Updating Preferences with Multiple Priors

We propose and axiomatically characterize update rules for the preferences with multiple priors of Gilboa and Schmeidler (J. of Math. Econ., 18, pp. 141-153, 1989) for decision making under ambiguity. These rules are the first, for any model of preferences over acts, to be able to reconcile typical behavior in the face of ambiguity (as exemplified by Ellsberg's paradox) with dynamic consistency for all appropriately non-null events. Updating takes the form of applying Bayes' rule to subsets of the set of priors, where the specific subset depends on the preferences, the conditioning event, and the choice problem (i.e., a feasible set of acts together with an act chosen from that set).


Introduction
In dynamic choice under uncertainty, preferences are updated as new information is gathered. Updated preferences are viewed as governing choice upon the realization of a conditioning event. Whereas for expected utility preferences simple conditions on dynamic behavior imply Bayesian updating, there is no such consensus on how to update preferences under ambiguity. In this paper we propose and characterize rules for updating preferences of the max-min expected utility (MMEU) form (Gilboa and Schmeidler [1989]). MMEU preferences have been widely used in modeling ambiguity averse behavior, as exemplified by the famous Ellsberg (Ellsberg [1961]) paradoxes. 1 The major new feature of the rules we characterize is that they are dynamically consistent when updating on any appropriately non-null event, and as a result, depend on prior choices and/or the feasible set for the problem.
To motivate our approach, consider a version of Ellsberg's three-color problem. There is an urn containing 120 balls, 40 of which are known to be black (B) and 80 of which are somehow divided between red (R) and yellow (Y), with no further information on the distribution. A ball is to be drawn at random from the urn, and the decision maker (DM) faces a choice among bets paying off depending on the color of the drawn ball. Any such bet may be written as a triple (u B , u R , u Y ) ∈ R 3 where each ordinate represents the payoff if the respective color is drawn. Typical preferences have (1, 0, 0) Â (0, 1, 0) and (0, 1, 1) Â (1, 0, 1), reflecting a preference for the less ambiguous bets. Notice that these preferences entail a preference to bet on black over red when no bet is made on yellow and a preference to bet on red over black when a bet is also made on yellow. Now consider a simple dynamic version of this problem. In the dynamic version, there is an interim information stage, where the DM is told whether or not the drawn ball was yellow. The DM is allowed to condition her choice of bets on this information. 2 The two relevant choice problems are depicted in Figure 1.1. In each tree, the nodes marked with circles represent the (partial) resolution of uncertainty about the color of the drawn ball, while the nodes marked with squares are choice nodes for the DM. In choice pair 1, the choice "Bet on B" leads to the payoff vector (1, 0, 0) while the choice "Bet on R" leads to payoffs (0, 1, 0). Similarly, in choice pair 2, the choice "Bet on B" leads to the payoff vector (1, 0, 1) while the choice "Bet on R" leads to payoffs (0, 1, 1). Observe that in choosing between the bets (1, 0, 0) and (0, 1, 0), the opportunity to condition one's choice on the new information does not change the problem in an essential way: if a yellow ball is drawn, either choice gives 0, thus it is only conditional on the event {B, R} that the choice changes payoffs. A similar statement applies to the choice between (1, 0, 1) and (0, 1, 1). Therefore, we expect typical preferences to remain (1, 0, 0) Â (0, 1, 0) and (0, 1, 1) Â (1, 0, 1) as in the original three-color problem.
Observe that these preferences are inconsistent with backward induction, which would require the DM to snip the tree at the node following the event {B, R} and to choose as if this were the entire problem. But then the choice between (1, 0, 0) and (0, 1, 0) must be the same as the choice between (1, 0, 1) and (0, 1, 1) since the snipped trees for the two choice pairs are identical, rendering the Ellsberg choices impossible. It follows that no model of dynamic choice under ambiguity implying backward induction can deliver the Ellsberg preferences in this example. This critique encompasses a wide variety of dynamic models in the ambiguity literature, including those of Epstein and Schneider [2003], Hayashi [2005], Klibanoff, Marinacci and Mukerji [2006], Maccheroni, Marinacci and Rustichini [2006], Sarin and Wakker [1998], Siniscalchi [2004] and Wang [2003].
If models implying backward induction cannot capture this behavior, what are the alternatives? Another possibility is to adopt a model in which behavior is naively inconsistent, in the sense that an ex-ante choice or plan of what to do contingent on the event {B, R} may differ from what will actually be chosen when and if that event occurs. The existing literature on updating rules for MMEU preferences (or for any other ambiguity preference model) falls into this category, in that all of the rules examined, if used to define conditional preferences, result in such inconsistent behavior. This literature includes many well-known rules, such as full (or generalized) Bayesian updating 3 and maximum likelihood updating 4 . Applying any of these rules to the dynamic Ellsberg problem will result in a reversal of ex-ante contingent choice in at least one of the two choice pairs. For example, applying full Bayesian updating to MMEU preferences delivering the Ellsberg choices ex-ante will result in a conditional preference for black over red, reversing the ex-ante contingent choice in choice pair 2.
How then may such choices be modeled? Our approach is to look for update rules for MMEU preferences that do not suffer from naive reversals. We show that not only do such rules exist, but they have nice characterizations. All of the rules take the simple form of applying Bayes' rule to a subset of the set of probability measures used in representing the DM's unconditional preferences. An explicit illustration of our rules applied to the above example appears in Section 2.4. All of the dynamically consistent rules we derive deliver the Ellsberg choices. Such rules are shown to necessarily have the feature that the updated preferences will vary with the choice problem encountered: past choices and the choice sets from which they were selected matter for updating. This approach is in the spirit of McClennen [1990] and Machina [1989], who recognized that consistency in choices may be achieved through having later choices influenced by earlier ones. In sum, we provide the only update rules in the ambiguity literature, MMEU or otherwise, that permit consistent dynamic behavior while maintaining the characteristic features of behavior under ambiguity as exemplified by the Ellsberg example.
The remainder of the paper is organized as follows. Section 2.1 introduces the formal framework. Section 2.2 presents the update rules and the preference axioms that characterize them. All of the update rules are shown to take the form of applying Bayes' rule to subsets of the set of measures representing preferences. In Section 2.3, we address the question of how large those subsets may be by identifying the unique ambiguity maximizing rule for each of the sets of rules described in Section 2.2. This delimits precisely the extent to which consistency requires departures from the full Bayesian rule of updating every measure in the set. Additionally, the ambiguity maximizing rules are shown to imply full Bayesian updating for the special case of events with respect to which the set of measures is rectangular. 5 Thus, 3 This rule calls for updating each measure in the representing set of measures according to Bayes' rule. See Jaffray ([1992], [1994]), Fagin and Halpern [1989], Wasserman and Kadane [1990], Walley [1991], Sarin and Wakker [1998], Pires [2002], Siniscalchi [2001], Wang [2003] and Epstein and Schneider [2003] for papers suggesting, deriving or characterizing this update rule in various settings. 4 See e.g., Gilboa and Schmeidler [1993] for a definition of this rule for MMEU preferences. 5 See Epstein and Schneider [2003] or the text of our Proposition 2.13 for the definition of rectangularity these update rules may be used to extend the recursive multiple-priors model (Epstein and Schneider [2003]) in a way that allows updating of all MMEU preferences conditional on all appropriately non-null events. This expansion of the domain on which updating may be carried out is of critical importance if one is interested in ambiguity-sensitive behavior. Section 2.4 applies our rules to the example given above. Section 3 contains further discussion and results concerning key axioms, the nature of the dependence of our rules on the choice problem, the scope of updating and related literature. Section 4 concludes. An Appendix contains lemmata and proofs not appearing in the main text.
2 Characterizing Update Rules

