Pre-probabilistic sophistication Osamu Kada Faculty of Engineering, Yokohama National University 79-5 Tokiwadai, Hodogaya, Yokohama 240-850, Japan (E-mail: smkada@ynu.ac.jp) July, 2009 We consider pre-probabilistic sophistication where an agent s two-consequence act is based on a probabilistic belief over a finite λ-system of events. We identify the likelihood relation on the events such that if an agent is pre-probabilistically sophisticated, then its probability measure must represents the likelihood relation. We give conditions on the relation that characterize pre-probabilistic sophistication. We also give a minimal condition that extends pre-probabilistic sophistication to probabilistic sophistication when events constitute an algebra. Keywords: Subjective probability, uncertainty, likelihood consistency, nonexpected utility, pre-probabilistic sophistication, probabilistic sophistication.
. Introduction Savage (954) characterized choice under uncertainty that is represented by mathematical expectation of utility with respect to probabilistic belief: subjective expected utility (SEU). Starting with Machina and Schmeidler (992), there has been a string of papers that attempted to characterize choice under uncertainty that does not conform to SEU, and yet that choice is based on probabilistic belief. An agent is called probabilistic sophisticated if she makes such choice patterns: she has a unique subjective probabilities such that two acts that generate the same distribution on consequences must come from the same indifferent class. This notion of probabilistic sophistication is composed of two parts: (a) the agent has a unique probabilistic belief over events which is characterized by her two-consequence acts, and (b) her (many-consequence) acts are based on this probabilistic belief. These two parts are conceptually different, and it seems to be considered separately. We call pre-probabilistic sophistication which is a notion corresponding to the first part (a). That is, an agent is pre-probabilistic sophisticated if two-consequence act is based on a probabilistic belief over events. It seems that there are differences between an agent s preferences on acts of two consequences and three or more consequences. She may have probabilistic belief and behave according to the belief in a simple betting, but in a more complicated betting, she may not behave according to the probabilistic belief. For instance, consider an agent who computes probabilities of events according to an information available and according to these probabilities she makes a choice, but she has a bounded capacity to calculate them. In the choice of simple betting, say, betting to two events A or B, she can calculate two probabilities that A occurs or B occurs, so that she can choose A or B according to its probabilities. But in the choice of more complicated betting, say, betting to [$30 if A, $20 if A 2, $0 if A 3 ] or [$30 if B, $20 if B 2, $0 if B 3 ], she cannot calculate six probabilities of A i, B i, i =, 2, 3 because of her limited ability. So that she makes a choice not by probabilistic belief but according to another one, say, by labelling; names or numbers. In this case, she may be pre-probabilistic sophisticated but not probabilistic sophisticated. 2
It seems that there are many examples as the above which show this gap between pre-probabilistic sophistication and probabilistic sophistication, because of bounded rationality, or by other behavioral aspects. Considering pre-probabilistic sophistication has its importance per se, but also for probabilistic sophistication. First, to extract subjective probabilities, considering many-consequence acts is complicated, not simple in comparison with twoconsequence acts. Second, there arises a question as to what condition that distinguishes between pre-probabilistic sophistication and probabilistic sophistication. This is an interesting question, and serves as an understanding of probabilistic sophistication. For pre-probabilistic sophistication, first question which must be considered is as follows: What is the likelihood relation on events such that pre-probabilistic sophisticated agent must have? If we require stochastic dominance in probabilistic sophistication, then the relation by betting is the likelihood relation that we are seeking (see Sarin Wakker (2000)). But in general, it is not known what the relation it is. Chew and Sagi (2006) introduced exchangeability, and showed that exchangeability based likelihood relation must satisfy event non-satiation. Examples of delayed resolution of uncertainty and Machina s Mom (see Example 2.5) do not satisfy event non-satiation, therefore, their likelihood relations are not based on exchangeability. In this paper, we define more probable relation on events and show that it is the likelihood relation that pre-probabilistic sophisticated agent s probability measure represents. This is defined as follows. We say events A and B are equally evaluated if they are indifferent by simple betting on them. This equally evaluated binary relation on events is not equivalent to equally probable relation as the above example shows, but restricting to small events, whom we call minimal events, the equivalence holds. We define that an event A is more probable than an event B if A contains an event B such that B and B have finite partitions by minimal events and they are piecewise equally evaluated. This definition conforms to additivity of probability measure. The notion of more probable relation might seem to be complicated, but with other axioms, it is shown that it is not complicated (see Remark 2.). 3
