Scholarly article on topic 'Dynamic choice under ambiguity'

Dynamic choice under ambiguity Academic research paper on "Computer and information sciences"

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Academic research paper on topic "Dynamic choice under ambiguity"

Theoretical Economics 6 (2011), 379-421

1555-7561/20110379

Dynamic choice under ambiguity

Marciano Siniscalchi

Department of Economics, Northwestern University

This paper analyzes dynamic choice for decision makers whose preferences violate Savage's sure-thing principle (Savage 1954) and, therefore, give rise to violations of dynamic consistency. The consistent-planning approach introduced by Strotz (1955-1956) provides one way to deal with dynamic inconsistencies;how-ever, consistent planning is typically interpreted as a solution concept for a game played by "multiple selves" of the same individual.

The main result of this paper shows that consistent planning under uncertainty is fully characterized by suitable behavioral assumptions on the individual's preferences over decision trees. In particular, knowledge of ex ante preferences over trees is sufficient to characterize the behavior of a consistent planner. The results thus enable a fully decision-theoretic analysis of dynamic choice with dynamically inconsistent preferences.

The analysis accommodates arbitrary decision models and updating rules; in particular, no restriction needs to be imposed on risk attitudes and sensitivity to ambiguity.

Keywords. Ambiguity, consistent planing, value of information. JEL classification. D81, D83.

1. Introduction

In a dynamic-choice problem under uncertainty, a decision maker (DM henceforth) acquires information gradually over time, and takes actions in multiple periods and information scenarios. The basic formulation of expected utility (EU) theory instead concerns a reduced-form, atemporal environment, wherein preferences are defined over maps from a state space O to a set of prizes X (acts). Thus, to analyze dynamic-choice problems, it is necessary to augment the atemporal EU theory with assumptions about the individual's preferences at different decision points. The standard assumption is of course Bayesian updating: if the individual's initial beliefs are characterized by the probability q, her beliefs at any subsequent decision point h are given by the conditional probability q( |B), where the event B represents the information available to the

Marciano Siniscalchi: marciano@northwestern.edu

I have greatly benefited from many extensive conversations with Peter Klibanoff and Nabil al-Najjar. I also thank the co-editor, Bart Lipman, and two anonymous referees, as well as Pierpaolo Battigalli, Eddie Dekel, Larry Epstein, Paolo Ghirardato, Alessandro Lizzeri, Fabio Maccheroni, Alessandro Pavan, and Larry Samuelson for their valuable insights, and Simone Galperti for excellent research assistance. All errors are my own.

Copyright © 2011 Marciano Siniscalchi. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at http://econtheory.org. DOI: 10.3982/TE571

individual at h. Together with the assumption that the individual's risk preferences do not change, Bayesian updating ensures that the DM's behavior satisfies a crucial property, dynamic consistency (DC): the course of action that the individual deems optimal at a given decision point h, on the basis of the preferences she holds at h, is also optimal when evaluated from the perspective of any earlier decision point h' (and conversely, if h is reached with positive probability starting from h'). This implies in particular that backward induction or dynamic programming can be applied to identify optimal plans of action.

Bayesian updating and the DC property are intimately related to the cornerstone of Savage's axiomatization of EU, namely his Postulate P2; Section 2 discusses this tight connection and provides references. Sensitivity to ambiguity (Ellsberg 1961), or to the common-ratio or common consequence effects (Allais 1953, Starmer 2000) and other manifestations of non-EU risk attitudes typically leads to violations of Savage's Postulate P2. As a consequence, violations of DC are to be expected when such preferences are employed to analyze dynamic-choice problems; again, Section 2 elaborates on this point and provides illustrative examples. These violations of DC, and ways to address them, are the focus of the present paper.

Whenever a conflict arises among preferences at different decision points, additional assumptions are required to make clear-cut behavioral predictions (Epstein and Schneider 2003, p. 7). One approach, introduced by Strotz (1955-1956) in the context of deterministic choice with changing time preferences and tastes, is to assume that the DM adopts the strategy of consistent planning (CP). In Strotz's own words, at every decision point, a consistent planner chooses "the best plan among those that [s]he will actually follow" (Strotz 1955-1956, p. 173).

Formally, CP is a refinement of backward induction that incorporates a specific tie-breaking rule. Informally, CP reflects the intuitive notion that the DM is sophisticated: that is, she holds correct "beliefs" about her own future choices. The problem with this intuitive notion is that, of course, beliefs about future choices cannot be observed directly; they also cannot be elicited on the basis of the DM's initial and/or conditional preferences over acts.

The literature on time-inconsistent preferences circumvents this difficulty by suggesting that CP is best viewed as a solution concept for a game played by "multiple selves" of the same individual. Strotz himself (Strotz 1955-1956, p. 179) explicitly writes that "[t]he individual over time is an infinity of individuals"; see also Karni and Safra (1990, pp. 392-393), O'Donoghue and Rabin (1999, p. 106), and Piccione and Rubinstein (1997, p. 17). However, at the very least, this interpretation represents "a major departure from the standard economics conception of the individual as the unit of agency" (Gul and Pesendorfer 2008, p. 30). It certainly does not clarify what it means for an individual decision maker to adopt the strategy of consistent planning. It reinforces the perception that a sound, behavioral analysis of multiperiod choice requires some form of dynamic consistency (Epstein and Schneider 2003, p. 2). Finally, it provides very little guidance with regard to policy analysis.

This paper addresses these issues by providing a fully behavioral analysis of CP in the context of dynamic choice under uncertainty. In the spirit of the menu-choice literature

initiated by Kreps (1979), I assume that the individual is characterized by a single, ex ante preference relation over dynamic choice problems, modeled as decision trees. I then show the following conditions.

• Under suitable assumptions, conditional preferences can be derived from ex ante preferences over trees, regardless of whether preferences over acts satisfy Savage's Postulate P2 (see Section 4.2 and Theorem 1).

• Sophistication can be formalized as a behavioral axiom on preferences over trees, regardless of whether DC holds (see Section 4.3.2).

• The proposed sophistication axiom, plus auxiliary assumptions, provides a behavioral rationale for CP (Theorems 2 and 3), again regardless of whether P2 or DC holds.

Three features of the analysis in this paper deserve special emphasis. First, the approach in this paper is fully behavioral in the specific sense that the implications of CP are entirely reflected in the individual's ex ante preferences over trees, which are observable.

Second, by providing a formal definition of sophistication that does not involve multiple selves, this paper provides a way to interpret this intuitive notion as a behavioral principle—but one that applies to preferences over trees, rather than over acts. The analysis also indicates that seemingly minor differences in the way sophistication is formalized can have significant consequences in the context of choice under uncertainty; see Section 5.2.

Third, minimal assumptions are required on preferences over acts: the substantive requirements considered in this paper are imposed on preferences over trees. In particular, Postulate P2 and hence DC play no role in the analysis. This allows for prior and conditional preferences that exhibit a broad range of attitude toward risk and ambiguity—a main objective of the present paper.

The main results in this paper do not restrict attention to any specific model of choice or "updating rule." However, to exemplify the approach taken here, Theorem 4 specializes Theorem 2 to the case of multiple-priors preferences (Gilboa and Schmeidler 1989) and prior-by-prior updating. Furthermore, Section 4.4.2 leverages the framework and results in this paper to address what is often cited as a "paradoxical" implication of CP (e.g., Machina 1989, Epstein and Le Breton 1993): a time-inconsistent, but sophisticated DM may forego freely available information, if by doing so she also limits her future options. The analysis in Section 4.4.2 shows that this behavior actually has a simple rationalization if preferences over trees, rather than just acts, are taken into account.

Organization of the paper

Section 2 illustrates the key issues by means of examples. Section 3 introduces the required notation and terminology. Section 4 presents the main results, the special case of multiple-priors preferences, and the application to value-of-information problems. Section 5 discusses the main results. Section 6 discusses the important connections

Figure 1. A dynamic decision problem;* e{0,1}.

with the existing, rich literature on dynamic choice under ambiguity, as well as work on menu choice, intertemporal choice with changing tastes, and dynamic choice with nonexpected utility preferences.

The above assertion is that Bayesian updating and DC are intimately related to Savage's Postulate P2; this implies that failures of DC are not pathological, but rather the norm, when non-EU preferences are employed to analyze problems of choice under uncertainty. Savage himself provides an argument along these lines in Savage (1954, Section 2.7); Ghirardato (2002) formally establishes the equivalence of DC and Bayesian updating with P2, under suitable ancillary assumptions. Proposition 1 in the present paper provides a corresponding, slightly more general equivalence result in the framework adopted here.1 These results can be illustrated in simple examples that are also useful to describe the proposed behavioral approach to CP.

Example. An urn contains 90 amber, blue, and green balls; in the following discussion, I consider different assumptions about what the DM knows regarding its composition. A single ball is drawn; denote the corresponding state space by O = {a, ft, y}, in the obvious notation. At time 0, without knowing the prevailing state, the DM can choose a "safe" action s that yields a prize of 1 if the ball is amber or blue, and x e {0,1} otherwise. Alternatively, the DM can choose to place a contingent bet c. In this case, the DM receives x if the ball is green, and can place a bet on amber (a) or blue (b) at time 1 otherwise. The situation is depicted in Figure 1: solid circles denote decision points and empty circles denote points where Nature moves, or more properly reveals information to the DM.

Given the state space O and prize space X = {0,1, 1}, the atemporal choice envi-roment corresponding to the decision problem under consideration consists of all acts

1All versions of this argument incorporate the assumptions of consequentialism and (with the exception of Proposition 1 in this paper) reduction; the discussion of these substantive hypotheses is deferred until Section 3.3.

2. Heuristic treatment

Savage's P2 and DC

(functions) h e XO. Suppose first that the DM knows the composition of the urn and that she has risk-neutral EU preferences; her beliefs q e A(O) reflect the composition of the urn. Thus, in the atemporal setting, the DM evaluates acts h = (ha,h^,h.y) e XO according to the functional V(h) = Eq[h].

Now, as described above, augment this basic preference specification by assuming that the DM updates her beliefs q in the usual way. At the second decision node, she then conditionally (weakly) prefers a to b if and only if q({a}|{a, > q(|^}|(a, This is of course equivalent to q({a}) > q({^}), which is the restriction on ex ante belief that ensures that, from the point of view of the initial node, the course of action "c then a" is weakly preferred to "c then b". This is an instance of DC: the ex ante and conditional rankings of the actions a and b coincide. In turn, this provides a rationale for the use of backward induction: the plans of action available to the DM at the first decision node are "c then a", "c then b", and "s", but one of the two c plans can be eliminated by first solving the choice problem at the second node. A simple calculation then shows that "s" is never strictly preferred, regardless of the ratio of blue versus green balls. Hence, for instance, if q({a}) > q({b}), then "c then a" is the unique optimal plan.2

To provide a concrete illustration of the relationship between DC and P2, recall that the assumptions of ex ante EU preferences and Bayesian updating delivered two conclusions: (i) the ranking of a versus b at the second decision is the same as the ranking of "c then a" versus "c then b" at the first decision node; furthermore, (ii) the ranking of a versus b at the second node is independent of the value of x. Now assume that the modeler does not know that ex-ante preferences conform to EU or that conditional preferences are derived by Bayesian updating; however, he does know that (i) and (ii) hold. Clearly, the modeler is still able to conclude that the ranking of "c then a" and "c then b" must also be independent of x, so that

(1,0,0) ^ (0,1,0) ^ (1,0,1) ^ (0,1,1), (1)

where ^ denotes the DM's preferences over acts. This is an implication of Savage's Postulate P2 (cf. Savage 1954, p. 23, or Axiom 2 in Section 4.1 below). In other words, as claimed, (1) is also a necessary condition for dynamic consistency in Figure 1.

Ambiguity, DC, and CP

I now describe ambiguity-sensitive preferences that violate P2, and hence yield a failure of DC; see below for an analogous example based on the common-consequence effect. Assume that, as in the three-color-urn version of the Ellsberg paradox Ellsberg (1961), the DM is told only that the urn contains 30 amber balls. Assume that she initially holds multiple-priors (also known as maxmin-expected utility (MEU)) preferences (Gilboa and Schmeidler 1989), is risk-neutral for simplicity, and updates her beliefs

2As per footnote 1, this argument incorporates the substantive assumptions of consequentialism and reduction (see Section 3.3). In the tree of Figure 1, the relevant aspect of consequentialism is the fact that the ranking of a versus b at the second decision node is independent of the value of x. Reduction instead implies that the choice of c followed by, say, a is evaluated by applying the functional V(-) to the associated mapping from states to prizes, i.e., (1,0,x). I maintain both assumptions in this Introduction; the formal results in the body of the paper allow for arbitrary departures from reduction.

Figure 2. A dynamic Allais-type problem;* e {0,1M}.

prior-by-prior (e.g., Jaffray 1994, Pires 2002) on learning that the ball drawn is not green. Formally, her preferences over acts h e RO, conditional on either F = O or F = {a, ft}, are given by VF(h) = minqeC Eq[h|F], where C is the set of all probabilities q on O such that q({a}) = 1. Notice that such conditional preferences are independent of the value of x, as is the case for Bayesian updates of EU preferences.

Note first that, a priori (i.e., conditional on F = O), this DM exhibits the modal preferences reported by Ellsberg (1961): she prefers a bet on amber to a bet on blue, but she also prefers betting on blue or green rather than amber or green. Therefore, the DM's preferences violate (1) and hence Savage's Postulate P2. Furthermore, conditional on {a, ft}, this DM prefers (1,0,x) to (0,1,x) regardless of the value of x, and hence strictly prefers a to b.

If now x = 1, DC is violated: at the first decision node, the DM strictly prefers the plan "c followed by b" to "c followed by a", but at the second node, she strictly prefers a to b.

To resolve these inconsistencies, suppose that the DM adopts CP The intuitive assumption of sophistication implies that, at the first decision node, the DM correctly anticipates her future choice of a, regardless of the value of x. This is true despite the fact that, for x = 1, she really prefers to commit to choosing b instead. Hence, when contemplating the choices c and s at the first decision node, the DM understands that she is really comparing the plan "c then a" to "s". For x = 0, she strictly prefers the former, but, for x = 1, she strictly prefers the latter. This logic thus delivers unambiguous and coherent behavioral predictions.

Violations of DC can also arise when preferences are probabilistically sophisticated but not EU; again, CP provides a way to deal with them. Suppose that one ball is to be drawn from an urn containing 100 balls, numbered 1-100. Figure 2 depicts the choice problem and the payoffs, where M denotes one million (dollars), and 1,..., 11, 12,..., 100, and so forth refer to the number on the ball drawn.

