Familiarity Breeds Completeness

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1 Familiarity Breeds Completeness Edi Karni February 20, 2013 Abstract This is a study of the representations of subjective expected utility preferences that admits state-dependent incompleteness, and subjective expected utility preferences displaying noncomparability of acts from distinct sources. The notions familiar events and sources are defined and characterized. The relation greater familiarity on sources and increasing familiairity of a source are also defined and characterized. Keywords: Incomplete preferences, source familiarity, event familiarity, state-dependent incomplete preferences JEL classification: D81 Department of Economics, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA. karni@jhu.edu Ibenefited from comments and conversations with Tsogbadral Galaabaatar, Evren Ozgur, Tomasz Strzalecki, and especially Edward Schlee, for which I am grateful. 1

2 It is conceivable and may even in a way be more realistic to allow for cases where the individual is neither able to state which of two alternatives he prefers nor that they are equally desirable. von Neumann and Morgenstern (1947). Of all the axioms of utility theory, the completeness axiom is perhaps the most questionable. Like others of the axioms, it is inaccurate as a description of real life; but unlike them, we find it hard to accept even from a normative viewpoint. Aumann (1962). 1 Introduction In general, the representations of subjective expected utility theory with incomplete preferences have the form of multi-prior expected multi-utility. 1 The set of priors represents the incompleteness of beliefs and the set of utility functions that of tastes. Specific models admit complete preferences on a subset of acts leading to more restrictive representations. For example, in Bewley s (2002) model of Knightian uncertainty the restriction of the preference relation to constant acts is complete, giving rise to multi-prior expected utility representation. In Galaabaatar and Karni (2013) the axiom of complete beliefs implies that if the preference relation between bets and constant acts is complete then its representation takes the form of subjective expected multi-utility representation. This paper explores the underlying structures of incomplete preference relations under uncertainty that are complete over distinct classes of acts. The motivation for this inquiry is the presumption that some classes of acts share features that make them more readily comparable, while acts belonging to distinct classes are not comparable. To begin with I examine subsets of acts that agree outside a given event and display event-dependent incompleteness. The interest in this investigation stems from the perception that the conditional preferences may be incomplete due to lack of familiarity with the underlying events. Event unfamiliarity might impact the decision maker s confidence in his own tastes when facing a choice among acts that agree outside the said event. This type of situations is prevalent in medical decision making. Consider, for example, 1 See Seidenfeld, et. al. (1995), Nau (2006) and Galaabaatar and Karni (2013). 2

3 a person diagnosed as having prostate cancer and must choose between alternative treatments, say, surgery and radiation therapy. Suppose that the patient is informed about the likelihood of being cured and the probabilities of other potential outcomes, including incontinence and impotency, associated with each of these treatments. Conceivably, the patient believes the likelihoods of the different outcomes under the differenttreatments asfacts, and yet finds it difficult to express clear preferences among the treatments because the potential outcomes include states of health that he has never experienced. In other words, it is quite natural that the patient is not clear about his own preferences conditional on the unfortunate events in which one of the bad outcomes obtains. In this instance, the indecisiveness is due to incompleteness of tastes rather than that of beliefs. Situations that require choice among acts whose payoffs are contingent on events that are more and less familiar, are both important and prevalent. Other than medical decision making, these situations include decisions about health insurance, long-term care insurance, disability insurance, career choice and choice of education, to mention but a few examples. In all of these examples, it is reasonable to suppose that the probabilities of the relevant events are given by the relative frequencies of their occurrence in the population and the incompleteness of the preferences is attribute to incompleteness of tastes. Adifferenttypeofactsonwhichthepreferencerelationmaybecomplete, or incomplete to different degrees, are sets of acts belonging to a familiar source. 2 It has been suggested that source preference, or familiarity bias, might explain the observed tendency of investors to forego portfolio diversification in order to invest in what they perceive to be more familiar companies (Heath and Tversky (1991)) or more familiar institutional environment, leading to domestic bias in financial and other investments, (Huberman (2001)). Rigotti and Shannon (2005) explore the general equilibrium implications, including the indeterminacy of the equilibrium prices and the allocation efficiency, due to incomplete beliefs. 3 More recently, Chew et. al. (2012) provide genetic evidence in support of the view that a sense of familiarity or competence or lack of them, underlie both ambiguity aversion and familiarity bias. In these instances, it is reasonable to suppose that tastes regarding 2 See Chew and Sagi (2008) for a model of attitudes toward source uncertainty. 3 They describe familiar sources as risky events and unfamiliar sources as uncertain events. 3