Second question is: What conditions on the more probable relation ensure preprobabilistic sophistication? We give two conditions on the more probable relation. One is the likelihood consistency on the relation, which is a reminiscence of likelihood consistency by Abdellaoui and Wakker (2008) or of event non-satiation by Chew and Sagi (2006) (see section 3). The other is the Pre-probabilistic Belief (PB), which requires that two events have finite partitions by minimal events which are piecewise equally evaluated, then they are equally evaluated. This is a reminiscence of additivity of probability measure. With these axioms and Archimedean Property and solvability, we get a characterization of pre-probabilistically sophistication. Third question mentioned as before: What condition extends pre-probabilistic sophistication to probabilistic sophistication? We show that Probabilistically Sophisticated Belief, which is a notion stronger than PB, is the condition that extends pre-probabilistic sophistication to probabilistically sophistication. This paper proceeds as follows. In section 2, we characterize pre-probabilistic sophistication; Subsection 2. is the definition of pre-probabilistic sophistication. Subsection 2.2 gives examples. Subsection 2.3 gives the notion of more probable relation, showing that a pre-probabilistic sophisticated agent s probability measure must represent it. Subsection 2.4 characterizes pre-probabilistic sophistication. Subsection 2.5 gives a condition that extends pre-probabilistic sophistication to probabilistic sophistication. In section 3, we give relations between our notions of minimal events and likelihood consistency, event non-satiation by Chew and Sagi (2006), and ps-likelihood consistency by Abdellaoui and Wakker (2008). In the Appendix, some lemmas for our main theorem and proofs are given. 2. Pre-probabilistic Sophistication 2.. Definition Let S be a set of states. A collection A of subsets of S is called a finite λ-system if it is closed with respect to complements and finite disjoint unions, and contains S. We consider on a finite λ-system. Restricting a domain of probability measure to λ-system was considered by Zhang (999) and Epstein and Zhang (200) relating to unambiguity. Consider a domain of probability measure as the events the agent 4
knows or can conceive the probability based on her some information; then even if she has her subjective probabilities on two events, she may not have a probability of the event of their intersection. Thus, finite λ-system seems to be an appropriate notion to be considered. Elements of A are called events. For E A, we denote by E c its complement. Let C be a set of consequences. A (simple) act f from (S, A) to C is a mapping from ( S to C such ) that f (c) A for all c C and its range is finite. f = A A 2... A n = [c c c 2... c i on A i ] n i= denotes the act that yields c i C on A i n A, i =,..., n. Let F = F((S, A); C) be the set of all acts from (S, A) to C. For each consequence c C, we identify c as a constant act. For A A, acts f, g, we denote by f A g the act that yields f on A and g on A c. A preference relation over A is a binary relation on the set of acts F = F((S, A); C), and throughout this paper we assume that is a weak order; that is, it is complete (f g or g f for all f, g) and transitive (f g and g h implies f h). A function V : F R represents if: for all f, g F, f g iff V (f) V (g). Savage (954) defined that an event A is null if two acts are indifferent whenever they are the same outside of A. This notion is to mean that an agent consider A virtually impossible (Savage (954, p. 24)), and when she is probabilistically sophisticated, this is equivalent to her assessment of the probability of A as zero. We consider pre-probabilistically sophisticated agents, that is, her subjective probability is based only simple betting (two-consequence act), so the definition must be modified. That is, we define null to mean that its subjective probability is zero for an agent who is pre-probabilistically sophisticated. So that even if she has subjective probabilities, if she is not probabilistically sophisticated, a null event may not be null in the sense of Savage. Further, to accommodate with finite λ-system, we take a weaker condition as a definition of null. Definition 2.. Event A is null if for each event B A, consequences c, c, c B c c (see Lemma 4.7). Outcomes c, c C are equivalent to each other if for each event A, c A c c (see Remark 2.9, (b) (i)). Nondegeneracy holds if there are 5
non-equivalent consequences c, c. If nondegeneracy holds, then there is a nonnull event. Remark 2.2. Event A is Savage-null if f A h g A h for all acts f, g, h. Savagenull implies null, but the converse does not hold. Restricting Savage-null to two consequence acts, we get the following: for all B, B A and D A c, for all c, c, c (B D) c c (B D)c. Letting B = D =, this becomes our definition. In probability theory, events considered constitute an (σ-)algebra. Zhang (999), and Epstein and Zhang (200) considered more general class of events, λ-system, and Kopylov (2005) introduced more general mosaic. Earlier, consideration of general finite λ-system as the domain of probability measure was given relating quantummechanics by Krantz et al (97). We consider a finitely additive probability measure on the finite λ-system A. Let P be a finitely additive probability measure on A. P represents a binary relation l on A if: for all A, B A, A l B iff P (A) P (B). P is solvable if for all A, B A, P (A) P (B) implies that there exists B A such that B A and P (B ) = P (B). A nonnull event A is an atom if A cannot be partitioned into two nonnull events. P is purely and uniformly atomic if the union of all atoms has unit measure and all atoms have equal measure. An agent is pre-probabilistically sophisticated if she has a unique subjective probabilities on events such that if the probabilities on two events are the same, she equally evaluates the events. She is probabilistically sophisticated if she has a unique subjective probabilities such that two acts that generate the same distribution on consequences must come from the same indifferent class. Formally, we define as follows. Definition 2.3. (Pre-probabilistically Sophistication) Let be a preference relation over a finite λ-system A. (a) is pre-probabilistically sophisticated if there exists a unique finitely additive and solvable probability measure P on A such that P (A) = 0 whenever A is null, and for any pair of events A and B in A and consequences c and c in C, (2.) P (A) = P (B) implies c A c c B c. 6
