perceived ambiguity and relevant measures we axiomatize preferences that can be represented by a monotonic aggregation of subjective expected utilities generated by a utility function and some set of i i d probability measures over a product state space s for such preferences we define relevant measures show that they are treated as if they were the only marginals possibly governing the state space and connect them with the measures appearing in the aforementioned representation these results allow us to interpret relevant measures as reflecting part of perceived ambiguity meaning subjective uncertainty about probabilities over states under mild conditions we show that increases or decreases in ambiguity aversion cannot affect the relevant measures this property necessary for the conclusion that these measures reflect only perceived ambiguity distinguishes the set of relevant measures from the leading alternative in the literature we apply our findings to a number of well known models of ambiguity sensitive preferences for each model we identify the set of relevant measures and the implications of comparative ambiguity aversion

a paradox for the smooth ambiguity model of preference two ellsberg style thought experiments are described that reflect on the smooth ambiguity decision model developed by klibanoff marinacci and mukerji 2005 the first experiment poses difficulties for the model s axiomatic foundations and as a result also for its interpretation particularly for the claim that the model achieves a separation between ambiguity and the attitude toward ambiguity given the problematic nature of its foundations the behavioral content of the model and how it differs from multiple priors for example are not clear the second thought experiment casts some light on these questions

algebras for the two sphere and the three sphere groups of compact simple lie groups the infinite dimensional lie algebras corresponding to the lie groups of smooth maps from two and three spheres to compact simple lie groups are studied the problem of their central extension is solved the problem of the existence of semidirect sum algebras containing these and the algebras of the groups of diffeomorphisms on the two and three spheres is treated

two stage lotteries without the reduction axiom preference relations over two stage lotteries are analyzed empirical evidence indicates that decisionmakers do not always behave in accordance with the reduction of compound lotteries axiom but they seem to satisfy a compound independence axiom although the reduction and the compound independence axioms together with continuity imply expected utility theory each of them by itself is compatible with all possible preference relations over simple lotteries using these axioms the author analyzes three different versions of expected utility for two stage lotteries the author suggests several different compound dominance axioms as possible replacements of the reduction axiom which are strictly weaker than the reduction of compound lotteries axiom copyright 1990 by the econometric society

decision analysis under ambiguity abstract in selecting the preferred course of action decision makers are often uncertain about one or more probabilities of interest the experimental literature has ascertained that this uncertainty ambiguity might affect decision makers preferences then the decision maker might wish to incorporate ambiguity aversion in the analysis we investigate the modeling ambiguity attitudes in the solution of decision analysis problems through functionals well established in the decision theory literature we obtain the multiple event problems for subjective expected utility smooth ambiguity and maximin decision makers this allows us to establish the conditions under which these alternative decision makers face equivalent problems results for certainty equivalents and risk premia in the presence of both risk and ambiguity aversion are obtained a recent generalization of the classical arrow pratt quadratic approximation allows us to quantify the portions of a premium due to risk plxinsert and to ambiguity aversion the numerical implementation of the objective functions is addressed showing that all functionals can be estimated at no additional burden through monte carlo simulation the well known carter racing case study is addressed quantitatively to demonstrate the findings

expected uncertain utility theory we introduce and analyze expected uncertain utility euu theory a prior and an interval utility characterize an euu decision maker the decision maker transforms each uncertain prospect into an interval valued prospect that assigns an interval of prizes to each state she then ranks prospects according to their expected interval utilities we define uncertainty aversion for euu use the euu model to address the ellsberg paradox and other ambiguity evidence and relate euu theory to existing models

a smooth model of decision making under ambiguity we propose and axiomatize a model of preferences over acts such that the decision maker prefers act f to act g if and only if e e u f e e u g where e is the expectation operator u is a vn m utility function is an increasing transformation and is a subjective probability over the set of probability measures that the decision maker thinks are relevant given his subjective information a key feature of our model is that it achieves a separation between ambiguity identified as a characteristic of the decision maker s subjective information and ambiguity attitude a characteristic of the decision maker s tastes we show that attitudes towards risk are characterized by the shape of u as usual while attitudes towards ambiguity are characterized by the shape of we also derive x 1 e x as the special case of constant ambiguity aversion ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures this characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by epstein zhang 2001 and ghirardato marinacci 2002 one advantage of this model is that the welldeveloped machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes the model is also distinct from many in the literature on ambiguity in that allows smooth rather than kinked indifference curves this leads to different behavior and improved tractability while still sharing the main features e g ellsberg s paradox etc the maxmin eu model e g gilboa and schmeidler 1989 with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion two illustrative applications to portfolio choice are offered