The DM's beliefs are uniform on O = {1,..., 100} at the initial node and are determined via Bayes' rule at the second; her preferences are of the rank-dependent EU form (Quiggin 1982), with quadratic distortion function. If x = 1M, the plan "c followed by b"

A dynamic "common consequence" paradox (cf. Allais (1953))

is preferred to "c followed by a", whereas the opposite holds if x = 0: this corresponds to the usual violations of the independence axiom and hence of P2. Furthermore, the DM strictly prefers a to b at the second decision node if x = 1M, so preferences are dynamically inconsistent. Nevertheless, CP again delivers well defined behavioral predictions: if x = 1M, the DM correctly anticipates choosing a at the second node, and hence, by a simple calculation, opts for s at the initial node.

Karni and Safra (1989, 1990) illustrate applications of CP with non-EU preferences under risk.

Behavioral analysis of CP

As noted in the Introduction, this paper provides a fully behavioral analysis of CP. To illustrate the key ingredients of the analysis, refer back to the decision tree in Figure 1 and adopt a simplified version of the notation to be introduced in Section 3 (an analogous treatment can be provided for the tree in Figure 2). Denote the original tree in Figure 1 by /x; also denote by cx and sx the subtrees of /x, where c or, respectively, s is the only action available at the initial node. Finally, denote by cax, and cbx the subtrees of cx where a or, respectively, b is the only action available at the second decision node; note that sx, cax, and cbx can be interpreted as fully specified plans of action. Assume that, at time 0, the DM expresses the following strict preferences M and indifferences (~) over decision trees:

ca0 ~ c0 ~ /0 >- s0 >- cb0 and cb1 >- /1 ~ S1 >- ca1 ~ c1. (2)

The preferences in (2) exhibit two key features. First, preferences over plans are consistent with act preferences in the Ellsbergparadox and, more generally, with the assumed MEU preferences at the initial node. Specifically, ca0 >- s0 >- cb0 and ca1 -< S1 -< cb1 correspond to the DM's ranking of the acts (1,0,x), (0,1,x), and (2, 2, x) for x = 0,1 provided by the MEU utility index VO.

The remaining preference rankings involve nondegenerate trees and do not merely follow from the assumption of MEU preferences (even if augmented with prior-by-prior updating); rather, they reflect the intuition behind sophistication that is the focus of this paper. In particular, the indifference c1 ~ ca1 indicates that the DM does not value the option to choose b at time 1, when a is also available. This is not because she dislikes action b from the perspective of time 0: on the contrary, the ranking cb1 >- ca1 suggests that she prefers to commit to choosing b at time 1. Therefore, it must be the case that this DM correctly anticipates her future strict preference for a over b, and evaluates the tree c1 accordingly.

I emphasize that this argument relies crucially on the rankings of nondegenerate trees—e.g., c1 ~ ca1 in (2). Indeed, this pattern of preferences constitutes the behavioral definition of sophistication in Section 4.3.2. More generally, the proposed approach leverages preferences, over trees to elicit conditional preferences, and analyze sophistication and related behavioral traits, just as the literature on menu choice leverages pref-

erences over menus to investigate attitudes toward flexibility or commitment, as well as temptation and self-control (see Section 6).3

The preferences in (2) indicate how this particular DM resolves the conflict between her prior and posterior preferences. Furthermore, the rankings /0 ~ ca0 and /1 ~ s can be intepreted as the behavioral implications of sophistication: if x = 0, the DM chooses c and plans to follow with a; if x = 1, she chooses s instead—as predicted by CP.

The preceding argument is that if the DM is assumed to strictly prefer a to b at the second decision node, then the prior preferences in (2) reveal that she is sophisticated. But by reversing one's perspective, the following interpretation is equally legitimate: if the DM is assumed to be sophisticated, then the prior preferences in (2) reveal her ranking of a versus b at the second decision node. To elaborate, as noted above, the rankings cb1 >- ca1 ~ c1 suggest that the DM expects to choose a rather than b at the second decision node; if the DM is assumed to be sophisticated, this expectation must be correct, so she must actually prefer a to b at that node. In this respect, the DM's prior preference relation ^ over trees, partially described in (2), provides all the information required to analyze behavior in this example.

Details

Certain subtle aspects of CP in the context of choice under uncertainty require further analysis and are fully dealt with in the remainder of this paper. First, eliciting conditional preferences in general trees requires a more refined approach than the one just described; the details are provided in Section 4.2. Note that only a weak form of sophistication is required.

Second, ties must be handled with care. The sophistication axiom in Section 4.3.2 is purposely formulated so as to entail no restrictions in case multiple optimal actions exist at a node. Instead, a separate axiom captures the tie-breaking assumption that characterizes CP.

Third, this "division of labor" is essential in the setting of choice under uncertainty. Section 5.2 shows that under solvability conditions that are satisfied by virtually all known parametric models of non-EU preferences, strengthening the sophistication axiom so as to deal with ties as well has an undesirable side effect: it imposes a version of P2 on preferences over acts, and hence, for instance, rules out the modal preferences in the Ellsberg example.

3. Decision setting

Due to the approach taken in this paper, the notation for decision trees must serve two purposes. First, it must provide a rigorous description of the dynamic-choice problem; second, it must allow a precise, yet relatively straightforward, formalization of "tree-surgery" operations—pruning actions at a given node, replacing actions at a node with

3Although I assumed that reduction holds in this specific example, the notation and formal setup allow the DM to strictly rank two plans p and p' that can be reduced to the same act. This is orthogonal to the issue of sophistication; imposing reduction throughout neither simplifies nor hampers the analysis. See Section 3.3.

different ones, and more generally "composing" new trees out of old ones. The proposed description of decision trees is relatively familiar4; however, formally describing tree-surgery operations requires a level of detail that is not needed in other treatments of dynamic choice under uncertainty.

For simplicity, attention is restricted to finite trees associated with a single, fixed sequence of information partitions; see Section 5.3 for possible extensions.

3.1 Actions, trees, and histories

Fix a state space O, endowed with an algebra 2, and a connected and separable space X of outcomes. Information is modeled as a sequence of progressively finer partitions F0,..., FT of O for some 0 < T < to, such that F0 = {O} and Ft c 2 for all t = 1,...,T (sometimes referred to as a filtration). For every t = 0,...,T, the cell of the partition Ft containing the state w e O is denoted by Ft(w); also, a pair (t, w), where t e{0,...,T} and w e O, is referred to as a node.

Trees and actions can now be defined recursively as menus of contingent menus of contingent menus.... A bit more rigorously, define first a "tree" beginning at the terminal date T in state w simply as an outcome x e X. Inductively, define an action available in node (t, w) as a map associating with each state w' e Ft(w) a continuation tree beginning at node (t + 1,w'); to complete the inductive step, define a tree beginning at node (t, w) as a finite collection or menus of actions available at (t, w). The details are as follows.

Definition 1. Let FT(w) = FT = X for all w e O. Inductively, for t = T - 1,..., 0 and w e O, define the following terms.

(i) Let At(w) be the set of Ft+i-measurable functions a: Ft(w) ^ Ft+i such that for

all w' e Ft+i(w), a(w') e Ft+i(w').

(ii) Let Ft(w) be the collection of nonempty, finite subsets of At(w).

(iii) Let Ft = UweOFt(w).

The elements of At(w) and Ft are called actions and trees, respectively.

Observe that the maps w ^ At(w) and w ^ Ft(w) are Ft-measurable.

A tree is interpreted throughout as an exhaustive description of the choices available in a given decision problem; in particular, if two or more actions are available at a node, the individual cannot also randomize among them. Of course, randomization can be explicitly modeled by suitably extending the state space and the description of the tree.

A history describes a possible path connecting two nodes in a tree: specifically, it indicates the actions taken and the events observed along the path. Given the filtration F0,..., FT, the sequence of events observed is fully determined by the prevailing state

4Epstein (2006) and Epstein et al. (2008) adopt a similar notation for decision trees, although they are not motivated by (and do not define) tree-surgery operations. In the context of risk, the notation in Section 3 of Kreps and Porteus (1978) is similar, again except for tree surgery; see Section 6 for further details.

of nature; thus, formally, a history is identified by the initial time t, the prevailing state w, and the (possibly empty) sequence of actions taken. The details, and some related notation and terminology, are described as follows.

Definition 2. A history starting at a node (t, w) is a tuple h = [t, w, a], where either of the following equalities holds.

• a = (at,...,aT), with t < t < T — 1, at e At(w), and, for all ? = t + 1,...,t, a? e at—i(w).

• a = 0 (an empty list).

The cardinality of a is denoted |a|. Furthermore: make the following definitions.

(i) If h = [t, w, a], a = 0, and at e At(w), then a U at = (at), and if a = (at,...,aT), t < T — 1, and aT+i e aT(w), then aU aT+1 = (at,...,aT,aT+1).

(ii) A history [t, w, a] is terminal if and only if t +|a| = T and is initial if and only if a = 0.

(iii) A history h = [t',w', a] is consistent with a tree f e Ft(w) if t' = t, w' e Ft(w), and either a = 0 or the first action in a is an element of f; in this case, the continuation tree of f starting at h is f(h) = f if a = 0 and is f(h) = aT(w') if a = (at,...,aT).

Certain special trees play an important role in the analysis. First, a plan is a tree where a single action is available at every decision point. Formally, a tree f e Ft is a plan if, for every history h = [t, w, a] consistent with f, |f(h) | = 1. The set of plans in Ft and Ft(w) are denoted by Ff and Ff (w), respectively. Second, a constant plan yields the same outcome in every state of the world. Formally, ftxw e Ft(w) is the unique plan such that, for every terminal history h consistent with ftxw, ftxw(h) = x. If the node (t, w) can be understood from the context, the plan ftxw is denoted simply by x.

As an example, the tree in Figure 1, as well as its subtrees, can be formally defined as follows (recall that a simplified notation is used in the Introduction). Let T = 2, F1 = {{a, ft}, {y}}, and F2 = {{a}, (ft), {y}}. The two choices available at the second decision node in Figure 1 correspond to the time-1 actions a, b e A1(a) = A1(ft) defined by

a(a) = 1, a(ft) = 0 and b(a) = 0, b(ft) = 1. (3)

Next, define the time-0 actions cx,sx, cax, cbx e A0(a) = A0(ft) = A0(y) by, for w = a, ft, cx(w) = {a,b}, sx(w) = §, cax(w) = {a}, cbx(w) = {b} (4)

Cx(Y) = Sx(Y) = cax(y) = cbx(y) = x. (5)

Here, x and | denote the constant plans fx Y and f^^^, respectively.

Now the full tree in Figure 1 is formally defined as fx = {cx,sx}, the subtree beginning with the choice of c (respectively, s) is {cx} (respectively, {sx}), and the plans corresponding to the choice of c at the initial node, followed by a (respectively, b) at the second decision node are {cax} and {cbx}. Finally, there are three nonterminal histories consistent with fx: 0, [0,a, Cx], and [0,ft, Cx].

3.2 Composite trees

Fix / e Ft, a history h = [t, w, a] consistent with /, and another tree g e Ft+|a|(w). The composite tree gh/ is, intuitively, a tree that coincides with / everywhere except at history h, where it coincides with g. Formalizing this notion is somewhat delicate, so I first provide some heuristics.

Since h = [t, w, a] is consistent with / and a = (at,...,aT),with t > t, the last element aT of the action list a satisfies aT(w') = /(h) for all w' e FT+1(w). To capture the idea that /(h) is replaced with g, one prefers to replace aT in the list a with a new action aT such that aT(w') = g at such states and aT(w') = aT(w') elsewhere. However, recall that, by definition, aT-1 (w') must contain aT for all w' e FT(w); if aT is replaced with aT, it is also necessary to "modify" aT-1 so that it now contains aT rather than aT in such states. These modifications must be carried out inductively for all actions aT_1,aT_2,...,at; this yields a new, well defined action list a = (at,..., aT). Finally, recall that, by definition, the history h = [t, w, a] is consistent with / precisely when the first action at in the list a is an element of / (trees are sets of actions). Then the tree gh/ differs from / precisely in that the action at is replaced with at.

Now for the formal details. If a = 0, then let gh/ = g. Otherwise, write a = (at,...,aT), with t > t; let aT(w') = g for all w' e FT+1(w), and let aT(w') = aT(w') for w' e FT(w) \ FT+1(w). Inductively, for f = t - 1,...,t, let af(w') = {aj+1}U (af(w') \{a^+1}) for all w' e Ff+1(w) and let af(w') = aj(w') for w' e Ff(w) \ Ff+1(w). Finally, let gh/ denote the set {at} U (/ \ {at}).

As a special case, consider a node (t, w) and a plan / e Fp. Since, by definition, a single action is available in / at any node, there is a unique history consistent with / that corresponds to the node (t, w); it is then possible to define a tree that, informally, coincides with / everywhere except at time t, in case event Ft(w) occurs. Such a tree is denoted gt,„/.

Formally, since / is a t-period plan, there is a unique action list a = (a0,...,at-1) such that h = [0,w, a] is consistent with /. Then, for all g e Ft(w), let gt,w/ = gh/.5 The notation gt,w/ is modeled after gE/, which is often used to indicate composite Savage acts.

3.3 Preferences, reduction, and consequentialism

Definition 3. A conditional preference system (CPS) is a tuple (W,w)o<t<T>eft, such that, for every t and ы, is a binary relation on Ft(w) and, furthermore, ы' e Ft(w) implies = . The time-0 preference is also denoted simply by

Three aspects are worth emphasizing. First, preferences are assumed to be "adapted to F": for each t = 0,...,T — 1, is measurable with respect to Ft. This reflects the assumption that is the DM's ranking of trees conditional on observing the event Ft(w) at time t.

5List a is also the unique list such that, for any ы' e Ft(w), the history h = [0,ы', a] is consistent with f; furthermore, the definition of composite trees implies that ghf = gh' f, consistently with the intended interpretation of gt,wf.

Second, recall that, in the Introduction, preferences over plans are implicitly defined by first "reducing" plans to acts in the obvious way and then invoking the DM's preferences over acts, represented by the functional V. While this is the "textbook" approach to dynamic choice with EU preferences, there are compelling reasons to consider alternatives. For instance, the DM may display a preference for early or late resolution of the uncertainty, as in Kreps and Porteus (1978), Epstein and Zin (1989), Segal (1990), and, in a fully subjective setting, Klibanoff and Ozdenoren (2007). To allow for such preference models, reduction is not assumed in the main results of this paper, Theorems 1 and 2. The proposed approach takes the DM's preferences over plans as given, as part of her CPS, regardless of whether they are obtained from underlying preferences over acts by reduction.

