, q ) = c q ( p ) for all ( p, q ) ∈ ∆ × ∆ σ . Define the map J : B (Σ) × ∆ σ → R by J ( ϕ, q ) = I q ( ϕ ). Observe that, for each ( p, q ) ∈ ∆ × ∆ σ , c ( p, q ) = c q ( p ) = sup ϕ ∈ B (Σ) (cid:26) I q ( ϕ ) − Z ϕdp (cid:27) = sup ϕ ∈ B (Σ) (cid:26) J ( ϕ, q ) − Z ϕdp (cid:27) (42) Step 3. J is convex and lower semicontinuous in the second argument.Proof of the Step Note that for each ϕ ∈ B (Σ) and for each q ∈ ∆ σ J ( ϕ, q ) = I q ( ϕ ) = u ( x f,q ) where f ∈ F is s.t. ϕ = u ( f )Fix ϕ ∈ B (Σ) and t ∈ R . By Step 1 and since n % ∗ Q o Q ∈Q is lower semicontinuous on ∆ σ ,the set { q ∈ ∆ σ : J ( ϕ, q ) ≤ t } = { q ∈ ∆ σ : u ( x ) ≥ u ( x f,q ) } = { q ∈ ∆ σ : x % ∗ x f,q } is closed where x ∈ X and f ∈ F are such that u ( x ) = t as well as u ( f ) = ϕ . Since ϕ and t were arbitrarily chosen, this yields that J is lower semicontinuous in the second argument.Fix ϕ ∈ B (Σ), q, q ′ ∈ ∆ σ and λ ∈ (0 , n % ∗ Q o Q ∈Q is averse to model hybridizationand u is affine, J (cid:0) ϕ, λq + (1 − λ ) q ′ (cid:1) = u (cid:0) x f,λq +(1 − λ ) q ′ (cid:1) ≤ u (cid:0) λx f,q + (1 − λ ) x f,q ′ (cid:1) = λu ( x f,q ) + (1 − λ ) u (cid:0) x f,q ′ (cid:1) = λJ ( ϕ, q ) + (1 − λ ) J (cid:0) ϕ, q ′ (cid:1) where f ∈ F is such that u ( f ) = ϕ . Since ϕ , q , q ′ and λ were arbitrarily chosen, this yieldsthat J is convex in the second argument. (cid:3) Step 4. c is jointly lower semicontinuous and convex. Moreover, its q -sections are grounded,lower semicontinuous and convex.Proof of the Step By Step 3, the map ( p, q ) J ( ϕ, q ) − R ϕdp , defined over ∆ × ∆ σ , isjointly lower semicontinuous and convex. By (42) and the definition of c , we conclude that c is jointly lower semicontinuous and convex. By Step 2, the rest of the statement follows. (cid:3) Step 5. For each Q ∈ Q we have that f % ∗ Q g if and only if f % ∗ q g for all q ∈ Q . Inparticular, we have that f % ∗ Q g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ∀ q ∈ Q (43) Proof of the Step
Fix Q ∈ Q . Since n % ∗ Q o Q ∈Q is monotone, we have that f % ∗ Q g = ⇒ f % ∗ q g ∀ q ∈ Q n % ∗ Q o Q ∈Q is Q -separable, we can conclude that f % ∗ Q g if and only if f % ∗ q g for all q ∈ Q . By Step 2 and the definition of c , (43) follows. (cid:3) Step 6. % ∗ Q agrees with % Q on X for all Q ∈ Q . Moreover, % Q is represented by the function u of Step 1.Proof of the Step Fix Q ∈ Q . Note that % ∗ Q and % Q , restricted to X , satisfy weak order,continuity and risk independence. By Herstein and Milnor (1953) and since % Q is non-trivial, there exists a non-constant affine function v Q which represents % Q . By Step 1, % ∗ Q is represented by u . Since (cid:16) % ∗ Q , % Q (cid:17) jointly satisfy consistency, it follows that for each x, y ∈ X u ( x ) ≥ u ( y ) = ⇒ v Q ( x ) ≥ v Q ( y )By Corollary B.3 of