subjective probability without monotonicity or how machina s mom may also be probabilistically sophisticated if an agent s preferences over subjectively uncertain acts are consistent with his or her having a subjective probability distribution over the states of nature then those preferences can induce consistent preferences over objectively risky lotteries such probabilistically sophisticated behavior thus allows us to treat decision making under situations of uncertainty in an analogous manner to those under risk this paper first characterizes exactly what probabilistic sophistication entails for an agent s beliefs about the likelihood of states of nature secondly it presents characterizations of probabilistically sophisticated individuals whose induced lottery preferences obey neither the independence axiom unlike savage 1954 1972 nor a monotonicity property that shares some of the nature of independence unlike machina and schmeidler 1992

axiomatic foundations of multiplier preferences this paper axiomatizes the robust control criterion of multiplier preferences introduced by hansen and sargent 2001 the axiomatization relates multiplier preferences to other classes of preferences studied in decision theory some properties of multiplier preferences are generalized to the broader class of variational preferences recently introduced by maccheroni marinacci and rustichini 2006 the paper also establishes a link between the parameters of the multiplier criterion and the observable behavior of the agent this link enables measurement of the parameters on the basis of observable choice data and provides a useful tool for applications

vector expected utility and attitudes toward variation this paper proposes a model of decision under ambiguity deemed vector expected utility or veu in this model an uncertain prospect or savage act is assessed according to a a baseline expected utility evaluation a nd b anadjustment that reflects the individual s perception of ambiguity and her attitudes toward it the adjustment is itself a function of the act s exposure to distinct sources of ambiguity as well as its variability the key elements of the veu model are a baseline probability and a collection of random variables or adjustment factors which represent acts exposed to distinct ambiguity sources and also reflect complementarities among ambiguous events the adjustment to the baseline expected utility evaluation of an act is a function of the covariance of its utility profile with each adjustment factor which reflects exposure to the corresponding ambiguity source a behavioral characterization of the veu model is provided furthermore an updating rule for veu preferences is proposed and characterized the suggested updating rule facilitates the analysis of sophisticated dynamic choice with veu preferences

risk ambiguity and the separation of utility and beliefs we introduce a general model of static choice under uncertainty arguably the weakest model achieving a separation of cardinal utility and a unique representation of beliefs most of the nonexpected utility models existing in the literature are special cases of it such separation is motivated by the view that tastes are constant whereas beliefs change with new information the model has a simple and natural axiomatization elsewhere forthcoming we show that it can be very helpful in the characterization of a notion of ambiguity aversion as separating utility and beliefs allows us to identify and remove aspects of risk attitude from the decision maker s behavior here we show that the model allows us to generalize several results on the characterization of risk aversion in betting behavior these generalizations are of independent interest as they show that some traditional results for subjective expected utility preferences can be formulated only in terms of binary acts

deviations from the circular law consider ginibre s ensemble of n n non hermitian random matrices in which all entries are independent complex gaussians of mean zero and variance open image in new window as n the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane in this note we describe fluctuations about this circular law first we obtain finite n formulas for the covariance of certain linear statistics of the eigenvalues asymptotics of these objects coupled with a theorem of costin and lebowitz then result in central limit theorems for a variety of these statistics

affine realizations of sphere algebras the group map m g of smooth mappings from a compact manifold m to a simple lie group g is an infinite dimensional lie group the simplest case is when m s1 the central extension of the corresponding lie algebra map s1 g with g the lie algebra of g is an affine kac moody algebra the representations of map s1 g and of its central extension have been thoroughly investigated less work has been done for higher dimensional m we discuss the particular cases when m is the two and three spheres sn n 2 3 in this work we construct particular realizations of the centrally extended map sn g from a given realization of an affine kac moody algebra