Third, the assumption that the only "conditioning information" relevant to the preference relation is the event Ft(«) implies that our analysis is consequentialist: in particular, two actions a, b e At(«) are ranked in the same way in any decision tree where they may be available. To elaborate, if the actions a and b are available at a history h = [t, «, a] consistent with a tree /, their ranking of course depends on the realized event Ft(w); however, prior choices and discarded alternatives on the path to h—choices the DM has to make at counterfactual histories in /, or possible events that did not occur—are irrelevant. This is a standard property of EU preferences and Bayesian updating, and is preserved in most applications of non-EU and ambiguity-sensitive preferences. However, some alternative theoretical approaches to dynamic choice with non-EU preferences or under ambiguity relax consequentialism to salvage dynamic consistency; this important point is discussed in Section 6.

To conclude, recall that a binary relation is a weak order if and only if it is complete and transitive.

4. Main results

This section presents the main results of this paper. Theorem 1 in Section 4.2 shows that sophistication provides a way to elicit conditional preferences over acts and trees from prior preferences over trees. Section 4.3 then takes as primitive a CPS and provides a definition (Definition 5) and characterization (Theorems 2 and 3) of CP in the context of choice under uncertainty. Section 4.4.1 considers CP for MEU preferences and prior-by-prior updating, and Section 4.4.2 analyzes the value of information under CP. All proofs are in the Appendix. Further motivation and discussion is provided in Section 5.

As a preliminary step, Section 4.1 formalizes the connection between dynamic consistency, Bayesian updating, and Savage's Postulate P2 mentioned in the Introduction. This result constitutes a useful benchmark, and aids in the interpretation of Theorems 1-3.

4.1 Dynamic consistency, Bayesian updating, and Postulate P2

The main result of this subsection should be considered a "folk theorem": various versions of it exist in the literature, beginning with Savage's own (Savage 1954, Section 2.7]).

Its original statement concerns preferences over acts; I restate it in terms of preferences over plans merely to avoid introducing new notation.6 Also note that, while the definition of a CPS involves general trees, throughout this subsection axioms, definitions and results are explicitly restricted to preferences over plans.

For simplicity, assume that every event in the filtration F0,..., FT is not Savage-null: for every node (t, w), it is not the case that p ~ rt,wp for all p e Ff and r e Fp. I begin by formalizing DC, Savage's Postulate P2, and Savage's qualitative notion of Bayesian updating; I follow Savage (1954, Section 2.7) throughout, to which the reader is referred for interpretation (for DC, see also Epstein and Schneider 2003). Note however that Savage's postulates and definitions pertain to all possible conditioning events, whereas I restrict attention to the elements of the filtration F0,..., FT.

Axiom 1 (Dynamic consistency (DC)). For all nodes (t, w) with t < T and all actions a, b e At(w) such that {a}, {b} e Fp, if a(w') ^t+1,w' b(w') for all w' e Ft(w), then {a} {b}; furthermore, iftime-(t + 1) preferences are strict for some w* e Ft(w), then {a} >-t,w {b}.7

Axiom 2 (Postulate P2). For all plans p, q e Ff, all nodes (t, w), and all plans r, s e Ff(w),

rt,wP > st,wp ^ rt,wq > st,fflq.8

Note that DC relates preferences at different histories; on the other hand, P2 pertains to prior preferences alone. As asserted in Section 2, the MEU preferences specified in Section 2, jointly with the reduction assumption, yield a violation of Axiom 2: take r = {a}, s = {b}, p = {ca0}, and q = {cb0}. This is, of course, the main message Ellsberg (1961) conveys.

Finally, say that the restriction of to Ff is derived from ^ via Bayesian updating (cf. Savage 1954, p. 22) if, for all plans r, s e Fp,

r W,ws ^ rt,wf > st,wf for some plan p e F0.

For ex ante EU preferences, the above condition indeed characterizes Bayesian updating of the DM's prior. The following result is then straightforward.9

Proposition 1. Consider a CPS (W,w)t,w. The following statements are equivalent.

(i) The binary relation ^ is a weak order on Ff, Axiom 2 (Postulate P2) holds, and for every node (t, w), the restriction of to Ff is derived from ^ via Bayesian updating.

6That is, to further clarify, the resulting additional generality is inessential for my purposes.

7Variables a and b are actions, whereas the singleton sets {a} and {b} are trees; on the other hand, a(w') and b(w') are trees in Ft+i(w). Finally, Ff is a set of plans, i.e., special types of trees, and and are defined over trees.

8A plan is, a fortiori, a t-period plan, so , and so forth are well defined: cf. Section 3.2.

9The proof is similar to that of analogous results (e.g., Ghirardato 2002); it is available in a supplementary file on the journal website, http://econtheory.org/supp/571/supplement.pdf.

(ii) Every is a weak order on Fp and Axiom 1 (DC) holds.

This result depends on the assumption of consequentialism implicit in the framework: according to Definition 5, each preference is defined on (sub)trees with initial event Ft(w) (cf. Section 6). Foraversion of this result that relaxes consequentialism, see Epstein and Le Breton (1993).

Proposition 1 highlights the tension between dynamic consistency and ambiguity that was anticipated in the Introduction. However, I now wish to emphasize the implications of this result for Bayesian updating. If one assumes that prior preferences satisfy P2, then one can define conditional preferences via Bayesian updating and, in this case, Proposition 1 implies that DC holds. Conversely, if one assumes that DC holds, Proposition 1 implies that Bayesian updating provides a way to elicit conditional preferences; furthermore, ex ante preferences necessarily satisfy P2.

4.2 Eliciting conditional preferences

Turn now to the main results of the paper, beginning with the elicitation of conditional preferences. First, we adopt a standard requirement: the conditioning event should "matter."

Assumption 1 (Non-null conditioning events). For every node (t, w) and prizes x, y such that x > y, there exists a plan g e Fp such that xt,wg > yt,wg.

For general preferences over acts or plans, Assumption 1 is stronger than the requirement that every set Ft(w) not be Savage-null (cf. Section 4.1); however, the two notions coincide, for instance, for MEU (and of course EU) preferences. Assumption 1 is weaker than analogous conditions in the literature, e.g., the notion of "non-null" event in Ghirardato and Marinacci (2001), which requires that g = y.

4.2.1 Beliefs about conditional preferences I begin by proposing a procedure that elicits the DM's beliefs about her own future preferences; the details are given in Definition 4. To motivate it, refer to the decision tree in Figure 1 with x = 1; adopt the notation in (3)-(5). Since {cb1} >- {ca1} ~ {c1} ex ante, it is argued in Section 2 that {a} >1,a {b} [equivalently, {a} >1,0 {b}]: if the DM prefers to commit to b at the second decision node, but deems the tree {c1} just as good as committing to a, it must be the case that the DM expects to choose a at the second decision node if both a and b are available.

However, this argument fails if {ca1} ~ {cb1}: in this case, the indifference {c1} ~ {ca1} is not sufficiently informative as to the relative conditional ranking of a versus b. Detecting conditional indifferences is even more delicate. Thus, a different but related approach must be adopted.

Definition 4. For all nodes (t, w) and trees /, /' e Ft(w), / is conjecturally weakly preferred to /' given (t, w), written / w /', if and only if a prize z e X exists such that, for all plans g e F0p ,

Vy e X, y > z ^ (/'Uy)t g and z > y ^ (/Uy)t,wg ~ /t,wg. (6)

The superscript 0 in the notation w emphasizes that this conjectural conditional preference relation is defined solely in terms of the DM's time-0, i.e., prior preferences.

The logic behind Definition 4 is as follows. Suppose that the DM believes that / /'. Under suitable regularity (in particular, solvability) assumptions that are captured by the axioms in the next subsection, a prize z e X exists such that (the DM also believes that) / z /'. Now consider another prize y e X such that a priori y > z; if the DM does not expect her preferences over prizes to change, then (she believes that) y >t,w z as well, and hence that y >t,w /'. But this implies that she expects y to be chosen rather than /' in the tree (/' U y)t,wg at node (t, w).10 As in the example of Section 2, the ex ante indifference between (/' U y)t,wg and yt,wg now reflects this belief. The argument for the case z > y is similar.

Note that for every tree considered in (6), there is a unique path from the initial history to the node (t, w), because g is a plan; furthermore, the event Ft(w) is not Savage-null. Hence, the DM "cannot avoid" contemplating her choices at that node.

4.2.2 Axioms and characterization The axioms I consider are divided into two groups. Axioms 3-6 relate the DM's actual conditional preferences with her prior preferences; Axioms 7-9 instead concern the DM's prior preferences only and ensure that the definition of conjectural conditional preferences (in Definition 4) is well posed (that is, noncontradictory).

Axiom 3 (Stable tastes). For all x, x' e X and all nodes (t, w), x x' if and only if

Axiom 4 (Conditional dominance). For all nodes (t, w), all f e Ft(w), and all x',x'' e X, if x' ^ f(h) ^ x'' for all terminal histories h of f, then x' f x''.

Axiom 5 (Conditional prize-tree continuity). For all nodes (t, w) and all f e Ft(w), the sets {x e X: x f} and {x e X: x f} are closed in X.

Axiom 6 (Weak sophistication). For all nodes (t, w), plans g e F^, trees f e Ft(w), and prizes x e X,

x >t,wf ^ (f U x)t,wg ~ xt,wg and x ^,wf ^ (f U x)t,wg ~ ft,wg.

Axiom 3 states that tastes, i.e., preferences over prizes, are unaffected by condition-ing.11 Axioms 4 and 5 are standard, and ensure that conditional certainty equivalents exist (recall that X is assumed to be a connected and separable topological space).

10The relationship /' U y denotes the time-t tree that contains all actions in /' plus the unique initial action in the plan /(yw that leads to the prize y in every state of nature. In other words, the notation exploits (a) the simplified notation for prizes and (b) the fact that trees are just sets of acts; therefore, unions of trees are also well defined trees.

11 For the present purposes, it is sufficient to impose this requirement on a suitably rich subset of prizes. For instance, if X consists of consumption streams, it is enough to restrict Axiom 3 to constant streams.

x ^ x'.

Axiom 6 assumes just enough sophistication to ensure that conjectural and actual conditional preferences coincide: in particular, the logic of sophistication is applied only to comparisons between a tree and a constant prize, and then only if the DM has no other choice available on the path to the node (t, w). Preferences at times t > 0 are not required to be sophisticated.

Turn now to the second group of axioms.

Axiom 7 (Prize continuity). For all x e X, the sets {x e X: x ^ x} and {x e X: x ^ x} are closed in X.

Axiom 8 (Dominance). Fix a node (t, w), a tree f e Ft(w), a plan g e Ff, and a prize x e X.

(i) If f(h) >- x for all terminal histories h of f, then (f U x)t,wg ~ ft,wg.

(ii) If f(h) -< x for all terminal histories h of f, then (f U x)t,wg ~ xt,wg.

Axiom 8 reflects stability of preferences over outcomes. If the individual's preferences over X do not change when conditioning on Ft(w), then in (i) she expects not to choose x at node (t, w), because f yields strictly better outcomes at every terminal history; similarly for (ii). As in Section 2, the indifferences in (i) and (ii) capture the DM's expectations.

The next axiom is a "beliefs-based" counterpart to Axiom 6 (Weak sophistication).

Axiom 9 (Separability). Considera node (t, w), f e Ft(w), plans g, g' e Ff, and x, y e X. Then the following statements can be made.

(i) (f U y)t,wg / ft,wg and x ^ y imply (f U x)t,wg' ~ xt,wg';

(ii) (f U y)t,wg / yt,wg and x - y imply (f U x)t,wg' ~ ft,wg'.

To interpret, consider first the case g = g' and fix a prize y. According to the by now familiar logic of belief elicitation, (f U y)t,wg / ft,wg indicates that the DM believes that she does not strictly prefer f to y given Ft(w)—otherwise indifference obtains. Thus, if x >- y and the DM's preferences over X are stable, she also expects to strictly prefer x to f given Ft(w); again, the elicitation logic yields (f U x)t,wg ~ xt,wg. The interpretation of (ii) is similar.

Additionally, Axiom 9 implies that these conclusions are independent of the particular t-period plan under consideration, and hence of what the decision problem looks like if the event Ft(w) does not obtain. In this respect, Axiom 9 reflects a form of "separability." More generally, Axiom 9 essentially requires that (6) in Definition 4 holds for all plans g or for none. There is a close analogy with the role of Savage's Postulate P2: see Section 4.1 for details.

The main result of this section can now be stated.

Theorem 1. Suppose that Assumption 1 holds. Consider the CPS (^t,w) and assume that ^ is a weak order on F0. Then the following statements are equivalent.

(i) The binary relation v satisfies Axioms 7-9; furthermore, for all nodes (t, w),

vt,w = vt,w-

(ii) For every node (t, w), vt,w is a weak order and satisfies Axioms 3-6.

Theorem 1 and Proposition 1 in Section 4.1 are structurally similar: Axioms 7-9 play the role of Postulate P2 (but add solvability requirements), the definition of conjectural conditional preferences corresponds to Bayesian updating, and Axioms 3-6 correspond to DC (but again add solvability requirements). The interpretation is also similar: under Axioms 7-9, Definition 4 yields well behaved conditional preferences and hence can be taken as the definition of conditional preferences; in this case, Axioms 3-6 hold. Conversely, if Axioms 3-6 hold, the beliefs derived via Definition 4 from prior preferences are actually correct, so that Definition 4 can be seen as a way to elicit actual conditional preferences. The main differences are that, of course, Theorem 1 does not rely on P2 or DC and concerns preferences over nondegenerate trees.

4.3 A decision-theoretic analysis of consistent planning

4.3.1 Consistent planning under uncertainty As noted in the Introduction, consistent planning (CP) is a refinement of backward induction. If there are unique optimal actions at any point in the tree, the two concepts coincide. Otherwise, CP complements backward induction with a specific tie-breaking rule: indifferences at a history h are resolved by considering preferences at the history that immediately precedes h.

To illustrate, consider the tree in Figure 1 with x = 1, but assume MEU preferences with priors C = {q e A(O): 90 - q(a) - 90, 90 - 9(0) - 15}. Continue to assume prior-by-prior updating and reduction, and again adopt the notation in (3)-(5). It can then be verified that {a} {b}; however, {ca1} > {cb1}, so CP prescribes that the DM follows c with a. The corresponding plan {ca1} is strictly preferred to {S1}, so the unique CP "solution" of this tree is the plan {ca1}.

Algorithmically, CP operates as follows. For each history h = [t, w, a] in a tree /, consider first the set CP° (h) of actions b e At+|a|(w) that, for every realization w' e Ft+|a|(w), prescribe a continuation action at+|a|+1,w' that has survived prior iterations of the procedure. Intuitively, such actions b correspond to plans that the DM actually follows. Then, out of these actions, select the conditionally optimal ones: this completes the induction step and defines the set CP/(h). Definition 5 is modeled after analogous definitions in Strotz (1955-1956) and Gul and Pesendorfer (2005), except that it is phrased in terms of preferences, rather than numerical representations.

