How To Know The Optimal Order Of Probability In A Restricted Domain

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1 EXTENSION RULES OR WHAT WOULD THE SAGE DO? EHUD LEHRER AND ROEE TEPER January 23, 2013 Abstract. Quite often, decision makers face choices that involve new aspects and alternatives never considered before. Scenarios of this sort may arise, for instance, as a result of technological progress or from individual circumstances such as growing awareness. In such situations, simple inference rules, past experience and knowledge about historic choice problems, may prove helpful in determining what would be a reasonable action to take vis-a-vis a new problem. In the context of decision making under uncertainty, we introduce and study an extension rule, that enables the decision maker to extend a preference order defined on a restricted domain. We show that utilizing this extension rule results in Knightian preferences, when starting off with an expected-utility maximization model confined to a small world. In addition to the methodological question introduced, the results present a relation between Knightian ambiguity and partially-specified probabilities. Keywords: Extension rule, growing awareness, restricted domain, prudent rule, Knightian preferences, partially-specified probability. JEL Classification: D81, D83 The authors wish to thank Eddie Dekel, Tzachi Gilboa, Edi Karni, Juan Sebastián Lleras, Gil Riella, Luca Rigotti, Marie-Louise Vierø, audience of several seminars and conferences and an editor and two anonymous referees for their comments and remarks. Lehrer acknowledges the support of the Israel Science Foundation, Grant #762/045. Tel Aviv University, Tel Aviv 69978, Israel and INSEAD, Bd. de Constance, Fontainebleau Cedex, France. lehrer@post.tau.ac.il Department of Economics, University of Pittsburgh. rteper@pitt.edu.

2 EXTENSION RULES 1 1. Introduction Quite often, a decision maker faces choices involving new aspects and alternatives never considered before. Situations of this sort may arise, for instance, as a result of technological progress or from individual circumstances such as improved awareness. In these novel situations, the decision maker could harness simple inference rules that would help her determine present preferences and actions, based on past experience and historic choices. To illustrate the idea central to this paper, suppose you are trying to choose a book to take with you on vacation. You are hesitating between two books: X and Y. A week ago you read a newspaper column by your favorite literary critic L.C. (with whom you often agree) comparing two other books, Z and W. The critic wrote that (a) Z resembles X and is written in the style of the well-known author A, whereas W resembles Y and is written in author B s style. As both you and L.C. agree that (b) B s style is preferred over that of A, you were surprised to read that L.C. concluded that (c) Z is a better than W. Through a comparison between another pair of books, Z and W, you can infer what would be the judgment of L.C. regarding X and Y. You make the following inference: L.C. obviously prefers the style of B over that of A. Despite that Z is written in A s style, L.C. concludes that it is better than W. This suggests that the ingredients of Z associated with X are what makes Z better in L.C. s eyes, and clearly not A s style. In other words, if (c) is true despite (a) and (b), this must stem from the fact that L.C. prefers X over Y. In such a case we would say that Z and W, over which the preferences of L.C. are explicitly stated, testify that X is better than Y Inference rules. In this example, the rulings in some of the situations have been previously recognized, established, or canonized: B s style is clearly preferred to that of A and Z is better than W. Based on these preferences, the decision maker infers that between X and Y the former must be better. We refer to canonized, or already well established, preferences as those of a sage. A natural question arises as to what would be a reasonable inference rule that could enable one to project from past decisions to choices between alternatives that have never been considered before. In other words, what would the sage do had he faced a choice problem he never contemplated before.

3 2 EHUD LEHRER AND ROEE TEPER In order to answer this question, we take a decision theoretic approach and adopt the Anscombe and Aumann [1] framework of choice between alternatives called acts. An act assigns an outcome to every possible state of the world that might be realized. In our setup, as in the examples, the preference relation is specified only over some alternatives, but definitely not over all of them. Hence, the primitive is a preference relation over a subset of alternatives. 1 To exemplify, consider the case of a financial planner (FP) charged with the task of making investment decisions on behalf of her client. By meeting with the client, the FP may learn his preferences over some subset of alternatives currently available in the market, but not over the entire domain. Additionally, the domain of investments itself may also be expanding with the market. This complicates the task of the FP seeking to construct a portfolio consistent with the client s preferences, that will ultimately contain investments not included in the set for which the client s preferences are known. This necessarily requires the FP to extend the client s known preferences to a larger domain in order to make investment decisions the client would have made having faced the decisions herself. One particularly interesting instance of our framework is the case where the decision maker s growing awareness enables her to distinguish between states, which the sage could not distinguish between. Due to growing awareness, the decision maker encounters a greater state space and acts that she never considered before. 2 In this case, the sage s preferences could be interpreted as the decision maker s own preferences, prior to the world s expansion. After the expansion she wishes to project from past choices to decision problems involving new alternatives that belong to the expanded world. Consider a decision maker who encounters a choice problem she has never contemplated before; for instance a choice between f and h. Opting to benefit from the guidance of the sage, she recalls that the sage preferred f to g and that h yields a worse outcome than g in every possible state of the world. Employing monotonicity for example, she intelligently deduces that the sage would have preferred f over h, in 1 We make the assumption that this subset contains the constant acts. That is, the decision maker fully ranks all possible outcomes. 2 This scenario can be interpreted as in Karni and Vierø [10]: for new states to be discovered, new acts that assign distinct consequences to the different states must have been discovered.