4 the payoffs of the investments are complete and that source unfamiliarity underlies the incompleteness of beliefs. Building upon Galaabaatar and Karni (2013), the main results of this paper are representations of incomplete preferences whose restrictions to familiar events and sources are complete. The study of representations of incomplete preferences due lack of familiarity of events, is the subject matter of the next section. Section 3, explores the representations of incomplete preferences due to lack of familiarity of sources. Concluding remarks appear in section 4. The proofs are collected in section 5. 2 Conditionally Complete Preferences 2.1 The analytical framework and basic preference structure Invoking the analytical framework of Anscombe and Aumann (1963), let be a finite set of states and denote be a finite set of outcomes. Let ( ) denote the set of all probability measures on Let be the set of all functions from to ( ) Elements of are referred to as acts. For every ( ) denotes the constant act whose payoff is (that is, ( ) = for all ). For each 0 ( ) and [0 1] define +(1 ) 0 ( ) by ( +(1 ) 0 )( ) = ( )+(1 ) 0 ( ) for all For all 0 and [0 1], define +(1 ) 0 by ( +(1 ) 0 )( ) = ( )+(1 ) 0 ( ) for all where the convex mixture ( )+(1 ) 0 ( ) is defined as above. Under this definition is a convex subset of the linear space R Subsets of are referred to as events. Let  be a binary relation on. The set is said to be Â-bounded if there exist and in such that   for all { } The following axioms depict the basic structure of the preference relation  and are maintained throughout. These axioms are well-known and require no elaboration. (A.1) (Strict partial order) The preference relation  is transitive and irreflexive. (A.2) (Archimedean) For all if  and  then + (1 )  and  +(1 ) for some (0 1). 4

5 (A.3) (Independence) Forall and (0 1]  if and only if +(1 )  +(1 ) The difference between the preference structure above and that of expected utility theory is that the induced relation (  ) is reflexive but not necessarily transitive (hence, it is not necessarily a preorder). Moreover, (  ) and (  ) does not imply that and are indifferent, rather they may be incomparable. For every denote by ( ) :={  } and ( ) :={  } the (strict) upper and lower contour sets of respectively. The relation  is said to be convex if the upper contour set is convex. If is Â-bounded then for 6=, ( ) and ( ) have nonempty algebraic interior in the linear space generated by. A binary relation on  on satisfying (A.1) (A.3), is convex. Moreover, the lower contour set is also convex Event completeness and conditional dominance To formalize the idea of conditionally complete preferences I use the following notations. For every event and acts and 0 define the act 0 by 0 ( ) = ( ) if and 0 ( ) = 0 ( ) otherwise. Denote by  the restriction of  to the subset of act { 0 } Note that (A.1)- (A.3), implies that  is well-defined (that is, it is independent of 0 ). 5 Define the weak conditional preference relation < on as follows: For all { 0 } < if (  ) Familiarity of events is a subjective attribute, revealed by choice. I assume that familiarity of an event implies completeness of the (weak) preference relation over acts that agree on the unfamiliar event, := Definition 1: An event, is familiar if the conditional preference relation  is negatively transitive If is a familiar event then < is complete and transitive. 6 The next axiom requires that, conditional on familiar events, the preference relation satisfies state independence. Formally, 4 The proof is by two applications of (A.3). 5 See Galaabaatar and Karni (2013) for a proof that (A.1)-(A.3) implies that  has additive separable representation. The additive separability across states, implies that  is well defined. 6 See Kreps (1988) proposition (2.4). 5