Framework
Consider an Anscombe-Aumann [1963] framework, where X is the set of all simple (i.e., finitesupport) lotteries over a set of consequences Z, S is a set of states of nature endowed with an algebra Σ of events and A is the set of all acts, i.e. Σ-measurable bounded functions f : S → X. Abusing notation in the standard way, z ∈ Z is also used to denote the degenerate lottery δ z ∈ X assigning probability 1 to the prize z, and x ∈ X is also used to denote the constant act for which ∀s ∈ S, f (s) = x. Let P W be the set of all weak orders on A.
Let P MMEU ⊆ P W denote the set of non-degenerate max-min EU preference relations over A (Gilboa and Schmeidler [1989]). For any preference %∈ P MMEU , there exists a compact and convex set of (finitely-additive) probability measures, C, and a vonNeumann-Morgenstern expected utility function, u : If % is non-degenerate, C is unique and u is unique (among vN-M expected utility functions) up to positive affine transformations. As usual, ∼ and Â denote the symmetric and asymmetric parts of %. For E ∈ Σ and f, h ∈ A, we use f E h to denote the act equal to f on E and h on E c . Similarly, if a and b are real Σ-measurable bounded functions, we use a E b to denote the function equal to a on E and b on E c . Let N (%) denote the set of events E ∈ Σ for which ∀q ∈ C, q(E) > 0. In the main body of the paper, these are the only events we consider conditioning on. For a discussion of conditioning on events assigned positive weight by only some measures in C, see Section 3.5. For E ∈ Σ, let ∆ (E) denote the set of all finitely-additive probability measures on Σ giving weight 0 to E c . For any q ∈ ∆ (S) with q(E) > 0, we denote by q E ∈ ∆ (E) the measure obtained through Bayesian conditioning of q on E. Let B denote the set of all non-empty subsets of acts B ⊆ A such that B is convex of a set of measures.
(with respect to the usual Anscombe-Aumann mixtures) and compact (according to the norm taking the supremum over states and Euclidean distance on lotteries in X). Elements of B are considered feasible sets and their convexity could be justified, for example, by randomization over acts. Compactness is needed to ensure the existence of optimal acts. Assume a preference %∈ P MMEU , an event E ∈ N (%) and an act g ∈ A chosen according to % from a feasible set B ∈ B before the realization of E (i.e., g % f, for all f ∈ B). Denote by T the set of all such quadruples (%, E, g, B). An update rule is a function U : T → P W , producing a conditional preference denoted by % E,g,B . Such a conditional preference is viewed as governing choice upon the realization of the conditioning event E.
In any theory including both conditional and unconditional preferences, observability of these preferences may be an issue. Specifically, observing conditional preference requires, at a minimum, observability of the conditioning event relevant to the DM. In a theory such as ours, where conditional preference may depend not only on the conditioning event, but also on the unconditionally chosen act, g, and the feasible set, B, from which it was chosen, one needs to assume that these features are observable as well. In the context of the dynamic Ellsberg example, for instance, one needs to observe the tree within which the choice of betting on black versus red is made to properly interpret which conditional preference this choice belongs to.
Unconditional preferences, %, can in principle be observed, as usual, by having a DM make choices among acts and paying her according to the acts she chooses and the realized states of the world. Having elicited % in this manner, one may then face the DM with a given feasible set B and observe her choice, g, from this set. Then, to elicit conditional preference, % E,g,B , the same problem is presented but now the DM is told that if the state of the world lies in E she will be given an opportunity to revise her initial choice, and then, assuming the state of the world is in E, asked to make choices among acts at that point. Notice that eliciting the conditional preference between f and h where neither is conditionally optimal within B requires offering a different choice set at E than was specified in B. Just as the preference % does not depend on the choice set to which it is applied, it seems quite natural (and in the spirit of imposing the same requirements on conditional preferences as on unconditional) to assume the same for % E,g,B . In the absence of this assumption, one could only hope to elicit conditionally optimal choices from B, rather than the whole conditional preference relation.

Axioms and Update Rules
We characterize update rules U which satisfy a number of axioms. The first axiom requires that since unconditional preferences that we consider are MMEU, conditional preferences should also satisfy the axioms characterizing MMEU (Gilboa and Schmeidler [1989]). There is no reason to imagine different families of preferences at different stages of a decision and good reason to think that properties reasonable for a preference relation are also reasonable for its updates. In particular, the attractiveness of a given set of axioms characterizing a class of preferences is not usually thought to depend on whether the preferences at hand are conditional or unconditional preferences.
In light of CL, for each (%, E, g, B) ∈ T , conditional preferences % E,g,B will be MMEU preferences and if C is the set of measures in the representation of %, we will use C E,g,B to denote the set of measures in the representation of % E,g,B . The next axiom states that a preference conditional on an event E should not depend on the consequences outside of E, a basic requirement on conditional preferences.
Axiom 2.2 NC (Null Complement). For any (%, E, g, B) ∈ T and f, h ∈ A, f ∼ E,g,B f E h.
Next, an axiom is provided that requires the preservation of the ordering of constant acts. In particular this will yield a separation between attitudes towards risk, which are held constant by this unchanged ordering, and the remainder of the preference, which may be affected by updating.
The straightforward proposition below states that these first three axioms are equivalent to the conditional preferences having an MMEU representation using the same u as the unconditional preferences and all measures in C E,g,B placing zero weight on E c . Proposition 2.1 Any update rule satisfying CL, NC, and UT produces MMEU conditional preferences % E,g,B representable using the same (up to normalization) vN-M utility function u as % and non-empty, closed and convex sets of conditional measures C E,g,B containing only measures in ∆ (E). Any such % E,g,B define an update rule satisfying CL, NC, and UT.
Definition 2.1 Let U NC denote the set of all such update rules.
We now introduce an axiom that, together with axioms CL, NC and UT, requires the updated set of measures C E,g,B be generated only from conditionals of measures in C. In other words, no new sets of relative weights on E may suddenly appear in the conditional set of measures that were not already present in the convex hull of the unconditional relative weights on E.
Proposition 2.2 U F LB is the set of all update rules satisfying CL, NC, UT and FLB.
and FLB is satisfied. If C E,g,B contains a measure not in {q E | q ∈ C} then by separating hyperplane arguments there will exist a q * ∈ C E,g,B and an act f such that and thus, f ≺ E,g,B x in violation of FLB. This proves the result.
All the update rules we focus on will satisfy CL, NC, UT and FLB. However, it is worth noting that, with the exception of Proposition 2.13, all of the characterization results in the paper may be easily modified to ones that hold when FLB is dropped. Specifically, wherever the restriction q ∈ C appears in the constructions used below, replacing this with q ∈ ∆ (S) such that q (E) > 0 and replacing U ∈ U F LB with U ∈ U NC will give analogous results that hold without FLB. We choose to present the results with FLB rather than without because we find the restriction to measures in C natural for updating. Furthermore, FLB helps in relating our rules to those in the literature (e.g., Proposition 2.13) and is satisfied by all previously proposed update rules for MMEU preferences. For further discussion of this axiom, its name, and its relation with axioms from the literature, see Section 3.4.
The fifth and most important axiom in our analysis is dynamic consistency. Dynamic consistency of one form or another has often been put forward as a rationality criterion and thus, from a normative point of view, it is important to identify rules that satisfy some version of this property. Moreover, from the normative point of view, optimal acts are the most important for dynamic consistency to be satisfied on because those are the acts that are chosen. Contingent plans made at an initial time should be respected at later times.
An example of a specific normative argument in favor of dynamic consistency appearing in the literature is the argument, referenced in Section 3.1, that inconsistency may lead to dominated choices. Dynamic consistency is also needed to ensure that information has nonnegative value. This is easy to see in the dynamic Ellsberg example from the introduction -if conditional choices differ from what was desired unconditionally, the DM would ex-ante strictly prefer to face the problem without the information as to whether E or E c occurred as opposed to the situation in the introduction where she is allowed to choose after learning this information. Dynamic consistency also makes it easier to describe an individual planning ahead and to make welfare statements in dynamic models. From a more psychological point of view, dynamic consistency may be viewed as a rationalization property: dynamically consistent update rules are those that support earlier choices or plans. Our axiom requires that, for each feasible set of acts, B, if act g is chosen from B unconditionally, the update rule leads it to remain optimal conditionally.

Axiom 2.5 DC (Dynamic Consistency). For any
Observe that conditional optimality of g is checked against all feasible f such that f = g on E c . Why check conditional optimality only against these acts? Dynamic consistency is relevant only ceteris paribus, i.e., when exactly the same consequences occur on E c . To make clear why this is reasonable, consider an environment where the DM has a fixed budget to allocate across bets on various events. It would be nonsensical (and would violate payoff dominance on the realized event) to require that the ex-ante optimal allocation of bets remained better than placing all of one's bets on the realized event. This justifies the restriction of the conditional comparisons to acts that could feasibly agree on E c . See Sections 3.1 and 3.2 for further discussion and results concerning DC. Now we examine the implications of DC for updating. We start by defining a key subset of the measures in C for each quadruple (%, E, g, B). Definition 2.3 For (%, E, g, B) ∈ T , define the measures in C supporting the conditional optimality of g to be Denote by Q E,g,B E the set of Bayesian conditionals on E of measures in Q E,g,B .
There are two reasons why calling these sets "measures supporting the conditional optimality of g" makes sense. The first is obvious: if we consider a conditional expected utility preference with measure q E ∈ Q E,g,B E , then according to such a preference, g will be conditionally optimal. The second reason is deeper: as we will show next, assuming CL, NC, UT, and FLB, the existence of a measure in Q E,g,B E that is used to evaluate u • g conditionally is equivalent to the conditional optimality of g.
The following result completely characterizes the set of update rules satisfying the previous four axioms plus DC: Proposition 2.3 U DC is the set of all update rules satisfying CL, NC, UT, FLB and DC.
Proof. First we show that all rules in U DC satisfy DC. Let q g E be an element of Q E,g,B An argument very similar to the one in the proof of Lemma A.1, with % E,g,B in place of % and the measure r corresponding to the separating hyperplane in ∆ (E) rather than ∆ (S), may be used to show that D 1 ∩ D 2 = ∅ (and thus DC) implies Q E,g,B E ∩ arg min q∈C E,g,B R (u • g)dq 6 = ∅. Therefore, any rules in U F LB satisfying DC must be in U DC . Since Proposition 2.2 implies that any rules outside of U F LB violate the axioms, we have shown that U DC is the set of all update rules satisfying CL, NC, UT, FLB and DC. Note the distinction between the condition in the proposition, Q E,g,B E ∩arg min q∈C E,g,B R (u• g)dq 6 = ∅, and the stronger Q E,g,B E ⊇ arg min q∈C E,g,B R (u • g)dq. The axioms allow that some measures used to evaluate g conditionally do not support the conditional optimality of g -consider a simple example with S = {1, 2, 3}, E = {1, 2}, Z = R, g = (1, 1, 1), B = co {(0, 2, 1), (2, 0, 1)}, u the identity for degenerate lotteries, and C any set containing a measure giving the same weight to states 1 and 2. 7 Here C E,g,B may contain many more measures than ¡ 1 2 , 1 2 ¢ , as full Bayesian updating of C does not conflict with DC in this example. The following immediate corollary states a useful alternative representation of U DC , as it suggests an algorithm for constructing such update rules. Start with any r ∈ Q E,g,B and include its update r E in C E,g,B . Further members of U DC are found by adding to C E,g,B updates of some measures in C satisfying R (u • g)dq E ≥ R (u • g)dr E . Doing this for each r ∈ Q E,g,B traces out the entire U DC . 7 We use co to denote the convex hull operator. ) .
If one considers feasible sets that are smooth at g, then the algorithm becomes even simpler, as going through it with only one r is sufficient: Corollary 2.2 If the feasible set B does not have a kink on E at the optimal act g, Q E,g,B E is a singleton.
Proof. By the argument in the proof of Lemma A.1, since there is no kink on E in the feasible set at g, there exists a unique tangent hyperplane to B on E at g. Thus, Q E,g,B E consists of the unique measure q E ∈ ∆ (E) associated with that hyperplane.
Our next axiom is a form of dynamic consistency not implied by DC. It says that all feasible acts indifferent to g that agree with g on E c should remain optimal conditional on E. Under DC, all that is implied for such acts is g % E,g,B f . One motivation for this axiom is the view that dynamic consistency may not simply be about the unconditionally chosen act remaining optimal among those feasible acts agreeing with g on E c . It should also be robust to indifference being resolved conditionally differently than it was unconditionally, in the sense that all feasible acts agreeing with g on E c that were unconditionally indifferent to g should remain optimal choices conditional on E. See Sections 3.1 and 3.2 for further discussion and connections with conditions in the literature.