4 EXTENSION RULES 3 case he the sage had to choose between these two alternatives. Monotonicity is thus used to extend the preference relation from a known domain to terra incognita. The goal of the current paper is to introduce a simple and natural extension rule, weaker than monotonicity. 3 A decision maker is required to choose between two acts, say f and g, that she has never encountered before. She knows that the sage made his choice between two other acts say b the better act and w the worse act. She knows also that a coin toss (a mixture) between f and w yields an act that is universally (i.e., in every state of the world) better than the act resulting from tossing the same coin to decide between g and b. In other words, mixing w and b with f and g reverses the preference relation: while b is preferred to w, the mixture of w and f is dominant over the mixture of b and g. The acts b and w provide evidence that f is better than g. The reasoning is that if w is worse than b, but when mixed with f it becomes undoubtedly better than b when mixed with g, then there must be one fact responsible for this reversal: w is mixed with the better act, f. In light of this evidence it is reasonable to adjudicate that f is preferred to g. This rule serves us to extend an existing preference relation from a small set of alternatives to a larger one. This extension rule is dubbed the prudent rule The main results. Two primary questions are addressed in this paper: (1) In case the sage s preferences adhere to standard behavioral properties such as expected utility maximization, what are the resulting preferences when employing the prudent rule? (2) Do the resulting preferences and those of the sage always agree (on the partial domain over which the sage s preferences are well defined)? If they do not always agree, which properties of the sage s preferences guarantee agreement? To answer the first question, assume that the sage s preferences cohere to maximization of expected utility. Unlike the standard expected-utility maximization case, the prior that represents the sage s preferences need not be unique, because the domain of his preferences is restricted to a subset of acts. That is, there might be several prior 3 Monotonicity seems as the strongest requirement one can ask for in the context of extensions.

5 4 EHUD LEHRER AND ROEE TEPER distributions over the state space, with respect to which the expected-utility maximization yields the same preference order. Nevertheless, in terms of expected utility, all these distributions agree on the restricted domain of acts. It turns out that utilizing the prudent rule results in the following preference order: an act f is preferred to g if and only if according to every belief consistent with the sage s preferences, f is preferred to g. Such preferences are known also as Knightian preferences (Giron and Rios [7], Bewley [2]). Knightian preferences are typically incomplete, thus it is natural to ask how to accommodate the prudent rule in order to obtain a complete extension. We suggest a somewhat less strict extension the wary rule that hinges on the main idea behind the prudent rule. The constant equivalent of an act f is the most favorable constant act for which there is evidence (in the sense described above) that f is preferred to. In turn, the wary extension suggests that f is preferred to g if and only if the constant equivalent of f is preferred (according to the sage s preferences) to that of g. We show that the wary extension yields complete preferences and in particular maxmin expected utility (MMEU) preferences (Gärdenfors and Sahlin [6], Gilboa and Schmeidler [9]) with respect to the set of priors that are consistent with the sage s belief. The classical Knightian and MMEU models establish a set of priors, which is interpreted as the DM s perception of ambiguity. An explanation to the source of ambiguity is usually not provided. Here, however, we provide a plausible narrative to the origin of the set of priors employed by preferences under ambiguity such as Knightian and MMEU: ambiguity emanates from the attempt to extend expected-utility preferences over a restricted domain to preferences over a larger one. We commence with conservative preferences that adhere to expected-utility maximization over a restricted domain of acts. These, in turn, induce a set of consistent priors (namely, partially-specified probability (Lehrer [11])) over the larger domain, all agree on the restricted one. Extending the Anscombe Aumann (the sage s) preferences over the restricted domain to the larger one by utilizing the prudent (or wary) rule, results in Knightian (or MMEU) preferences based on the set of the consistent priors. As to the second question, the answer is negative, namely the resulting preferences and those of the sage do not always agree. For example, in case the sage s preference relation is incomplete even over a restricted domain, the extension could still be