(b) is probabilistically sophisticated if in the above (a), (2.) is replaced by the following: for any pair of act f and g (2.2) P (f (c)) = P (g (c)) for all consequences c implies f g. Taking f, g to be two-consequence acts c A c and c B c, respectively, (2.2) becomes (2.), so that pre-probabilistically sophistication is a notion weaker than probabilistically sophistication. As stated in the introduction, consider an agent who calculates probabilities of events according to an information available to her, but for its limited ability of its calculation who can only make a probabilistic belief on simple betting. For such an agent, her preference relation may be pre-probabilistically sophisticated but may not be probabilistically sophisticated. 2.2. Examples We give examples of pre-probabilistically sophistication. Example 2.4. (cf. Chew and Sagi (2006, Example )) Let A = {A, B, C} be a partition of a state space S, C = [0, ], and is represented by: ( ) A B C V = x + y + z (x y)(y z)(z x) x. x y z 8 Then is pre-probabilistically sophisticated with the uniform probability measure, and satisfies eventwise monotonicity. This follows by showing that( partial derivatives are positive except finite points. But for different x, y, z, V ) A B C x y z ( ) A B C V due to the asymmetry between x and y in the last term. Hence, y x z is not probabilistically sophisticated. The following Examples 2.5 do not satisfy event non-satiation (see section 3.) or eventwise monotonicity, and these are given by Machina and Schmeidler (992) (delayed resolution of uncertainty), and by Grant (995) (social ordering or ethical preference over subjectively uncertain acts). Example 2.5. (a) (Delayed resolution of uncertainty) Let S = [0, ], A = B [0,] : the Borel algebra on [0, ], and ( P is the Lebesgue ) measure. Let C = {x, x 2,..., x n }, A A and is represented by: V 2... A n d = max n x x 2... x c D i= u(x i, c)p (A i ), n 7
here D is a set of actions taken before the uncertainty is resolved, and u is von Neumann-Morgenstern utility function. Since auxiliary action c D must be made before the state is known, although an agent has an expected utility after she chooses c D, she is not an expected utility maximizer. For instance, consider the following story. Suppose you are going to a conference, and it is uncertain whether you can have your favorite coffee for free in the conference or not. You make a choice between going to a coffee house or not before the conference is held with the following preference ranking: (not coffee, coffee) (coffee, not coffee) (coffee, coffee) (not coffee, not coffee), here (not coffee, coffee) means you do not go to a coffee house having no coffee before the conference, and have your favorite coffee for free in the conference; similarly for others. Let p be your subjective probability that there is your favorite coffee for free in the conference. If p = 0, you may go to a coffee house and have a coffee, but as p increases in some degrees, your expected payoff will decrease, which means you are locally satiated at p = 0. Let n = m = 2, x = there is a favorite coffee for free in the conference, x 2 = there is no favorite coffee for free in the conference, ( c = go) to a( coffee ) house, c 2 = not go to a ( coffee house, ) a a u(x i, c j ) = a ij, 2 a b A A =, b c > a d. Then V 2 a 2 a 22 c d d = x x 2 max{ (c a)p + c, (b d)p + d}, here p = P (A ). Letting p = p = c d, we have the following Figure. b d c d and b+c a d b c a d 0 p p p Figure. Delayed resolution of uncertainty 8
(b) (Social ordering or ethical preference over subjectively uncertain acts) Consider a social planner as Machina s Mom, or an individual of ethical preference as Harsanyi (955) considered. Let S = [0, ], A = B [0,], P be as the above (a). Let C = {x,..., x n } and consider it as the set of social consequences, there are I individuals, and social preference or individual ( ethical preference ) over A A subjectively uncertain prospects (an act) f = 2... A n. Suppose x x 2... x n the preference is represented by: ( ) A A V 2... A n s = W (U x x 2... x (p),..., U I (p)), n here p = n j= P (A j)δ xj and U i (p) = n j= u i(x j )P (A j ) is an expected utility with von Neumann-Morgenstern utility function u i for each individual i. W is a social welfare function (SWF); (i) W is generalized utilitarian if W (u,..., u I ) = I i= g(u i) for an increasing and concave function g( ) (see Mas-colell et al. (995 Example 22.C.3)), and it is purely utilitarian if g is the identity. (ii) W is quadratic if it is a polynomial of order two (see Epstein and Siegel (992)). If W is not purely but generalized utilitarian, then it does not satisfy event non-satiation, therefore does not satisfy event monotonicity (see Figure 3). Consider the example of Machina s Mom, which is based on Diamond (967): a mother gives an indivisible treat to her daughter or her son. Suppose there is a subjective uncertainty upon which the mother submmit her choice whether she gives the good to her daughter or her son. Let p be the probability that the daughter gets the good. By consideration of fairness, p = is the best 2 for the mother, and the more the probability gets greater than, the less 2 becomes the mother s satisfaction. Therefore, at p ( =, the mother is satiated. This example is quadratic: n = I = 2, V 2 2 ) A A s = W (p x x, p 2 ) = 2 { if i = j p p 2, here p i = P (A i ), and U i (p δ x + p 2 δ x2 ) = p i, u i (x j ) = 0 if i j. ( ) A A More generally, let V 2... A n s = W (p x x 2... x,..., p n ) = i<j p ip j, p i = n P (A i ), U i ( n j= p jδ xj ) = p i, u i is as the above. This is an example of quadratic 9
form axiomatized by Epstein and Siegel (992). It is easy to see that the above W s has the maximum when p i = for all i. n.. 0 2. p W (p, p ) = p ( p ) 0 2 p W (p, p ) = p + p Figure 2. Machina s mom Figure 3. Generalized utilitarian The following Examples satisfy event non-satiation but do not satisfy eventwise monotonicity or other axioms, which are given by Chew and Sagi (2006). Example 2.6. Let S = [0, ], A = B [0,], P be as in Example 2.5 (a). (a) (Lexicographic preference) Let C = R. For f P = n i= p iδ xi, let E(f) = n i= p ix i, Var(f) = n i= p i(x i E(f)) 2. Define f g if E(f) > E(g), or E(f) = E(g) and Var(f) < Var(g), and, as usual. Then as stated by Chew and Sagi (2006), does not satisfy P6 (continuity). (b) Let C = R 2 +. For L = n i= p iδ xi (R 2 +), x i = (x i, x 2 i ), let U(L) = n i= p i( + x i )( + x 2 i ) Var(L), here Var(L) = n 2 i= p i x i E(L), is the Euclidean metric, E(L) = n i= p ix i. Let be represented by: for f F(A), V (f) = U(f P ). P (A) = P (B). Hence, Let x = (, 0), y = (0, 0). Then x A y x B y implies A ev B iff P (A) = P (B). In this example, x A y x B y for some x y does not imply P (A) = P (B). Let x = (4, 0), y = (0, 3), A, B be events such that P (A) = 3 5, P (B) = 5. Then x y and x A y x B y but P (A) P (B). As shown by Chew and Sagi (2006, Appendix), this example (b) violates P3 (event monotonicity), P4 (comparative likelihood) in Machina and Schmeidler 0