Definition 5 (Consistent planning). Consider a tree / e Ft(w). For every terminal history h = [t, w, a] consistent with /, let CP/(h) = {/(h)}. Inductively, if h = [t, w, a] is consistent with / and CP/([t, w', a U a]) is defined for every w' e Ft+|a|(w) and a e /(h), let

CP°(h) = {b e At+|a|(w): 3a e /(h) s.t. Vw' e Ft+N(w),

b(w') = {a+1,w'} for some a+1,w e CP/([t, w', aU a])}

CP/(h) = {b e CP0(h): Va e CP0(h), {b} W+jaj,« {a}}. A plan {a} e Ft(w) is a consistent-planning solution of / if a e CP/([t, w, 0]).12

Note that, to carry out the CP procedure, it is only necessary to specify the DM's preferences over plans. The output of the CP algorithm is also a set of plans.13 Moreover, it is straightforward to verify that if preferences over plans are complete and transitive, then Definition 5 is well posed: it always delivers a nonempty set of solutions that the DM deems equally good.

4.3.2 Behavioral characterization of consistent planning The behavioral analysis of CP takes as input the DM's CPS (^t>w). The key assumption of sophistication was introduced in Section 2; Axiom 6 applies the same principle to a small set of trees, with unique features. To capture the implications of Sophistication in general trees, it is assumed that pruning conditionally dominated actions leaves the DM indifferent. Formally, if g is a subset of actions available in tree / at history h, and every action b e g is strictly preferred to every action w that lies in /(h) but not in g, then ex ante the DM must be indifferent between / and the tree gh/ in which the inferior actions have been pruned.

Axiom 10 (Sophistication). For all / e Ft, all histories h = [t, w, a] consistent with / and such that a = 0, and all g c /(h), if, for all b e g and w e /(h) \ g, {b} >-t+jaj,w {w}, then

/ ~t,w gh/•

Observe that Axiom 10 is silent as far as indifferences at node (t + jaj,w) are concerned. For instance, if /(h) = {a, b} and {a} ~t+jaj,w {b}, the axiom does not require that / ~t,w {a}h/ ~t,w {b}h/. This allows for the possibility that, ex ante, the DM has a strict preference for commitment to a or b; Axiom 11 deals with these situations. Axiom 10 is also silent if h is the initial history of /: Axiom 12 below encodes the assumptions required in this case. This "division of labor" is crucial so as to avoid unduly restricting ambiguity attitudes; see Section 5.2.

The next axiom formalizes the tie-breaking assumption that characterizes CP within the class of backward-induction solutions: if the DM is indifferent among two or more actions at a history h, then she can precommit (more precisely, expects to be able to precommit) to any of them at the history that immediately precedes h. It is important to emphasize that no such precommitment is possible in case the individual has strict preferences over actions at h: in such cases, the full force of Axiom 10 (Sophistication) applies.

12To help parse notation, a, a+i,w/, and b in this definition are acts; b(w') must, therefore, be a tree and, in particular, the definition requires that it be the tree {a+i,w/} that has a single initial action a+i,w/ taken from the set CP/([t, w', aU a]). Finally, braces in {b} ^i+^w {a} are required because "^t+\a\,w is defined over trees, not actions.

13Formally, CP/([t, w, 0]) is a set of actions, not plans; however, if a e CP/([t, w, 0]), then {a} is a plan.

To formalize this assumption, the notion of a next-period commitment version of a tree is required. Again, refer to the tree fx in Figure 1; as it turns out, the notation in (3)-(5) greatly simplifies the exposition. Consider a modified version of the tree fx = {cx,sx}, where the action cx at the initial history 0 is replaced with the actions cax and cbx. Recall that while cx allows a choice between a and b at the second decision node, cax and cbx enforce a commitment to a and, respectively, b: cf. (4). The resulting tree {cax, cbx,sx}, referred to as the next-period commitment version of fx, is depicted in Figure 3.

To reflect the DM's ability to precommit in case of future indifferences, it is assumed that if {a} ~i,a {b}, the DM is indifferent ex ante between fx = {cx , sx} and its next-period commitment version {cax, cbx,sx}. Intuitively, if {a} ~i,a {b}, the DM regards the original tree as if it affords the same "physical" ability to commit as its next-period commitment version.

In the tree fx, nontrivial future choices must be made only following cx and only if m e {a, ft}; this simplifies the construction of its next-period commitment version. For a general tree, proceed as follows. Given a tree f at a node (t, m), fix an initial action a in the tree f; in every state m' e Ft(w), a leads to a continuation tree a(w'), which by definition is a set of time-(t + 1) actions (in the intended application of this definition, i.e., Axiom 11, such actions are mutually indifferent, but the following definition does not require this). Out of the time-(t +1) actions in a(M'), pick a distinguished one a+i,M'. Finally, construct a new action b available at time t that, for any state m' e Ft(a>), leads to the time-(t +1) tree containing the single initial action a+i,M. Each possible choice of initial action a and subsequent actions a+i,M' leads to a different initial action b in the next-period commitment version of f. Definition 6 formalizes this idea.

Definition 6. Fix a tree f e Ft(a>). The next-period commitment version of f is the tree

g = {b e At(M): 3a e f s.t. Vm' e Ft(M)3a+iM' e a(M;) s.t. b(M') = {a+i,M'}}.

Now consider a tree f at a node (t, m) and a history h consistent with f; suppose that every action a e f(h) and every realization of the uncertainty m' e Ft(a>) leads to a new history where the DM is indifferent among all available actions. Then replacing

the continuation tree /(h) with its next-period-commitment version g leaves the DM indifferent at (t, w).

Axiom 11 (Weak commitment). For all / e Ft and all histories h = [t, w, a] consistent with /, if, for all a e /(h), all w' e Ft+|a|(w), and all a+1,b+1 e a(w'), it is the case that {a+1} ~t+|a|+1,w' {b+1}, then / ~t,w gh/, where g is the next-period commitment version of /(h).

Finally, sophistication allows for the possibility that actions at future histories are tempting for future preferences, even though they are unappealing for initial preferences (or vice versa). The following, standard axiom ensures that, in contrast, the availability of choices at the initial history of / that are deemed inferior given the same initial preference relation vt,w is considered neither harmful (as might be the case if the DM was subject to temptation) nor beneficial (as is the case for a DM who has a preference for flexibility). This rules out deviations from standard behavior that are not due to differences in information and perceived ambiguity at distinct points in time.

Axiom 12 (Strategic rationality). Forall /, g e Ft(w) such that / c g, if, for all b e / and

w e g, {b} Vt,w {w}, then / ~t,w g.

It is now possible to state the main result of this section. Two related characterizations of CP are provided. The first is better suited to the analysis of specific preference models and updating rules (as in Section 4.4.1), and applications (as in Section 4.4.2). The second emphasizes that all behavioral implications of CP can be identified on the basis of prior preferences alone (as noted in the Introduction) and also has implications for policy evaluation (cf. Section 5.3).

Begin by specifying the DM's prior and conditional preferences over plans. Next, assume that this DM employs CP to determine her course of action in any given tree. Then the DM's CPS should indicate indifference between a tree / and any one of its CP solution(s). The following theorem shows that this is the case precisely when Axioms 10-12 hold.

Theorem 2. Consider a CPS (Vt,w)0<t<r,ws0 such that, for every t = 0,...,T and w e O, Vt,w is a weak order on Fp(w). The following statements are equivalent.

(i) Every vt,w is a weak order on all of Ft(w); furthermore, Axioms 10-12 hold.

(ii) For every node (t, w), every tree / e Ft(w), and every action a e CP/([t, w, 0]), / ~t,w {a}.

Suppose instead that the axioms of this section are applied to the CPS (V0 w)0<t<r,ws0 derived from the DM's prior preference V via Definition 4. In this case, Axioms 10-12 are effectively assumptions on the DM's prior preferences; formulating them in terms of the revealed conditional preferences V0 w is merely a matter of nota-tional convenience. Leveraging Theorems 1 and 2, one then obtains Theorem 3.

Theorem 3. Consider a weak order V on F0 that satisfies Assumption 1, Axioms 7-9, and the CPS (V?j0<t<r, weO obtained from V via Definition 4. The following statements are equivalent.

(i) Axioms 10-12 hold.

(ii) For every tree / e F0 and action a e CP/([0,w, 0]), / ~ {a}.

4.4.1 Consistent planning for MEU preferences and prior-by-prior updating To illustrate the results of Section 4.3.2, this subsection specializes Theorem 2 to the MEU decision model and prior-by-prior Bayesian updating, assuming reduction of plans to acts. It is straightforward to adapt the analysis to different representations of preferences and different updating rules (cf., e.g., Gilboa and Schmeidler 1993 or Eichberger et al. 2007 and Horie 2007).

Begin by noting that, if / e Ft is a plan, every state w determines a unique path through /: formally, for every w' e Ft(w), there is a unique list of actions a such that [t, w', a] is terminal and consistent with /. Throughout this subsection, for every node (t, w) and plan / e Ft(w), the notation /(w) indicates the prize /(h), where h = [t, w, a] is the unique terminal history consistent with /. The required assumption on preferences can now be stated.

Assumption 2 (MEU). There exist a weak*-closed, convex set C of finitely additive probabilities on (0,2) and a continuous function u: X ^ R such that, for all plans /, g e F0,

Moreover, (i) there exist plans /, g e F0 such that / > g, and (ii) for every node (t, w) and all q e C, q(Ft(w)) > 0.

Note that the MEU decision rule is often seen as embodying "pessimistic" expectations; by contrast, Axiom 11 in Section 4.3.2 is "optimistic" about one's ability to carry out ex ante preferred courses of action (provided one does not have opposite strict preferences in the future).14

Assumption 2(i) states that ex ante preferences over acts are not trivial. Assumption 2(ii) is a strengthening of the assumption that every conditioning event Ft(w) is not Savage-null; it ensures that prior-by-prior Bayesian updating is well defined (cf. Pires 2002, Proposition 1 and p. 150).

Assumption 2 pertains solely to prior preferences (over plans); Axiom 13 below provides a link with conditional preferences over plans and, in particular, is shown to characterize prior-by-prior updating. This axiom (see Siniscalchi 2001) recasts the main axiom in Pires (2002) andJaffray (1994) in a form that is more easily compared with Axiom 1 (DC) of Section 4.1.15

14I thank a referee for this observation.

15For a non-decision-theoretic analysis, see Walley (1991).

4.4 Applications

/ V g u(/(w))q(dw) >] u(g(w))q(dw).

Axiom 13 (Constant-act dynamic consistency). For all plans p e Fp, prizes x e X, and non-terminal histories h = [0,m, a] consistent with p,16

(p(h) ^|a|,ffl x) A (Vm'e F|a| (mX p(M') ^ x) p ^ x

(p(h) >|a|,M x) A (VM'e F|a| (m), p(m') ^ x) p > x

(p(h) ^|a|,«x) A (Vm' e F|a|(M), p(m') ^ x) p ^ x (p(h)

^ |a|,M x) A (Vm'e F|a|(M), p(m') ^ x) p -< x.

Axiom 13 differs from Axiom 1 (DC) in two respects. First, Axiom 13 considers only conditional comparisons between a plan p and a prize x,17 whereas Axiom 1 (DC) has implications whenever two arbitrary plans are compared conditional on F|a|(M). Second, dominance, rather than conditional preference, is required outside of the conditioning event F|a| (m) . The motivations for these restrictions are discussed in the sources cited above (especially Pires 2002 and Siniscalchi 2001).

I now specialize the definition of CP to reflect the assumption that preferences over plans at a node (t, m) are derived from an ex ante MEU preference via prior-by-prior updating. Let u and C be as in Assumption 2; consider a tree f e Ft(M). For every terminal history h = [t, m, a] consistent with f, let CPMEUf (h) = {f(h)}. Inductively, if h = [t, m, a] is consistent with f, and CPMEUf ([t, m', a U a]) is defined for every m' e Ft+|a|(M) and a e f(h), let

CPMEUf (h) = {{b} c At+|a|(M): 3a e f(h) s.t. Vm' e Ft+N(M),

b(M') = pM' for some pM' e CPMEUf ([t, m', aU a])}

CPMEUf (h) = jp e CPMEUf (h): Vp' e CPMEUf (h),

min / w(p(m'))q(dM'|Ft+|a|(M))

qeC JFt+|a| (m)

> min / w(p'(M'))q(dM'|Ft+|a|(M)U.

?S^F,+|a|(M) J

Note that the assumption of prior-by-prior updating is embodied in the second line in the definition of CPMEUf (h). The counterpart to Theorem 2 can then be stated.

Theorem 4. Considera CPS ffl)o<t<r «sü • Suppose that Assumption 2 holds and that

every event E e UL0 Ft is non-null. Then the following statements are equivalent.

16Note that the history h reaches node (|a|,M); hence the notation F|a|(M), ^|a|,M, and so forth.

17This is why the four cases p(h) ^t,M x, p(h) >t,m x, p(h) =^t,w x, and p(h) -<t,m x must all be explicitly considered.

(i) For every node (t, w), Vt,w is a weak order on Ft; also, Axioms 10-13 hold.

(ii) For every node (t, w), tree / e Ft(w), and action a e CPMEU/([t, w, 0]), / ~t,w {a}.

Unlike Theorem 2, the above result (see statement (ii) and the definition of CPMEU/) has a specific implication for the way plans are evaluated at a node (t, w): prior-by-prior updating is employed. Again, this follows from Axiom 13, which appears in Theorem 4(i).

4.4.2 Sophistication and the value of information This subsection analyzes a simple model of information acquisition and addresses the concern noted in the Introduction regarding the implications of CP: a sophisticated DM may reject freely available information. I argue that this behavior reflects a basic trade-off between information acquisition and commitment; this trade-off is difficult to uncover when preferences over acts only are considered, but becomes transparent in the richer setting of this paper.18

Consider an individual facing a choice between two alternative actions, a and b (the term "action" is used informally here). Uncertainty is represented by a state space O = O1 x O2, where O1 = 02 = {a, 0}. The individual receives H dollars if she chooses action a and the second coordinate of the prevailing state is a or if she chooses action b and the second coordinate of the prevailing state is 0; otherwise, she receives L < H dollars. Finally, prior to choosing an action, the DM can observe the first coordinate of the prevailing state.

The DM has risk-neutral MEU preferences over acts and evaluates plans by reduction. Her set of priors is C = {AP + (1 - A)Q: A e [0,1]}, where P, Q e A(O) are defined by

P(a, a) = Q(0, 0) = 1 - 2e, P(a, 0) = P(0, a) = e = Q(a, 0) = Q(0, a) P(0,0) = Q(a, a) = 0.