Ghirardato et al. (2004), v Q and u are equal up to an affine and positivetransformation. So we can set v Q = u , proving the statement. (cid:3) Step 7. For each Q ∈ Q we have that f % Q g ⇐⇒ min p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + min q ∈ Q c ( p, q ) (cid:27) (44) Moreover, the function c Q : ∆ → [0 , ∞ ] , defined by c Q ( p ) = min q ∈ Q c ( p, q ) for all p ∈ ∆ , iswell defined, grounded, lower semicontinuous and convex.Proof of the Step Fix Q ∈ Q . By Cerreia-Vioglio et al. (2011) and since % Q is a rationalpreference relation, there exists a normalized, monotone and continuous functional I Q : B (Σ) → R such that f % Q g ⇐⇒ I Q ( u ( f )) ≥ I Q ( u ( g )) (45)We next show that I Q ( ϕ ) ≤ inf q ∈ Q I q ( ϕ ) for all ϕ ∈ B (Σ). Consider ϕ ∈ B (Σ). Since each I q is normalized and monotone and u is onto, we have that I q ( ϕ ) ∈ [inf s ∈ S ϕ ( s ) , sup s ∈ S ϕ ( s )] ⊆ Im u for all q ∈ Q . Since ϕ ∈ B (Σ), it follows that there exists f ∈ F such that ϕ = u ( f )and x ∈ X such that u ( x ) = inf q ∈ Q I q ( ϕ ). For each ε > x ε ∈ X such that u ( x ε ) = u ( x ) + ε . Since inf q ∈ Q I q ( ϕ ) = u ( x ), it follows that for each ε > q ∈ Q such that I q ( u ( f )) = I q ( ϕ ) < u ( x ε ) = I q ( u ( x ε )), yielding that f % ∗ Q x ε . Since (cid:16) % ∗ Q , % Q (cid:17) jointly satisfy caution, we have that x ε % Q f for all ε >
0. By (45), this impliesthat u ( x ) + ε = u ( x ε ) = I Q ( u ( x ε )) ≥ I Q ( u ( f )) = I Q ( ϕ ) ∀ ε > q ∈ Q I q ( ϕ ) = u ( x ) ≥ I Q ( ϕ ). We next prove that I Q ( ϕ ) ≥ inf q ∈ Q I q ( ϕ ) for all ϕ ∈ B (Σ). Consider ϕ ∈ B (Σ). We use the same objects of before. Note that for each q ′ ∈ Q I q ′ ( u ( f )) = I q ′ ( ϕ ) ≥ inf q ∈ Q I q ( ϕ ) = u ( x ) = I q ′ ( u ( x ))48hat is, f % ∗ Q x . Since (cid:16) % ∗ Q , % Q (cid:17) jointly satisfy consistency, we have that f % Q x . By (45),this implies that I Q ( ϕ ) = I Q ( u ( f )) ≥ I Q ( u ( x )) = u ( x ) = inf q ∈ Q I q ( ϕ )proving that I Q = inf q ∈ Q I q . Since c is jointly lower semicontinuous and convex, we canconclude that I Q ( ϕ ) = inf q ∈ Q min p ∈ ∆ (cid:26)Z ϕdp + c ( p, q ) (cid:27) = min q ∈ Q min p ∈ ∆ (cid:26)Z ϕdp + c ( p, q ) (cid:27) = min p ∈ ∆ min q ∈ Q (cid:26)Z ϕdp + c ( p, q ) (cid:27) = min p ∈ ∆ (cid:26)Z ϕdp + min q ∈ Q c ( p, q ) (cid:27) ∀ ϕ ∈ B (Σ)By (45), this implies that (44) holds. Finally, by Lemma 4, we have that the function p min q ∈ Q c ( p, q ) is well defined, grounded, lower semicontinuous and convex. (cid:3) Step 8. c − Q (0) = Q for all Q ∈ Q . Moreover, c ( p, q ) = 0 if and only if p = q .Proof of the Step Fix Q ∈ Q . Since % Q is subjectively Q -coherent, it follows that c − Q (0) = Q .In particular, when Q = { q } for some q ∈ ∆ σ , we have that c ( p, q ) = 0 if and only if c Q ( p ) = 0 if and