(992), and P 3 CU, P 3 CL, and P 4 CE in Grant (995). But the above examples (a) and (b) satisfy Chew and Sagi s (2006) Axioms, and also our Axioms. 2.3. What is the Likelihood Relation that a Pre-probabilistically Sophisticated Agent Must Have? To extract probability measure from preference, Savage-De Finetti betting method is such that it considers how an agent bets on events. That is, an event A is more likely than an event B, denoted by A l B, if c A c c B c for all c c, and considers conditions on l for the existence of a probability measure that represents l. If we require stochastic dominance in probabilistic sophistication, then l is the likelihood relation that the probability measure of a pre-probabilistically sophisticated agent represents as noted by Sarin Wakker (2000, pp. 92)). But generally, it is not the case. Thus we have a question: For a pre-probabilistically sophisticated agent, what is the likelihood relation that she has? We define a more probable relation pr and shows that it is the likelihood relation that the probability measure of pre-probabilistically sophistication represents. First we introduce an equally evaluated relation on events. This is a notion that an agent s evaluation on the two events which is represented by simple betting on the events are the same. Definition 2.7. Event A is revealed equally evaluated as event B, denoted by A ev B, if for all consequences c, c c A c c B c. Note that two events iff their compliments are equally evaluated, and ev transitive. Next we define minimal events, and equally probable relation on events, which does not necessarily coincide with the equally evaluated relation. Then we define more probable relation. For a pair of events A and B, if we divide them fine enough to satisfy monotonocity on each piece, and they are piecewise equally evaluated, then by additivity of probabilistic belief, A and B will be equally probable. And if A contains an event which is equally probable with B, then A will be more probable is
than B. Thus, we define equally probable and more probable relation on events as follows. Definition 2.8. (a) Event A A is minimal if: for each event B A, B ev A implies A \B is null. A is strongly minimal if each event B A is minimal. (b) Event A is revealed equally probable as event B, denoted by A pr B, if there exist finite partitions by minimal events {A i } n i= and {B i } n i= of A and B, respectively, such that A i ev B i for all i. (c) Event A is revealed strictly more probable than event B, denoted by A pr B, if there exists an event B A with nonnull A \ B such that B pr B; and A is revealed more probable than event B, denoted by A pr B, if A pr B or A pr B. Hereafter, we assume that for each event A, there exists a finite partition by strongly minimal events of A. Therefore, Example in Remark 2.9 (c) (see Appendix) is excluded. If we consider sufficient many times of adequate coin flips, this assumption will be reasonable. Remark 2.9. Suppose that A is an algebra. (a) Let A be the equivalence class by the relation that setwise difference is null. Define A B for A, B A if there exists A A, B B such that A B and A ev B. Then A A is minimal iff A is minimal by the above relation. (b) Savage s P3 (Eventwise monotonicity) states that for all consequences c, c, Savage-nonnull event A, c A f c Af iff c c. Assume P3. Then, (i) c, c are non-equivalent to each other iff c c or c c; (ii) all events are minimal; and (iii) event non-satiation defined by Chew and Sagi (2006) holds (see section 3.2). (c) There is an example that there are no nonnull minimal events. (d) Consider Example 2.5 (a). Let A be an event and let p be its probability. Then A is not minimal if p (p, p], A is minimal if p [0, p] (p, ], and A is strongly minimal if p [0, p]. (e) We consider Example 2.5 (b) of Machina s Mom. Let A be an event such that P (A). Then there exists a subevent 2 A of A such that P (A ) = P (A). 2
Since V m (x A y) = P (A )[ P (A )] = [ P (A)]P A = V m (x A y), we have A ev A with nonnull A\A. Hence, A is not minimal. For each event A with P A, A is strongly minimal. Let A = (0, 3], B = (0, ], C = ( 3, ]. Then A 2 4 4 4 is not minimal, and A ev B but A C ev B C. Therefore, ev cannot be a symmetric part of a likelihood relation in this example (cf. Chew and Sagi (2006, 2.2. Lemma )). If consequence space is enlarged in this example as Chew and Sagi (2006, pp 779) argues, then event non-satiation holds. But it seems that the question is: If restricted to two consequences, is it possible to characterize Mom s probabilistic belief by her preference? The following proposition shows that the more probable relation is the likelihood relation that the probability measure of pre-probabilistically sophistication represents. In particular, the above holds for probabilistically sophistication. Proposition 2.0. Let be pre-probabilistically sophisticated with its probability measure P. Assume that there is a finite partition by minimal events for each event. Then P represents pr. Remark 2.. Although the more probable relation pr is the only binary relation on events whom the probability measure of pre-probabilistic sophistication represents, it may be argued that it is somewhat complicated. It appears that in order to check A pr B, we may have to apply Definition 2.8 (b) to all finite partitions by minimal events of A and B, and in order to check whether an event is minimal, apply Definition 2.8 (a) for all subsets of this event; therefore it might seem to have little descriptive value. But in characterization of pre-probabilistic sophistication in section 2.4, it is shown in Lemma 4.0 that with some assumptions (Axioms S, LC and PB), A pr B iff for each partition {B i } n i= of B by minimal events, there exists a partition {A i } n i= of A by minimal events such that A i ev B i for all i. Therefore, we need not to apply Definition 2.8 to all finite partitions by minimal events but to some finite partition by minimal events. Moreover, we assume that there exists a finite partition by strongly minimal events for each event, which may be reasonable by considering, for instance, by sufficient many times of adequate coin flips. Therefore, it is not complicated. If we assume eventwise monotonicity, then all events are minimal (Remark 2.9 (b) (ii)). 3