The parameter e lies in the interval (0,1) and should be thought of as being "small." In other words, this individual believes that the signal (w1) is most likely equal to the payoff-relevant component of the state (w2), but the relative likelihood of w2 = a versus w2 = 0 is ambiguous; furthermore, she assigns a (small and unambiguous) probability e to each state where the signal is "wrong" (i.e., different from the payoff-relevant component). Finally, assume prior-by-prior updating. Note that the resulting conditional preferences over acts are dynamically inconsistent (they violate Axiom 1 in Section 4.1).

The objective is to determine the value of the information conveyed by the signal w1; this value turns out to depend on whether the DM has the opportunity to commit to subsequent, w1 -contingent choices (e.g., by writing a binding contract or by delegating choices to an agent or machine). To adopt the formal framework of Section 3, it is useful to consider four plans, denoted paa, pab, pba, and pbb (the formal definitions are omitted for brevity). For instance, pab is the plan that prescribes the choice a after seeing w1 = a

18Footnote 35 in Machina (1989) attributes a similar observation, albeit expressed in the language of multiple selves, to Edi Karni.

and the choice b after observing mi = ft; the DM evaluates it by "reducing" it to the act that yields H if m e {(a, ft), (ft, a)} and L elsewhere. Under the assumed preferences,

pab > pba > paa ~ pbb. (7)

If the individual acquires information and can commit, then she effectively faces the tree fI,C = pflfl U pflb U pba U pbb. Her preferred ex ante choice is pflb, so fI,C ~ pflb.

If the DM does not acquire information, her feasible choices are the plans paa and pbb: thus, she can trivially "commit" to either a or b regardless of the realization of mi, which she does not observe. Formally, she faces the tree fNI = paa U pbb. By (7), CP implies that fNI ~ paa ~ pbb. The value of the signal mi under commitment is then the difference between the MEU evaluation of pab and that of paa, namely (i — 3e)(H — L).

If the individual acquires information but cannot commit, then she faces a tree fI,NC wherein the choice of a versus b is made after observing mi .19 If the individual is sophisticated (as well as strategically rational), she determines her willingness to pay for the information by taking into account the choices she actually makes after observing mi: in other words, she evaluates the tree fI,NC according to its CP solution.

Under prior-by-prior updating, one can verify that the DM strictly prefers b after observing mi = a and prefers a after observing mi = ft; therefore, by CP, fI,NC ~ pba. The value of the signal mi is then the difference between the MEU evaluations of pba and paa, namely e(H — L); since e e (0, 4), this is positive, but smaller than in the commitment case.

To summarize, if the DM can exogenously commit, information is valuable, as usual: the DM has more options in the tree fI,C than in the tree fNI (formally, fNI c fI,C) and this is of course beneficial. Furthermore, and symmetrically, if the DM "exogenously" receives information, then commitment is also valuable: it expands the effective choice set from just pba, the CP solution of fI,NC, to fI,C. Finally, there is a trade-off between information and commitment: the CP solution pba of fI,NC is not a subset or superset of fNI, so one cannot say a priori whether this sophisticated but dynamically inconsistent DM should acquire information. For the preferences considered here, information is valuable; however, in other settings, the commitment problem may be so severe that the DM may rationally choose to pay a price so as to avoid information: for an interesting example, see Eichberger et al. (2007, p. 892). Similar patterns of behavior also emerge in related contexts featuring time-inconsistent but sophisticated decision makers; see, e.g., Carrillo and Mariotti (2000) and references therein.

5. Discussion of Theorems 1-3

5.1 Counterfactuals and conditional preferences20

Any treatment of dynamic choice involves statements about preferences at potentially counterfactual decision points. In the tree of Figure 1, the second node is not reached if

19Formally, fI,NC = {c}, where the action c satisfies, for instance, c(a) = {aa,ba}, with aa,ba: {m' : m'i = a}^ {0, i}, where aa(M') = i if m2, = a and aa(M') = 0 otherwise, and similarly for ba.

20I thank a co-editor and the referees for several observations that guided and motivated this discussion.

the ball drawn is green; in such case, one cannot directly observe the DM's preferences at that node. Consequently, substantive assumptions about conditional preferences are required.

This issue arises even with dynamically consistent preferences (e.g., under EU). As noted in Section 4.1, one may employ Bayesian updating to define conditional preferences based on prior ones, thereby ensuring that DC holds per Proposition 1; however, the preferences thus defined need not be the DM's "actual" conditional preferences. As noted in Section 4.1, one may equivalently assume that DC holds and employ Bayesian updating to elicit conditional preferences; however, the DM's "actual" conditional preferences may be dynamically inconsistent, in which case Bayesian updating elicits a spurious object. In other words, the Bayesian updating and DC assumptions may well be incorrect from a descriptive point of view.

Theorem 1 is subject to the same qualifications. Whether one views Definition 4 as a way to define or elicit conditional preferences, a substantive assumption about conditional preferences must be made: one either stipulates directly that Vt,w = V0 w "by fiat" or else stipulates that Axioms 3-6 hold, so beliefs are correct and hence Vt,w = V0 w, as Theorem 1 shows. Again, either of these substantive assumptions may be incorrect from a descriptive standpoint.

On the other hand, Theorem 3 can be "safely" interpreted as a behavioral characterization of CP in terms of the DM's prior preferences over trees alone. The conjectural preferences V0 w can be interpreted as reflecting the DM's prior beliefs about her future behavior, and Axioms 10-12 then ensure that such beliefs "support" or "explain" her ex ante choices.

5.2 An important caveat: Strong sophistication

Recall that the sophistication axiom has no implications in the case of indifferences at future nodes. This is crucial to avoid unduly restricting preferences over plans. If Axiom 10 is strengthened by replacing strict preferences at future nodes with weak preferences, one obtains the following axiom.

Axiom 14 (Strong sophistication). For all / e Ft, all histories h = [t, w, a] with a = 0 consistent with /, and all g c /(h), if, for all b e g and w e /(h) \ g, {b} Vt+|a|,w {w}, then

/ ~t,w gh/.

Refer to the tree in Figure 1, with x = 1 and notation as per (3)-(5); consider the MEU preferences described in Section 4.3.1, so in particular {a} ~1,a {b} and {ca1} >- {cb1}, and the history h = [1,a, c1], i.e., the second decision point: under Axion 14 (Strong sophistication), {a} ~1,a {b} implies that {ca1} = {a}h{c1} ~ {c1} ~ {b}hfa} = {cb1}, which is inconsistent with the preferences over acts (and plans) specified at the beginning of Section 4.3.1.

The example points out the key problematic implication of Axion 14 (Strong sophistication): it implies that, loosely speaking, if the DM is indifferent between two actions at a given history, she must also be indifferent between them at any earlier history. Furthermore, unlike the Axiom 10 (Sophistication) adopted here to characterize CP, Axiom 14

(Strong sophistication) does impose restrictions on preferences over acts (or, more generally, plans).

Indeed, it turns out that these restrictions are overly strong for a very broad class of preferences. Say that preferences admit certainty equivalents if, for every (t, m) and p e Fp(m), there is x e X with p ~t;M x; this includes virtually all parametric models of preferences over acts, assuming reduction. Also say that Ft(M) is strongly non-null (for if, for all x, y e X and p e Fp(m), x > y implies xt,Mp > yt,Mp.21

Proposition 2. If each M is a weak order that admits certainty equivalents, each Ft(M) is strongly non-null, and Axioms 3 and 14 hold, then ^ satisfies Axiom 2 (Postulate P2), the restriction of each to Fp is derived from ^ via Bayesian updating, and Axiom 1 (DC) holds.

The basic intuition behind this result is fairly straightforward (although the actual proof takes a different approach). Given a plan p and a history h consistent with p, if x e X is the certainty equivalent of p(h) at h, then Axion 14 (Strong sophistication) implies that, loosely speaking, x is also the "value" that the DM attaches to p(h) at any history immediately preceding h. In other words, the DM can evaluate the plan p by recursion, as with standard EU preferences. This recursive structure implies dynamic consistency.

Proposition 2 thus implies that, in particular, Axiom 14 can preclude ex anteEllsberg preferences in the tree of Figure 1.22 More broadly, Axiom 14 rules out precisely the kind of behavior that is the focus of the present paper. Axioms 10 and 11 are formulated so as to avoid this.

5.3 Miscellaneous

Policy evaluation Welfare analysis is problematic in the presence of dynamic inconsistency. Refer to the tree in Figure 1, with x = i and preferences as in the Introduction; consider a "policy" that removes action a. Suppose that an irreversible decision to implement the policy must be made at time 0; can a definite recommendation be made, despite the noted inconsistency of the DM's preferences over acts? If the DM is sophisticated, then at the initial node she strictly prefers that a be removed, even though she anticipates being unhappy at the second node if a is indeed deleted. Gul and Pesendor-fer (2008, p. 31) observe that "standard economic models identify choice with welfare"; from this point of view, the argument just given supports a policy that removes a (again, assuming an irreversible decision must be made at time 0).

The crux of the argument is that, in view of the results of this paper, there is no ambiguity as to whose "choice" and "welfare" one is concerned with: there is a single individual, characterized by her time-0 preferences over decision trees, who in particular

21For MEU preferences with priors C, this corresponds to minqec q(Ft(«)) > 0.

22In Section 2, modify the ex ante MEU preferences so C = {q({a}) = 3 ,q({ft}) e[e, 3 — e]} for some small e > 0: then {a, ft} and {7} are strongly non-null and Proposition 2 applies. Of course, the analysis in Section 2 does not change.

strictly prefers the subtree with a removed to the tree /1. By way of contrast, if the decision problem is interpreted as a game played by multiple selves, it is no longer clear whose choices and welfare one should focus on; clear-cut policy prescriptions thus necessitate the introduction of an exogenous welfare criterion—perhaps one that trades off the well-being of the different selves.

Extensions In the decision framework adopted in this paper, all trees are defined with respect to a fixed filtration F0,..., . However, if the DM holds well defined preferences over plans conditional on arbitrary (non-null) events, Theorem 2 suggests that she can also compare two trees / and g that are each adapted to a different filtration. This is required, for instance, so as to model a DM who, at time 0, faces a choice between two different information structures, of which neither is finer than the other. Intuitively, the DM can apply CP to / and g separately, and then rank these trees according to her preferences over their respective CP solutions. By Theorem 2, this is equivalent to the assumption that the axioms in Section 4.3.2 apply "filtration by filtration"; the straightforward details are omitted.

The previous version of this paper (Siniscalchi 2009a), provides results analogous to those in Section 4 for a more general class of decision trees that allows the agent's actions at any point in time to determine the information she can receive at subsequent nodes. It also discusses the extension to a class of infinite decision trees.

6. Related literature and alternative approaches

Kreps and Porteus (1978)

As was noted above, the notation adopted here for decision trees is closely related to the formalization of a "decision problem" under risk in Section 3 of Kreps and Porteus (1978), KP henceforth. Specifically, a time-t tree in the sense of the present paper is a (finite) set of state-contingent menus of time-(t +1) trees, whereas in KP, a time-t decision problem is a (closed) set of lotteries over (contemporaneous payoffs and) time-(t + 1) continuation problems. The key difference with this paper is that KP propose a model of recursive preferences, which satisfies dynamic consistency: see their central Axiom 3.1. Byway of contrast, the present paper is concerned precisely with violations of dynamic consistency.

Segal (1990)

Again in the setting of risky choice, Segal (1990) studies preferences over two-stage lotteries that do not satisfy the reduction axiom; he also allows for non-EU risk attitudes. As Segal states explicitly (Segal 1990, p. 353), decisions are made only ex ante in his framework (i.e., before first-stage lotteries are resolved); therefore, the issue of dynamic (in)consistency simply does not arise. By way of contrast, the present paper focuses on nondegenerate decision situations in which choices are made at two or more histories.

Karni and Safra (1989, 1990)

As noted in Section 2, Karni and Safra (1989, 1990) study economic applications of CP (which they call behavioral consistency) to choice under risk with non-EU preferences. These papers employ CP as a solution concept of a game played by agents, or selves, of the decision maker; by way of contrast, the present paper is concerned with the decision-theoretic foundations for CP.

The menu-choice literature and Gul and Pesendorfer (2005)

The approach in the present paper is influenced by the menu-choice literature initiated by Kreps (1979) and further developed by Dekel et al. (2001) and Gul and Pesendorfer (2001) in the context of certainty. The work of Epstein (2006) and Epstein et al. (2008), which deals with non-Bayesian updating for EU preferences but does not allow for broader risk or ambiguity attitudes, and does not focus on CP, is already mentioned (cf. footnote 4).

Gul and Pesendorfer (2005) axiomatize a version of CP in the setting of dynamic choice under certainty. Their Theorems 1 and 2 are loosely related to Theorem 1 in Section 4.2 of this paper; they axiomatize ex ante preferences that admit a "weak Strotz representation," i.e., roughly speaking, a system of conditional preferences that generates it via CP. However, multiple conditional preference systems can generate the same ex ante preferences (Gul and Pesendorfer 2005, p. 437); by way of contrast, Theorems 1 and 3 relate ex ante preferences to a unique CPS. Gul and Pesendorfer's Theorem 3 (Gul and Pesendorfer 2005, p. 439) is the closest counterpart to Theorem 2 in this paper: it characterizes CP for a given time-0 preference on decision problems and a given collection of conditional choice correspondences.23 However, their key axiom Independence of Redundant Alternatives (IRA) corresponds to Axiom 14 (Strong sophistication) discussed in Section 5.2. As noted in Section 5.2, this is too strong an assumption for the purposes of this paper, as it implies a strong form of dynamic consistency.

Hammond (1988) and related contributions

Hammond (1988) also takes the DM's behavior in decision trees as given. He proposes a notion of "consequentialism" that differs significantly from the one discussed in Section 3.3 (cf. Machina 1989, p. 1641); call this property Hconsequentialism. Hammond's main result shows that a behavioral rule satisfies H consequentialism, consequential-ism in the sense of Section 3.3, and continuity if and only if it is consistent with EU. In other words, Hammond emphasizes that assumptions about dynamic-choice behavior can provide a foundation for (atemporal) EU preferences over acts. In contrast, the assumptions on dynamic-choice behavior considered in this paper are specifically designed not to restrict preferences over acts in any way. Rather, Hammond's result is related to Propositions 1 and 2 (loosely speaking, with H consequentialism playing the role of DC or strong sophistication).

23In the statement of Theorem 3 in Gul and Pesendorfer (2005), the phrase "weak Strotz representation" should actually read "canonical Strotz representation," as I have confirmed with the authors (the term "canonical" is formally defined in their paper).