6 EXTENSION RULES 5 complete. However, given standard assumptions regarding the sage s preferences, the extension according to the prudent rule coincides with them if and only if the the sage s preferences are Knightian. In this case, the prudent rule yields Knightian preferences (over the entire domain of acts) that are based on this set of priors representing the sage s preferences Related literature. Nehring [12], Gilboa et al. [8] and Karni and Vierø [10] are the most closely related papers to the current one. All three, even though very different from each other, consider two preference relations, where one is the completion of the other, and study consistency axioms that tie the two preferences together. Furthermore, several different assumptions on the completion itself are postulated in each of these papers. The current line of research and its spirit are different from those mentioned above; We present an extension rule and study its implications without posing conditions or requirements on the preferences resulting from the employment of that rule. In particular, we do not assume completeness or any further assumptions on the extension. Gilboa et al. [8] consider a pair of preference relations, Knightian and MMEU, defined over the entire domain of Anscombe-Aumann acts. The former being more reliable but incomplete (referred to as objective ), while the latter is more precarious, but complete ( subjective ). 4 The consistency axioms presented relate the two preferences together, in the sense that both are based on an identical set of priors. The results presented here are related to Gilboa et al. in two aspects. First, by carefully considering how preferences over a restricted domain can serve as a benchmark for preferences over the entire domain, the prudent extension provides an explanation as to how Knightian preferences may arise. Second, by considering the more lenient wary rule, our approach yields a more general result than Gilboa et al without posing any assumptions on the resulting preferences, our approach allows one to obtain MMEU preferences over the entire domain, with the set of priors being the collection of beliefs consistent with the sage s Knightian preferences over the partial domain. Karni and Vierø [10] consider a Bayesian agent and allow the set of alternatives to expand, that is, they let the agent become aware of new alternatives, outcomes 4 See also Cerreia-Vioglio [3].

7 6 EHUD LEHRER AND ROEE TEPER and states of the world. Karni and Vierø postulate that the agent remains Bayesian. In our framework a Bayesian agent need not stay Bayesian proceeding the expansion. Consider, for instance, an Ellsberg type decision maker, who obtains information about the probability of certain events, but not of all events. Over a restricted domain, namely over those acts about which she is fully informed, the decision maker maximizes expected utility. Upon letting the world of alternatives expand to the entire domain, however, she may present a behavior pattern other than Bayesian. It is very likely that she exhibits Knightian uncertainty. Lastly, Nehring [12] presents a framework with a pair of binary relations and, similar to the aforementioned work, studies the relation between these two preferences. The main distinction between the papers mentioned above and Nehring [12] is that one of his primitives (the objective one, in the jargon of Gilboa et al.) is a likelihood relation over events which is assumed to be incomplete, while the other is a preference relation over acts which is assumed to be complete Organization. The following Section 2 presents the framework and primitives of our model. Section 3 then introduces the problem we consider and formally defines the prudent rule. Section 4 studies the example in which the sage is an expectedutility maximizer. Section 5 discusses how the present framework can be interpreted as growing awareness on the decision maker s part and the implications the main results have in this particular case. The issue of consistency is dealt with in Section 6. Finally, proofs are presented in Section The set-up 2.1. Domain. Consider a decision making model in an Anscombe Aumann [1] setting, as reformulated by Fishburn [4]. Let Z be a non empty finite set of outcomes, and let L = (Z) be the set of all lotteries, 5 that is, all probability distributions over Z. In particular, if l L then l(z) is the probability that the lottery l assigns to the outcome z. Let S be a finite non empty set of states of nature. Now, consider the collection F = L S of all functions from states of nature to lotteries. Such functions are referred 5 Given a finite set A, (A) denotes the collection of all probability distributions over A.

8 EXTENSION RULES 7 to as acts. Endow this set with the product topology, where the topology on L is the relative topology inherited from [0, 1] Z. By F c, we denote the collection of all constant acts. Abusing notation, for an act f F and a state s S, we denote by f(s) the constant act that assigns a lottery f(s) to every state of nature. Mixtures (convex combinations) of lotteries and acts are performed pointwise. In particular, if f, g F and α [0, 1], then αf + (1 α)g is the act in F that attains the lottery αf(s) + (1 α)g(s) for every s S Preferences. In our framework, a decision maker is associated with a binary relation over F representing his preferences. We say that f, g F are -comparable (to each other) if either f g or g f. is the asymmetric part of the relation. That is f g if f g but it is not true that g f. is the symmetric part, that is f g if f g and g f. Any such binary relation induces a monotonicity relation S over F. For two acts f, g F, we denote f S g if f(s) g(s) for every s S. Our main focus throughout this paper will be preferences over a subset of acts, which we will denote by H. In particular, if f, g are -comparable, then f, g H. Assumption 1. H is a convex set of acts that contains the constants F c. 3. Sage s preferences and the prudent extension rule Consider a sage s preferences, where the only relevant alternatives at the time of determination were those in H, a sub-collection of acts. In particular, -comparable acts consist only of couples from H. Also consider a decision maker who currently needs to specify his preferences over the entire domain of acts F. Opting to use the guidance of the sage, whose preferences were limited, over H their preferences coincide. But how can he decide between alternatives that are not in H? The decision maker may try to guess what a reasonable preference between such alternatives would be, given its specification over H. In other words, the decision maker may try to assess what would have been plausible if the sage had to state his preferences over the entire domain. Leading example. As in Ellsberg s urn, there are three colors and the probability of drawing red is known to be 1. In order to make this example fit our sage s story, 3