only if p ∈ Q if and only if p = q . (cid:3) Step 9. dom c ( · , q ) ⊆ ∆ ( Q ) for all q ∈ Q and for all Q ∈ Q .Proof of the Step By the previous part of the proof, we have that % ∗ q coincides with % q on F for all q ∈ ∆ σ . By Lemma 15 and since % ∗ q is objectively { q } -coherent, we can concludethat dom c ( · , q ) ⊆ ∆ ( q ) ⊆ ∆ ( Q ) for all q ∈ Q and for all Q ∈ Q . (cid:3) Steps 4, 7, 8 and 9 prove that c is a statistical distance which is jointly lower semicon-tinuous and convex such that dom c ( · , q ) ⊆ ∆ ( Q ) for all q ∈ Q and for all Q ∈ Q , yieldingthat dom c Q ⊆ ∆ ( Q ) for all Q ∈ Q . Steps 1, 5 and 7 prove, respectively, (26) and (27). Asfor uniqueness, assume that the function ˜ c : ∆ × ∆ σ → [0 , ∞ ] is a statistical distance whichis jointly lower semicontinuous and convex and such that dom c Q ⊆ ∆ ( Q ) for all Q ∈ Q and that satisfies (26) and (27). By Proposition 6 of Maccheroni et al. (2006) and sinceIm u = R and % ∗ q is a variational preference for all q ∈ ∆ σ , it follows that ˜ c ( · , q ) = c ( · , q ) forall q ∈ ∆ σ , yielding that c = ˜ c . (cid:4) A.4 Other proofs
Proof of Proposition 2
First, note that min q ∈ Q R ( p || q ) = 0 if and only if p ∈ Q . Indeed,we have thatmin q ∈ Q R ( p || q ) = 0 ⇐⇒ ∃ ¯ q ∈ Q s.t. R ( p || ¯ q ) = 0 ⇐⇒ ∃ ¯ q ∈ Q s.t. p = ¯ q λ n = n for all n ∈ N . For each n ∈ N , we have λ n min q ∈ Q R ( p || q ) = 0 if and only if p ∈ Q . So, for each p ∈ ∆, lim n λ n min q ∈ Q R ( p || q ) = ( p ∈ Q ∞ if p Q Since λ n min q ∈ Q R ( p || q ) = 0 for each n ∈ N if and only if p ∈ Q , by Proposition 12 ofMaccheroni et al. (2006) we havelim n min p ∈ ∆ (cid:26)Z u ( f ) dp + λ n min q ∈ Q R ( p || q ) (cid:27) = min q ∈ Q Z u ( f ) dq ∀ f ∈ F Finally, by (19), we have that for each f ∈ F min q ∈ Q Z u ( f ) dq ≤ lim n min p ∈ ∆ (cid:26)Z u ( f ) dp + λ n min q ∈ Q R ( p || q ) (cid:27) ≤ lim λ ↑∞ min p ∈ ∆ (cid:26)Z u ( f ) dp + λ min q ∈ Q R ( p || q ) (cid:27) ≤ min q ∈ Q Z u ( f ) dq yielding the statement. (cid:4) Proof of Proposition 3
Note that c ( · , q ) = λD φ ( ·|| q ) is Shur convex (with respect to q )for all q ∈ Q . Consider A, B ∈ Σ. Assume that q ( A ) ≥ q ( B ) for all q ∈ Q . Let q ∈ Q .Consider x, y ∈ X such that x ≻ y . It follows that Z v ( u ( xAy )) dq ≥ Z v ( u ( xBy )) dq for each v : R → R strictly increasing and concave. By Theorem 2 of Cerreia-Vioglio et al.(2012) and since q was arbitrarily chosen, it follows thatmin p ∈ ∆ (cid:26)Z u ( xAy ) dp + λD φ ( p || q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( xBy ) dp + λD φ ( p || q ) (cid:27) ∀ q ∈ Q yielding that xAy % ∗ xBy and, in particular, xAy % xBy . (cid:4) Proof of Proposition 4