For two-period decision problems under risk, Gul and Lantto (1990) propose weakenings of H consequentialism and dynamic consistency, and show that, under reduction of compound lotteries, the properties they propose are equivalent to the assumption that preferences over lotteries satisfy Dekel's betweenness axiom (Dekel 1986). Grant et al. (2000) focus on value-of-information problems; they do not require reduction of two-stage lotteries, and assume suitable versions of DC and strong sophistication (cf. Section 5.2). They identify conditions on ex ante preferences over two-stage lotteries that are necessary and sufficient for the DM to prefer more information to less. Thus, like Hammond (1988), both papers relate dynamic-choice behavior to properties of ex ante preferences over lotteries; in contrast, the present paper does not impose or derive restrictions on ex ante preferences over acts or plans.

Recursive preferences under ambiguity

In an influential paper, Epstein and Schneider (2003) characterize the class of MEU preferences over acts that are recursive, and hence dynamically consistent, in all decision trees consistent with a given filtration. This permits the application of standard dynamic-programming techniques even in the presence of ambiguity. Maccheroni et al. (2006) and Klibanoff et al. (2009) adapt this approach to different preference models.

Epstein and Schneider's dynamic consistency requirement corresponds to Axiom 1 (DC) in Section 4.1; while this assumption does allow for nontrivial manifestations of ambiguity aversion at each decision node, by Proposition 1 it implies that prior preferences must satisfy Savage's Postulate P2 relative to every conditioning event in the filtration under consideration. In particular, as noted in Section 4.1, in the tree of Figure 1, their requirement rules out the modal (ambiguity-averse) preferences at the initial node. In contrast, the approach in this paper does not restrict preferences over acts.

Nonconsequentialist choice

An alternative approach to dynamic choice with non-EU preferences, advocated in the context of risky choice by Machina (1989) and McClennen (1990), among others, es-sentially24 allows conditional preferences at a history h to depend on the "context" of the entire decision tree, so as to preserve optimality of the ex ante optimal plan. Thus, this approach espouses dynamic consistency, but drops consequentialism. Hanany and Klibanoff (2007, 2009) implement this approach for a broad class of ambiguity-averse preferences.

When consequentialism is relaxed in the presence of ambiguity, interpreting the effect of information on preferences can be problematic. In particular, the same information may appear to eliminate ambiguity (or perception thereof) in one decision tree and preserve it in another: I provide an example in Siniscalchi (2009b). This conclusion stands in sharp contrast with the prevailing view of ambiguity as an informational phenomenon.

Al-Najjar and Weinstein (2009) discuss further difficulties with violations of consequentialism.

24This necessarily brief summary overlooks nuances among different proponents of this approach.

A ppendix

A.1 Proof of Theorem 1 (eliciting conditional preferences)

Remark 1. Fix a node (f, «) and let > be a weak order on Ff(«) such that (i) for all x, y e X, x > y if and only if x ^ y, (ii) if x e X satisfies x ^ f(h) [resp• x ^ f(h)] for all terminal histories h consistent with f e Ff(«), then x > f [resp• f > x], and (iii) the sets U = {x e X: x > f} and {x e X: f > x} are closed in X• Then there are two conditions:

(a) For every f e Ff,«, there exists x e X such that x > f and f > x (abbreviated x = f)•

(b) If f > g, there is x e X such that f > x > g

The proof of this remark is routine, hence it is omitted.

Turn to Theorem 1. For f, f e Ff(«) and g e F°, write f «|g f to denote that (6) holds for g and for a suitable z e X. Thus, f « f' if and only if f «|g f' for all g e F^.

Assume that part (ii) holds and consider a node (f, «). Suppose that f f and let z e X be such that z ~f;« f': such a prize exists by Remark 1. Fix g e F° arbitrarily: I claimthatf «|g f, so f « f'. To see this, suppose first y >- z,so y >-f,« z by Axiom 3; then y f by transitivity and Axiom 6 implies that (f U y)f,«g ~ yf,«g. Next, suppose that y -< z: again invoking Axiom 3 and transitivity, we get y -f,« f, and Axiom 6 implies (f U y)f,«g ~ ff,«g. This proves the claim. In the opposite direction, consider f, f e Ff («) and suppose that f « f; let z e X be such that (6) holds for all g e F^. Suppose, to the contrary, that z >-f,« f ,so there exist y',y'' e X such that z >-f,« y' >%« y'' f (by Remark 1). Now Definition 4 implies (f U y')f,«g ~ ff,«g ~ (f U y'')f,«g for all g e F^, but Axiom 6 and the assumption that Ff («) is not null imply (f U y')f,«g ~ y' «g >- y''«g ~ (f U y'')f,«g for some such g: this is a contradiction. Hence, f z; similarly, z f and it follows that f W,« f'.

It remains to be shown that ^ satisfies Axioms 9 and 8 (Axiom 7 is immediately implied by Axioms 3 and 5). Consider first Axiom 9: fix a node (f, «), f e Ff,«, g, g' e F^, and x,y e X. For (i), suppose that (f U y)f,«g / ff,«g and x >- y. By Axiom 6, the first relation implies that f y, so by transitivity f -f,« x, and Axiom 6 implies that (f U x)f,«g' ~ xf,«g'. The argument for (ii) is similar. Finally, consider Axiom 8. If f(h) >- x for all terminal histories h consistent with f, then, since f is finite, there is y e X such that y >- x and f(h) ^ y for all terminal h. Now Axiom 4 implies f y and, hence, f >-f,« x; then Axiom 6 implies (f U x)f,«g ~ ff,«g, as required. The argument for (ii) is similar.

Now assume that (i) holds. To streamline the exposition, for any node (f, «) and

f, f e Ff,«, call any z e X with the properties in Definition 4 for all g e F^ a cutoff for

Claim 1. For every node (f, w), /s transitive.

Consider f, /',/'' e Ff,w such that f f' and f' f", and let x, x' e X be the respective cutoffs (which, remember, must apply for all g e F^). Then it must be the case that x ^ x'; otherwise, consider y',y'' e X such that x' >- y' >- y'' >- x (which exist by

Remark 1): by Assumption 1, for some g e Fp, y' Mg >- yi'Mg, and since f ^0 M|g f' must hold, we conclude that (f U y')<;„g - y^g > y^g - (f U /')f,„g; but f' f'' must also hold and it implies (f U y')f,Mg — f Mg — (f U y'')f,Mg, so a contradiction results.

Now consider y e X and fix an arbitrary g e Fp. If y >- x', then f' ^0 M|g f'' implies (y U f ''), g; if instead y -< x', then y -< x and f ^0,M|g f' implies (f U y)<;fflg — ff,«g.

Hence, x' is a cutoff for f ^0 M f''.

Claim 2. Fix a node (f, m) and x, y e X. Then x ^ y if and only if x ^0 M y. In particular, x ^ y implies (x U y)i;„g — x<;„g for all g e Fp.

Suppose x ^ y and fix an arbitrary g e F0. For all x' >- y, Axiom 8 implies that (x' U y)f,Mg — xf Mg; similarly, for all x' - y, also x' - x, andAxiom 8 implies (x U x')f,Mg — xf,Mg. Hence, y is a cutoff for x M y.

Conversely, suppose x ^0 M y and let y' be a cutoff. If y' - z - y, then for any g e Fp, Axiom 8 implies (z U y)<;„g — y<;„g, but Definition 4 requires (z U y)<;„g — z<;„g: since Ff(w) is non-null, this is a contradiction. Hence, y' ^ y and, similarly, x ^ y'. By transitivity, x ^ y.

Claim 3. Fix a node (f, m), f e Ff,M and x e X. Then either f M x or x M f (or both). In particular, if x, x' e X satisfy x ^ f(h) ^ x' for all terminal histories h consistent with f, then x ^0,„f and f x'.

Suppose that it is not the case that f ^0 M x. Then in particular x is not a cutoff; by Claim 2, for all y >- x, (y U x)f,Mg — yf,Mg for all g e Fp, so there must be y - x and g* e Fp such that (f U y)f,Mg* — ff,Mg*. Then Axiom 9 implies that for all y' ^ y and all g e Fp, (f U y')f,Mg — y' Mg. On the other hand, for all y' - y, also y' - x, so Claim 2 implies (x U y')f

g for all g e Fp. Hence, y' is a cutoff for x M f.

If x, x' are as above, then Axiom 8 implies that for every y - x' and g e F0 , (f U y)f,Mg — ft,Mg, and Claim 2 implies that for every y ^ x' and g e Fp, (y U x')<;„g — yf,Mg. Thus, f M x' and the other relation follows similarly.

Claim 4. Fix a node (f, m) and f e Ff(w). Then there exists x e X such that x —0 M f (i.e., x ^0 M f and f ^0 M x both hold). Hence, ^0 M is complete on Ff(w).

Let L = P| x. x^0 f {y: x ^ y}. Notice that L is an intersection of closed sets by Axiom 7 and hence is closed. Also, the last part of Claim 3 shows that there always exists x e X such that x M f. Since M is transitive by Claim 1, iff M y, then x M y (and hence x ^ y) for every x e X such that x ^0 M f: thus, f ^0 M y implies y e L. On the other hand, suppose f ^0 M y: then, in particular, y cannot be a cutoff and, as in the proof of Claim 3, Claim 2 implies that there must exist x - y and g* e Fp such that (f U x)f,Mg* — ff,Mg*. Then Axiom 9 implies that for all x' ^ x and g e Fp, (f U x')f,Mg — xf Mg; also, by Claim 2, for all x' - x and g, (x U x')f,Mg — xf,Mg. Thus, x is a cutoff for x M f, and since y e {y': x ^ y'}, y e L. Thus, L = {y: f M y}; as noted above, this set is nonempty. Similarly, the set U = {y: y ^0 M|g f} is nonempty and closed.

By Claim 3, U U L = X,so there exists x e U n L, which by definition satisfies x M f.

To complete the proof of Theorem 1, note first that M is complete and transitive on Ft(M) by Claims 4 and 1, respectively; by Claim 2, it satisfies Axiom 3; by Claim 3, it satisfies Axiom 4; by Claim 4 and Axiom 7, it satisfies Axiom 5.

Finally, we verify that it also satisfies Axiom 6. Fix a node (t, m), f e Ft(M), and x e X. Suppose f >°M x; if (f U x)t,Mg* / ft,Mg* for some g* e F^, then Axiom 9 and Claim 2 imply that, for all g e F^, y > x implies (f U y)t,Mg ~ yt,Mg and y -< x implies (x U y)t,Mg ~ xt,Mg. Thus, by definition x f, which is a contradiction. Similarly, suppose x >°,M f: if (f U x)t,Mg* / xt,Mg* for some g*, then for all g e F^, y - x implies (f U y)t,Mg ~ ft,Mg and y > x implies (x U y)t,«g ~ xt,«g, i.e., x is a cutoff for f „ x, which is a contradiction.

A.2 ProofofTheorems 2 and 3 (consistent planning)

Say that history h = [t, m, a] precedes history h' = [t',m', a'] if and only if t = t', J"t+|a|(M) = Ft+N(M'), and either a = 0, orelse a = (at,...,aT) and a' = (at,...,aT,a'T+i, ...,a'T+T,) for some t' > 0. In this case, write h <H h'. The notation h <H h' means h <H h' and not h' <H h; h =H h' instead means that h <H h' and h <H h'. Observe that h =h h' if and only if h = [t, m, a] and h' = [t, m', a] for some m' e Ft+|a|(M). Finally, if h, h' are consistent with f, then h immediately precedes h', written h <H h', if h <H h' and there is no history h'' such that h <H h'' <H h'.

Begin with two preliminary remarks. First, the set of actions that CP associates with a given history h in a tree f depends only on the continuation tree f(h).

Remark 2. Foreverynode (t, m) with t< T, tree f e Ft(M), and nonterminal history h = [t, m', a] consistent with f, CP° (h) = CPf(h)([t +|a|,M', 0]) and CPf(h) = CP/(h)([t +|a|, m', 0]).

Proof. Suppose first t + |a| = T — i. Note that for every a e f(h) and m'' e Ft+|a|(M'), [t, m'', a U a] is consistent with f and [t + |a|,M'', 0 U a] is consistent with f(h); furthermore, f([t, m'', a U a]) = a(M'') = (f(h))([t + |a|,M'', 0 U a]). Therefore, for every a e f(h) and m'' e Ft+|a|(M'), CPf([t, m'', aU a]) = {a(M'')} = CPf(h)([t + |a|,M'', 0 U a]). This immediately implies that CP° (h) = CPf(h)([t + |a|,M', 0]) and thus also CPf(h) = CP/(h)([t +|a|,M', 0]).

The induction step is immediate: if, for every a e f(h) and m'' e Ft(M'), it is the case thatCPf ([t, m'', a U a]) = CPf(h)([t + |a|,M'', 0 U a]), thenDefinition 5 readily implies that CP° (h) = CPf(h)([t + |a|,M', 0]) and thus also CPf(h) = CP/(h)([t + |a|,M', 0]). □

Second, CP solutions are measurable with respect to the partitions F°,..., Ft—i. Hence, only finitely many histories of any given tree need to be considered to evaluate it.

Remark 3. For every node (t, m) with t < T, tree f e Ft(M), and history h consistent with f, if h' =h h, then h' is consistent with f and f(h') = f(h). Consequently, CPf(h) = CPf(h').

Proof. Since h =H h', one canwrite h = [f, w', a] and h' = [f, w'', a], with w'' e Ff+jaj(w') and w' e Ff(w). Since Ff+jaj(w') c FT(w') for f = f,...,f +jaj — 1, it is the case that w'' e Fj(w') for such f; in particular, w'' e Ff(w') = Ff(w), so h' is consistent with f.

If a = 0, then f(h') = f = f(h). Otherwise, let a = (af,...,aT), with aT e AT(w'); since f + jaj = f + (t — f + 1) = t + 1, then by assumption, FT+i(w'') = FT+i(w') and aT is Ft+i-measurable; hence, f(h) = aT(w') = aT(w'') = f(h'). The last implication follows from Remark 2. □

Next, a technical issue must be addressed. Recall from Section 3.2 that if h = [f, w, a] is a history consistent with the tree f, with a = [af,...,aT], the composite tree ghf is obtained by iteratively constructing actions af,..., at and replacing the initial action af in f with a f. One consequence is that the history h itself is no longer consistent with ghf: rather, in a natural sense, it corresponds to the history [f, w, (af,..., at)]. Indeed, any history h' = [f, w', a'] consistent with f, with a' = (af ,...,a'T/ ) = 0 and af = af, is no longer consistent with ghf. Thus, it is necessary to construct the history corresponding to such h' in ghf and to define notation for it.