9 8 EHUD LEHRER AND ROEE TEPER suppose that S = {R, B, W } and H consists of acts of the form (a, b, b). 6 This restricts the sage s domain to acts whose expected utility is known. Assume that the sage prefers (a, b, b) over (a, b, b ) if and only if (a a ) 3 + 2(b b ) 3 0. In other words, (a, b, b) is preferred over (a, b, b ) if and only if the expected value of the former is greater than that of the latter, provided that the probability of the first state is 1/3. The sage s preferences over H conform to expected-utility maximization with respect to the prior p = ( 1, 1, 1 ). Of course, any other prior that assigns the probability 1/3 to the first state could represent the sage s preferences as well. The question is how could one extend the sage s preferences so as to compare, for instance, g = (.5,.4,.2) and h = (.7,.2,.1), which are beyond his restricted domain. Definition 1. We say that is an extension of if f 1 f 2 implies f 1 f 2. Here we suggest an extension rule to a binary relation. The rule we introduce uses only pointwise dominance, better known as monotonicity, and the relation itself. Definition 2. Fix two acts g, h F. Two -comparable acts, f 1 and f 2, testify that g is preferred over h if there exists α (0, 1) such that αg +(1 α)f 1 S αh+(1 α)f 2 and f 2 f 1. The -comparable acts f 1 and f 2 testify that g is preferred over h, if despite the fact that f 2 is preferred over f 1, the mixture αg + (1 α)f 1 (state-wise) dominates αh + (1 α)f 2. That is, f 1 and f 2 testify that this reversed dominance (note, f 1 is on the dominating side), must stem from g being preferred over h. The following definition is the heart of the paper. Starting with defined over some pairs of acts, we expand it and define the preference order P R. The rule that enables us to expand the domain of the preference order is called an extension rule. The prominent extension rule is the prudent rule that follows. PR: The prudent rule. g P R h if and only if there are -comparable acts testifying that g is preferred over h. 6 Here a and b already stand for utils. That is, if R is the realized state then the DM exerts a utils, while if states B or W are realized the DM exerts b utils.

10 EXTENSION RULES 9 According to the prudent rule, 7,8 in order to declare that g is preferred to h it is necessary to have an explicit testimony that this is the case. Restating the prudent rule, g P R h if and only if for every two -comparable acts f 1 and f 2 and α (0, 1), αg + (1 α)f 1 S αh + (1 α)f 2 implies f 2 f 1. That is, when αg + (1 α)f 1 S αh + (1 α)f 2 always implies f 2 f 1, it is an evidence that g P R h. Proposition 1. P R is an extension of. Proof. If f g then 1 2 f g = 1 2 g f. In particular, f and g testify that f is preferred to g and, according to the prudent rule, we have that f P R g. Leading example, continued. The acts f 1 = ( 0.2, 0.1, 0.1) and f 2 = (0, 0, 0) are in H, and their expected values are 0. That is, the sage is indifferent between f 1 and f 2. Mixing g with f 2 and h with f 1, with equal probabilities yields 1g+ 1f = (0.25, 0.2, 0.1) and 1h + 1f = (0.25, 0.15, 0.1). The former dominates the latter. In particular, f 1, f 2 testify for the preference of g over h. The prudent rule in this case dictates that the decision maker prefers g to h. We will see below that there is no pair from H that testifies for the preference of h over g (although it is merely an exercise to write the inequalities that are given by the definition of PR and verify that no solution exists), and thus, the conclusion of the prudent rule in this case is that there is a strict preference of g over h. 7 What seems as a possible alternative to Definition 2 is resorting to the preference itself instead of state-wise dominance. However, extending according to this alternative definition will not satisfy monotonicity, for example, which seems a natural property required from an extension rule. 8 Definition 2 and subsequently the prudent rule as they are currently formulated do not fit a framework of risky alternatives. However, replacing state-wise dominance with first order stochastic dominance could yield a candidate for an extension rule in the von Neumann-Morgenstern environment. More formally, assume that is a preference relation over a subset of lotteries. We say that l L first order stochastic dominates l L if l(x Z : x z) l (x Z : x z) for every z Z. Now, in the spirit of Definition 2, l, l L and α (0, 1) testify for the preference of p L over q L if l l but αp + (1 α)l first order stochastic dominates αq + (1 α)l. A full analysis of the extension rules in this framework is left for future investigation.