Axiom 2.6 PFI (Preservation of Feasible Optimal Indifference). For any
Proposition 2.4 U DC∩P F I is the set of all update rules satisfying CL, NC, UT, FLB, DC and PFI.
Proof. By inspection, U DC∩P F I ⊆ U DC and so these rules satisfy the first five axioms. We now show they satisfy PFI.
for all f ∈ B with f = g on E c and f ∼ g and PFI is satisfied.
Consider an update rule U satisfying the axioms, with associated sets of measures C U E,g,B . Since U satisfies the first five axioms, Corollary 2.1 implies that U ∈ U F LB and there exists an Therefore, it also implies that for any q E ∈ C U E,g,B , This shows that C E,g,B must have the stated form.
We now introduce an additional condition which, in the presence of DC, implies PFI and is stronger than PFI only for infeasible acts. This strengthening is not easily connected with dynamic consistency considerations, however the condition does not seem unreasonable. In this axiom, the part of the indifference curve through g agreeing with g on E c is picked out for a special role in updating. This part of the indifference curve may be thought of as the portion where g is being used as a reference act. The axiom requires that g occupy an extremal position in the conditional preference relative to the other elements of this part of the (unconditional) indifference curve.
Axiom 2.7 RA (Chosen Act as a Reference Act). For any (%, One way of viewing RA is as saying that updating must preserve or increase ambiguity affecting g more than ambiguity affecting any f ∼ g with f = g on E c .

Definition 2.6 Given
and T E,g E be the set of Bayesian conditionals on E of measures in T E,g . 8 ) 8 Our results for rules satisfying RA can be simplified if there is no best or worst outcome or if one is willing to restrict attention to cases where u • g is interior to the utility boundary. In these circumstances, we can replace the set T E,g everywhere with the set arg min p∈C R (u • g)dp. Notice that the former depends on E while the latter does not.
Proposition 2.5 U RA is the set of all update rules satisfying CL, UT, NC, FLB and RA.
Proof. Suppose the axioms are satisfied. Then the rule must be in Therefore, it also implies that for any It remains to show that any element of arg min p∈C E,g,B R (u • g)dp is also the update So, the axioms imply the form given above. Conversely, fix any update rule in U RA with associated set of measures C E,g,B . By definition, such an update rule is in U F LB and thus satisfies the first four axioms.
This proves RA is satisfied as well.
An alternative strengthening of PFI is obtained by replacing f % E,g,B g with g % E,g,B f in RA. However, a small modification of the example in the proof of Proposition 3.4 can be used to show there is no update rule satisfying the alternative condition, CL, NC and UT. We now consider the implications of RA in the presence of DC.
and R E,g,B E be the set of Bayesian conditionals on E of measures in R E,g,B .
Proposition 2.6 U DC∩RA is the set of all update rules satisfying CL, UT, NC, FLB, DC and RA.
Proof. By inspection, U DC∩RA ⊆ U DC ∩ U RA and thus the axioms are satisfied. Consider an update rule U satisfying the axioms, with associated set of measures C U E,g,B . Then (u • g)dp and U violates RA, a contradiction. Thus r ∈ R DC ∩ R RA and this implies U ∈ U DC∩RA . One natural question that has not yet been addressed is whether U DC∩RA or any of the larger classes of rules defined above is non-empty. In the next section, we examine a number of specific update rules, including a rule denoted U DC∩RAmax that is shown to be an element of U DC∩RA . The existence of this rule suffices to prove that all of the sets of rules we have defined above are non-empty.