6 (A.4) (Conditional state independence) For every familiar event, and all 0 and ( )  if and only if 0  0 If is familiar and itself is not then, for some 0 00 the acts 0 and 00 are non-comparable. The incompleteness of the conditional preference relation  may itself be due to incompleteness of beliefs regarding the likelihoods of the different states in the unfamiliar event and/or to the decision maker s lack of confidence in his own tastes in this event. To represent preference relations that are incomplete on some events, I adopt a modified version of the dominance axiom of Galaabaatar and Karni (2013). Let be a familiar event. For each let denote the constant act whose payoff is ( ) in every state (that is, ( 0 )= ( ) for all 0 ). The axiom asserts that if an act, is strictly preferred over every act, for every then is strictly preferred over. To grasp the intuition underlying this assertion, note that for any possible consequence of in the event the act is an element of the lower contour set of Convexity of the lower contour sets implies that any convex combination of the consequences of is dominated by Think of as representing a set of such combinations whose elements correspond to the implicit set of conditional subjective probability distributions on that the decision maker might entertain. Since any such combination is dominated by so is Formally, (A.5) (Conditional dominance) For the maximal familiar event, and all if  for every then  Theorem 1 below shows that a preference relation satisfies the axioms (A.1) (A.5) if and only if there is a non-empty set of utility functions on and, corresponding to each utility function, a set of probability measures on such that, when facing a choice between two acts, the decision maker prefers the act that yields higher expected utility according to every utility function and every probability measure in the corresponding set. Moreover, if is a familiar event then the conditional preference relation,  has a subjective expected utility representation and for the corresponding unfamiliar event, the conditional preference relation,  has a multi-prior expected multiutility representation. To state the theorem I use the following notations: Given a familiar event, let Φ ( ) :={( ) V ( ) Π } where V ( ) is a nonempty 6

7 set of real-valued functions on and, for each V ( ) Π is a set of full-support probability measures on For every real-valued function, on and probability measure ( ) := P ( ) ( ) Theorem 1 (Representation: Existence) Let  be a binary relation on, then the following conditions are equivalent: ( ) is Â-bounded and  satisfies (A.1) (A.5). ( ) For every familiar event, there exists a real-valued function ˆ on and a set, Φ ( ) of probability-utility pairs such that, for all { } and ( ) Φ ( ), X ( ) ( ) ˆ +X ( ) ( ) X ( )( ( ) ˆ )+ X ( )( ( ) ) X ( )( ( ) ˆ )+ X ( )( ( ) ) and, for all 0  0 if and only if X ( )( ( ) ˆ )+X ( )( ( ) ) X ( )( 0 ( ) ˆ )+ X ( )( 0 ( ) ) (1) for all ( ) Φ ( ) Moreover, for all 0 V Π and every familiar event, the conditional probability measures, ( ) and 0 ( ), satisfy ( ) = 0 ( ) The preference relation depicted above may be thought of as displaying event-dependent incompleteness. This interpretation entails an equivalent formulation of the representation in (1). Let U ( ) :{ ( ; ) : R } be a set of state-dependent utility functions, each of which depends only on the event (that is, ( ; ) = ( ; 0 ) for all 0 and likewise for the complementary event ). Moreover, all the elements of U agree on (that is, ( ; ) = 0 ( ; ) for all 0 U ( ) and ). Then,  0 X ( )( ( ) ( ; )) X ( )( 0 ( ) ( ; )) (2) for all ( ) {( ) U ( ) Π } for each U ( ) Π is a set of full-support probability measures on 7

8 The idea of the proof is as follows: First, use (A.1) - (A.4), the completeness of the weak preference relation on familiar events, and the theorem of Anscombe and Aumann (1963) to obtain a subjective expected utility representation of the preference relation conditional on the familiar event, Second, invoke (A.5) and modify the arguments in the proof of Theorem 1 in Galaabaatar and Karni (2013) to obtain a multi-prior expected multi-utility representation of the preference relation conditional on the complementary event,. Third, use the set of probability measures to link the two representations. To state the uniqueness properties of the utilities and probabilities that figure in the representation in Theorem 1, I introduce the following additional notations. Let U be a set of real-valued functions on. Fix 0 and for each U define a real-valued function, ˆ on R by ˆ ( ) = ( ) ( 0 ) for all and Let U b be the normalized set of functions corresponding to U (that is, bu = {ˆ U}). Let be the vector in R such that ( ) =0for all if 6= and ( ) =1for all if =. Define = { R} Denote by h bui the closure of the convex cone in R generated by all the functions in U b and. For each ( ) Φ ( ) that figure in the representation, define a vector := ( ( ) ( )) ( ) in R Denote by W the set of all these vectors. Define h Φ \ ( )i = hwi. c Theorem 2 (Representation: Uniqueness) If Φ 0 ( ) ={( 0 0 ) 0 V, 0 Π 0 } and ˆ 0 represent  in the sense of (1), then h \ Φ 0 ( )i = h \ Φ ( )i and ˆ 0 is a positive affine transformation of ˆ. The uniqueness result is implied by Lemma 2 in Galaabaatar and Karni (2013). Clearly, the conditional probability measures satisfy ( ) = ˆ ( ) = 0 ( ) for all and 0 W Thus, ( ) = 0 ( ) = ˆ ( ) Σ 0 ˆ ( 0 ) for every familiar event, The uniqueness of ˆ is implied by the uniqueness part of the von Neumann-Morgenstern expected utility theorem. In general, ˆ is not an element of V ( ) However, if  implies that  for all ( ) then, it is easy to see that ˆ V ( ) 8