some examples of the algebra of flows the algebra of the group of smooth maps from a manifold m to a compact simple lie group g is studied for two cases the first is when m is the double coset so d r so d 1 r so d r the corresponding maps are those from a d sphere to g that are invariant under left translations by elements from so d r in the second example m is a two dimensional torus the problem of central extension of these algebras is solved for the first example no central extension is possible for the second the number of independent central extensions is infinite

the dual theory of choice under risk in this essay a new theory of choice under risk is being proposed it is a theory which in a sense that will become clear is dual to expected utility theory hence the title dual theory risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility except that the roles of payments and probabilities are reversed this theme the reversal of the roles of probabilities and payments will recur throughout the paper i should emphasize that playing games with probabilities masquerading as payments and payments masquerading as probabilities is not my object rather i hope to convince the reader that the dual theory has intrinsic economic significance and that in some areas its predictions are superior to those of expected utility theory while in other areas the reverse will be the case two reasons have prompted me to look for an alternative to expected utility theory the first reason is methodological in expected utility theory the agent s attitude towards risk and the agent s attitude towards wealth are forever bonded together at the level of fundamental principles risk aversion and diminishing marginal utility of wealth which are synonymous under expected utility theory are horses of different colors the former expresses an attitute towards risk increased uncertainty hurts while the latter expresses an attitude towards wealth the loss of a sheep hurts more when the agent is poor than when the agent is rich a question arises therefore as to whether these two notions can be kept separate from each other in a full fledged theory of cardinal utility the dual theory will have this property the second reason that leads me to look for an alternative to expected utility theory is empirical behavior patterns which are systematic yet inconsistent with expected utility theory have often been observed two prominent references among many others are allais 1953 and kahneman tversky 1979 so deeply

risk aversion in cumulative prospect theory this paper characterizes the conditions for strong risk aversion and second order stochastic dominance for cumulative prospect theory strong risk aversion implies a convex weighting function for gains and a concave one for losses it does not necessarily imply a concave utility function the latter does follow if the weighting functions are continuous by investigating the exact relationship between loss aversion and strong risk aversion a natural index for the degree of loss aversion is derived

characters and a verlinde type formula for symmetric hopf algebras abstract we study certain aspects of finite dimensional non semisimple symmetric hopf algebras h and their duals h we focus on the set i h of characters of projective h modules which is an ideal of the algebra of cocommutative elements of h this ideal corresponds via a symmetrizing form to the projective center higman ideal of h which turns out to be h where is an integral of h and is the left adjoint action of h on itself we describe h via primitive and central primitive idempotents of h we also show that it is stable under the quantum fourier transform our best results are obtained when h is a factorizable ribbon hopf algebra over an algebraically closed field of characteristic 0 in this case h is also the image of i h under a translated drinfel d map we use this fact to prove the existence of a steinberg like character the above ingredients are used to prove a verlinde type formula for h

atoms and dobrakov submeasures in effect algebras the present paper deals with the study of the notion of dobrakov submeasure m defined on an effect algebra l with values in 0 we have established the equivalence of the following three properties for a dobrakov submeasure m defined on a s complete effect algebra l i m is atomless ii m has saks property iii m has darboux property we have also studied the concept of an atom of a function m defined on an effect algebra l with values in 0 and finally we have proved for a monotone m continuous and semicontinuous function m defined on a s complete effect algebra l that m is non atomic if and only if m is atomless

on the smooth ambiguity model a reply epstein 2009 describes three ellsberg style thought experiments and argues that they pose difficulties for the smooth ambiguity model of decision making under uncertainty developed by klibanoff marinacci and mukerji 2005 we revisit these thought exeperiments and find to the contrary that they either point to strengths of the smooth ambiguity model compared to other models such as the maximum expected utility model gilboa and schmeidler 1989 or in the case of one thought experiment raise criticisms that apply equally to a broad range of current ambiguity models