11 10 EHUD LEHRER AND ROEE TEPER 4. Expected-utility maximization on a restricted domain In this section, we consider the case in which the sage is an expected-utility maximizer, and wish to study the implication of our prudent extension rule Expected-utility maximization and partially-specified probabilities (PSP). Assume that the sage s preferences over H adhere to expected-utility maximization. In particular, there exists an affine 9 vn M utility u over lotteries and a prior distribution p (S) over the state space such that, for every f, g H, f g if and only if E p (u(f)) E p (u(g)). 10 Since H is a subset of F, the prior distribution p that represents the preferences over H need not be unique. Nevertheless, each such prior identically projects on H. To make this statement formal, denote P(u, H, p) the collection of priors over S that are consistent with the ranking of p over the acts in H. That is, (1) P(u, H, p) = {q (S) : for every f, g H, E q (u(f)) E q (u(g)) E p (u(f) E p (u(g))}. For example, consider the simple case where H consists only of constants. Then, P(u, H, p) = (S) and, for every l F c and prior p, we have that E p (u(l)) = u(l). Proposition 2. E q (u(f)) = E q (u(f)) for every q, q P(u, H, p) and every f H. Proposition 2 states that although several preference representing priors may exist, they all cardinally agree on H. This implies that expected utility maximization over H is identical to partially-specified probabilities (Lehrer [11]), where the information available about the probability is partial and is given by the expected values of some, but typically not all, random variables. Formally, a partially-specified probability is given by a list of pairs X = {(X, E p (X))} where X are random variables and E p (X) is the expectation of X with respect to a probability distribution p. By X 1 we denote the set of random variables X such that (X, E p (X)) X. X 1 consists of all random variables the expectations of which 9 u : L R is said to be affine if u(l) = z Z l(z)u(z) for every l L. 10 Such representation is guaranteed given the standard Anscombe-Aumann assumptions (see, for example, Proposition 2.2 in Siniscalchi [13]).

12 EXTENSION RULES 11 become known. That is, the information regarding p is obtained only through the expectations of the random variables in X 1. For instance, if the variables in X 1 are all constants, then the expectations are necessarily the corresponding constants, meaning that no information is obtained about p. However, if X 1 is the set of all random variables, then p is fully specified. Note that in case p is not fully specified, there exist other distributions q that are consistent with the information obtained. That is, E q (X) = E p (X) for every X X 1. Getting back to the sage s preferences over H, due to Proposition 2 we have that {(u(f), E p (u(f))) : f H} (where p is any prior representing over H) is a partiallyspecified probability (henceforth, PSP). Throughout the paper, we denote by (u,h,p) a binary relation over H that coheres to expected-utility maximization with respect to u and p, or alternatively conforms to a PSP Knightian preferences. Before investigating the implications of the prudent extension rule to a preference relation that conforms to PSP, we need to make formal the notion of Knightian preferences. Giron and Rios [7] introduces a status quo approach towards uncertainty and axiomatizes Knightian preferences. Under Knightian preferences the decision maker prefers g to h if g dominates h in the sense that according to all priors in a given multiple-prior set, the expected utility induced by g is greater than that induced by h. Formally, for every two acts g, h F, g is preferred over h, with respect to the utility function u and the convex and closed set P of probability distributions over S, if and only if q P, E q (u(g)) E q (u(h)). Leading example, continued. Denote the collection of priors, consistent with the sage s information, by P = {q ({R, B, W }) : q(r) = 1 }. Let the utility function 3 u be the identity function. Thus, u(g) u(h) = g h. Note that E q (g h) = E q ( 0.2, 0.2, 0.1) 0 for every q P. The minimum over all such q s is 0, and is obtained when q P assigns probability 2 to the state W. On the other hand, there is 3 a prior in P, for instance q = ( 1, 2, 0), such that E 3 3 q(h g) = E q (0.2, 0.2, 0.1) < 0. Thus, it is not true that for every q P, E q (h g) 0.

13 12 EHUD LEHRER AND ROEE TEPER 4.3. The lower integral. Let P be a closed set of probability distributions and A be a set of real-valued functions defined over the state space S. The following functional is defined on the set of random variables over S: (2) U (A,P) (X) = min max q P Y A X Y E q (Y ). In words, 11 U (A,P) (X) is the best possible evaluation of X by random variables in A that are dominated from above by X itself. In case of PSP, denote by A(u, H) the linear subspace of random variables spanned by {u(f) : f H}. Y A(u, H) and every q P(u, H, p), E q (Y ) = E p (Y ). Thus, U (A(u,H),P(u,H,p)) (X) = min q P(u,H,p) max E q (Y ) = Y A(u,H) X Y max E p (Y ). Y A(u,H) X Y Then, for every Leading example, continued. To illustrate the lower integral, let A = span{(1, 1, 1), (1, 0, 0)}. That is, A is the collection of all linear combinations of alternatives that have a know expected value. The largest element in A dominated by g h = ( 0.2, 0.2, 0.1) is ( 0.2, 0.1, 0.1), the expected value of which is 0. Thus, U( 0.2, 0.2, 0.1) = E p ( 0.2, 0.1, 0.1) = 0. On the other hand, the largest element in A dominated by h g = (0.2, 0.2, 0.1) is (0.2, 0.2, 0.2), the expected value of which is strictly smaller than 0. U(0.2, 0.2, 0.1) = E p (0.2, 0.2, 0.2) < 0. Thus, 4.4. What would the sage do? In the leading example we have seen that (a) the prudent rule yields a preference for g over h (b) the lower integral of g h is nonnegative, and (c) the Knightian preferences, with respect to the set of priors consistent with the sage s information, prefers g to h as well. The main result of this section shows that the fact that (a), (b) and (c) occur simultaneously is not a coincidence. To formulate this result, fix a preference relation (u,h,p) that conforms to a PSP as described in Section 4.1. Theorem 1. Consider a binary relation (u,h,p). Then, (1) For every g, h F, g P R h if and only if U (A(u,H),P(u,H,p)) (u(g) u(h)) 0; 11 By X Y we mean that X(s) Y (s) for every s S.