9 2.3 Complete beliefs There are situations in which decision makers rely on experts assessment of the likelihood of events. For example, in the medical decision problem described in the introduction, the likelihoods of the different outcomes under alternative treatments are provided by the physician. Similarly, accident risks (e.g., airplane crash) are depicted by their empirical distributions. It is reasonable to suppose that, in such cases, the decision maker s beliefs coincide with the empirical distributions and are complete. At the same time, the decision maker may feel unable to compare certain acts whose payoffs are contingent on events outside his realm of experience. For instance, a decision maker who enjoyed good health all his life, may find it difficult to assess the relative merits of long term care insurance policies that include payoffs inthe events in which he is disabled and needs professional care. To model situations of involving complete beliefs and incomplete tastes regarding payoffs contingent on unfamiliar events, I invoke the axiom, dubbed complete beliefs, introduced by Galaabaatar and Karni (2013). Denote by the constant act whose payoff is +(1 ) in every state. (A.6) (Complete beliefs)forallevents and [0 1] and constant acts and such that  either  or  0 for every 0 The next theorem characterizes the representations of preference relations displaying complete beliefs in the presence of familiar events. Theorem 3 Let  be a binary relation on, then the following conditions are equivalent: ( ) is Â-bounded and  satisfies (A.1) (A.6). ( ) There is a unique probability measure on a real-valued function, ˆ on and for every familiar event, a nonempty set, V ( ) of real-valued functions on such that, for all { } and V ( ), X ( ) ( ) ˆ +X ( ) ( ) X ( )( ( ) ˆ )+ X ( )( ( ) ) X (3) ( )( ( ) ˆ )+ X ( )( ( ) ) 9

10 and, for all,  if and only if X ( )( ( ) ˆ )+X ( )( ( ) ) X ( )( ( ) ˆ ( ))+ X ( )( ( ) ) (4) for all V ( ). Moreover, ˆ is unique up to positive affine transformation, and if U ( ) is another set of functions that represent  in the sense of (4) then hu b ( )i = hbv ( )i. The proof of Theorem 3 follows from Theorem 1 above and Theorem 4 of Galaabaatar and Karni (2013), and is omitted. 3 Source Familiarity 3.1 Familiar sources There is a substantial body of literature dedicated to the proposition that decision makers prefer acts that are measurable with respect to more familiar sources over acts measurable with respect to less familiar sources. To formalize the idea of source familiarity, let T = { 1 } be a partition of Each such partition is a source. 7 For every partition T = { 1 } let (T ) be the subset of acts that are T -measurable. 8 For every partition T, (T ) with the usual mixture operation is a convex subset of a linear space. 9 Denote by P the set of all partitions of Definition 2: A familiar source is a partition, T, of the state space such that  on (T ) is negatively transitive. A partition that is not a familiar source is an unfamiliar source. Note that, like familiar events, what constitutes a familiar source is subjective and is revealed by the completeness of the weak preference relation of the subset of acts, (T ) that are measurable with respect to T. Let T 0 := { } denote the trivial partition, then (T 0 ) is the subset of all constant acts. 7 Equivalently, a source may be regarded as the algebra generated by a partition. 8 That is, (T )={ = ( ) for all =1 }) 9 The mixture operation is defined as follows: for every (T ) and [0 1] +(1 ) (T ) is defined by ( +(1 ) )( ) = ( )+(1 ) ( ) for all 10