Ambiguity maximizing update rules
Among the update rules we characterize, one may wonder if there are rules that are most conservative, in the sense of maintaining the most ambiguity in the process of updating. Examining such rules is particularly illuminating because they reveal the precise extent to which dynamic consistency and the other axioms force the DM to eliminate measures present in the unconditional set when updating. If, for example, one views full Bayesian updating (updating all measures in the initial set) as "the right thing to do" then examining these rules shows how far one must depart from this to maintain consistency. To begin our exploration, we first define what it means for a preference to display more ambiguity than another.
Definition 2.11 Suppose % 1 , % 2 ∈ P MMEU . Say that % 1 displays more ambiguity than This definition is essentially the comparative ambiguity aversion of  (and is also closely related to Epstein [1999]) in our setting. 9 The way to understand this is as follows. MMEU preferences reflect an aversion to ambiguity, and so when ambiguity increases, this should be bad news for acts which may be affected by ambiguity relative to acts, such as constant acts, that are evaluated unambiguously. Thus, more ambiguity should correspond to general acts falling in the preference order relative to constant acts.
We correspondingly define an update rule to entail more ambiguity than another if it always produces an updated preference displaying more ambiguity than the preference updated by the other rule. This leads to the following notion of an ambiguity maximizing update rule: 9 It is known that changes in ambiguity and in ambiguity attitude are potentially entangled in MMEU models. Some formal support for the notion that, to the extent one separates them, it is ambiguity rather than ambiguity attitude that changes across the MMEU class is in Ghirardato, Maccheroni and Marinacci [2004] and Klibanoff, Marinacci and Mukerji [2005].
Definition 2.12 We say that an update rule U is ambiguity maximizing among a set U of update rules if for all U 0 ∈ U, (%, E, g, B) ∈ T , % E,g,B = U(%, E, g, B) and % 0 E,g,B = U 0 (% , E, g, B), % E,g,B displays more ambiguity than % 0 E,g,B .
We begin by stating two simple results showing that, when combined with our first three and first four axioms respectively, ambiguity maximization characterizes well known rules: conditional maxmin utility and full Bayesian updating. These results follow immediately from inspection of the families U NC and U F LB derived in the previous section.
Definition 2.13 The maxmin utility update rule is defined by U MU ∈ U NC with sets of measures C E,g,B = ∆ (E).
Proposition 2.7 U MU is the unique ambiguity maximizing update rule satisfying axioms CL, NC, and UT.
Definition 2.14 The full Bayesian update rule, denoted U F B , is the update rule in U NC such that Proposition 2.8 U F B is the unique ambiguity maximizing update rule satisfying axioms CL, NC, UT and FLB.
Neither of the above rules are dynamically consistent. After imposing DC we obtain the following rule (where Q E,g,B E is as defined in Definition 2.3 of the previous section) that eliminates exactly those measures in C that do not conditionally evaluate the chosen act g highly enough.
Definition 2.15 U DCmax is the update rule in U F LB such that Proposition 2.9 U DCmax exists and is the unique ambiguity maximizing update rule satisfying CL, NC, UT, FLB and DC.
Proof. To show U DCmax exists, it suffices to show that min q∈Q E,g,B E R (u • g)dq always exists, as C E,g,B will then be non-empty. If Q E,g,B E is non-empty and (weak * -)compact then That Q E,g,B (and thus Q E,g,B E ) is non-empty follows from Lemma A.1. We now show that intersection of two closed sets, and so is closed. It is bounded because it is a set of measures.
Thus it is compact. So, min q∈Q E,g,B E R (u • g)dq exists and thus so does U DCmax .
conditionals included in anyĈ E,g,B associated with rules in U DC , U DCmax is the unique ambiguity maximizing rule satisfying the five axioms.
To define the ambiguity maximizing rule in U DC∩P F I , it is useful to define a subset, denoted K E,g,B , of the measures supporting the conditional optimality of g. Measures in K E,g,B are required to satisfy an additional condition beyond those determining Q E,g,B : Thus, measures in K E,g,B may be thought of as supporting not only the conditional optimality of g, but also the conditional optimality of all unconditionally optimal acts agreeing with g on E c .
Definition 2.16 For (%, E, g, B) ∈ T , Definition 2.17 U DC∩P F Imax is the update rule in U F LB such that .
Proposition 2.10 U DC∩P F Imax exists and is the unique ambiguity maximizing update rule satisfying CL, NC, UT, FLB, DC and PFI.
Proof. First we show that U DC∩P F Imax exists and U DC∩P F Imax ∈ U DC∩P F I and so satisfies the axioms. Then we show it is uniquely ambiguity maximizing within U DC∩P F I . We begin proving existence by noting that Lemma A.1 shows K E,g,B (and thus with f = g on E c and f ∼ g}, and Q E,g,B E was shown to be closed in the proof of Proposition 2.9, K E,g,B E is the intersection of two closed sets, and so is closed. It is bounded because it is a set of measures. Thus it is compact. So, and, by inspection, Thus, taking r E ∈ arg min p∈K E,g,B E R (u • g)dp and an associated unconditional r ∈ K E,g,B , we see that C E,g,B is non-empty and U DC∩P F Imax ∈ U DC∩P F I . Given r E , it is obvious that anyĈ E,g,B associated with an update rule in U DC∩P F I is made largest by replacing ⊆ with = in the definition of U DC∩P F I . It only remains to show that such a choice of r E leads to the smallest value of min p∈Ĉ E,g,B R (u • g)dp for rules in Without loss of generality, assume min p∈Ĉ E,g,B R (u • g)dp = R (u • g)dr E . If and only if such anr E ∈Ĉ E,g,B will there be a rule in U DC∩P F I displaying more ambiguity than U DC∩P F Imax .
, a contradiction. This shows C E,g,B is the maximal set of measures among all sets associated with rules in U DC∩P F I , and so U DC∩P F Imax is uniquely ambiguity maximizing. We now derive the unique ambiguity maximizing rules for the collections of update rules satisfying RA (both with and without DC).
Definition 2.18 U RAmax is the update rule in U F LB such that Proposition 2.11 U RAmax is the unique ambiguity maximizing rule satisfying CL, UT, NC, FLB and RA.
Proof. For any q ∈ arg min p∈T E,g R (u • g)dp E , q ∈ T E,g and therefore Thus U RAmax ∈ U RA and so satisfies the axioms.
Given such a q, it is obvious that anyĈ E,g,B associated with an update rule in U RA is made largest by replacing ⊆ with = in the definition of U RA . Choosing q ∈ arg min p∈T E,g R (u•g)dp E leads to the smallest value of min p∈Ĉ E,g,B R (u • g)dp for rules in U RA and thus C E,g,B is the largest set of conditional measures for rules in U RA .
Definition 2.19 U DC∩RAmax is the update rule in U F LB such that Proposition 2.12 U DC∩RAmax exists and is the unique ambiguity maximizing update rule satisfying CL, UT, NC, FLB, DC and RA.
Proof. The proof follows by an argument almost identical to the one in the proof of Proposition 2.10 with R E,g,B playing the role of K E,g,B .
We conclude this section by exploring the form of these ambiguity maximizing rules in an interesting special case that has received attention in the literature. The following proposition shows the set C E,g,B in the case of rectangular preferences (Epstein and Schneider [2003]), i.e., when the representing set of measures C satisfies a type of stochastic independence given an event and given its complement. For these preferences, each act is evaluated by minimizing separately the conditional expectation on E and on E c and then minimizing the expectation of these conditional expectations. Our result is that, for any of the ambiguity maximizing rules we have identified satisfying at least the first four axioms, the conditional preference is defined using the Bayesian update of all measures. This result shows that these update rules agree with that of Epstein and Schneider's recursive multiple priors model on the domain of that model. 10 Thus, when updating on events with respect to which preferences are rectangular, the ambiguity maximizing rules identified by our approach satisfy the same properties as recursive multiple priors. At the same time, our rules are able to allow for updating on all other non-null events in a manner satisfying desirable axioms including DC, while the recursive multiple priors approach rules out and therefore does not address such events (e.g., the event that the ball drawn is not yellow in the Ellsberg example of the Introduction and Example 2.1 in Section 2.4).
Definition 2.20 Suppose E, E c ∈ N (%). If ∀q 1 , q 2 , q 3 ∈ C, ∃q 4 ∈ C such that ∀F , q 4 (F ) = q 3 (E) q 1 (F ∩E) Proposition 2.13 Any of the ambiguity maximizing update rules defined above satisfying at least CL, NC, UT and FLB coincides with U F B for (%, E, g, B) ∈ T such that E, E c ∈ N (%) and C is rectangular with respect to {E, E c }.
Proof. Suppose f ∈ J g . Then min p∈C R (u •f )dp = min p∈C R (u• g)dp. By rectangularity and f = g on E c , the minimized expected utilities of f and g conditional on E c must be equal. Therefore, to maintain the overall equality in the minimization, again invoking rectangularity, they must also be evaluated equally on E (i.e., min p∈C R (u • f )dp E = min p∈C R (u • g)dp E ).
10 Note that Epstein and Schneider's framework is not identical to ours in that they place more temporal structure on the acts and outcomes than we do. For example, their ultimate outcomes are vectors of consumption over time while we have unstructured ultimate outcomes. Thus, literally speaking, with a rectangular set of measures, the two models are identical in their sets of measures and how they are updated, but not in all the structural details.
Therefore, under rectangularity, arg min p∈C R (u • g)dp E ⊆ T E,g ⊆ C and so, min p∈T E,g R (u • g)dp E = min p∈C R (u • g)dp E . By inspection of the definition of U RAmax (Definition 2.18), the just proven equality, and [f ∈ J g =⇒ min p∈C R (u • f )dp E = min p∈C R (u • g)dp E ], U RAmax = U F B . Furthermore, under rectangularity, it is straightforward to show that U F B satisfies DC and therefore U RA∩DCmax = U RAmax . Observe that U DC∩RAmax is the ambiguity maximizing rule defined over the most restrictive set of axioms we examine, and thus has sets of updated measures that are weakly smaller (in the sense of set inclusion) than those for all other ambiguity maximizing rules we consider. This together with Proposition 2.8 proves the result.

A dynamic Ellsberg example
Example 2.1 Recall the three-color Ellsberg problem described in the introduction. Consider %∈ P MMEU with C = co{( 4 12 , 3 12 , 5 12 ), ( 4 12 , 5 12 , 3 12 )}, Z = R, and u (z) = z for z ∈ Z. The states correspond to the color of the ball drawn, (black, red, yellow) respectively. Consider U DC , the set of dynamically consistent update rules we derived earlier. Suppose that the conditioning event is E = {1, 2} (i.e., the DM learns that the ball drawn is not yellow). When faced with the choice set B = co {(1, 0, 0), (0, 1, 0)}, since g = (1, 0, 0) is the unique optimum in B, and, applying Corollary 2.1, any closed convex subset of is the updated set of measures corresponding to some rule in U DC . Notice that all these measures place more weight on black than red. In particular, the rule U DCmax updates exactly all such measures, resulting in Similarly, when the choice set is B 0 = co {(1, 0, 1), (0, 1, 1)}, since the corresponding g 0 = (0, 1, 1), conditional on E rules in U DC would update a subset of measures that put at least as much weight on red as black. So, and conditionally the DM maintains the preference for betting on red over black. This provides an explicit illustration of how update rules satisfying DC reconcile the Ellsberg choices with dynamically consistent updating.
In this example, one may check that additionally imposing PFI or RA does not change the resulting rules.