the midweight method to measure attitudes toward risk and ambiguity this paper introduces a parameter free method for measuring the weighting functions of prospect theory and rank dependent utility these weighting functions capture risk attitudes subjective beliefs and ambiguity attitudes our method called the midweight method is based on a convenient way to obtain midpoints in the weighting function scale it can be used both for risk known probabilities and for uncertainty unknown probabilities the resulting integrated treatment of risk and uncertainty is particularly useful for measuring ambiguity i e the difference between uncertainty and risk compared to existing methods to measure weighting functions and attitudes toward uncertainty and ambiguity our method is more efficient and can accommodate violations of expected utility under risk an experiment demonstrates the tractability of our method yielding plausible results such as ambiguity aversion for moderate and high likelihoods but ambiguity seeking for low likelihoods as predicted by ellsberg this paper was accepted by george wu decision analysis

uncertainty aversion with second order utilities and probabilities subjective expected utility theory does not distinguish between attitudes toward uncertainty ambiguous probabilities and attitudes toward risk unambiguous probabilities both are explained in terms of nonlinear utility for money rather than properties of events per se hence the decision maker displays the same attitude toward all sources of risk and uncertainty there is ample evidence that real decision makers do not always behave or even wish to behave in this way and instead they often distinguish between risk and uncertainty as in ellsbergs 1961 paradox this paper presents a simple axiomatic model of nonneutral attitudes toward uncertainty and a behavioral test for uncertainty aversion that is applicable even if utility is state dependent the decision maker may display different degrees of aversion toward gambles on different kinds of events e g being systematically more averse toward gambles on events whose probabilities are more ambiguous for such a decision maker the elicitation of preferences among objective gambles may not yield the correct measure of risk aversion for modeling real world decisions

imprecise data sets as a source of ambiguity a model and experimental evidence in many circumstances evaluations are based on empirical data however some observations may be imprecise meaning that it is not entirely clear what occurred in them we address the question of how beliefs are formed in these situations the individual in our model is essentially a frequentist he first makes a subjective judgment about the occurrence of the event for each imprecise observation this may be any number between zero and one he then evaluates the event by its subjective frequency of occurrence our model connects the method of processing imprecise observations with the individual s attitude toward ambiguity an individual who in imprecise observations puts low high weight on the possibility that an event occurred is ambiguity averse loving an experiment supports the main assertions of the model with precise data subjects behave as if there were no ambiguity whereas with imprecise data subjects turn out to be ambiguity averse this paper was accepted by brad barber teck ho and terrance odean special issue editors

alpha as ambiguity robust mean variance portfolio analysis we derive the analogue of the classic arrow pratt approximation of the certainty equivalent under model uncertainty as described by the smooth model of decision making under ambiguity of klibano marinacci and mukerji 2005 we study its scope by deriving a tractable meanvariance model adjusted for ambiguity and solving the corresponding portfolio allocation problem in the problem with a risk free asset a risky asset and an ambiguous asset we nd that portfolio rebalancing in response to higher ambiguity aversion only depends on the ambiguous asset s alpha setting the performance of the risky asset as benchmark in particular a positive alpha corresponds to a long position in the ambiguous asset a negative alpha corresponds to a short position in the ambiguous asset and greater ambiguity aversion reduces optimal exposure to ambiguity the analytical tractability of the enhanced arrow pratt approximation renders our model especially well suited for calibration exercises aimed at exploring the consequences of model uncertainty on equilibrium asset prices

subjective probabilities on subjectively unambiguous events this paper suggests a behavioral definition of subjective ambiguity in an abstract setting where objects of choice are savage style acts then axioms are described that deliver probabilistic sophistication of preference on the set of unambiguous acts in particular both the domain and the values of the decision maker s probability measure are derived from preference it is argued that the noted result also provides a decision theoretic foundation for the knightian distinction between risk and ambiguity

on policy capturing with fuzzy measures policy capturing methods generally apply linear regression analysis to model human judgment in this paper we examine the application of fuzzy set and fuzzy measure theories to obtain subjective descriptions of cue importance for policy capturing at the heart of the approach is a method of learning fuzzy measures the shapley values associated with the fuzzy measures provide a basis for comparison with the results of linear regression however the fuzzy measure theoretical approach provides additional insight into interaction effects corresponding to the nonlinear noncompensatory nature of the underlying decision model to illustrate the methodology we estimated the importance of factors and the interactions among them that influence decisions related to strategic investments in telecommunications infrastructure and compared the results from the fuzzy approach to those obtained from traditional statistical methods