11 3.2 Axioms and representation Consider the axiom of (unconditional) dominance, of Galaabaatar and Karni (2013). Dominance For all if  for every then Â. Note that dominance implies that the preference relation satisfies (unconditional) state independence. Then, adding dominance to the (A.1) - (A.3) implies the following multi-prior expected utility theorem. Theorem 4 Let  be preference relation on and suppose that T 0 is a familiar source, then the following conditions are equivalent: ( ) is  bounded and  satisfies axioms (A.1) - (A.3) and dominance. ( ) There is a real-valued function on unique up to positive affine transformation, such that = = and there is a unique, nonempty, convex set, M of probability measures on such that, for each source, T, and all (T ) { } ( ) X ( )( ( ) ) ( ) M (5) T For all 0  0 X ( )( ( ) ) X ( )( 0 ( ) ) M (6) Moreover, for every familiar source, T, and all T, ( ) = 0 ( ) for all 0 M P For any unfamiliar source, T and all (T ) isrepresented by T ( )( ( ) ) M For any familiar source, T there is a unique source-dependent probability measure T on such that, for all (T ) is represented by P T T ( )( ( ) ) For every element, of a partition that defines a familiar source, the subjective probabilities of the states are not unique. However, the sum of the probabilities of all the states in is unique. 11

12 3.3 Comparative source familiarity Distinct sources may be unfamiliar but not necessarily unfamiliar to the same degree. To formalize this idea, I define a relation more familiar than on the set of sources in terms of the preferences and characterize it in terms of the set of probabilities. Following Galaabaatar and Karni (2013), define a binary relation < on by: For all, < if  implies  for all It is possible to show that all the representation results in this paper apply to < with weak inequalities replacing the strict inequalities. To define the relation more familiar than I use the following notations and definitions: Let B be a binary relation on defined as follows: for all, B if < and (  ) For all such that  and define the act ( )( ) = if and ( )( ) = otherwise. This act is referred to as a bet on For every let ( ) :=sup{ ( ) M} and ( ) :=inf{ ( ) M} The idea captured by the definition of B is that, for each constant act and bets and ˆ such that B B ˆ, the pair of outcomes ( ˆ ) spans of the gulf between the bets on an thatarejustcomparableto The presumption is that the less familiar is the source, the bigger is the gulf that needs to be spanned, for every bet on event 0 belonging to that source. To make sure that the measurements of these gulfs is based on the same scale, the upper bet,, isfixed. Formally, Definition 3: Asource T is more familiar than a source T 0 if for each T and 0 T 0 and all 0 ( ) B B ˆ and B 0 B ˆ 0 0 imply ˆ 0 ˆ The following theorem characterizes the relation more familiar than in terms of the probability measures that figure in the representation. This characterization entails a monotone likelihood ratio property, namely,one source is more familiar than another if the likelihood ratio of the highest and lowest probabilities of each event belonging to the more familiar source is smaller than the corresponding likelihood ratio of every event belonging to the less familiar source. Theorem 5. Let  be an Archimedean strict partial order on satisfying independence and dominance Suppose that is Â-bounded and that T 0 is a familiar source. Then source T is more familiar than another source, 12

13 T 0 if and only if, for each T and 0 T 0 ( ) ( ) ( 0 ) ( 0 ) 3.4 Increasing source familiarity Presumably, familiarity grows with experience. As sources become more familiar the preference relation become less incomplete. The following definition captures this idea by providing a choice-based formalization of a source being more familiar according to one preference relation than according to another. Definition 4: A source T is more familiar according to the preference relation  than according to the preference relation  0 if  0 implies  for all (T ) The next theorem characterizes the notion that a source is more familiar according to one preference relation than according to another. It asserts that, under the aforementioned axioms, this relation is equivalent to the set of probability measures that figure in the evaluation of acts corresponding to the former preference relation is contained in that of the latter. Let M  denote the set of probability measures on that figure in the representation of  in Theorem 4. Theorem 6. Let  and  0 be Archimedean strict partial orders on satisfying independence and dominance Suppose that is  and  0 bounded, and that T 0 is a familiar source. Then T is more familiar under  than under  0 if and only if M  M Â0 For each source, T,and T,define T  ( ) ={ R = ( ) for some M  } It is natural to suppose that as T becomes more familiar, the range T  ( ) shrinks, for all T This implies that M must be trimmed to accommodate the tighter ranges T  ( ). In the extreme, that is, if ˆT is a familiar source then, for each ˆT, ( ) reduces to a singleton ˆT set This implies that M  must consist of probability measures that agree on the events in ˆT. Building on Gilboa and Schmeidler (1989) and departing from their uncertainty aversion axiom, Ghirardato, Maccheroni and Marinacci (2004) advance a model of choice in which they separate ambiguity from ambiguity attitude. One of the primitives of their model is a complete preference relations on Anscombe and Aumann (1963) acts. From this they derive an incomplete 13