14 EXTENSION RULES 13 (2) For every g, h F, g P R h if and only if E q (u(g)) E q (u(h)) q P(u, H, p). Corollary 1. Applying the prudent extension rule to (u,h,p) generates Knightian preferences, where the set of priors representing the preferences consists of all priors consistent with the PSP (i.e., P(u, H, p)). In particular, when applying the prudent extension rule, the decision maker would prefer g over h if and only if all priors consistent with the sage s preferences agree that g should be ranked higher than h. In (2) the Knightian preference order is expressed in the traditional way: g is preferred to h if and only if there is a consensus among all priors in P that this is the case. In statement (1), however, the Knightian preference order is expressed differently. It is expressed in terms of the lower integral: g is preferred to h if and only if the lower integral of u(g) u(h) is at least zero. Recall that the lower integral is defined by the set of priors P(u, H, p) and the set of variables A(u, H). However, in the case of a binary relation (u,h,p), all priors in P(u, H, p) agree on the variables in A(u, H). Thus, there is no need to consider all priors; one prior, for instance p, is enough; meaning that g is preferred to h if and only if (3) max Y A(u,H) u(g) u(h) Y E p (Y ) 0. Statement (1) of Theorem 1 and Eq. (3) imply that the Knightian preference order can be expressed in terms of the variables in A(u, H), while the set of priors is playing no role. In other words, when starting up with Anscombe Aumann preferences over a restricted domain (or alternatively, with PSP specified only on the acts in H), there is no need to find all the priors that agree with p on H. Rather, only the expected utility associated with every act in H, like in the preferences themselves, are needed in order to find out whether or not g is preferred over h (i.e., to determine whether or not Eq. (3) is satisfied) Complete extension and ambiguity aversion. The term prudent hints to the nature of the suggested solution; for many pairs of acts it is impossible to find a testimony for a preference and thus the extension is, in general, incomplete. We

15 14 EHUD LEHRER AND ROEE TEPER suggest a somewhat more lenient extension, which is a modification of the prudent rule but yields complete preferences over the entire domain. If f is not in H it is impossible to rank f and a(ny) constant act according to (u,h,p). For an act f F, denote by C(f) the subset of constants c F c such that f P R c. That is, C(f) contains all those constants for which there is evidence, in the sense of Definition 2, that f is preferred to. Now, define c(f) to be the maximum over C(f) according to (u,h,p). 12 Given this notion of constant equivalence we extend the preferences (u,h,p) to F: for every f, g F, f W R g if and only if c(f) (u,h,p) c(g). We refer to this extension rule as the wary rule. The following result is an immediate corollary of Theorem 1 and implies that the wary rule yields a complete extension. Proposition 3. Consider a binary relation (u,h,p). Then, for every f, g F, f W R g if and only if min E q(u(f)) min E q(u(g)). q P(u,H,p) q P(u,H,p) The proposition states that in order to rank the different acts, for a decision maker who maximizes expected utility prior to the expansion of the domain, following the wary rule is equivalent to resorting to MMEU preferences a la Gärdenfors and Sahlin [6] and Gilboa and Schmeidler [9], where the collection of priors representing the preferences consists of all priors that agree with p over H. 5. Growing awareness So far we presented a model having a fixed state space. This setup, however, can accommodate informational acquisition and growing awareness. For example, it allows for an expanding state space that might result from an improved ability to distinguish between different states, as we demonstrate in this section. 13 A growing awareness structure is a triple (S 1, S 2, γ), where S 1 and S 2 are two state spaces and γ : S 1 S 2 is a correspondence 14 between S 1 and S 2 that satisfies the two following conditions: (i) if s, s S 1 are two different states, then γ(s) and γ(s ) 12 While the max over C(f) need not be a singleton, every two such elements in C(f) obtain the exact same expected value. 13 See Karni and Vierø [10] for a detailed discussion on the different instances of growing awareness. 14 A correspondence associates a set to each point. Here, γ(s) S 2 for every s S 1.

