Generalized Preference for Flexibility

Generalized Preference for Flexibility Madhav Chandrasekher 1 Current Draft: November 8, 2007 Abstract: This paper provides a new representation theorem for the class of menu preferences that satisfy only the order and monotonicity axioms. The utility function we axiomatize nests both the model for preference for flexibility in Kreps (1979) and a model of local preference for flexibility introduced in a companion paper, Discrete Models of Unforeseen Contingencies. 1 Introduction This paper provides a new representation theorem for menu preferences that satisfy only the order and monotonicity axioms. The result serves two objectives. First, it properly nests the model of preference for flexibility in Kreps (1979) and the model of local preference for flexibility in a companion paper, Chandrasekher (2007). This allows us to clarify the distinction between the two models by showing how each utility functional can be obtained as a specialization of a more general utility form. Second, the theorem gives an interpretation to the set of monotone menu preferences that is distinct from the costly contemplation model in Ergin (2003). The canonical framework, after Kreps (1979), is a two period choice problem structured as follows. In the first period, the ex ante phase, a decision maker (DM) must choose to commit to a menu from a collection of menus (e.g. make a dinner reservation at a restaurant). In the second period, the ex post phase, the DM selects an item from the menu she committed to in the ex ante phase. This choice problem is interesting precisely when the DM faces ex ante uncertainty regarding her ex post consumption preferences. Taking the first period choice (i.e. menu choice) as the only observable, the goal of this literature is to identify, through axioms on menu choice, plausible utility models that illustrate the decision process of the DM. For example, Kreps (1979) axiomatizes the following four step procedure. First, the DM conceives of the set of plausible ex post tastes (e.g. states). Second, she considers what her utility over consumption items will be in each of these realized states. Third, she attaches weights to each of these states according to the likelihood with which she believes they will occur. Finally, she computes ex ante utility of a menu by summing its weighted ex post values across each of the hypothesized ex post states. Labeling the collection of states as S and the state-dependent utilities as u s ( ) this procedure yields a utility of the following form, U KP F (A) := s S p s max A u s (x) 1 Acknowledgements: I would like to thank Andrea Cann Chandrasekher and Jai Amari Chandrasekher for inspiration and discussions. Any errors are my own. Correspondence should be sent to: madhav@econ.berkeley.edu. 1

where the superscript on the utility stands for Kreps Preference for Flexibility. In our companion paper we argue that the existence of the state space S imposes restrictions on choice that preclude certain menu preferences from consideration that nevertheless seemed to express preference for flexibility. For example, the above utility implies that the question of whether an ex post choice x is preferred to y is independent of the menu to which x and y belong. This rules out situations where the ambient menu containing x and y acts as a frame of reference for the choice problem, so that the set of ex post states is itself a function of the menu. To illustrate, consider the menu choice problem facing a DM who is making dinner plans. That is, in the ex ante phase she chooses a menu to commit to (e.g. makes a reservation at a restaurant). In the ex post phase, she picks items from the selected menu. In this setting, the question of which among two consumption elements x vs. y has higher value ex post may well depend on the ambient menu A to which x and y belong. If x is white wine and y is red wine, then the value of red vs. white in those ex post states where the DM has a taste for wine may depend on the consumption items in A that are coupled with the wine choice. The Kreps framework is not rich enough to model these preferences. 2 To accommodate these considerations we switch the domain of ex post choice to the collection of menu-dependent choices X 2 X. The ex post utility u s ( ) in the Kreps model is replaced with a menu-dependent utility kernel u : X 2 X R + with the property that u(x, A) u(x, B) whenever A B. Moreover, the state space S in the Kreps model is replaced with the following commitment correspondence, C(A) := {x A : A\x A}. The primitives u( ), C( ) are used to define the following menu utility, U LP F (A) = C(A) u(x, A) where the superscript stands for Local Preference for Flexibility. In contrast to the Kreps decision rule, a DM with an LPF utility uses the following modified four-step procedure. First, for the given menu A, she conceives of a relevant set of (menu-dependent) ex post states. Second, she formulates the set of ex post actions (x, A) she anticipates taking in those states (this is the set C(A)). Third, she assigns weights to each of the states. Fourth, she computes ex ante utility of a menu A by taking the weighted sum of the ex post kernel values u(x, A) across the elements of C(A). There is no containment between the classes of preferences generated by the LPF and KPF utility models. Thus, it isn t always possible to write one utility form as a specialization of the other. This makes it difficult to clarify the distinction between the two models beyond the qualitative descriptions given above. Instead, to place both 2 For more discussion on this point, the reader is referred to our companion paper Chandrasekher (2007). 2