14 preference relations representing unambiguous preferences. According to their approach, an act is (weakly) unambiguously preferred over another act if and only if the subjective expected utility of the former is at least as great as that of the latter, for every probability measure belonging to a convex and closed set of priors. A derived preference relation < 1 is said to reveal more ambiguity than another derived preference relation < 2 if < 1 implies < 1 for all acts The resemblance of this to Definition 4 above is apparent. Proposition 6 of Ghirardato et. al. (2004) says that < 1 reveals more ambiguity < 2 if and only if C 2 C 1 where C is the set of priors corresponding to <, =1 2 Clearly, except of context and interpretation, Theorem 6 and Proposition 6 of Ghirardato et. al. (2004) are essentially the same. 4 Proofs 4.1 Proof of theorem 1 (outline) The proof invokes results and arguments from the proof of Theorem 1 in Galaabaatar and Karni (2013), properly modified to accommodate the conditional preferences. ( ) ( ) For every event, the sets of acts that agree on the complementary event (that is, the sets {{ } }) areconvex. 10 For familiar events, the conditional preference relation, < is a weak order (i.e., transitive and complete binary relation) satisfying the weak version of (A.3). Moreover, since  6=, by (A.4) and the theorem of Anscombe and Aumann (1963), there is a real-valued function, ˆ on, and a unique (conditional) probability measure, ˆ on such that, for all < if and only if P ˆ ( ) P ˆ ( ) ( ; ) P ˆ ( ) P ˆ ( ) ( ; ) Moreover, ˆ is unique up to positive affine transformation and ˆ is unique. Define an auxiliary binary relation 3 on as follows: For all, 3 if  implies  for all 11 Let := { ( 0 ) } Each is a point in R. However, because the weights on consequences in each state add up to 1 can also be seen as a 10 The convex combination + (1 ) is define, as usual, by ( +(1 ) )( ) is ( )+(1 ) ( ) if and ( ) if 11 This relation was first introduced in Galaabaatar and Karni (2013), and was further studied in Karni (2011). 14

15 point in R ( 1). For any act thecorrespondingactinr ( 1) is denoted by ( ). Thus, : R R ( 1) is a one-to-one linear mapping. Define ( ) :={ ( ) 3 and and 0} Then ( ) is a closed convex cone with non-empty interior in ( 1). By theorem V.9.8 in Dunford and Schwartz (1957), there is a dense set, in its boundary such that each point of has a unique tangent. Let W be the collection of all the supporting hyperplanes corresponding to this dense set. Without loss of generality, we assume that each function in W has unit normal vector. Then W represents 3 Henceforth, I denote ( )( ) by ( ) W With this convention, for all 3 X ( ( ) ) X ( ( ) ) W For every ( ) let be the constant act such that ( ) = Σ 0 0 ( 0 ) for all By an argument analogous to Galaabaatar and Karni (2013), it can be shown that (A.5) is equivalent to the following condition. For all  ( ) for every ( ) implies that  The proof that the component functions, { } of each essential function, W are positive linear transformations of one another (that is, if ˆ W then for all non-null, ˆ ( ) and ˆ ( ) are positive linear transformations of one another) is analogous to that of Lemma 6 in Galaabaatar and Karni (2013). The representation is implied by the following arguments: For each W define ( ) = ( ) for some and for all let ( )= ( ) + 0 Define ( ) =ˆ ( ) (1 + Σ 0 0) for all and ( ) = (1 + Σ 0 0) for all Let V ( ) be the collection of distinct and, for each V ( ) define Π = { such that = } 12 ( ) ( ) The  boundedness. of and axioms (A.1) - (A.3) are implied by Lemma 2 in Galaabaatar and Karni (2013) and (A.4) and (A.5) are immediate implications of the representation. 12 If there are kinks in so that there are more than one supporting hyperplane then, by the same argument as in Galaabaatar and Karni (2013), there is at least one that can be expressed as a limit point of sequence { } from W Then the component functions of are positive linear transformation of one another. Add all those s to W then the new set of functions will represent  15