16 EXTENSION RULES 15 are disjoint; and (ii) S 2 = s S1 γ(s). Conditions (i) and (ii) imply that γ induces a partition of S 2, which we denote by A(γ). The growing awareness structure is interpreted as follows. Initially, the decision maker was aware of the state space S 1. Due to growing awareness, she realizes that the state space is in fact S 2 and what she previously conceived as state s is in fact an event γ(s) in S 2. The reader should be aware of the fact that although S 2 emerges as a result of a growing awareness from S 1, typically, S 1 is not a subset of S 2. In case the underlying state space is S i, the set of all acts is denoted F i, i = 1, 2. The set F 1 can be naturally embedded in F 2. An act f F 1 is identified with an act γ(f) F 2 as follows. For every s S 1, let (γ(f))(t) = f(s) for every t γ(s). This definition implies that the act γ(f) is constant over all states in a partition element of A(γ). Moreover, the value that γ(f) attains over γ(s) coincides with f(s). We denote the image of F 1 under γ by γ(f 1 ). The growing awareness structure describes a particular case of the model presented in Section 2. In order to see this, one may take F = F 2 and H to be γ(f 1 ). In other words, H is the set of all acts defined over S 2 that are measurable with respect to the partition A(γ). In order to illustrate this point, consider a case where a decision maker s preferences adhere to expected-utility maximization over a state space S 1 that consists of two states, R and BW. Suppose that the probability of R is 1 3, while that of BW is 2 3. As a result of growing awareness the decision maker realizes that BW is in fact two separate states, B and W. She is now in the position of the decision maker in the leading example. She is facing the state space S 2 = {R, B, W } with the probability of R being 1 3. In terms of the growing awareness structure model, γ(r) = {R}, γ(bw ) = {B, W } and A(γ) = {{R}, {B, W }}. Upon having improved awareness, the decision maker is facing a larger state space S 2 and she is able to compare between any two acts in γ(f 1 ), namely between acts of the type (a, b, b), but not between other acts. The preference order over γ(f 1 ) conforms to PSP and is precisely the situation dealt with in Theorem 1. By applying the prudent extension rule, the decision maker can extend her preferences, currently defined only on a restricted domain of acts, and obtain Knightian preferences over the set of all acts in F 2.

17 16 EHUD LEHRER AND ROEE TEPER 6. Consistency Proposition 1 states that no matter what the preference over H is, the prudent rule generates preferences P R that extend. Nevertheless, it might be the case that there exist f, g H such that f g but f P R g. An important question is if and under what assumptions P R coincides with (over H). Throughout this section we will make the following assumption. Assumption 2. over H is non-degenerate, reflexive, transitive, complete over constants, monotonic, continuous and constant-independent. 15 Definition 3. is a consistent extension of if for every f, g H, f g if and only if f g. Before attempting to answer the question regarding the consistency of PR, consider the following observation. If satisfies constant-independence, then for any f 1, f 2 H, e F, l F c and α, γ [0, 1] we have that αg + (1 α)f 1 S αh + (1 α)f 2 if and only if γ(αg + (1 α)f 1 ) + (1 γ)(αe + (1 α)l) S γ(αh + (1 α)f 2 ) + (1 γ)(αe + (1 α)l) That is, given constant-independence and transitivity, f 1 and f 2 in H testify (recall Definition 2) that g is preferred to h if and only if γf 1 + (1 γ)l and γf 2 + (1 γ)l testify that γg + (1 γ)e is preferred to γh + (1 γ)e. Proposition 4. If satisfies constant-independence and transitivity, the extension P R satisfies independence. 15 These properties are standard, but here applied only to H. is: non-degenerate if there are f, g such that f g; reflexive if f f for every f; complete over constants if l, l are -comparable, for every l, l F c ; transitive if for every f, g, h, f g and g h imply f h; monotonic if f, g such that f S g then f g; continuous if the sets {f H : f g} and {f H : f g} are closed, for every g H; constant-independence if for every f, g and α [0, 1] and l F c, f g if and only if αf + (1 α)l αg + (1 α)l.