To this end, continue to denote by (af,..., at) the action sequence constructed in the definition of the composite act ghf. Assume first that h' is not initial and af = af. Let a be the largest number in (í,...,t) such that af = af and w' e Ff(w) for f = f,...,a. Then define the image of h' in ghf, denoted h'jghf, as [f, w',(af,..., aa,a/a+i,...,a/T,)]. If instead h' is initial or af = af, then simply let h'jghf = h'.

While h'jghf is formally defined for all histories h' consistent with f, it is not a history (let alone one consistent with ghf) in case h strictly precedes h' (except for g = (a'T+i}). However, h'jghf is a history consistent with ghf if h does not strictly precede h' (the case of interest in the proof of Theorem 2). The following remark establishes this and other useful facts.

Lemma 1. Let h, h' be histories consistent with f e Ft, with h h'. Then there

(i) A history h'jghf is consistent with ghf.

(ii) If h'' is another history consistent with f and such that h h'', then h' <H h'' if and only if h'jghf <h h''jghf.

(iii) If also h' =h h, then there is a surjection a : f(h') ^ (ghf)(h'jghf) such that if a' e f(h'), h' = [f, w', a'], and h'jghf = [f, w', b'], then [f, w'', a' U a']jghf = [f, w'', b'U a(a')] for all w'' e Ft+jа'j(w').

(iv) If neither h <H h' nor h' <H h, then (ghf)(h'jghf) = f(h').

Proof. Write h = [f, w, a] and let notation be as in the construction of h'jghf. The assumption that h < h' rules out the possibility that a = t < t'. Also, the first claim is immediate if a = t' < t. Thus, assume that a < min(T, t'). It must be shown that a'a+i e aa(w'); there are two cases.

If w' e Fa+i(w), then, accordingto the definition of (af,..., aT), aa(w') = aa(w') and by assumption a' i e a'a(w'); furthermore, by definition, a'a = aa. Thus, a' i e ¿¿a(w').

If instead w' e F^+1(w), then a^(w') = a^(w) = {a^+1} U (a^(w) \ {a^+1}); furthermore, by definition, a'^+1 = a^+1 and a' = aa. Because h' is a history, ¿'+1 e ¿'(w') = a^(w') = ¿'(w); but since ¿'+1 = a^+1, it follows that ¿'+1 e a^(w) \ {¿^+1} and, therefore, ¿'+1 e ¿'(w) = a^(w').

The second claim is immediate if h' is initial; otherwise, write h' = [t, w',(a't,...,a'T/)] and h'' = [t, w'',(a/t',...,a"¥,)], let '' be the largest index in {t,...,r} such that aj = af and Fj(w) = Fj(w') for f = t,...,', and define ''' analogously for h''. Since af = af for f = t,...,r' and FT +1(w') = FT +1(w''), it must be the case that '' < '''; furthermore, the argument for the first claim indicates that a' < t'. The second claim then follows immediately if '' = t'; moreover, if '' < t', then we must also have '' = ''', because ¿''+1 = a'^ and F''+1(w') = Fа/+l(w''): but then h'|gh/ = [t, w',(at,..., aa,a/a/+1,...,a/т/)] <h [t, w'' ,(at,..., ¿',¿0.'+!,...^', ¿"'+1 ,...,a'T„)] = h''|gh/, as required. The converse implication follows by reversing the roles of / and gh/, and, correspondingly, the roles of h' and h'', and h'|gh/ and h''|gh/, because / = [/(h)]h|gh/[gh/].

For the third claim, again refer to the notation in the construction of h'|gh/. If h' is not initial and a't = at, then h'|gh/ = h' and, furthermore, (gh/)(h') = a^(w') = /'(h'), so a can be taken to be the identity map. Next, if h' is initial, a surjection a with the required properties can be obtained by letting a(a) = a for every a e /(h') \ {at} and a(at) = at. Finally, if h' is not initial, let ' be as in the construction of h'|gh/. As in the proof of the first claim, it cannot be the case that ' = t < t', because h <h h', so either ' < min( t, t') or ' = t' < t. In the first case, the last action in both h' and h'|gh/ is ¿T, so (gh/)(h'|gh/) = ¿T(w') = /'(h') and a can be taken to be the identity map. In the second case, the last action in h'|gh/ is aT', whereas the last action in h' is a^ = aT'. There are two subcases: if w' e FT+1(w), then we must have t' < t because h =H h': then (gh/)(h'|gh/) = aT'(w') = {a^+1} U (a^(w') \ a^+1), /'(h') = a'^(w') = a^(w'), and a suitable a is given by a(a) = a for a = aT'+1 and a(a^+1) = aT'+1. If, instead, w' / FT'+1(w), then (gh/)(h'|gh/) = aT'(w') = aT'(w') = a'T,(w') = /'(h') and a can again be taken to be the identity.

Finally, let h and h' be as in the fourth claim. Since h and h' are unranked by <H, neither can be initial, so a' = [a't,...,a' J; it also cannot be the case that w' e Fj-'+^w), because otherwise h' <H h. Then the proof of the third claim shows that a:/(h') ^ (gh/)(h'|gh/) can be taken to be the identity map, and the result follows. □

For sufficiency, assume part (i) in Theorem 2. I will show that, for all (t, w) and / e

Ft(w),

/ ~t,w CP/([t,w, 0]); (8)

to interpret, recall that for every history h consistent with /, CP/(h) is a set of acts and, hence, can itself be viewed as a tree in Ft(w). By Axiom 12 and the definition of CP/(h), for every a e CP/([t, w, 0]), {a} ~t,w CP/([t, w, 0]); transitivity then implies that Theorem 2(ii) holds.

Fix a node (t, w) and a tree / e Ft(w). Now construct a sequence /0,...,/N of trees by iteratively replacing continuation trees /(h) with the corresponding CP solutions CP/(h). To do so, two issues must be addressed. First, if O is infinite, there are infinitely

many histories consistent with f ; however, by Remark 3, these can be partitioned into equivalence classes, each element of which yields the same continuation tree and set of CP solutions. Second, as the tree f is iteratively modified, one must keep track of the image of its histories in the modified trees; however, the notation developed above and in Lemma 1 makes this relatively straightforward.

To address the first issue, for every t = t.....T, fixa collection ÛT c Û such that,

for every Et e Ft, there is a unique w(ET) e Ot such that w(ET) e Et. Then let H0 be the collection of all nonterminal histories h = [t, w, a] consistent with f and such that w e Ot+|a| (the reason for the superscript 0 is clarified momentarily). Since f(h) is finite for every history h consistent with f and every Ot is finite, the set H0 is also finite.

Next, to address the second issue, define a sequence of iteratively modified trees as follows. First, let f0 = f. Then enumerate the elements of H0 as h0,1,...,h0,N in such a way that, for all n, m e{0,...,N}, n < m implies h0,n <H h0,m: that is, since, by construction, h0,n =h h0,m, either h0,m strictly precedes h0,n or the two histories are not ordered by the precedence relation.

The induction step consists of the following two substeps. Let n > 0 and assume that the tree fn-1 is already defined, along with the collection {hn-1,1,...,hn-1,N}.

• Let fn = CPf(h°,n)hn-1,nfn-1.

• For m = 1,...,N, let hn,m = hn-1,m|ghn-1,nfn-1 if hn-1,n <H hn-1,m; else, let hn,m =

hn-1,m

To elaborate, the tree fn is obtained from fn-1 by replacing the current continuation at the history hn-1,n, which intuitively corresponds to h0,n, with the set of consistent-planning solutions of f at h0,n. Then, for each history in fn-1 that does not strictly follow hn-1,n, the image in fn is constructed (formally, hn-1,m is defined for all m = 1,...,N, but the particular assignment chosen is irrelevant if hn-1,n < hn-1,m). Inductively, this ensures that the structure of actions and histories in fn reflects that of the corresponding actions and histories in f = f0:

Lemma 2. For all i = 0,...,N, there are two conditions:

(i) For all n, m e {i, ...,N}, h^ <H h£,m if and only if h0,n <H h0,m.

(ii) For all n e{i + 1,...,N}, there is a surjection ai,n : f(h0'n) ^ f£(h£'n) such that, for all a e f(h0>n), h0>n = [t, wn, a0>n], h£>n = [t, wn, a^n], and h0>m = [t, wm, a0>n U a] for some m e{1,...,n} imply h£'m = [t, wm, a£'n U a£'n(a)].

Proof. The first statement is obviously true for I = 0. Inductively, suppose it holds for some I < N and consider n, m > I + 1. By the induction hypothesis, h£'n <H h£'m if and only if h0,n <H h0,m; furthermore, byconstruction, h0>£+1 <H h0,n and h0>£+1 <H h0,m, so again by the induction hypothesis, h£'£+1 <H h£'n and h£'£+1 <H h£'m. Apply Lemma 1 (ii) to conclude that h£'n <H h£'m if and only if h£+1,n <H h£+1,m; the assertion then follows.

The second claim is trivially true for I = 0. Inductively, suppose it holds for some i < N and consider n > i + 1 (if i = N — 1, there is nothing to show). Since n > £, the induction hypothesis yields a surjection a£,n : f(h0'n) ^ fi(hi'n) with the properties stated

in the lemma. By the choice of ordering on H0, h°'m <H h0'"; thus, by the first claim, h£'£+1 <H h^". Moreover, by assumption, n = £ + 1, so also h^^1 =H h^". Lemma 1 (iii) then yields a surjection a: /£(h£,n) ^ /£+1(h£+1,n) such that h£,m = [t, wm, a£,n U a] implies h£+1,m = [t, wm, a£+1," U a(a)]. Thus, fix a e /(h0,"), h0," = [t, w", a0,"], h£," = [t, w", a£,"], h£+1," = [t, w", a£+1,"], and h0,m = [t, wm, a0," U a]. Then h£,m = [t, wm, a£," u a£,"(a)] and, therefore, h£+1,m = [t, wm, a£+1," U a(a£,"(a))]. Thus, a£+1," = a o a£," has the required properties; furthermore, it is onto, as are both a and a^". □

Next, there is a unique w* e Ot n Ft(w) and, hence, a unique initial history in H0; since this history necessarily precedes every other history consistent with /, it must be h0,N. Then, by construction, h",N = h0,N = [t, w*, 0] is the only initial history in /" for all ": in particular, this is true for " = N - 1, so CP/th0^)^-^^-1 = CP/(h0,N), i.e., /N = CP/([t, w, 0]), the right hand side of (8). Thus, to prove sufficiency, it is enough to show that /"-1 ~t,w /" for all" = 1,...,N.

Thus, consider " e{1,...,N}. By construction, the history h"-1^ is consistent with /"-1 and intuitively corresponds to h1^". I now show that, at every history h"-1,m that immediately follows h"-1^ in /"-1, the continuation tree /"^(h"-1^) is CP/(h0,m).

Lemma 3. For every m e{1,...,"} such that h0," <H h0,m, /"^(h"-1^) = CP/(h0,m).

Proof. It must be the case that m < ". I claim that, for £ = m + 1,...," - 1 and for k = 0,...,£ - 1, and h^m are unordered, and, furthermore, hk,k+1 <H hk,£ for £ = m,...," - 1.

To see this, consider first k = 0: for £ = m +1,...," -1, h0,£ <H h0,m, because h0'n <H h0,m by assumption. Furthermore, by construction, h0,m <H h0,£, so h0,£ and h0,m are unordered. Finally, by construction, h0,1 <H h0,£ for £ = m,...," - 1.

Inductively, assume the claim is true for k - 1 <£ - 1. By the inductive hypothesis, hk-1,k <H hk-1,£ for £ = m,...," - 1, and hk-1,£ and hk-1,m are unordered for £ = m + 1,...," - 1. By Lemma 1 (ii), hk,£ and hk,m are also unordered for £ = m + 1,...," - 1. Moreover, since k < £, k + 1 < £, so h0,k+1 <H h0,£; the last part of the claim then follows from Lemma 2.

The claim implies, in particular, that for £ = m + 1,...," - 1, h£-1,£ and h£-1,m are unordered; Lemma 1 (iii) now implies that /£(h£,m) = /£-1(h£-1,m) for such £. Therefore, /"-1 (h"-1,m) = /m(hm,m) and the result follows from the construction of /m. □

Lemma 4. CP0(h0," ) is the next-period commitment version of /" 1(h" 1'n).

Proof. Write h0," = [t, w", a0,"]. Consider b e CP°(h0,"); by definition, there exists a e /(h1^") such that, for every w' e Ft+|a0,"|(w"), b(w') = {a+^k/} for some a+^z e CP/([t, w', a0," U a]).

Now, for every w' e Ft+|a0'n|(wn), let m(w') e{1,...,"} be such that h0^^ =H [t, w', a0," U a]. By Lemma 2(ii), there is a e /"^(h"-1,") such that h"-1^^ = [t, wm(w), a"-1," U a] for all w' e Ft+|a0,"(w") = Ft+|an-l'n|(wn). By Lemma 3 and Remark 3, /"-^h"-1^^)) = CP/(h0'm(ffl/)) = CP/([t, w', a0," U a]). Conclude that, for

every b e CP0(h0'"), there exists a e /"-1 (h"-1'") such that, for all w' e Ft+|a0'B|(w"), there exists (a+^J e CP/([t, w', a0'" U a]) = CP/(h0'm) = /"-1(h"-1'm) = a (w') such that b(w') = (a+1,w'}: that is, b is an element of the next-period commitment version of

/"-1(h"-1'").

In the opposite direction, let b be an element of the next-period commitment version of /"-1'", so there is a e /"-1(h"-1'") such that, for all w' e Fi+|a«-1,«|(w"), there is a+1,w' e a(w') such that b(w') = (a+1,w'}. As above, for every such w', let m(w') be such that h"-1'm =H [t, w', a"-1'" U a]. By measurability, a(w') = a(wm(w )), so by Lemma 3, a+1,w' e CP/(h0,m(w )); furthermore, by Lemma 2, there is a e /(h0'") such that a"-1'"(a) = a and h0'm(ffl,) = [t, wm(w'\ a0'" U a] =H [t, w', a0'" U a]. Hence, for every w' e Ft+|a0,"|(w"), b(w') = (a+1,w'} for some (a+1,w'} e CPf([t, w', a0'" U a] = CPf(h0'm), where the equality follows from Remark 3: thus, b e CP0(h0'"). □

The proof of sufficiency can now be completed. By Lemma 4, the set CP0(h0'"), viewed as a tree, is the next-period commitment version of /"-1 (h"-1'"), so byAxiom 11, /"-1 ~t,w CP0 (h0'")h«-1'«/"-1. If now " = N, then hN-1'N is initial, so actually /N-1 ~t,w CP°(h0'"); furthermore, Axiom 12 implies that CP°(h0'") ~t,w (b e CP°(h0'"): Va e g, (b} (a}} = CP/(h0'N) = /N. If, instead, " < N, then Axiom 10 implies that CP°(h°'")h«-1,'«/"-1 CP/(h0'N)h„-1'„/"-1 = /". Thus, in either case, /"-1 ~f,w /", as required.