Dynamic consistency
Given that DC and the related PFI are key axioms in our analysis, in this and the subsequent section we present remarks and results designed to help better understand these conditions and their relation to the previous literature. We start by discussing possible modifications of these axioms.
Recall that DC and PFI impose conditions only on feasible acts equal to g on E c . This was justified in the text immediately following the statement of DC. Note, however, that a similar justification might also admit slightly stronger versions of the axioms where the qualification f = g on E c is replaced by the weaker statement u (f (s)) = u (g (s)) for all s ∈ E c . We wish to point out that if this change is made to these axioms, then all of the results involving the axioms continue to hold when the same replacement is made in the statements of the results and, furthermore, the same arguments can be used in the proofs. It is also worth noting that in cases where the feasible set has a special structure such that f, f 0 ∈ B =⇒ f E f 0 ∈ B, the restriction f = g on E c in the statement of DC is superfluous. Such feasible sets arise, for example, whenever one starts from a decision tree with branches corresponding to events E and E c and derives B by specifying what is feasible conditional on E, denoted B E , and what is feasible conditional on E c , denoted B E c , and then combining the two so that B While this may hold in some settings, there are many common and important applications, such as any problem involving allocating resources across events subject to a budget constraint, in which feasible sets will not have this form.
DC checks only conditional optimality of g. Adding PFI extends the checking of conditional optimality to any f ∼ g and equal to g on E c . Why not check that the conditional ordering of other acts agreeing with g on E c is preserved? The reason is threefold. First, the verbal essence of dynamic consistency involves reversals, which will only ever have the opportunity to occur when they involve unconditionally optimal acts. As Machina [1989] writes (pp. 1636-7) ". . . behavior. . . will be dynamically inconsistent, in the sense that . . . actual choice upon arriving at the decision node would differ from . . . planned choice for that node." Second, many normative arguments in support of dynamic consistency, such as arguments showing how lack of consistency may lead to payoff-dominated outcomes (see e.g., Machina [1989], McClennen [1990], Seidenfeld [2004], and Segal [1997]), require only the conditional optimality of g. Third, and perhaps most importantly, if we expand DC to cover comparisons of other acts, we have the following impossibility result.
Proposition 3.1 There does not exist an update rule satisfying CL, NC, UT and DC1. Epstein and Schneider [2003], when discussing differences between recursive multiple priors and the robust control model of Hansen and Sargent [2001] point out that the robust control model satisfies a version of dynamic consistency that checks only optimality of g. Aside from minor differences in the framework, we write this axiom below as DC2. The difference from DC is that comparisons with g are not restricted to acts in the feasible set. Why restrict comparisons of g to feasible acts? First, again we point out that the essence of dynamic consistency involves reversals, which are only relevant if they involve ex-ante feasible acts; second, if we expand DC to cover comparisons with acts outside of B, we again have an impossibility result.
Axiom 3.2 DC2 For any (%, E, g, B) ∈ T , if f ∈ A with f = g on E c , then g % f implies g % E,g,B f . Proposition 3.2 There does not exist an update rule satisfying CL, NC, UT and DC2.
Another way in which DC is weak is that it requires only weak conditional preference of g over f . Therefore, it is compatible with the axiom to unconditionally have g Â f for some f = g on E c while conditionally g ∼ E,g,B f . In such a circumstance, it is true that the DM is willing to continue with the initially chosen act g, but this is only weakly so. If we were to strengthen DC to rule out such shifts from strict preference to indifference, however, we would again have impossibility. Finally, we offer a condition stronger than PFI in that it omits reference to feasibility. The condition is related to DC3 but is neither implied by it nor does it imply it. The condition requires the entire part of the indifference curve containing g that agrees with g on E c to be preserved upon updating. We show that this condition also leads to impossibility in our framework.
Axiom 3.4 PI (Preservation of Optimal Indifference). For any (%, E, g, B) ∈ T , if f ∈ A with f = g on E c and f ∼ g, then f ∼ E,g,B g.
Proposition 3.4 There exists no update rule satisfying CL, NC, UT and PI.
Collectively, these results show that many directions of strengthening DC and PFI are unsuitable for our purposes. In the next section, we discuss related notions of dynamic consistency appearing in the literature. It turns out that not only are these conditions distinct from ours, but the existing conditions lead to impossibilities in our framework.

Related consistency concepts in the literature
Though dynamic consistency appears not to have been formally stated in the literature in the way we do, it is clearly related to some existing notions. McClennen [1990] provides an excellent and deep analysis of the problem of rational dynamic choice and as a key part of this defines dynamic consistency (p. 120) within a formal framework of decision trees. 11 In our terms, this definition seems closest to the following: f % h for all h ∈ B if and only if f % E,g,B h for all h ∈ B such that f = h on E c . The 'only if' direction (running from unconditional to conditional preference) is stronger than DC and PFI together. Specifically, DC and PFI together imply that direction only for those f such that f = g on E c , where g was chosen unconditionally. The 'only if' direction of McClennen's condition is what we would get if we were to impose, in addition to DC and PFI, the axiom ICA described in Section 3.3. There we also describe the unique ambiguity maximizing update rule that can satisfy these three axioms in our setting. To us, ICA concerns the extent to which conditional preference may depend on past choices and is not about dynamic consistency. Moreover, the 'if' direction of McClennen's definition is not implied by any combination of our axioms. In fact, when we strengthen DC to imply even a weaker version of this condition, we get an impossibility result (see Proposition 3.3 above). Machina [1989] discusses dynamic consistency in the context of preferences over lotteries in decision trees with known probabilities. As reflected in the quote we used in the previous section, Machina's verbal explanation goes quite well with our definition. However, when later in the same paper Machina proposes a rule for conditional preferences, his rule imposes a much stronger consistency property; namely, having initially chosen (p, x; 1 − p, y), 12 conditional on going down the branch given probability p, z % p,(p,x;1−p,y) z 0 if and only if (p, z; 1 − p, y) % (p, z 0 ; 1 − p, y). Machina and Schmeidler [1992] adopt the subjective analogue of Machina's rule. They require that if g is the initially chosen act and the event E occurs, then conditional preferences are defined by f % E,g h if and only if f E g % h E g. Epstein and Le Breton [1993] show that Machina and Schmeidler's update rule together with Savage's [1954] axioms minus P2 (the sure-thing principle) applied to both unconditional and conditional preferences implies preferences must be probabilistically sophisticated. This implies that using Machina and Schmeidler's update rule to define dynamic consistency leads to nonexistence in our setting, as MMEU preferences are not generally probabilistically sophisticated. Our Proposition 3.1 above, shows that even a weaker version of Machina and Schmeidler's condition is sufficient to generate impossibility in our problem. Segal [1997] also discusses dynamic consistency in the context of preferences over lotteries in decision trees. The weakest dynamic consistency property he states (his axiom 1) says that any conditionally optimal choice must also be part of an unconditionally optimal plan. Since this axiom requires strict unconditional preference for an optimal act to remain strict conditionally, it leads to nonexistence in our setting (this follows from our Proposition 3.3). Segal [1997] goes on to propose an even stronger consistency axiom (his axiom 3) which weakens Machina's update rule to apply only to the indifference curve containing the initially chosen lottery. Specifically, having initially chosen (p, x; 1 − p, y), conditional on going down the branch given probability p, x % p,(p,x;1−p,y) z if and only if (p, x; 1 − p, y) % (p, z; 1 − p, y). All of the analysis in Segal [1997] is carried out using an extension of the stronger axiom 3 rather than axiom 1. Gul and Lantto [1990] propose an axiom (called weak consequentialism) on preferences over lotteries in two-stage decision trees with known probabilities. At the broadest level, they motivate their axiom by the principle that all feasible choices that are optimal at the root, should remain optimal when and if the DM arrives at a subset of the original feasible set. This may sound similar to our axioms DC and PFI together. However, from our earlier analysis and the discussion to follow, it will be apparent that their axiom has strikingly different implications for preferences. Their axiom says the following: Suppose that at the root of the tree the feasible set of lotteries is {x, y, px + (1 − p)q, py + (1 − p)q}. More specifically, the DM faces an initial choice between branches leading to x, y and a chance node; if the chance node is chosen, with probability p the DM gets the opportunity to choose from a subset, {x, y}, of the original choice set, while with probability 1 − p, he gets q. Furthermore, suppose that according to the preference at the root of the tree x ∼ py + (1 − p)q and both are optimal within the feasible set. Then it must be that px + (1 − p)q ∼ x. The idea is that if the DM imagines selecting py + (1 − p)q from the root and the "p" branch is realized, then if the DM has the option of choosing x instead of y at that point, they should find this weakly desirable, since x was optimal at the root when y was present. They show that this axiom is equivalent to preferences over lotteries satisfying betweenness (i.e., v ∼ z implies αv + (1 − α)z ∼ v for all α ∈ [0, 1]). There appears to be no clear analogue to weak consequentialism in the context of preferences over acts -taking mixtures over lotteries is quite different than combining acts across events. In a direct translation to combining acts across events the axiom would seem to be: If {f, h, f E r, h E r} are feasible and f ∼ h E r are optimal within this set then f E r ∼ f . This is completely unsatisfactory -take f = 0 E 1, h = 1 E 0, and r = 0 E 0 and one gets 0 E 1 ∼ 1 E 0 implies 0 E 0 ∼ 0 E 1. An alternative would be to translate mixtures over lotteries to mixtures over acts, so the axiom would say: If {f, h, αf + (1 − α)r, αh + (1 − α)r} are feasible and f ∼ αh + (1 − α)r are optimal within this set then αf + (1 − α)r ∼ f . Letting h = r implies the mixture analogue of betweenness for optimal acts: f ∼ h implies αf + (1 − α)h ∼ f for all α ∈ [0, 1]. In fact, by adapting the proof of Gul and Lantto to apply to mixtures over acts, one can show that betweenness for all acts is implied. Fixing an MMEU preference, this betweenness condition is satisfied if and only if preferences are expected utility. In contrast, the conjunction of DC and PFI places no restrictions on unconditional MMEU preferences. Compared to this literature, then, DC and PFI are weak requirements.