characterizations of smooth ambiguity based on continuous and discrete data in the anscombe aumann setup we provide conditions for a collection of observations to be consistent with a well known class of smooth ambiguity preferences klibanoff p marinacci m mukerji s 2005 a smooth model of decision making under ambiguity econometrica 73 6 1849 1892 each observation is assumed to take the form of an equivalence between an uncertain act and a certain outcome we provide three results that describe these conditions for data sets of different cardinality our findings uncover surprising links between the smooth ambiguity model and classic mathematical results in complex and functional analysis

fuzzy logic based generalized decision theory with imperfect information the existing decision models have been successfully applied to solving many decision problems in management business economics and other fields but nowadays arises a need to develop more realistic decision models the main drawback of the existing utility theories starting from von neumann moregnstern expected utility to the advanced non expected models is that they are designed for laboratory examples with simple well defined gambles which do not adequately enough reflect real decision situations in real life decision making problems preferences are vague and decision relevant information is imperfect as described in natural language nl vagueness of preferences and imperfect decision relevant information require using suitable utility model which would be fundamentally different to the existing precise utility models precise utility models cannot reflect vagueness of preferences vagueness of objective conditions and outcomes imprecise beliefs etc the time has come for a new generation of decision theories in this study we propose a decision theory which is capable to deal with vague preferences and imperfect information the theory discussed here is based on a fuzzy valued non expected utility model representing linguistic preference relations and imprecise beliefs

ellsberg revisited an experimental study an extension to ellsberg s experiment demonstrates that attitudes to ambiguity and compound objective lotteries are tightly associated the sample is decomposed into three main groups subjective expected utility subjects who reduce compound objective lotteries and are ambiguity neutral and two groups that exhibit different forms of association between preferences over compound lotteries and ambiguity corresponding to alternative theoretical models that account for ambiguity averse or seeking behavior

quantum groups and deformations of preprojective algebras abstract let be a quiver containing finite cyclic blocks and be the deformed preprojective algebra of it is proved that a suitable quotient uq of for an appropriate vector is a finite dimensional quantum group some properties of uq are also obtained an isomorphism as hopf algebras is given between uq and a restricted quantum group u q sl 2

order statistics and ginibre s ensembles the moduli of the eigenvalues at the edge of ginibre s complex and quaternion gaussian random matrix ensembles are shown to respond to a limit theorem identical in nature to that for independent identically distributed sequences this is a companion work to ref 15 in which the limit law for the scaled spectral radius of these ensembles was identified

probabilistic sophistication and multiple priors we show that under fairly mild conditions a maximin expected utility preference relation is probabilistically sophisticated if and only if it is subjective expected utility this abstract was borrowed from another version of this item

pseudo atoms atoms and a jordan type decomposition in effect algebras abstract the aim of this paper is to introduce and investigate the concept of pseudo atoms of a real valued function m defined on an effect algebra l a few examples of pseudo atoms and atoms are given in the context of null additive null null additive and pseudo null additive functions and also some fundamental results for pseudo atoms under the assumption of null null additivity are established the notions of total variation m positive variation m and negative variation m of a real valued function m on l are studied elaborately and it is proved for a modular measure m which is of bounded total variation defined on a d lattice l that m is pseudo atomic or atomic if and only if its total variation m is pseudo atomic or atomic finally a jordan type decomposition theorem for an extended real valued function m of bounded total variation defined on an effect algebra l is proved and some properties on decomposed parts of m such as continuity from below pseudo atomicity or atomicity and being measure are discussed a characterization for the function m to be of bounded total variation is established here and used in proving above mentioned jordan type decomposition theorem

pseudo d lattices and separating points of measures in 1986 a basile and h weber proved that every countable family m of additive measures on a complete boolean ring r admits a dense g set of separating points with respect to a suitable topology in the following two cases either i each element of m is s bounded and whenever m are distinct the quotient of r modulo n is infinite or ii each element of m is continuous in this paper we consider modular measures on lattice ordered pseudo effect algebras using topological methods we extend the result of basile and weber in a way that allows to unify the above two cases i and ii this gives a new contribution also in the classical setting of algebras of sets