For necessity, begin with a preliminary

Lemma 5. Consider a node (t, w*) with t < T, a tree / e Ft(w*), and a nonterminal history h = [t, w, a] consistent with /. Then the following statements are valid:

(i) CP0(h) = Uae/(h) CP0a}([t +|a|,w, 0]).

(ii) If g e Ft+|a|(w) is such that CPg([t +|a|,w, 0]) = CP/(h), then CPgh/([t, w, 0]) = CP/([t, w, 0]).

Proof. Claim (i) holds because CP|0(h) = CP/(h)([t +|a|,w, 0]) by Remark2 and Definition 5.

To prove the second claim, it is shown that, for any history h' = [t, w', a'] consistent with / and suchthat t + |a'| e(t,...,t + |a|}, CP/(h') = CPgh/(h'|gh/). The claim is obviously true for h' = h; also, if t + |a'| = t +|a| and h = h', then neither h <H h' nor h' <H h, so Lemma 1(iv) implies that /(h') = (gh/)(h'|gh/) and Remark2 then implies that CP/(h') = CPgÄ/(h'|gh/).

Now consider h' = [t, w', a'] such that t +|a'| < t + |a| and assume that the claim has been proved for all histories that immediately follow h'. Pick b e CP/(h') and let a e /(h') be such that, for all w'' e Ft+|a/|(w'), b(w'') = (a+1,w//} c CP/([t, w'', a' U a]). Write h'|gh/ = [t, w', b']. Since h <H h', Lemma 1 yields a surjection a:/(h') ^ (gh/)(h'|gh/) such that, for all w'' e Ft+|a'|(w'), [t, w'', a' U a]|gh/ = [t, w'', b' U a(a)]. The induction hypothesis implies that for all such w'', CP/([t, w'', a' U a]) = CPgh/([t, w'', b' U a(a)]). Therefore, b e CPg/(h'|gh/).

Conversely, if b e CPg /(h'|gh/), there is a e (gh/)(h'|gh/) such that, for all m'' e Fi+ja'iC«'), b(M'') = (a+i,M") c CPgh/([t, m'', b'U a]). Again by Lemma 1, thereis a e /(h') suchthat [t, m'', b'U a] = [t, m'', a'U a]|gh/ for all such m'' and, therefore, by the induction hypothesis, CPgh/([t, m'', b' U a]) = CP/([t, m'', a' U a]). Therefore, b e CP0(h').

Thus, CP0(h') = CPg y(h'|gh/), which implies that CPy(h') = CPgh/(h'|gh/), as required. □

Now assume that Theorem 2(ii) holds. Since each is complete and transitive on Ff (m), and CP/([t, m, 0) = 0 for all / e Ft(M), is also complete and transitive on all of Ft(M). Next, the three axioms in Theorem 2(i) are considered in turn.

Axiom 10 (Sophistication). Let / e Ft(M) and fix a history h = [t, m', a] consistent with / that is neither initial nor terminal, and finally take g c /(h) as in the axiom. By the first claim in Lemma 5, CPg([t + |a|,M', 0]) c CP/(h)([t + |a|,M', 0]). Now fix bCP e CPg([t + |a|,M', 0]) and wCP e CP/(h)([t + |a|,M', 0]). By the definition of consistent planning and Lemma 5, there are b, w e /(h) such that b e g, bCP e CP0b}([t + | a|,M', 0]) and wCP e CP0w}([t +|a|,M', 0]).

Suppose that w e /(h) \ g: then, by the assumption in the axiom, {b} >t+|a|;M' (w}, and by Theorem 2(ii), (bCP} >t+|a|,M (wCP}. But since bCP e CPg([t + |a|,M'', 0]) c CP/(h)([t + |a|,M', 0]) and wCP e CP/(/0([t + |a|,M', 0]), (wCP} W+|a|,u' (bCP}, there is a contradiction.

It follows that w e g. That is, for every wCP e CP/(h)([t + |a|,M', 0]), wCP e CP0w}([t + |a|,M', 0]) c CPg([t + |a|,M', 0]). Since, furthermore, wCP W+|a|,M' aCP for all aCP e CP/(h)([t + |a|,M', 0]), hence a fortiori for all aCP e CPg([t + |a|,M', 0]), one can conclude that wCP e CPg([t + |a|,M', 0]); that is, CP/(h)([t + |a|,M', 0]) c CPg([t + |a|,M', 0]).^

' " pPg([i + |a|,m , 0]), men b~~ e CP/(h)(

gumentjust given implies that also bCP ~t+|a|,M' wCPforany wCP e CP/(h)([t + |a|,M', 0]): thus, it is also the case that CPg([t + |a|,M', 0]) c CP/(h)([t + |a|,M', 0]).

Lemma5(ii) then implies that CP/([t, m, 0]) = CPgh/([t, m, 0]), and the definition of in Theorem 2(ii) then implies that / ~t;M gh/, as required.

Axiom 11 (Weak commitment). Let / e Ft(M) and fix a history h = [t, m', a] with the properties indicated in the axiom. In particular, if a e /(h), m'' e Ft+|a|(M'), and b+i,b+i e a(M'') = /([t, m'', a U a]), then (b+1} ~t+|a|+i,M" (b+1}; by Theorem 2(ii), for all a+i e CP(b+1}([t + |a| + 1,m'', 0]) and a+1 e CP(b+i}([t + |a| + 1,m'', 0]), (a+i} ~t+|a|+i,M'' (a+i}. By Lemma 5(i),

CP0(M'')([t + |a| + 1,m'', 0]) = J CP(b+!}([t + |a| + 1,m'', 0]).

b+isa(M'')

Now let g be the next-period commitment version of /(h); I claim that CP°(h)([t + |a|, m', 0]) = CPg([t + |a|,M', 0]): by Lemma 5(ii), this implies that CP/([t, m, 0]) = CPgh/([t, m, 0]) and hence, by Theorem 2(ii), / ~t;M gh/, as required.

Fix a0 e CP/(h)([t + |a|,M', 0]) and let a e /(h) be suchthat, for every m'' e Ft+|a|(M'), a0(M'') = (a+i,M''} for some a+i,M'' e CP/(h)([t + |a|,M'', 0 U a]) = CP0(M")([t + |a| + 1,

Conversely, if bCP e CPg([t + |a|,M', 0]), then bCP e CP/(h)([t + |a|,M', 0]) and the ar-

m'', 0]). Then, by the above argument, a+1,M» e CP(b+1 M„}([t + |a| + 1,m'', 0]) for some e a(m''). Now let b e At+|a|(M') be such that, for all m'' e Ft+|a|(M'),

) = |b+i;M"}; then b e g and, furthermore, a° satisfies the following property: for every m'' e Ft+|a|(M'), a°(M'') = (a+1;M«} for some a+1;M« e CPg([t + |a|,M'', 0 U b]) = CP{b+1 M„}([t + |a| + 1,m'', 0]), where the equality follows from Remark 2 and the fact that g([t + |a|,w", 0 U b]) = b(«") = (b+1;M"}. Therefore, a0 e CPg([t + |a|,w', 0]).

Conversely, suppose a0 e CPg([t + |a|,M', 0]), so there is b e g such that, for every m" e Ft+|a|(M'), a0(m") = (a+1;M«} for some a+1;M« e CPg([t + |a|,M'', 0 U b]). But by the definition of g, there is a e /(h) such that, for all such m'', b(M'') = (b+1,M»} for some b+1;M» e a(m''). Hence, CPg([t + |a|,M''0 U b]) = CP(b+1 M„}([i + |a| + 1, m'', 0]) c CPa(M//)([f + |a| + 1,m'', 0]) = CP/(h)([t + |a|,M", 0 U a]). It follows that a0 e CP/(h)([f + |a|,M', 0]), as claimed.

Axiom 12 (Strategic rationality). Let / and g be as in the axiom. Arguing as for Axiom 10 (Sophistication), Lemma 5 implies that CPg([t, m, 0]) c CP?([t, m, 0]). Fix bCP e CPg([t, m, 0]) and wCP e CP/([t, m, 0]). By the definition of consistent planning and Lemma 5, there are b, w e / such that b e g, bCP e CP0b} ([t, m, 0]) and wCP e CP0w}([t, m, 0]).

Suppose that w e / \ g: then, by the assumption in the axiom, {b} ^t;M (w} and, by the way ^t;M is defined in Theorem 2(ii), (bCP} (wCP}. Since bCP e CPg([t, m, 0]) c CP0([t, m, 0]) and wCP e CPf([t, m, 0]), (wCP} (bCP}: thus, (wCP} ~t,M (bCP}. If, instead, w e g, then wCP e CPg([t, m, 0]), so (bCP} ^t;M (wCP}; but since (wCP} ^t;M (bCP} as well, again (wCP} ~t;M (bCP}. The definition of ^t;M in Theorem 2(ii) now implies that / g, as required.

A.2.1 Proof of Theorem 3 The result follows readily from Theorems 1 and 2, exceptthat to show that (ii) implies (i), we must show that Theorem 3(ii) implies Theorem 2(ii). In doing so, since ^ satisfies Axioms 7-9, Theorem 1 implies that (^°M) satisfies Axioms 3-6 and each relation is a weak order; we make use of this fact. Thus, assume that (ii) holds, fix a node (t, m), / e Ft(M), and a e CP/([t, m, 0]); we apply Definition 4 to show that / (a}.

As in the proof of Theorem 1, there exists z e X such that z M (a}. Fix g e and choose the unique sequence of t actions a in g such that h = [0,m, a] is a history of g. Note that for any y e X, Lemma 5(i) implies that CP?/Uy)t (h) = CP?Uy([t, m, 0]) = (ay M } U CP? ([t, m, 0]), where we explicitly represent y as the plan /tyM = (ay M} e Fp(M). Similarly, CP°(a}Uy)( Mg(h) = CP°a}Uy([t, m, 0]) = (a, ay M}, because (a} is a plan. It is also clear that CPy([t, m, 0]) = CP/y ([t, m, 0]) = (a^J andCP(a}([t, m, 0]) = (a}. I use these calculations below without explicit notice.

First, we claim that (a} M|g /, using z as cutoff. If y > z, so also y >°,M z by Axiom 3, by transitivity y = (ayM} >°M (a} ~°M (a'} for every a' e CP/([t, m, 0]), and hence y >°M (a"} for every a'' e CP? ([t, m, 0]): therefore, CP(/Uy)tMg(h) = (a^J = CPfy ([t, m, 0]). Then, Lemma 5(ii) implies that CP(/Uy)t Mg([0,M, 0]) = CPyt Mg([0,M, 0]) and so, by Theorem 3(ii), also (/ U y)t,Mg ~ yt,M g. If instead y ^ z, then (a} >°M y; then CP((fl}Uy)([t, m,

0]) = (a} = CP(a}([t, m, 0]) and, therefore, CP((a}Uy)(>ug([0,M, 0]) = CP(O}(>„g([0,M, 0]) by Lemma 5(ii): thus, ((a}Uy)t,Mg ~ (a}t,Mg. This proves the claim.

Next, we claim that / ^0 M|g (a}. If y ^ z, then y ^ M (a}. Then CP(a}Uy([t, m, 0]) = (a]'M} = CPy([t, m, 0]), and so ((a} U y)t,Mg ~ yt,Mg by Lemma 5(ii) and Theorem 3(ii). If, instead, y -< z, then also (a} >0 M y and hence (a'} >0 M y for all a' e CP/([t, m, 0]). Then CP/Uy([t, m, 0]) = CP/([t, m, 0]). Lemma 5(ii) now implies that CP(/Uy)i Mg([0,M, 0]) = CP/t,Mg([0,M,0]) and so (/ U y)t,„g - /t,„g. Thus, / ~°,M|g (a} for all g e Fp, which concludes the proof.

A.3 Other results

Proof of Theorem 4. Recall that all relevant conditional preferences are well defined. Moreover, Axiom 13 holds if and only if for all nodes (t, w), prizes x e X, and plans p e Fp (w), p x if and only if pt,wx ^ x. The "only if" direction is immediate; for the converse, let p, x, h be as in the axiom; by assumption p x implies pt,wx ^ x andfor all w' e Ft(w), p(w') ^ x: thus, by monotonicity of MEU preferences, p ^ pt,wx ^ x. The other cases are similar.

Now suppose (i) holds. This implies that Axiom A9 in Pires (2002) holds, and the results therein imply that is derived from ^ via prior-by-prior Bayesian updating. Hence, CPMEU and CP coincide, and (ii) follows from Theorem 2.

Conversely, assume that (ii) holds. Consider a plan p e F0(w) and a prize x e X such that u(x) = minqeC ft w u(p(w))q(dw|E); one such prize must exist because X is connected and u is continuous. Now consider the tree (p U x) e Ft(w); clearly, CPMEU(pUx}(0) is precisely the set containing p and x; by (ii), we have p ~t,w (p U x) ~t,w x, i.e., p ~t,w x. Thus, is consistent with MEU and prior-by-prior Bayesian updating of C. This implies that CPMEU and CP coincide, so Theorem 2 ensures that each preference is complete and transitive, and that Axioms 10, 11, and 12 hold. Finally, prior-by-prior updating implies that the above restatement of Axiom 13 holds. □

Proof of Proposition 2. By Axiom 14, for all (t, w), all r, s e Ff (w), and all p e F^, r ~t,w s implies rt,wp ~ (r U s)t,wp ~ st,wp. Thus, by Axiom 3, for all x, y e X and p e Fp, x ~ y implies xt,wp ~ yt,wp (for x = y, the claim is true by reflexivity), and since Ft(w) is strongly non-null, x >- y implies xt,wp >- yt,wp. By assumption, for all r e Fp(w) there is x e X such that x ~t,w r and so also xt,wp ~ rt,wp for all p e F0\

Now suppose that for r, s, p, and q as in Postulate P2, it is the case that rt,wp ^ st,wp. Let x, y e X be such that x ~t,w r and y ~t,w s. If r -t,w s, then x -< y and so rt,wp ~ xt,wp - yt,wp ~ st,wp, which is a contradiction: thus, r s. Therefore, x ^ y and so also rt,wq ~ xt,wq ^ yt,wq ~ st,wq. Thus, P2 holds; furthermore, this argument also shows that the restriction of to Fp(w) is derived from ^ via Bayesian updating. □

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Submitted 2009-6-1. Final version accepted 2010-10-28. Available online 2010-10-28.