We prove the “only if”, the converse being obvious. Define & ∗ by f & ∗ g if and only if R u ( f ) dq ≥ R u ( g ) dq for all q ∈ Q . By hypothesis, the pair ( & ∗ , % )satisfies consistency. Let f & ∗ x . Then, there exists q ∈ Q such that u ( x qf ) = R u ( f ) dq
For each q ∈ Q define I q : B (Σ) → R by I q ( ϕ ) = min p ∈ ∆ (cid:26)Z ϕdp + c ( p, q ) (cid:27) ∀ ϕ ∈ B (Σ)Recall that f ≻≻ ∗ g if and only if for each h, l ∈ F there exists ε > − δ ) f + δh ≻ ∗ (1 − δ ) g + δl ∀ δ ∈ [0 , ε ] (46)Moreover, given f, g ∈ F , define k ∗ = inf s ∈ S u ( f ( s )) and k ∗ = sup s ∈ S u ( g ( s )).“Only if.” Assume that f ≻≻ ∗ g . Let ˆ ε >
0. Consider u ( x ) = k ∗ − ˆ ε and u ( y ) = k ∗ + ˆ ε .By definition, there exists ε > − δ ) f + δx ≻ ∗ (1 − δ ) g + δy ∀ δ ∈ [0 , ε ]Note that for each q ∈ Q and for each δ ∈ [0 , I q ( u ((1 − δ ) f + δx )) = I q ((1 − δ ) u ( f ) + δu ( x )) = I q ( u ( f ) − δu ( f ) + δu ( x )) ≤ I q ( u ( f ) − δk ∗ + δ ( k ∗ − ˆ ε )) = I q ( u ( f )) − δ ˆ ε and I q ( u ((1 − δ ) g + δy )) = I q ((1 − δ ) u ( g ) + δu ( y )) = I q ( u ( g ) − δu ( g ) + δu ( y )) ≥ I q ( u ( g ) − δk ∗ + δ ( k ∗ + ˆ ε )) = I q ( u ( g )) + δ ˆ ε It follows that for each q ∈ Q and for each δ ∈ [0 , ε ] I q ( u ( f )) − I q ( u ( g )) − δ ˆ ε ≥ I q ( u ((1 − δ ) f + δx )) − I q ( u ((1 − δ ) g + δy )) ≥ δ = ε >
0, then I q ( u ( f )) ≥ I q ( u ( g )) + 2 ε ˆ ε , proving the statement.“If.” Let f, g ∈ F . Assume there exists ε > I q ( u ( f )) ≥ I q ( u ( g )) + ε forall q ∈ Q . Without loss of generality, we can assume that ≥ holds with strict inequality. Consider h, l ∈ F . Define k ⋆ = inf s ∈ S u ( h ( s )) and k ⋆ = sup s ∈ S u ( l ( s )). Define also k ∼ =sup s ∈ S u ( f ( s )) and k ∼ = inf s ∈ S u ( g ( s )). Note that for each q ∈ Q and for each δ ∈ [0 , I q ( u ((1 − δ ) f + δh )) = I q ((1 − δ ) u ( f ) + δu ( h )) = I q ( u ( f ) − δu ( f ) + δu ( h ))= I q ( u ( f ) + δ ( u ( h ) − u ( f ))) ≥ I q ( u ( f ) + δ ( k ⋆ − k ∼ )) = I q ( u ( f )) + δ ( k ⋆ − k ∼ ) It is enough to replace ε with ε/ I q ( u ((1 − δ ) g + δl )) = I q ((1 − δ ) u ( g ) + δu ( l )) = I q ( u ( g ) − δu ( g ) + δu ( l ))= I q ( u ( g ) + δ ( u ( l ) − u ( g ))) ≤ I q ( u ( g ) + δ ( k ⋆ − k ∼ )) = I q ( u ( g )) + δ ( k ⋆ − k ∼ )It follows that for each q ∈ Q and for each δ ∈ [0 , I q ( u ((1 − δ ) f + δh )) − I q ( u ((1 − δ ) g + δl )) ≥ I q ( u ( f )) + δ ( k ⋆ − k ∼ ) − I q ( u ( g )) − δ ( k ⋆ − k ∼ ) ≥ ε + δ ˆ ε where ˆ ε = k ⋆ − k ∼ − k ⋆ + k ∼ . We have two cases:1. ˆ ε ≥
0. In this case, I q ( u ((1 − δ ) f + δh )) − I q ( u ((1 − δ ) g + δl )) > δ ∈ [0 , q ∈ Q , proving (46).2. ˆ ε <
0. In this case, I q ( u ((1 − δ ) f + δh )) − I q ( u ((1 − δ ) g + δl )) > δ ∈ [0 , − ε/ ε ] and all q ∈ Q , proving (46).This completes the proof of the result. (cid:4) Proof of Lemma 3