Dependence of conditional preference on the past choice and feasible set
A key feature of our notion of dynamic consistency is that it allows for updating to vary depending on the choice of g and the feasible set B. In contrast, more restrictive definitions of dynamic consistency that rule out such dependence are not uncommon in the literature (see e.g., Ghirardato [2002]). It is well-known that this more restrictive dynamic consistency makes it impossible to capture non-expected utility behavior under the assumption that the DM cares only about the acts induced by their strategy (see Ghirardato [2002] among many others). Others in the literature on subjective uncertainty have allowed, as we do, dependence on g (e.g., Machina and Schmeidler [1992], Epstein and Le Breton [1993]). One may wonder, however, if it is really necessary to allow dependence of the conditional preference on the feasible set B as we do. The following proposition shows that it is necessary.
Proposition 3.5 There does not exist an update rule satisfying CL, NC, UT, DC and IFS.
Our next result shows that it is really only dependence on part of the feasible set that is needed.
Definition 3.1 Given (%, E, g, B) ∈ T , the undominated-on-E feasible set is Axiom 3.6 IDFS (independence from the dominated part of feasible sets). For every (% , E, g, Proposition 3.6 There exist update rules satisfying CL, NC, UT, DC and IDFS.
Note that a similar proposition is true even when our additional axioms, such as PFI and RA are also imposed. Generally speaking, if one wanted to build the whole paper while considering only the undominated-on-E parts of the feasible sets, very little would change. Specifically, the only thing that would change would be that the option to vary update rules with other aspects of the feasible set would be eliminated. The point is that this option is never used to prove existence, ambiguity maximality or any other important properties of our rules.
The necessity of dependence on the feasible set is related to the lack of smoothness characteristic of MMEU preferences. The next proposition shows that in the special case when there is no preference kink at g, one may dispense with dependence on the feasible set.
Proposition 3.7 If we restrict attention to % such that there is no kink in the indifference curve at g, then there exist update rules satisfying CL, NC, UT, DC and IFS for such %. U DC∩RAmax is such a rule.
Given that it is generally necessary to allow conditional preference to depend on the feasible set, one may wonder whether if it is also necessary to have further dependence on g? We show below that the answer is no. We also show that while the U DC∩RAmax and U DC∩P F Imax rules do depend on g, this dependence is on only the part of g on E c .
First, the possibility result. The result makes use of the following lemma: We define what it means to be independent of g and identify the ambiguity maximizing rule satisfying this condition and our first six axioms.
Recall that K E,g,B E was defined in Definition 2.16 of Section 2.3.
Proposition 3.8 U DC∩ICAmax exists and is the unique ambiguity maximizing update rule satisfying CL, NC, UT, FLB, DC, PFI and ICA.
We note that DC and ICA together imply PFI, thus U DC∩ICAmax is also the unique ambiguity maximizing update rule satisfying CL, NC, UT, FLB, DC, and ICA. The following example shows that U DC∩ICAmax can yield much different updating than when ICA is not imposed. In the example, while the other axioms allow a substantial amount of ambiguity after conditioning no matter which optimal act is chosen as g, adding ICA removes the ambiguity entirely. 13 Example 3.1 Suppose there are three states. Consider %∈ P MMEU with C = co{(0.3, 0.2, 0.5), (0.1, 0.7, 0.2), (0.7, 0.2, 0.1)}, Z = R, and u (z) = z for z ∈ Z. When faced with the choice set B = co{(1, 16.75, 1), (2, 15.25, 1),(8.4, 5.9, 0.9), (7.4, 7.4, 0.9)}, the DM is indifferent among all acts in B and all are evaluated by (0.3, 0.2, 0.5). Suppose that the conditioning event is E = {1, 2} (i.e., the DM learns that the true state is one of the first two) and that g = (1, 16.75, 1). According to the update rule U DC∩P F Imax , the updated set of measures is If, instead, g = (8.4, 5.9, 0.9) then the updated set of measures according to U DC∩P F Imax is From the definition of U DC∩ICAmax , the updated set of measures according to that rule (and thus according to any rule satisfying the axioms CL, NC, UT, FLB, DC and ICA) is the singleton set {(0.6, 0.4, 0)} .
The next result shows that U DC∩RAmax and U DC∩P F Imax do depend on g.
Proposition 3.9 U DC∩RAmax and U DC∩P F Imax do not satisfy ICA.
The reason for these violations of ICA is quite intuitive. When the chosen act g differs on E c from all other feasible acts, the consistency conditions DC and PFI have no bite, as there are no relevant comparisons. Thus, in U DC∩P F Imax there is nothing to prevent using the updates of all measures in such cases, while for other possible g, DC and PFI often do have bite. In addition to this effect, for U DC∩RAmax , the restrictions imposed by RA vary with the set of acts f ∈ A such that f = g on E c and f ∼ g which in turn varies with g on E c .
Next we show that despite failing ICA, the U DC∩RAmax and U DC∩P F Imax rules satisfy a weaker independence condition. Specifically, given the feasible set and unconditional preferences, they depend on only the part of g on E c .
Proposition 3.10 U DC∩RAmax and U DC∩P F Imax satisfy ICA-E.

The fixed point constant lower bound condition
In this section, we supplement the discussion of the FLB axiom that followed Proposition 2.2 by relating FLB to an axiom from the literature. The following is an axiom that has been used to characterize full Bayesian updating (U F B ) for MMEU preferences in Pires [2002]. 14 Axiom 3.9 BFE (Backward Fixed Point Constant Equivalent) For any (%, E, g, B) ∈ T , We call this condition "Fixed Point" because it requires the conditional constant equivalent x to satisfy the fixed point condition f E x ∼ x or, in terms of the MMEU representation, . We call it "Backward" because it runs backward in time from the conditional preference to the unconditional. The following is the characterization result in our setting: Proposition 3.11 (Pires [2002]) U F B is the unique update rule satisfying axioms CL, NC, UT and BFE.
This is a slight strengthening of the main result in Pires [2002], the only difference being that the possible update rules in Pires [2002] were not allowed to depend on g or B. The same proof as in Pires [2002], however, shows that this expansion does not affect the result. The next result shows that BFE may be written as an equivalent condition going "forward" from the unconditional to the conditional preference.
Proposition 3.12 Given axioms CL, NC and UT, BFE is equivalent to FE.
Given these results and the fact that U F B does not satisfy DC, neither FE nor BFE will yield any dynamically consistent update rules. One may think of FE as having two components: If Together with axioms CL, NC and UT, Proposition 2.2 shows that the first condition, which is axiom FLB, requires that the updated set of measures C E,g,B be generated only from conditionals of measures in C. In other words, no new sets of relative weights on E may suddenly appear in the conditional set of measures that were not already present in the convex hull of the unconditional relative weights on E. The second condition requires the opposite -that the convex hull of conditionals (weakly) expands compared to that in the unconditional set. Thus, FLB may be thought of as a weakening of FE (or BFE) that is compatible with dynamic consistency.

Updating on events not always given positive probability
Throughout the paper we have limited attention to updating on events that are non-null in the strong sense that q (E) > 0 for all q ∈ C. In common with standard Bayesian updating of subjective probabilities, we have nothing to say about updating on events that are obviously null in that q (E) = 0 for all q ∈ C. What about events for which there is some q ∈ C with q(E) > 0? In this section we extend our characterizations of rules obeying the first four axioms, both with and without imposing dynamic consistency, to apply to updating on such events. Though the characterization results extend, we will see that the set of events cannot be expanded too far without running into existence problems under dynamic consistency.
The result below shows that Proposition 2.2 extends to events where we only suppose that there exists a q ∈ C with q(E) > 0. The only difference from the original result is an explicit limitation to those measures in C giving positive weight to E and the need to take the closure since {q E | q ∈ C and q(E) > 0} may not be closed.
Denote byQ E,g,B E the set of Bayesian conditionals on E of measures inQ E,g,B .
Similarly, let b U DC be the modification of U DC to refer toQ E,g,B instead of Q E,g,B and b U F LB instead of U F LB . We can now extend Proposition 2.3 to events E such that there exist a q ∈ C with q (E) > 0.

Proposition 3.14 b
U DC is the set of all update rules satisfying CL, NC, UT, FLB and

DC.
However, without further assumptions, b U DC might be empty.
The reason this may occur is that the separation arguments in Lemma A.1, while guaranteeing that the set Q E,g,B is non-empty, do not imply the existence of such elements that also assign positive weight to E. In the example, the only measure in Q E,g,B is (0, 0, 1).
A sufficient condition to ensure the existence of rules in b U DC is that all the measures that unconditionally minimize the expected utility of g place positive weight on E. This follows because Lemma A.1 shows that Q E,g,B ∩ arg min q∈C R (u • g)dq 6 = ∅. Is it enough to require that some measure in arg min q∈C R (u • g)dq places positive weight on E? An example shows that the answer is no: .1, .8, 0), (0, 0, 0, 1)}, u(z) = z, B = co{(4, 7, 0, 1), (6, 4, 0, 1)}. Let g = (6, 4, 0, 1). Note that arg min q∈C R (u • g)dq = C and all but one measure in C places positive weight on E. However Q E,g,B = {(0, 0, 0, 1)} and thuŝ Q E,g,B is empty.
In sum then, our Proposition 2.3 characterizing dynamically consistent update rules extends in a natural way to any events assigned positive probability by some measure in C. However, existence of such rules is not guaranteed for all these events. Still, existence can be guaranteed for events assigned positive weight by all minimizers of u • g, a larger set than our set of non-null events, N (%). Presumably, similar exercises can be explored for the other, more restrictive sets of update rules we characterize as well. Whether there is a nice characterization of the largest sets of events for which existence of the various rules may be guaranteed is a question we leave for future work.