on the structure of nonsemisimple hopf algebras the left module structure of finite dimensional quantum algebras is analysed using the theory of primitive idempotents in particular a complete structural result in terms of principal projective indecomposable modules p i m is given in the case of q a root of 1 by finding a complete set of primitive idempotents the structure of p i m is analysed in detail the jacobson radical of the algebra is investigated and its significance in the study of nonsemisimple symmetries in physical systems is discussed

ambiguity learning and asset returns we develop a consumption based asset pricing model in which the representative agent is ambiguous about the hidden state in consumption growth he learns about the hidden state under ambiguity by observing past consumption data his preferences are represented by the smooth ambiguity model axiomatized by klibanofi et al 2005 2006 unlike the standard bayesian theory this utility model implies that the posterior of the hidden state and the conditional distribution of the consumption process given a state cannot be reduced to a predictive distribution by calibrating the ambiguity aversion parameter the subjective discount factor and the risk aversion parameter with the latter two values between zero and one our model can match the flrst moments of the equity premium and riskfree rate found in the data in addition our model can generate a variety of dynamic asset pricing phenomena including the procyclical variation of price dividend ratios the countercyclical variation of equity premia and equity volatility and the mean reversion and long horizon predictability of excess returns

quantum double of uq sl2 0 abstract let u q sl 2 be the quantized enveloping algebra associated to the simple lie algebra sl 2 in this paper we study the quantum double d q of the borel subalgebra u q sl 2 0 of u q sl 2 we construct an analogue of kostant lusztig z v v 1 form for d q and show that it is a hopf subalgebra we prove that over an algebraically closed field every simple d q module is the pull back of a simple u q sl 2 module through certain surjection from d q onto u q sl 2 and the category of finite dimensional weight d q modules is equivalent to a direct sum of k copies of the category of finite dimensional weight u q sl 2 modules as an application we recover in a conceptual way chen s results h x chen irreducible representations of a class of quantum doubles j algebra 225 2000 391 409 as well as radford s results d e radford on oriented quantum algebras derived from representations of the quantum double of a finite dimensional hopf algebras j algebra 270 2003 670 695 on the quantum double of taft algebra our main results allow a direct generalization to the quantum double of the borel subalgebra of the quantized enveloping algebra associated to arbitrary cartan matrix

a simple axiomatization of nonadditive expected utility textabstractthis paper provides an extension of savage s subjective expected utility theory for decisions under uncertainty it includes in the set of events both unambiguous events for which probabilities are additive and ambiguous events for which probabilities are permitted to be nonadditive the main axiom is cumulative dominance which adapts stochastic dominance to decision making under uncertainty we derive a choquet expected utility representation and show that a modification of cumulative dominance leads to the classical expected utility representation the relationship of our approach with that of schmeidler who uses a two stage formulation to derive choquet expected utility is also explored our work may be viewed as a unification of schmeidler 1989 and gilboa 1987

ambiguity in the small and in the large this paper considers local and global multiple prior representations of ambiguity for preferences that are i monotonic ii bernoullian that is admit an affine utility representation when restricted to constant acts and iii locally lipschitz continuous we do not require either certainty independence or uncertainty aversion we show that the set of priors identified by ghirardato maccheroni and marinacci s 2004 unambiguous preference relation can be characterized as a union of clarke differentials we then introduce a behavioral notion of locally better deviation at an act and show that it characterizes the clarke differential of the preference representation at that act these results suggest that the priors identified by these preference statements are directly related to local optimizing behavior

tensor product decomposition and cg coefficients for nonsemisimple hopf algebras the tensor product of two modules q a root of 1 is decomposed into indecomposable summands for both irreducible and indecomposable modules clebsch gordan coefficients in the general case are computed an apparently new identity is derived and some possible applications are conjectured