Given q ∈ Q , if c ( p, q ) = ∞ for all p / ∈ Q , then c Q ( p ) = ∞ for all p / ∈ Q . Since c Q ( q ) = 0 for all q ∈ Q , we conclude that c Q ( p ) = δ Q ( p ) for all p ∈ ∆.Conversely, for each q ∈ Q we have c ( p, q ) ≥ c Q ( p ) = δ Q ( p ) = ∞ for all p / ∈ Q . (cid:4) Proof of Proposition 6 (i) implies (ii). By Proposition 2 of Cerreia-Vioglio (2016) andsince % ∗ is unbounded, there exists a compact and convex set C ⊆ ∆ and an affine and ontomap u : X → R such that f % ∗ g ⇐⇒ Z u ( f ) dq ≥ Z u ( g ) dq ∀ q ∈ C (47)and f % g ⇐⇒ min q ∈ C Z u ( f ) dq ≥ min q ∈ C Z u ( g ) dq (48)By Lemma 14 and since % is subjectively Q -coherent and % ∗ and % coincide on X , wecan conclude that C = Q . If we set c : ∆ × Q → [0 , ∞ ] to be c ( p, q ) = δ { q } ( p ) for all( p, q ) ∈ ∆ × Q , then it is immediate to see that c is a statistical distance. By (47) and (48)and since C = Q , (13) and (14) follow.(ii) implies (i). It is trivial. (cid:4) Proof of Proposition 7 (i) Let ˆ f ∈ F be optimal. By (23), if there is g ∈ F such that g ≻≻ ∗ Q ˆ f , then g ≻ Q ˆ f , a contradiction with ˆ f being optimal. We conclude that ˆ f is weakly52dmissible. A similar argument proves that there is no g ∈ F such that g ≻ ∗ Q ˆ f when (24)holds.(ii) Suppose ˆ f ∈ F is the unique optimal act, that is, ˆ f ≻ Q f for all f ∈ F \ n ˆ f o . If g ∈ F is such that g ≻ ∗ Q ˆ f , then g = ˆ f and g % Q ˆ f . In turn, this implies g % Q ˆ f ≻ Q g , acontradiction. We conclude that ˆ f is admissible. (cid:4) Proof of Proposition 8
Since Q ⊆ Q ′ , it follows that min q ∈ Q c ( p, q ) ≥ min q ∈ Q ′ c ( p, q ) forall p ∈ ∆. We thus havemin p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u ( f ) dp + min q ∈ Q ′ c ( p, q ) (cid:27) ∀ f ∈ F yielding that v ( Q ) ≥ v ( Q ′ ). Next, fix Q and assume that the sup in (29) is achieved. Let¯ f ∈ F be such that min p ∈ ∆ (cid:26)Z u (cid:0) ¯ f (cid:1) dp + min q ∈ Q c ( p, q ) (cid:27) = v ( Q )By contradiction, assume that ¯ f ∈ F/F ∗ Q . By Proposition 5 and since ¯ f F ∗ Q and ¯ f ∈ F ,there exists g ∈ F such that g ≻≻ ∗ Q ¯ f , that is, there exists ε > p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ≥ min p ∈ ∆ (cid:26)Z u (cid:0) ¯ f (cid:1) dp + c ( p, q ) (cid:27) + ε ∀ q ∈ Q This implies that v ( Q ) ≥ min p ∈ ∆ (cid:26)Z u ( g ) dp + min q ∈ Q c ( p, q ) (cid:27) = min p ∈ ∆ min q ∈ Q (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ≥ inf q ∈ Q min p ∈ ∆ (cid:26)Z u ( g ) dp + c ( p, q ) (cid:27) ≥ inf q ∈ Q min p ∈ ∆ (cid:26)Z u (cid:0) ¯ f (cid:1) dp + c ( p, q ) (cid:27) + ε ≥ min p ∈ ∆ min q ∈ Q (cid:26)Z u (cid:0) ¯ f (cid:1) dp + c ( p, q ) (cid:27) + ε = min p ∈ ∆ (cid:26)Z u (cid:0) ¯ f (cid:1) dp + min q ∈ Q c ( p, q ) (cid:27) + ε = v ( Q ) + ε a contradiction. (cid:4) References [1] F. J. Anscombe and R. J. Aumann, A definition of subjective probability,
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