2.4. Characterization of Pre-probabilistic Sophistication We consider conditions on the more probable relation pr, and characterize preprobabilistic sophistication. If there is a probability measure representing pr, the probability measure is necessarily solvable (see Proposition 2.0). In other words, in the definition of pr, solvability is implicitly assumed. Our first assumption is that if there exists a sequence of nonnull events that are equally probable, then they are finite. Axiom A (Archimedean Property): There exists no infinite sequence of disjoint nonnull events {A j } j= such that A j pr A j+ for all j. The second assumption ensures that for every two events, one event is more probable than the other. If the agent s behavior is based on a unique probability measure, then the above assumption is reasonable. Axiom S (Solvability): For all strongly minimal events A and B, there exists A B such that A ev A, or there exists B A such that B ev B. Lemma 2.2. Assume that there exists a finite partition by strongly minimal events for each event. Then Axiom S holds iff pr is complete. We give a sufficient condition for Solvability. We assume that A is an σ-algebra. The following axiom weakens P4 (Weak Comparative Probability) by Savage (954). P4 If A and B are strongly minimal events, c and c are non-equivalent consequences, such that c A c c B c, then d A d d B d for all consequences d and d. For consequences c, c, define a binary relation c,c on events by: A c,c B if c A c c B c. Define c,c and c,c as usual. Axiom C (Continuity): (a) (Small Event Continuity on Minimal Events) There exists a sequence of refining finite non-null partitions {S n } n=, S n = {Si n } k n i= such that if S n+ j Si n then S n+ j+ Sn i or S n i+ and the following holds: For all strongly minimal events A, B and consequences c, c with A c,c B, 4
(i) for each n, m m, A ( m i=si n ) c,c A ( ) m i=si n, and there exist n and m such that A c,c A ( m i=s n i ) c,c B, or (ii) for each n, m m, B ( ) m i=si n c,c B ( m i=si n ), and there exist n and m such that (b) (Monotone Continuity) A c,c B ( m i=s n i ) c,c B. If {E n } n= is a decreasing sequence of events, A, B are strongly minimal events, c, c are consequences, such that A c,c E n c,c B for all n, then A c,c n=e n c,c B. Roughly speaking, the above (a) shows that for strongly minimal events A, B, c,c is monotone with respect to set inclusion on E A,B := {E A c,c E c,c B}. The example 2.5 (b) Machina s Mom is the case of (i), [P (B), P (A)] [0, ], and c,c 2 is increasing on E A,B. The example 2.5 (a) Delayed resolution of uncertainty is the case of (ii), [P (A), P (B)] [0, p ], and c,c is decreasing on E A,B. Considering {S n i } kn i= as the events of flips of a coin n times, Axiom C may be a reasonable assumption of continuity. In this case, define as follows. Suppose S = Ω {H, T }, and let H n and T n be events of head and tail in n-th coin flips, respectively. We define a sequence of partitions {S n } n=(s n = {S n i } 2n i=) of S inductively as follows: S = {H }, S 2 = {T }, and S n+ 2i = {Sn i H n+ }, S n+ 2i = {S n i T n+ }, i =,..., 2 n. Axiom C (a) and (b) are translations of P6 of Savage (954) and the condition of Villegas (964), respectively. The following lemma shows that P4 and Axiom C (Continuity) is a sufficient condition for solvability. Lemma 2.3. Suppose A is a σ-algebra. implies Axiom S (Solvability). Then P4 and Axiom C (Continuity) The next assumption excludes inconsistencies between comparative measurements of two pair of events. Axiom LC (Likelihood Consistency): A pr B and A pr B for no events A, B. 5
LC implies that there does not exist two pairs of non-null partitions of A and B as follows: one is such that strict subset of the partition of A is piecewise equally evaluated with the partition of B itself, and the other is such that the two partitions of A and B are piecewise equally evaluated. For instance, consider the case of ambiguous averse agent as in the Ellsberg paradox (so that she does not have subjective probabilities). Consider a partition of A by unambiguous events, and two partitions of B by ambiguous events and by unamgiuous ones, respectively. Then by ambiguity aversion, a strict subset of the partition of A may be piecewise equally evaluated with the partiton of B by ambiguous events, but the partition of A itself may piecewise equally evaluated in comparison with the partition of B by unambiguous events. This implies that LC does not hold. Hence, LC excludes such an agnet. The next assumption rephrase the condition of pre-probabilistic sophistication if there exists a probability measure representing equally probable relation pr. Axiom PB (Pre-probabilistically Sophisticated Belief ): For all events A and B, A pr B implies A ev B. Again, consider the case of ambiguous averse agent. Assume that there is a partition of an event A by unambiguous events and a partition of an event B by ambiguous events such that they are piecewise equally evaluated. Since she is ambiguous averse, she may strictly prefer A to B, which means that PB does not hold. Hence, PB excludes such an agnet. Note that even if two events are equally evaluated, it does not necessarily mean that they are equally probable. For instance, in Example of Machina s Mom 2.5 (b), to give the indivisible good to her son with probability is equally evaluated as to give him with probability 0, because she considers equality between her son and daughter. Remark 2.4. If we assume Savage s P3 (Eventwise monotonicity), then PB implies LC. Therefore in this case, LC does not needed. To consider more concretely, we give the following exampl. Example 2.5. Consider the following two-coin example due to Schmeidler, which is equivalent to two-color urn example of Ellsberg (96) (see Gilboa (2004)): One 6