18 EXTENSION RULES 17 This statement implies that when the original order satisfies constant independence but not independence, the prudent rule does not yield a consistent extension. The following proposition summarizes this conclusion. Proposition 5. If satisfies constant-independence but not independence, 16 then P R is not consistent with. Proof. If does not satisfy independence, then there are f, g, h H and α [0, 1] such that αf + (1 α)h, αg + (1 α)h H, were f g but αf + (1 α)h αg + (1 α)h. f P R g since P R extends, and since P R satisfies independence we have that αf + (1 α)h P R αg + (1 α)h. Thus, independence is a necessary condition for consistency. The question is whether independence is a sufficient condition. The following theorem answers this question on the affirmative. Theorem 2. Given Assumption 2, P R is consistent with if and only if satisfies independence. Theorem 2 is implied by Proposition 5 above and Propositions 6 and 7 below. Proposition 6. Let be a preference relation over H satisfying Assumption 2. Then the following are equivalent: 1. satisfies independence; 2. There exist a non-constant affine utility function u over L and a closed convex set of priors P (S), such that, for every f, g H, f g if and only if p P, E p (u(f)) E p (u(g)). Furthermore, (i) u is unique up to a positive linear transformations; and (ii) if P represents as well, then P P. The preferences obtained by independence are standard Knightian preferences, but restricted to H. Unlike the case were the preferences are defined over the entire set of acts, here the representing set of priors need not be unique. However, this case 16 satisfies independence if for every f, g H, α [0, 1] and h H, f g if and only if αf + (1 α)h αg + (1 α)h.

19 18 EHUD LEHRER AND ROEE TEPER owns a maximal set (with respect to inclusion) that contains all the sets of priors that represent the preferences. 17 Proposition 7. Consider Knightian preferences over H represented by a utility u and maximal P. Then, 18 (1) For every g, h F, g P R h if and only if U (A(u,H),P) (u(g) u(h)) 0; (2) For every g, h F, g P R h if and only if E q (u(g)) E q (u(h)) q P. Remark 1. Proposition 7 implies that whenever the sage s preferences are Knightian (and in particular, expected-utility), the prudent rule has a full effect after the first iteration. A second iteration would leave the preferences unchanged. 7. Proofs 7.1. Proof of Proposition 2. Consider two prior distributions p, q over S that represent over H. For every f, g H, we have that (4) E p (u(f)) E p (u(g)) E q (u(f)) E q (u(g)). Now, to the contrary of the postulate, assume that E p and E q do not coincide over H. That is, there exists h H such that, without loss of generality, E p (u(h)) > E q (u(h)). In particular, there exists c {u(l) : l L}, such that E p (u(h)) > c > E q (u(h)). Pick any l L with u(l) = c. Then, since H contains the constants, we have that E p (u(h)) > c = E p (u(l)) and E q (u(l)) = c > E q (u(h)), in contradiction to Eq. (4). 17 To conceptualize that, consider the example in which H = F c. Every subset of priors induces the same preferences which, in fact, adhere to expected-utility maximization. 18 A similar result holds for multi-utility multi-prior incomplete preferences as in Galaabaatar and Karni [5]. The proof follows the lines of that of Theorem 1.

20 EXTENSION RULES Proof of Theorem 1. Before turning to the proof of Theorem 1 we need the following lemma. Lemma 1. Consider a Knightian preference order over H represented by a utility u and a maximal set of priors P. Denote A = A(u, H). Then, U (A,P) is a concave function and U (A,P) (X) = min q P E q (X). Proof. Denote U = U (A,P). The fact that U ia a concave function is easy to prove. We prove that U(X) = min q P E q (X). P is closed and convex. For every p P, E p (X) is linear. Thus, V (X) := min p P E p (X) is well defined and concave. For every q P, E q U. To see this claim, fix a random variable X and suppose that U(X) = E p (Y ), where X Y, Y A, and E p (Y ) E q (Y ) for every q P. Thus, E q (X) E q (Y ) E p (Y ) = U(Y ) for every q P. Hence, E q U, implying that V U, while V (Y ) = U(Y ) for every Y A. We show now that V = U. Fix a random variable X. There is a probability distribution r such that E r supports U at U(X). That is, E r (X) = U(X) and E r U. In particular, E r (Y ) U(Y ), for every Y A. The reason why such E r exists is that U is concave. Thus, it is the minimum of linear functions. But since U is homogenous, these functions are of the form r X (the dot product of r and X), where r is a vector in R S. Since U is monotonic w.r.t. to, r is non-negative. Furthermore, since the constants (positive and negative) are in A, and U(c) = c, and therefore r is a probability distribution. We claim that r P. Otherwise there are f, g H such that f g, but E r (u(g)) > E r (u(f)). Thus, 0 > E r (u(f)) E r (u(g)) = E r (u(f) u(g)). But since E r U, E r (u(f) u(g)) U(u(f) u(g)). We obtain, (5) U(u(f) u(g)) < 0. However, u(f) u(g) A and f g. Thus, for every p P, E p (u(f) u(g)) 0, implying that U(u(f) u(g)) = min p P E p (u(f) u(g)) 0, in contradiction to Eq. (5). We conclude that r P. Thus, U(X) = E r (X) min p P E p (X) = V (X), implying U(X) = V (X). Hence, V = U, as desired. Proof of Theorem 1. To see (1), define a binary relation over all acts as follows: for every two acts g and h, g h if and only if U(u(g) u(h)) 0. Let us show that