Differentiating ambiguity and ambiguity attitudeIntroductionThe revelation of ambiguity and ambiguity attitudeDiscussionThe related literatureOutline of the paperPreliminaries and notationInvariant biseparable preferencesPriors and revealed ambiguityUnambiguous preferenceRevealed ambiguityEnter ambiguity attitude: the representationCrisp actsThe representation theoremAn index of ambiguity aversionRevealed ambiguity is a differentialSome theoretical consequencesAn operational consequenceA special case: alpha-MEU preferencesConclusionsAcknowledgementsFunctional analysis mini-kitConic preordersClarke derivatives and differentials: preliminary propertiesClarke differentials: representation on compact metric SProofs of the results in the main textProof of Proposition 4Proof of Proposition 5Proof of Remark 1Proof of Proposition 6Proof of Proposition 7Proof of Lemma 8Proof of Proposition 10Proof of Theorem 11Proof of Proposition 12Proof of Theorem 14Proof of Proposition 16Proof of Proposition 18Proof of Theorem 19Proof of Proposition 20ReferencesJournal of Economic Theory 118 (2004) 133–173Differentiating ambiguity and ambiguity attitudePaolo Ghirardato,a,Fabio Maccheroni,bandMassimo MarinacciaaDipartimento di Statistica e Matematica Applicata and ICER, Universita`di Torino, ItalybIMQ and IGIER, Universita`Bocconi, ItalyReceived 22 April 2003; final version received 1 December 2003We dedicate this paper—an extended version of which was previously circulated with the title ‘‘Ambiguityfrom the Differential Viewpoint’’—to Erio Castagnoli on the occasion of his 60th birthday.AbstractThe objective of this paper is to show how ambiguity, and a decision maker (DM)’sresponse to it, can be modelled formally in the context of a general decision model. Weintroduce a relation derived from the DM’s preferences, called ‘‘unambiguous preference’’,and show that it can be represented by a set of probabilities. We provide such set with a simpledifferential characterization, and argue that it is a behavioral representation of the‘‘ambiguity’’ that the DM may perceive. Given such revealed ambiguity, we provide arepresentation of ambiguity attitudes. We also characterize axiomatically a special case of ourdecision model, the ‘‘a-maxmin’’ expected utility model.r 2003 Elsevier Inc. All rights reserved.JEL classification: D80; D81Keywords: Ambiguity; Unambiguous preference; Clarke differentials; a-maxmin expected utilityIntroductionWhen requested to state their maximum willingness to pay for two pairs ofcomplementary bets involving future temperature in San Francisco and IstanbulARTICLE IN PRESSCorresponding author.E-mail addresses: [email protected] (P. Ghirardato), [email protected](F. Maccheroni), [email protected] (M. Marinacci).URL: http://web.econ.unito.it/gma.0022-0531/$ - see front matter r 2003 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2003.12.004(and identical prize of $ 100 in case of a win) 90 pedestrians on the University ofCalifornia at Berkeley campus were on average willing to pay about $ 41 for the twobets on San Francisco temperature, and $ 25 for the two bets on Istanbultemperature. That is, on average they would have paid almost $ 16 more to bet onthe (familiar) San Francisco temperature than on the (unfamiliar) Istanbultemperature (Fox and Tversky [15, Study 4]).This striking pattern of preferences is by no means peculiar to the inhabitants ofthe Bay Area. Ever since the seminal thought experiment of Ellsberg [11], it has beenacknowledged that the awareness of missing information, ‘‘ambiguity’’ in Ellsberg’sterminology, affects subjects’ willingness to bet. And several experimental papers,the cited [15] being just one of the most recent ones, have found significant evidenceof ambiguity affecting decision making (see [25] for a survey). Though Ellsbergemphasized the relevance of aversion to ambiguity, later work has shown that thereaction to ambiguity is not systematically negative. Examples have been producedin which subjects tend to be ambiguity loving, rather than averse (e.g., Heath andTversky [23]’s ‘‘competence hypothesis’’ experiments). However, the availableevidence does show unequivocally that ambiguity matters for choice.The benchmark decision model of subjective expected utility (SEU) maximizationis not equipped to deal with this phenomenon: An agent who maximizes SEUexhibits no care about ambiguity. Therefore, theory has followed experiment.Several decision models have been proposed which extend SEU in order to allow arole for ambiguity in decision making. Most notable are the ‘‘maxmin expectedutility with multiple priors’’ (MEU) model of Gilboa and Schmeidler [22], whichallows the agent’s beliefs to be represented by a set of probabilities, and the‘‘Choquet expected utility’’ (CEU) model of Schmeidler [34], which allows theagent’s beliefs to be represented by a unique but nonadditive probability. Thesemodels have been employed with success in understanding and predicting behaviorin activities as diverse as investment [13], labor search [32] or voting [16].The objective of this paper is to show how to model formally ambiguity, and adecision maker (DM)’s response to it, in the context of a general decision model(that, for instance, encompasses MEU and CEU). It is an objective that in our viewhas not yet been fully achieved. In fact, as we discuss below, the existing literaturehas either focused on narrower models, or has not—within the limits of a traditionaldecision-theoretic setting—produced a description of ambiguity as complete as theone offered here.The intuition behind our approach can be explained in the context of the ‘‘3-color’’ experiment of Ellsberg. Suppose that a DM is faced with an urn containing 90balls which are either red, blue or yellow. The DM is told that exactly 30 of the ballsare red. If we offer him the choice between a bet r that pays $ 10 if a red ball isextracted, and the bet b that pays $ 10 if a blue ball is extracted, he may display thepreferencergb:On the other hand, let y denote the bet that pays $ 10 if a yellow ball is extracted, andsuppose that we offer him the choice between the ‘‘mixed’’ act ð12Þr þð12Þy and theARTICLE IN PRESSP. Ghirardato et al. / Journal of Economic Theory 118 (2004) 133–173134‘‘mixed’’ act ð12Þb þð12Þy: Then, we might observe12r þ12y!12b þ12y;a