coin is such that an agent is familiar with and she believes to be fair, another one unfamiliar, and bet on sides of the either coin. She may consider fifty-fifty on each side of the unfamiliar coin as well as to the familiar coin, but bet on the head (tail) of the familiar coin rather than the head (tail) of the unfamiliar coin. Let A f = {H f, T f } be the events that flip of familiar coin of known probability is head and tail, and similarly A uf = {H uf, T uf } for unfamiliar, ambiguous coin. Let S = {H f H uf, H f T uf, T f H uf, T f T uf }, A = 2 S. By symmetry, we may assume H f T uf ev T f H uf. Then we have H f = (H f T uf ) (H f H uf ) pr (T f H uf ) (H f H uf ) = H uf. But H f ev H uf. Thus, Head of the familiar coin and Head of the unfamiliar coin is equally probable but not equally evaluated, which implies that PB does not hold. On the other hand, LC holds. But in this case, the preference does not satisfy Axiom S, solvablity. So, enlarge the state space S to S := S [0, ]. Since H uf [0, ] is less likely than H f [0, ], assuming continuity and monotonicity, there may exist an interval J such that H uf [0, ] ev H f J, and similarly, T uf [0, ] ev T f J. Then we have S = (H uf [0, ]) (T uf [0, ]) pr (H f J) (T f J) S, hence S pr S. Together with S pr S, LC does not hold either. We can consider similarly that in three urn example of Elsberg, it does not hold PB or LC. In the above example, which satisfies monotonicity, neither LC nor PB is satisfied. So, the question is: Are there examples of continuous preferences (so that solvability holds) that (a) LC holds but PB does not hold, and (b) PB holds but LC does not hold? The following examples give these two cases, which does not satisfy event non-satiation (see section 3.). Example 2.6. Let S = [0, ] [0, ], C = {x, y}, A = B [0, ], A 2 = [0, ] B, here B is the Borel algebra on [0, ], and A := A A 2 is a λ-system. Let f( ) and g( ) be some functions on [0, ] such that f(0) = g(0) and f() = g(), and let V ( x (A B) y ) { f(p (A)) if B = [0, ] = here P is the Lebesgue measure. Let g(p (B)) if A = [0, ], be represented by the above V. We consider the following f( ) and g( ) s. (a) Let f( ) and g( ) be as in Figure 4 and 5, respectively. That is, 7
f(x) = { 2x if 0 x 2 and g(x) = 2(x ) if x, 2 { 2x if x 4 2(x ) 3 if x. 4 f( ) g( ) A A 2 B B 2 0 4 2 3 4 0 4 2 3 4 Figure 4. Figure 5. Let E = [0, 4 ], E 2 = ( 4, 2 ], A i = E i [0, ] A, B i = [0, ] E i A 2, i =, 2. Then for each i =, 2, A i and B i are minimal, A i ev B i, hence, A := A A 2 pr B B 2 =: B. Since V (x A y) > V (x B y), we have A ev B. Hence, PB does not hold. On the other hand, LC holds. (b) Let { f( ) and g( ) be functions on [0, ] of Figure 6 and 7, respectively. That is, 2x if 0 x 3 f(x) = 4 and g(x) = x. 2(x ) + if 3 x, 4 f( ) g( ) 0 A A 2 4 2 Figure 6. 3 4 0 B B 2 2 Figure 7. Let E = [0, 4 ], E 2 = ( 4, 2 ], E 3 = ( 2, 3 4 ], E 4 = ( 3 4, ], F = E E 2, F 2 = E 3 E 4, A i = E i [0, ], i =,..., 4, B j = [0, ] F j, j =, 2. Then A i, B i are minimal, A i ev B i, i =, 2, hence, S A := A A 2 pr B B 2 = S, and A ev S. 8
This implies S pr S. Since S pr S, LC does not hold. On the other hand, PB holds. In the above example (a), if the events whose probabilities are less that 4, since f( ) and g( ) coincide on these probalities, there does not exist two different measurements, and Likelyhood Consistency holds. But if probabilities are greater than and not equal to, then f( ) and g( ) are different on these probalities, so that, A 4 and B have partitions such that they are piecewise equally evaluated does not mean A and B are equally evaluated. In the above example (b), probabilities of events are less than 3, then f( ) is two times of g( ), so that events A and B have partitions 4 such that they are piecewise equally evaluated means that 2P (A) = P (B), and Likelyhood Consistency does not hold. But f( ) and g( ) coincides on the probabilities of A and S, so that PB holds. The following theorem characterize pre-probabilistically sophistication. Theorem 2.7. Let A be a finite λ-system on S, and the preference relation on the set of acts over A is a weak order, nondegeneracy holds, and there exists a finite partition by strongly minimal events for each event. Then Axioms A, S, LC and PB hold iff is pre-probabilistically sophisticated. Here its probability measure P is solvable and represents pr, and it is either atomless or purely and uniformly atomic. Remark 2.8. In the Only if part of the above Theorem, assuming A is a σ- algebra, replacing Axiom S (Solvability) by P4 and Axiom C (Continuity) (so that Axiom S holds), pre-probabilistic sophistication is replaced by pre-probabilistic sophistication with the probability measure which is stomless, solvable, and countably additive (see Villegas (964)). 2.5. What Condition Extends Pre-probabilistic Sophistication to Probabilistic Sophistication? As the example of bounded rational agent in the introduction shows, an agent who is pre-probabilistically sophisticated needs not be probabilistically sophisticated. Then, what condition ensures pre-probabilistically sophisticated agent to be probabilistically sophisticated? Similar question was considered by Sarin Wakker (2000), 9
where they showed that cumulative dominance is what extends qualitative probability to probabilistically sophistication assuming monotonicity. In this subsection, we show that PB (Probabilistically Sophisticated Belief) is the condition that extends pre-probabilistic sophistication to probabilistically sophistication without assuming monotonicity. We assume that A is an algebra. Consider the exchangeable relation on disjoint events defined by Chew and Sagi (2006): Events A and ( B are exchageable, ) A B (A B) denoted by A B, if for all consequences c, c c and acts f, c c f ( ) A B (A B) c c. We extend exchangeable relation to general events by the following: [A B] if [A \B B \A]. Consider the following c f condition. Axiom PB (Probabilistically Sophisticated Belief ): For all pair of events A and B, A pr B implies A B. That is, if events are equally probable, then they are exchangeable. This is a notion weaker than P 4 (strong comparative probability) of Machina and Schmeidler (992). The following is our theorem. Theorem 2.9. Assume A is an algebra, is a weak order, nondegeneracy holds, there exists a finite partition by strongly minimal events for each event, and assume that Axioms A, S and LC hold. Then the following two statements are equivalent. (a) Probabilistic sophistication holds. (b) PB holds. In particular, assuming pre-probabilistic sophistication, probabilistic sophistication holds iff PB holds. And in Theorem 2.7, assuming A is an algebra and replacing Axiom PB by PB, pre-probabilistic sophistication is replaced by probabilistic sophistication. 3. Likelihood Consistencies, Minimal Events, and Event Non-satiation In this section, we compare our notions of likelihood consistency, minimal events to event non-satiation by Chew and Sagi (2006) and ps-likelihood consistency by Abdellaoui and Wakker (2005). 20