state dependent utilities several axiom systems for preference among acts lead to the existence of a unique probability and a state independent utility such that acts are ranked according to their expected utilities these axioms have been used as a foundation for bayesian decision theory and the subjective probability calculus in this paper we note that the uniqueness of the probability is relative to the choice of what counts as a constant outcome although it is sometimes clear what should be considered constant there are many cases in which there are several possible choices each choice can lead to a different unique probability and utility by focusing attention on state dependent utilities we determine conditions under which a truly unique probability and utility can be determined from an agent s expressed preferences among acts suppose that an agent s preference can be represented in terms of a probability p and a utility u that is the agent prefers one act to another if and only if the expected utility of the one act is higher than that of the other there are many other equivalent representations in terms of probabilities q which are mutually absolutely continuous with p and state dependent utilities v which differ from u by possibly different positive affine transformations in each state of nature an example is described in which two different but equivalent state independent utility representations exist for the same preference structure what differs between the two representations is which acts count as constants the acts involve receiving different amounts of one or the other of two currencies and the states are different exchange rates between the currencies it is easy to see how it would not be possible for constant amounts of both currencies to simultaneously have constant values across the different states savage foundations of statistics john wiley new york 1954 sec 5 5 discovered a situation in which two seemingly equivalent preference structures are represented by different pairs of probability and utility savage attributed the phenomenon to the construction of a small world we show that the small world problem is just another example of two different but equivalent representations treating different acts as constants finally we prove a theorem similar to one of karni decision making under uncertainty harvard university press cambridge 1985 that shows how to elicit a unique state dependent utility and does not assume that there are prizes with constant value to do this we define a new hypothetical kind of act in which both the prize to be awarded and the state of nature are determined by an auxiliary experiment

dilation for sets of probabilities stable url http links jstor org sici sici 0090 5364 28199309 2921 3a3 3c1139 3adfsop 3e2 0 co 3b2 vthe annals of statistics is currently published by institute of mathematical statistics your use of the jstor archive indicates your acceptance of jstor s terms and conditions of use available athttp www jstor org about terms html jstor s terms and conditions of use provides in part that unless you have obtainedprior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content inthe jstor archive only for your personal non commercial use please contact the publisher regarding any further use of this work publisher contact information may be obtained athttp www jstor org journals ims html each copy of any part of a jstor transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission the jstor archive is a trusted digital repository providing for long term preservation and access to leading academicjournals and scholarly literature from around the world the archive is supported by libraries scholarly societies publishers and foundations it is an initiative of jstor a not for profit organization with a mission to help the scholarly community takeadvantage of advances in technology for more information regarding jstor please contact support jstor org http www jstor orgtue mar 4 10 39 53 2008

on the determination of subjective probability by choices the paper explores the uniqueness properties of the subjective probabilities in two axiomatizations of state dependent preferences karni schmeidler and vind s ksv 1983 system depends on selecting an arbitrary auxiliary probability and as such does not guarantee the uniqueness of the derived subjective probability however an axiom system initially designed by karni and schmeidler ks 1981 and further elaborated upon here does guarantee the desired uniqueness as well as a useful property of stability of the derived solution when the preference relation displays state independence even the ks probabilities may not agree with those derived from the classic anscombe aumann aa 1963 theorem however we claim that in this case the ks rather than the aa probabilities are the appropriate representation of the agent s beliefs

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

experiments on compound risk in relation to simple risk and to ambiguity we conduct experiments measuring individual behavior under compound risk simple risk and ambiguity we focus on 1 treatment of compound risks relative to simple risks and 2 the relationship between compound risk attitudes and ambiguity attitudes we find that compound risks are valued differently than corresponding reduced simple risks these differences measure compound risk attitudes these attitudes display more aversion as the reduced probability of the winning event increases like halevy halevy y 2007 ellsberg revisited an experimental study econometrica 75 503 536 we find an association between compound risk reduction and ambiguity neutrality however in contrast to the almost perfect identification in halevy s data we find a substantially weaker relation in both directions first a majority of our ambiguity neutral subjects fail to reduce compound risk second almost a quarter of our subjects who reduce compound risk are nonneutral to ambiguity all of the latter come from the more quantitatively sophisticated part of our subject pool data as supplemental material are available at http dx doi org 10 1287 mnsc 2014 1953 this paper was accepted by peter wakker decision analysis

ambiguity and second order belief anscombe and aumann 1963 wrote a classic characterization of subjective expected utility theory this paper employs the same domain for preference and a closely related but weaker set of axioms to characterize preferences that use second order beliefs beliefs over probability measures such preferences are of interest because they accommodate ellsberg type behavior