Further discussion of related literature
There are a number of papers that have examined update rules for MMEU preferences. As mentioned in the Introduction, full Bayesian updating (applying Bayes' rule to each measure in the set of measures) has been suggested and explored by Jaffray ([1992], [1994]), Fagin and Halpern [1989], Wasserman and Kadane [1990], and Walley [1991]. Sarin and Wakker [1998], Pires [2002], Siniscalchi [2001], Wang [2003] and Epstein and Schneider [2003] formally characterize this update rule using preference axioms in various settings. Maximum likelihood updating (applying Bayes' rule to only those measures assigning the largest probability to the observed event) has been explored in terms of preferences in Gilboa and Schmeidler [1993]. Gilboa and Schmeidler [1993] also define, but do not characterize in terms of preferences, a large class of rules for updating sets of priors that they call "classical" update rules. This class includes both maximum likelihood and full Bayesian updating, among others, and turns out to be exactly the set of rules in U F LB that also are independent of g and B. Thus our axioms CL, NC, UT, FLB, ICA and IFS provide a preference characterization of the entire set of classical update rules. Hansen and Sargent [2001] investigate a rule that, like Maximum likelihood updating, applies Bayes' rule to only a subset of measures in the set of measures, but determines this subset through a procedure that may depend on the feasible set and that produces relative entropy neighborhoods surrounding a reference measure. Their rule is only defined and explored by them for the subset of MMEU preferences that have sets of measures taking the form of relative entropy neighborhoods of a given measure. For preferences that lie in the intersection of MMEU and Choquet expected utility (Schmeidler [1989]), a variety of rules that have been explored in the context of updating non-additive measures may be applied to the sets of measures in the preferences we consider. The most prominent of these rules is the Dempster-Shafer rule. This rule was developed in Dempster [1968] and Shafer [1976], and has been characterized in terms of preferences a number of times (see e.g., Gilboa and Schmeidler [1993], Wang [2003], and Nishimura and Ozaki [2003]). Other rules for updating non-additive measures are also explored in Gilboa and Schmeidler [1993] and Lehrer [2004]. Many update rules we describe and characterize in this paper, for example, U DCmax , U DC∩P F Imax , and U DC∩RAmax , are distinct from any previously mentioned in the literature. In particular, all of the previous rules for updating MMEU preferences that satisfy our first three (very basic) axioms, also are independent of the feasible set B. Proposition 3.5 then implies that these rules must violate DC. Thus, no previously proposed update rules for the class of MMEU preferences satisfy our first three axioms plus DC.
There are a number of papers related to updating multiple priors preferences that impose dynamic consistency on more restricted domains. For example, Epstein and Schneider [2003], Sarin and Wakker [1998], Hayashi [2005], Maccheroni, Marinacci, Rustichini [2006] and Eichberger, Grant and Kelsey [2005a] all impose versions of dynamic consistency only for a given information filtration (as opposed to all filtrations simultaneously). Siniscalchi [2001], Eichberger, Grant and Kelsey [2005b] and Ghirardato, Maccheroni and Marinacci [2002] apply dynamic consistency only to preference comparisons that involve constant acts in a particular way. Hansen and Sargent's [2001] rule is only guaranteed to deliver dynamic consistency when applied to the subset of MMEU preferences that have sets of measures that are relative entropy neighborhoods of a given reference measure. 15 In addition to dynamic consistency, two other conditions important to justifications of updating in the decision theory literature are consequentialism and reduction of compound acts/decision trees. In our setting, consequentialism means that preference conditional on an event E depends only on the unconditional preference, the event E, and satisfies NC. Reduction means that preferences are defined over acts, and thus the DM is assumed to care only about the mapping from states to (lotteries over) outcomes induced by his actions. Our approach maintains reduction but relaxes consequentialism. Two papers on updating multiple priors that relax reduction while maintaining consequentialism are Wang [2003] and Hayashi [2005]. 16 It has been shown in various contexts that consequentialism, reduction and some version of dynamic consistency (together with standard assumptions) imply expected utility and Bayesian updating (see e.g., Karni and Schmeidler [1991], Ghirardato [2002]). As a result, at least one of these properties must be relaxed if updated preferences are not to be of the expected utility form.
Our focus on relaxing consequentialism rather than reduction has at least three justifications. First, maintaining consequentialism rules out the Ellsberg pattern of choices over acts in the dynamic Ellsberg example of the Introduction. Second, relaxing reduction means that uncertainty is treated differently at different decision nodes, a property which may not be attractive in terms of a pure updating interpretation where the role of time is not, in and of itself, an important consideration. Third, viewing the current state of the literature, the only previous theory of updating MMEU preferences while at the same time maintaining dynamic consistency with respect to updating on all appropriately non-null events is in Wang [2003]. As mentioned above, Wang's theory relaxes reduction. In light of this, one might have thought that relaxing reduction was the only way to maintain this scope of dynamic consistency. Thus, it makes sense for us to focus on relaxing consequentialism because there is more to be learned in that direction. Furthermore, there is an important sense in which relaxing reduction breaks completely the connection between dynamic consistency and the form of updating. In contrast, our approach of relaxing consequentialism maintains important and interesting links between the two, as we have shown.
There are approaches to dynamic decision making that take dynamic inconsistency of 15 It is not hard to show, using examples like the Ellsberg ones presented earlier, that natural attempts to generalize Hansen and Sargent's [2001] updating procedure to all MMEU preferences may fail to satisfy DC. They justify their rule on hueristic and robust-control related grounds, rather than in terms of the properties of the underlying preferences. 16 Both of these papers relax reduction in a manner influenced by the seminal work of Kreps and Porteus [1978], which was done in the context of decision trees with objective probabilities. preferences over acts as given. Dynamic behavior is derived by coupling the underlying preferences with an assumption about how the different "selves" deal with the inconsistency. The best known approach here is due to Strotz [1955-6], and assumes that behavior is sophisticated in the sense that the DM correctly anticipates her future conditional desires and chooses taking the results of those future decisions as a constraint. Observe that this is quite a distinct treatment of dynamic choice from the one put forward here, where conditional preferences agree with the optimal choices of the unconditional preference. Siniscalchi [2004] investigates the preferences over decision trees that are compatible with the Strotzian type of backward induction approach. As we pointed out in the Introduction, no such preferences can deliver behavior in the dynamic modification of Ellsberg's example that agrees with the typical preferences in Ellsberg's original problem (namely, (1, 0, 0) Â (0, 1, 0) and (0, 1, 1) Â (1, 0, 1)). For a critique of the sophistication approach based on value of information considerations see e.g., Epstein and Le Breton [1993] (pp. 11-12).
Three alternative approaches that, like sophistication, accept that preferences over acts may be dynamically inconsistent, are McClennen's [1990] resolute choice, the cooperative decision processes of Jaffray [1999], and the game-theoretic approach of Peleg and Yaari [1973]. In resolute choice, it is assumed that conditional choices are in agreement with an unconditionally optimal plan, even when those conditional choices may conflict with some underlying conditional preference. McClennen [1990] does not specify how this agreement is to be obtained. The resulting behavior is as if the conditional choices were governed by dynamically consistent underlying conditional preferences that depend on earlier choices and the choice problem, such as those in our paper. 17 In this sense, adopting one of our update rules satisfying DC is a way of implementing resolute choice for MMEU preferences, while also preserving the property that conditional choices are based solely on conditional preferences. Jaffray [1999] suggests that the inconsistency between unconditional and conditional preferences might be resolved in a way that is more of a compromise between the different preferences. He examines a selection criterion that chooses a plan that is "not too bad" in a utility sense according to any of these preferences and is not dominated in that no feasible plan is better according to all the preferences. Peleg and Yaari [1973] propose treating the conflict generated by dynamic inconsistency as a non-cooperative game between the different "selves" of the DM and solving for an equilibrium. Ozdenoren and Peck [2005] use extensive form games of conflict with nature to illustrate how varying the game the DM thinks she is playing is an alternative modeling strategy for generating variation in conditional choices in Ellsberg-like problems. Tallon and Vergnaud [2003] show that it may be important in terms of dynamic consistency in the Ellsberg problem whether one updates on information about the color of the ball drawn or the color composition of the urn.
Finally, Grunwald and Halpern [2004] and Seidenfeld [2004] are some other recent papers that have considered max-min decision rules and noted the dynamic inconsistency of full Bayesian updating. Each of these papers argues that, at least in some cases, e.g., the "Ignorance is Bliss" phenomenon discussed in Section 2 of Grunwald and Halpern [2004], the ex-ante or "global" max-min preference seems more reasonable than the inconsistent ones generated by full Bayesian updating of non-rectangular sets of priors. Since the exante choice is exactly what is maintained through dynamic consistency in our approach, the examples in their papers could also be used as support for this dynamic consistency in the face of multiple priors.