3.. Event Non-satiation First we consider event non-satiation (EN) introduced by Chew and Sagi (2006), which is a necessary assumption for their axiomatization of probabilistic sophistication. We assume that A is an algebra in this subsection. Chew and Sagi (2006) defined event non-satiation as follows: (EN) : If E E, then E A F for no Savage-nonnull event A with E A = and F E. We consider in a more general setting. Let l be a binary relation (likelihood relation) on events that is reflexive (A l A) and symmetric (A l B B l A), which is not necessarily an equivalent relation. Let l -null ( A) be a property of an event such that If A is l -null (A l -null), A B = and B l C, then A B l C. The following give examples of being l -null. Example 3.. (a) Let l be the exchangeability relation, l -null be Savagenull. (b) Define that A is l -null if A B = and B l C, then A B l C. (c) Let l be the equally evaluated relation ev, l -null be null as defined in this paper. Since ev is transitive, this is well defined by Lemma 4.7 (a) (b). Define l -event non-satiation as follows: ( l -EN) : If E l E, then E A l F for no l -nonnull event A with E A = and F E. -event non-satiation is the event non-satiation (EN) defined by Chew and Sagi (2006). Define A l B if there exists an subevent B of A with l -nonnull A \ B such that B l B, and A l B if A l B or A l B. Consider the following l -likelihood consistency: ( l -LC) : A l B and B l A for no events A and B. The following proposition shows that event non-satiation can be considered as a likelihood consistency. 2
Proposition 3.2. l -event non-satiation ( l -EN) is equivalent to l -likelihood consistency ( l -LC). Next we consider on minimal events. Define that event A is l -minimal if there exists no subevent A of A with l -nonnull A \A such that A l A. We have the following: Lemma 3.3. l -event non-satiation implies that all events are l -minimal. In particular, -event non-satiation implies that all events are -minimal. 3.2. Ps-likelihood Consistency Abdellaoui and Wakker (2005) introduced likelihood method for various models. Especially, they defined ps-likelihood consistency, and assumed it in axiomatization of probabilistic sophistication. We consider a relation to our ps-likelihood consistency. They defined binary relations ps and ps on the set of events A as follow: A ps B (resp. A ps B) holds if: c A f c B g, d A f d B g (resp. d A f d B g), and c Af c Bg imply d Af d Bg (resp. d Af d Bg ), whenever d c, d c, f, g F((S, A); {c, d}), f, g F((S, A); {c, d }). Axiom LC AW (Ps-likelihood consistency by Abdellaoui and Wakker (2005)): A ps B and A ps B for no events A, B. They assumed monotonicity, that is, f g whenever f(s) g(s) for all s. The following lemma shows that with some assumptions, ps-likelihood consistency by Abdellaoui and Wakker (2005) implies our likelihood consistency. Lemma 3.4. (a) Assume monotonicity and solvability (Abdellaoui and Wakker (2005, Definition 5.2)). Then LC AW implies LC and P B. (b) In Example 2.6 (b), LC and P B hold, but LC AW does not hold. 4. Appendix 4.. Proofs Proof of Remark 2.9: (b) (i) Suppose c c. (Only if): Let A be such that c A c c. Then by P3, c c A c c. (If): c S c = c c. 22
(ii): We may assume that there exist c c. Suppose A is not minimal, and let A ( A be such that ) ( A \A is nonnull )(so that Savage-nonnull) and A ev A. Then A \A A A c A \A A c c c A c c c c, which contradicts P3. (iii): We may assume that there exist c c. Let {E, A, E, G} be a partition of S, ( A is Savage-nonnull, ) E E and E A F E. Denote (c c 2 c 3 c 4 c 5 ) by E A F E \F G. Then c c 2 c 3 c 4 c 5 which contradicts P3. (cc c c c ) (c c ccc ) (by E E ) (c c cc c ) (by P 3) (ccc c c ) (by E A F ), (c) Let S = [0, ], A = B [0,], C = {c, c }, P the Lebesgue measure. is represented by: V (c A c ) = { sin π if P (A) > 0 2 P (A) 0 if P (A) = 0. Let A A be such that P (A) > 0. Then for any n N, any events A n A such that P (A n ) =, we have A 4n+ n ev A. Therefore, each nonnull event is not P (A) minimal in this example. We exclude the case in our main theorem. To prove Proposition 2.0, we give a lemma. Lemma 4.. Let be pre-probabilistically sophisticated with its probability measure P. (a) For each event A, P (A) = 0 iff A is null. (b) Let A, B A be such that P A = P B. Then A is minimal iff B is minimal. (c) Let A, B be minimal events. Then P (A) = P (B) iff A ev B. Proof. (a) If P (A) = 0, then for each A A, P (A ) = 0 = P ( ) implying A ev. Hence, A is null. (b) Suppose A is minimal and B is not minimal. Then there exists B B with nonnull C := B \ B such that B ev B. By the solvability of P, there exist disjoint events A, C A such that P (A ) = P (B ) and P (C ) = P (C). Then 23
A ev B. Hence, A ev B ev B ev A. Since C is nonnull by the above (a), this contradicts the assumption that A is minimal. (c) (If): Suppose A ev B and P (A) > P (B). By the solvability of P, there exists B A such that P (B ) = P (B). Then B ev B ev A. Since P (A \B ) > 0, A \B is nonnull. This contradicts the assumption that A is minimal. Proof of Proposition 2.0: Suppose P A P B. We show A pr B. By the solvability of P, there exists B A such that P (B) = P (B ). Let {B i } n i= be a partition of B by minimal events. By the solvability of P, there exists a partition {B i} n i= of B by minimal events such that P (B i ) = P (B i) for all i. Then B i ev B i for all i. By Lemma 4. (b) each B i is minimal. Hence, B pr B, which implies A pr B. Next, suppose A pr B. We show P A P B. Then there exists B A such that B pr B. Let {B i } n i=, {B i} n i= be partitions by minimal events of B and B, respectively, such that B i ev B i for all i. By Lemma 4. (c), P (B i ) = P (B i) for all i. Hence, P (B) = P (B ), which