16 Let 0 V Π then, ( ) =ˆ ( ) = 0 ( ) Thus, ( ) = 0 ( ) =ˆ ( ) Σ 0 ˆ ( 0 ) for every familiar event 4.2 Proof of theorem 4 (Sufficiency) Since T 0 is a familiar source, the restriction of < to the subset of constant acts is complete. Hence, by Galaabaatar and Karni (2013), Theorem 3, there is a real-valued function, on unique up to positive affine transformation and a convex nonempty set, M of probability measures on such that, for all X ( ) ( ) X ( )( ( ) ) X ( )( ( ) ), (7) for all M. Let arg max ( ), arg min ( ) = and = Then P ( ) ( ) = ( ) and P ( )( ( ) ) = ( ) Hence, (5) is implied by (7) and (6) is an implication of Galaabaatar and Karni (2013). Let T be a familiar source then (T ) is a convex subset of a linear space. The restriction of < to (T ) is a weak order satisfying (A.2), the weak version of (A.3) and, by Lemma 2 in Galaabaatar and Karni (2013), it satisfies state independence. Hence, by Anscombe and Aumann (1963) there exist a unique T such that T,forall, (T )  if and only if P T T ( )( ( ) ) P T T ( )( ( ) ) But, by (6),  if and only if P T ( )( ( ) ) P T ( )( ( ) ) for all M Hence, it must be the case that, for all T, ( ) = T ( ) for all M. (Necessity) The necessity is implied by Theorem 3 in Galaabaatar and Karni (2013). 4.3 Proof of theorem 5 Let  be preference relation on satisfying the conditions in the hypothesis of the theorem Suppose that is Â-bounded and that T 0 is a familiar source. Suppose that T is a more familiar source than T 0 Let T and 0 T 0, fix abeton Without loss of generality, normalize the utility function so that ( ) =0 Let ( ) =Σ ( ) ( ) Then, by Theorem 16

17 4, B B ˆ if and only if and and 0 B 0 B ˆ 0 0 if and only if But (8) and (9) imply ( ) ( ) = ( ) = ( ) (ˆ ) (8) ( 0 ) ( ) = ( 0 )= ( 0 ) (ˆ 0 ) (9) ( ) ( ) = ( ) (ˆ ) and ( 0 ) ( 0 ) = ( ) (ˆ 0 ) (10) Now, ˆ 0 ˆ if and only if (ˆ 0 ) (ˆ ) Thus (10) holds if and only if 4.4 Proof of theorem 6 ( ) ( ) ( 0 ) ( 0 ) (11) Let  and  0 be preference relations on satisfying the conditions in the hypothesis of the theorem Suppose that is  and  0 bounded and that T 0 is a familiar source. (Necessity) Fix a source T and let  and  0 be preference relations on such that T is more familiar according to  than according to  0 Take any (T ) such that  0 By Theorem 4, there exists an real-valued affine function 0 on ( ) and a convex set of probability measures, M Â0 such that X ( )( ( ) 0 ) X ( )( ( ) 0 ) M Â0 (12) T T By Definition 4, (12) and Theorem 4, there exists a real-valued affine function on ( ) and a convex set of probability measures, M  such that (12) implies X ( )( ( ) ) X ( )( ( ) ) M  (13) T T 17

18 But (T 0 ) (T ) Hence, for all ( ) and (T 0 )  0 if and only if 0 0 and  if and only if By Definition 4, for all ( )  0 implies  Hence, 0 is a positive linear transformation of and can be normalized to be equal to Let T,and T  ( ) ={ R = ( ) for some M } Define 0 =sup T Â0 ( ), 0 =inf T Â0 ( ), =sup  T ( ) and =inf  T ( ) Fix ( ) such that  Define a bet on to be the act, such that ( ) = if and ( ) = otherwise. Consider the bets, 0 0 (T ) where 0 0 ( ) satisfy  0  0  Suppose that  then, by (12) ( )+(1 ) ( ) ( 0 )+(1 ) ( 0 ) Â0 T Suppose further that 0 and 0 arechosensoastosatisfy ( ) (14) 0 ( )+(1 0 ) ( ) = 0 ( 0 )+(1 0 ) ( 0 ) (15) Then, by Definition 4, (15) implies ( )+(1 ) ( ) ( 0 )+(1 ) ( 0 )  T ( ) (16) Moreover, (15) and (16) imply that 0. For a differentchoiceofbets,bythesameargument,itcanbeshown that 0 Hence, for all T, { ( ) M  } { ( ) M Â0 } Hence, M  M Â0 (Sufficiency) Suppose that M  M Â0. For all (T )  0 implies X ( )( ( ) ) X ( )( ( ) ) M Â0 (17) T T Since M  M Â0 (17) implies X ( )( ( ) ) X T T ( )( ( ) ) M  (18) By Theorem 4, (18) implies  Thus, by Definition 4, T is more familiar under  than under  0 18

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