Conclusion
We have proposed and axiomatically characterized novel update rules that apply to MMEU preferences over acts. These preferences are important in analyzing dynamic behavior in the presence of ambiguity. This paper provides the first theories of ambiguity sensitive preferences that maintain, in dynamic extensions (as in our introductory example), typical behavior under ambiguity as exemplified by the classic Ellsberg problem. In particular, no theories based on recursion or backward induction may deliver this behavior, and neither can any theory that naively updates in a dynamically inconsistent manner.
The major new feature of the rules we characterize is that they are dynamically consistent when updating on any appropriately non-null event, and as a result, depend on prior choices and/or the feasible set for the problem. Nonetheless, all of the rules take the simple form of applying Bayes' rule to a subset of measures used in representing the DM's unconditional preferences. In addition to deriving the entire set of rules compatible with our axioms, we identify, in Section 2.3, the unique ambiguity maximizing rules, thus delimiting the extent to which dynamic consistency and other axioms we examine force the DM to reduce the set of measures she considers in the face of new information. In the special case of events with respect to which the DM's set of measures is rectangular (the updating domain in Epstein and Schneider [2003]), we show that all measures in the set may be updated without conflicting with our axioms.
The dynamic consistency axioms we use, DC and PFI, are weaker than those appearing elsewhere, yet effectively capture the fundamental idea that choices optimal unconditionally remain optimal according to the updated preference. As we discussed in Section 3.1, strengthening these axioms in various ways leads to non-existence. A variation that is possible, and that we explore, is replacing PFI by the stronger axiom RA.
While MMEU is a well-known and important class of preferences for modeling behavior under ambiguity, it is far from the only such model. In ongoing work, we are investigating the implications of similar axioms for other preference models, such as the model in Klibanoff, Marinacci and Mukerji [2005].
A Appendix: some lemmata and selected proofs A.1 Lemmata used in Section 2 Proof. The first part of the proof shows ∅ 6 = Q E,g,B ∩ arg min p∈C R (u • g)dp. Let m and ≈ be the asymmetric and symmetric parts of v. Consider the convex sets D 1 ≡ {a | a ∈ I with a m u • g} and D 2 ≡ {u • f | f ∈ B with f = g on E c }. Unconditional optimality of g implies D 1 ∩ D 2 = ∅. D 1 is non-empty by inspection and also has a nonempty interior. D 2 is non-empty since it contains u • g. By a separating hyperplane theorem (e.g., Aliprantis and Border [1999], Thm. 5.50, p. 190), there must exist a hyperplane separating D 1 and D 2 . Without loss of generality, such a hyperplane may be defined by © a ∈ I | R adr = α ª for a finitely additive measure r ∈ ∆ (S) and real α such that Then by the event-wise continuity of R (·) dr and monotonicity of v, there would exist an a ∈ D 1 such that R adr < α, a contradiction. Thus, α = R (u • g)dr. Therefore, The same argument by contradiction shows for all a ∈ I. Now, by the argument in the "only if" direction of the proof of Lemma A.2 applied to v, r ∈ arg min p∈C R (u • g)dp. From the definition of Q E,g,B and the fact that R (u • g)dr ≥ R bdr for all b ∈ D 2 , r ∈ Q E,g,B . Thus, r ∈ Q E,g,B ∩ arg min p∈C R (u • g)dp 6 = ∅.
Finally, suppose q ∈ Q E,g,B ∩ arg min p∈C R (u • g)dp.
Therefore, q ∈ R E,g,B . So, Q E,g,B ∩ arg min p∈C R (u•g)dp ⊆ R E,g,B . Since R E,g,B imposes more restrictions than K E,g,B , R E,g,B ⊆ K E,g,B . Since K E,g,B imposes more restrictions than Q E,g,B , K E,g,B ⊆ Q E,g,B .
Lemma A.2 Suppose there is no best or worst consequence in Consider h 0 ∈ A such that h = αh 0 + (1 − α) x for some α ∈ (0, 1). Such an h 0 exists because the assumption of no best or worst consequence in Z ensures that an open neighborhood of h (using the norm defined by the supremum over state utility differences) exists. Observe that h 0 ∼ It remains to show that q ∈ C. For all f ∼ h, h)dp = min p∈C is the unique unconditionally optimal choice in B 1 . Similarly, g is uniquely unconditionally optimal in B 2 . By Proposition 2.1, % E,g,B ∈ P MMEU and % E,g,B can be represented using the same u as % and using a set of measures contained in ∆ (E). Thus, since all acts in B 1 give a weakly higher payoff in state 1 than in state 2, according to % E,g,B 1 , all acts in B 1 should be evaluated using the same measure, namely the measure in the updated set of measures that places the most weight on state 2. Denoting this measure (α, 1 − α, 0), DC implies that α = 1 2 . To see this, observe that α < 1 2 implies ( 3 2 , 3 2 , 1) Â E,g,B 1 g and α > 1 2 implies ( 5 2 , 1 2 , 1) Â E,g,B 1 g, both violations of DC. Therefore the updated set of measures in the representation of % E,g,B 1 contains no measure placing weight more than 1 2 on state 2 and contains the measure placing exactly weight 1 2 on state 2. Now suppose that U satisfies IFS. Therefore % E,g,B 2 =% E,g,B 1 . From the facts about the representation of % E,g,B 1 derived above, one can see that all acts in B 2 are conditionally evaluated using the measure ( 1 2 , 1 2 , 0). But then ( 14 9 , 14 9 , 1) Â E,g,B 2 g, contradicting DC. Thus U cannot satisfy IFS. Proof of Proposition 3.6. The rule U DCmax defined in Definition 2.15 satisfies these axioms. To see that IDFS is satisfied, observe that the only place B enters the definition of the rule is through the set Q E,g,B . It is clear that adding or removing acts dominated on E from B does not affect which measures satisfy the inequalities defining Q E,g,B (Definition 2.3).
Proof of Proposition 3.7. Consider the U DC∩RAmax update rule (see Definition 2.19). By Proposition 2.12, it satisfies CL, NC, UT, and DC. Moreover, if there is no kink in the indifference curve at g, then arg min p∈C R (u • g) dp is a singleton, and therefore equals the set R E,g,B used in defining U DC∩RAmax . Thus U DC∩RAmax does not depend on B (except through g) and U DC∩RAmax (%, E, g, B 1 ) = U DC∩RAmax (%, E, g, B 2 ) for any (%, E, g, B 1 ), (%, E, g, B 2 ) ∈ T with no kink in the indifference curve at g.
Proof of Lemma 3.1. Let (%, E, g, B) ∈ T . By an argument similar to the proof of Lemma A.1, there exists v ∈ arg min p∈C R (u • g)dp such that Proof of Proposition 3.8. Consider v ∈ C as identified in Lemma 3.1. From the proof of that lemma and the definition of K E,g,B E , it follows that v E ∈ C E,B . Therefore C E,B is non-empty. It is closed and convex since it is the intersection of closed convex sets. Thus U DC∩ICAmax exists. U DC∩ICAmax ∈ U DC∩P F I because for each g, v ∈ K E,g,B ⊆ Q E,g,B and C E,B ⊆ {q E | q ∈ C and R (u • f )dq E ≥ R (u • g)dv E for all f ∈ B with f = g on E c and f ∼ g}. Now, suppose that for some E and B a rule in U DC∩P F I uses a setĈ E,B , independent of g, that contains measures not in C E,B . Then, by definition of C E,B , there must be someq ∈Ĉ E,B and some g such thatq / ∈ C U DC∩P F Imax E,g,B . But this implies that one of the first six axioms in the statement of the result would be violated, because U DC∩P F Imax is the ambiguity maximizing rule satisfying these six axioms. This proves that U DC∩ICAmax is the unique ambiguity maximizing rule satisfying the seven axioms in the proposition.
Proof of Proposition 3.10. Fix any (%, E, g 1 , B), (%, E, g 2 , B) ∈ T . If there is a unique optimal act in B according to % then g 1 = g 2 and the result is trivial. Similarly, if there are multiple optima, but none agree on E c , then ICA-E has no bite. Therefore, suppose that there are multiple optima and at least some of them agree on E c . Fix any such optima g 1 and g 2 with g 1 = g 2 on E c . From the definition of K E,g,B (Definition 2.16), since g 1 , g 2 ∈ B with g 1 ∼ g 2 and g 1 = g 2 on E c , K E,g 1 ,B = K E,g 2 ,B . Furthermore, q ∈ K E,g 1 ,B implies R (u • g 1 )dq E = R (u • g 2 )dq E . Therefore, by inspection of the definition of U DC∩P F Imax , U DC∩P F Imax (%, E, g 1 , B) = U DC∩P F Imax (%, E, g 2 , B). Similarly, one can show that R E,g 1 ,B = R E,g 2 ,B and q ∈ R E,g 1 ,B implies R (u • g 1 )dq E = R (u • g 2 )dq E . Since J g 1 = J g 2 , this is enough to show that U DC∩RAmax (%, E, g 1 , B) = U DC∩RAmax (%, E, g 2 , B This is a contradiction since it would imply min q∈C R (u • f E x)dq < u (x), and thus f E x ≺ x.) Suppose f E x ∼ x. If C E,g,B is the closure of a subset of {q E | q ∈ C and q (E) > 0} then min q∈C E,g,B R (u • f )dq ≥ inf q∈{p E |p∈C and p(E)>0} R (u • f )dq ≥ u (x) and FLB is satisfied. This shows that all update rules in b U F LB satisfy the axioms.
Next suppose C E,g,B contains a measure not in the closure of {q E | q ∈ C and q (E) > 0}. By separating hyperplane arguments there will exist a q * ∈ C E,g,B and an act f such that R (u • f )dq * < inf q∈{p E |p∈C and p(E)>0} R (u • f )dq. We claim that there always exists an x * ∈ X such that f E x * ∼ x * and inf q∈{p E |p∈C and p(E)>0} R (u • f )dq = u (x * ). (To see this, first note that by taking an appropriate mixture over the best and worst outcomes that f gives on E, we can find an x * such that inf q∈{p∈C|p(E)>0} R (u • f )dq E = u (x * ). It remains to show that this implies f E x * ∼ x * . If argmin q∈C R (u • f E x * )dq contains a measure assigning zero weight to E, then this follows trivially. Otherwise, letting r ∈ argmin p∈C R (u•f E x * )dp, since Thus f E x * ∼ x * .) Therefore f E x * ∼ x * and, since R (u • f )dq * < inf q∈{p E |p∈C and p(E)>0} R (u • f )dq = u(x * ), f ≺ E,g,B x * in violation of FLB. This shows that no update rules outside of b U F LB obey the axioms.
Proof of Proposition 3.14. Modify the proof of Proposition 2.3 by replacing Q E,g,B E withQ E,g,B E and replacing the reference to Proposition 2.2 by a reference to Proposition 3.13.