implies P (A) P (B). Proof of Lemma 2.2: (Only if): Let A, B be events, and let {A i } m i=, {B i } n i= be partitions by strongly minimal events of A and B, respectively. By Axiom S, (a) there exists A A such that B ev A, or (b) there exists B B such that A ev B. Thus we have partitions (a) {A \ A, A 2,..., A m } and {B 2,..., B n } of A \ A and B \ B, respectively, or (b) {A 2,..., A m } and {B \ B, B 2,..., B n } of A \ A and B \ B, respectively. Continuing this procedure, we have partitions {A i} k i= and {B j} l i= of A and B, respectively, such that (i) k l and A i ev B i for i =,..., l, or (ii) k l and A i ev B i for i =,..., k. Then, we have A pr B in case (i) or B pr A in case (ii). Proof of Lemma 2.3: Let A, B be strongly minimal events. We suppose that Axiom C (a) holds. The case Axiom C (b) holds is similar. We define a decreasing sequence of events {A nk } k= such that A n k A and A nk c,c B, inductively by the following: Let A n := A and suppose A nk Define A nk := A nk ( m k is defined. Let n k n k and m k be such that A nk c,c A nk ( m k i= Sn k i ) c,c B. i= Sn k i ) = A ( m k i= Sn k i ). Let A := k= A n k. We prove A c,c B. By Axiom C (b), A c,c B. Suppose A c,c B. Then there exist n and 24
m such that Since A ( m i=s n i ) = A ( i=s n k i A c,c A ( m i=s n i ) c,c B. ) ( m i=si n ) = A ( i=s n k i ) = A, we have A c,c A, which is a contradiction. Thus we get A A such that A c,c B, which implies A ev B by P4. (If): Evident. Proof of Remark 2.4: Suppose PB holds but LC does not hold. Then there exist A, B such that A pr B and A pr B. Therefore there exists A A with nonnull A \A such that B pr A. By PB, A ev B ev A, which implies that A is not a minimal event. This is a contradiction. Proof of Example 2.6: (a) We prove LC. Note that A A (resp. B A 2 ) is minimal iff P (A) ( resp. P (B) ); and for minimal events A A 2 4 and B A 2, A ev B iff P (A) = P (B). Let A A and B A 2 be such that A pr B, and let {A i } n i=, {B i } n i= be partitions of A and B, respectively, such that A i ev B i for all i. Then P (A i ) = P (B i ) for all i, so that P (A) = P (B). This implies A pr B does not hold. Indeed, if there exists B A with non-null A \ B such that B pr B, then P (B) = P (B ) < P (A), a contradiction. Hence, LC holds. (b) We prove PB. Note that A A is minimal iff P (A) 3, and each B 4 A 2 is minimal. Let A A and B A 2 be such that A pr B, and let {A i } n i=, {B i } n i= be partitions of A and B, respectively, such that A i ev B i for all i. Then 2P (A i ) = P (B i ) for all i, so that 2P (A) = P (B), implying P (A) 2. Thus A ev B holds. Proof of Proposition 3.2: ( l -EN) ( l -LC): Suppose A l B and B l A. By A l B, there exists B A such that A \ B is l -nonnull and B l B. By B l A, we consider the following two cases: (i) Case : If A l B, then we have B l B and B (A \ B ) = A l B, which contradicts l -EN. (ii) Case 2: If B l A, then A B with l -nonnull B \ A such that A l A. Then B l B and B (A \B ) = A l A B, which contradicts l -EN. ( l -EN) ( l -LC): Suppose E l E and E A l F E with E A = and 25
l -nonnull A. We consider the following two cases: (i) Case : If E \F is l -null, then E A l F E \ F = E implying E l E A. Since A is l -nonnull and E l E we have E A l E, which contradicts l -LC. (i) Case 2: If E \ F is l -nonnull, then E l E A. Since E A l E we have E A l E, which contradicts l -LC. Proof of Lemma 3.3: Suppose A is not l -minimal. Then there exists A A with l -nonnull A \ A such that A l A. Since A (A \ A ) = A l A, this contradicts EN. Proof of Lemma 3.4: (a) Suppose LC AW holds. Define A b B (resp. A b B) if there exists c c such that c A c c B c (resp. c A c c B c ). Note that by LC AW, A b B (resp. A b B) iff A ps B (resp. A ps B). First we show (4.3) A ev B iff A b B. (Only if) is obvious. (If): Suppose A ev B. We may assume that there exist c, c such that c A c c B c. Case. c c ; By monotonicity, we have c c, a contradiction. Case 2. c c ; We have A b B implying A ps B. By LC AW, we have A b B. Case 3. c c; We have A c b B c. Suppose A b B. Then by Abdellaoui and Wakker (2005, Lemma D2), S = A A c b B B c = S, implying d d for some d, a contradiction. Hence, (4.3) holds. Next we show Axiom PB holds. Suppose A pr B. Let {A i } n i= and {B i } n i= be finite partitions by minimal events of A and B, respectively, such that A i ev B i for all i. Then A i b B i for all i, and by Abdellaoui and Wakker (2005, Lemma D2), we have A b B. We have A ev B by (4.3), so that PB holds. Next we show Axiom LC holds. Suppose A pr B and A pr B. Let B A be such that A\B is nonnull and B pr B. Then by the above result we have A ev B. Therefore, for all outomes c, c, c A c c B c, which implies that (4.4) C := A \B ps. ( ) D C \D C Since C is nonnull, there exist D C and c c c such that c c c c. This implies that D ps. Hence, D b. By Abdellaoui and Wakker 26
(2005, Lemma D2), C = D (C\D) b C\D. Since C\D b by monotonicity, we have C b. This implies C ps by LC AW. This contradicts (4.4) by LC AW. Thus LC holds. (b) Suppose LC AW holds. Let x = (4, 0), y = (0, 3), x = (, 0), y = (0, 0), and A, B be events such that P (A) = 3, P (B) =. Then x y, x 5 5 Ay x B y, and x y, x Ay x By. Therefore, A b B and A b B, which implies A ps B and A ps B by LC AW. This contradicts LC AW. 4.2. Lemmas for Proof of the Main Theorems Throughout this subsection, we assume that there exists a finite partition by strongly minimal events for each event. So that Axiom S implies that pr is complete by Lemma 2.2. Lemma 4.2. Let A B = C D =. (a) (i) A pr C and B pr D implies A B pr C D. (ii) A pr C and B pr D (resp. B pr D) implies A B pr C D (resp. A B pr C D). (b) Under Axioms S and LC; (i) A B pr C D and A pr C implies B pr D. (ii) A pr B iff A c pr B c. Proof. (a) These assertions hold by Definition 2.8 directly. (b) (i) Suppose B pr D. By completeness of pr, we may assume that B pr D. Then there exists D B with nonnull B \ D such that D pr D. By (a) A D pr C D. Since A B \ (A D ) is nonnull, this implies A B pr C D, which contradicts LC. (ii) Suppose A pr B and A c pr B c. By completenss of pr, we may assume that A c pr B c. Then by (a), S = A A c pr B B c = S, which contradicts LC. Lemma 4.3. Under Axioms S and LC; let A pr B and {C, D} be a partition of B. Then there exists a partition {C, D } of A such that C pr C and D pr D. 27