models on common ground consider the following more general menu utility U GP F (A) = s S max A u s (x, A) where the superscript stands for Generalized Preference for Flexibility. In the remaining sections of this paper we undertake a closer study of this model. In particular, we show that (i) any monotonic preference admits a GPF utility representation, (ii) most monotonic preferences admit a GPF utility representation with non-trivial state space, and (iii) both the KPF and LPF utilities are special cases of the GPF model. Section 2 lays out the model and states the main result. Section 3 contains the proof of this result and section 4 concludes. 2 Model and Statement of Result We begin with a description of the choice domain. The set-up is the same as in Chandrasekher (2007). Let X = {x 1,.., x n } be a set with n elements (the prize space). Let 2 X := {M M X} be the space of menus of prizes drawn from X. Let P(X) := the set of complete, rational preference relations on 2 X. Recall the following axioms from Kreps (1979). Axiom 1 : (Monotonicity) If A, B 2 X and A B, then B A. Axiom 2 : (Modularity) If A, B 2 X, A B and A B, then for any C 2 X, A C B C. Let Σ KP F denote the set of preferences in P(X) that satisfy Axioms 1 and 2. The following is the main result in Kreps (1979). Theorem 1. (Kreps) Let P(X). Then Σ KPF if and only if there exists a pair (S, u(s, )) consisting of an index set S of subjective states and a state-dependent utility function u(s, ) : X R such that the function U(A) := s S max x A u(s, x) represents. In our companion paper, we argued that this utility function could not represent certain classes of preferences that nevertheless seemed to fit the intuitive notion of preference for flexibility. Consider the map C( ) : 2 X 2 X defined as follows: { {x A : A\x A}, if A > 1 C(A) = {x}, if A = {x}. 3

This plays the role of the Kreps state space S in the model of local preference for flexibility. Chandrasekher (2007) introduces an alternative to Modularity by requiring the commitment correspondence, C( ), to satisfy a natural monotonicity condition: Axiom 2:(Commitment Monotonicity) A B, C(B) A C(A) C(B). The following additional restriction on preferences was also introduced. Viewing the elements of C(A) as the set of implicit ex-post consumption choices in A, the axiom says that in every menu there is at least one ex-post commitment element. Axiom 3: (Non-Emptiness) C(A), A. We now describe the elements of the utility functional that characterizes monotonic preferences that satisfy Axioms 2 and 3. Definition 1: A non-empty valued correspondence I : 2 X 2 X is called a flexibility index if it satisfies (i) I(A) A, (ii) I(A) I(B) whenever A B and I(B) A, and (iii) I(A) = I(A \ y), y A\I(A). Definition 2: A utility kernel u : X 2 X R + is said to be a flexibility kernel if u(x, A) u(x, B) whenever x A B and A B. Call the latter property upwards set monotonicity. Definition 3: A utility function U : 2 X R is said to be an LPF utility if there is a flexibility index I and a flexibility kernel u such that U(A) = u(x, A) x I(A) and, in addition, the following properties hold: 1. x I(A\y) u(x, A\y) < x I(A) 2. x I(A\y) u(x, A\y) x I(A) u(x, A) whenever y I(A). u(x, A) whenever y A\I(A). Label the class of monotonic menu preferences that satisfy Axioms 2 and 3 as Σ LP F. The following theorem is one of the main results in our previous paper. Theorem 2. Σ LPF if and only if it admits a representation by a LPF utility. Since there is no containment between the classes Σ LP F and Σ KP F, it is not generally possible to write an LPF utility as a special case of a Kreps utility, or vice versa. Thus, we introduce a utility which is naturally the hull of the preferences in Σ LP F and Σ KP F and check that both the LPF and Kreps utilities are specializations of this more general 4

utility. Given a pair (S, u s (, )) consisting of a state space and a collection of statedependent set monotonic flexibility kernels, u s : X 2 X R + consider the menu utility defined by U(A) := max x A u s (x, A) s Call such a utility a Generalized Preference for Flexibility (hereafter, GPF) utility. The following example demonstrates that a preference for flexibility cannot necessarily be represented by either a Kreps or an LPF utility. Example 1. (State-Dependent Complementarity) Let X = {t, sc, c, e} (t = tea, sc = sugar cookie, c = cookie, e = egg). Consider the following preference, {e, c} {e}, {e, c, t} {e, t}, {e, sc, t} {t, sc, c, e} {e, c, t}, {sc, c, t} Note that both Commitment Monotonicity and Modularity are violated so this preference does not admit either a Kreps or LPF representation. On the other hand, the preference is monotonic so it can be represented by a GPF utility. Let S = {s 1, s 2, s 3 }, where s 1 is a state where the DM has a taste for eggs, s 2 is a state where the DM has a taste for tea, s 3 is a state in which the DM has a craving for something sweet to have with tea (in this state, the sweeter the complement the more desirable it is). Consider the following table of utilities, u s1 ( ) u s2 ( ) u s3 ( ) (e, A) 1 0 0 (t, A) 0 1 0 (c, A) 0 0.5 (1 t A ) (sc, A) 0 0 1 (1 t A ) The GPF utility defined by U GP F (A) = 3 i=1 max A u si (x, A) represents the above preference. Note that the interpretation of a state is somewhat non-standard in the above example. In the Kreps framework and in the model in Ergin (2003), for example, the state space is just an index of a collection of ex post rankings over the elements of X. In the above example, the state s 3 does not correspond to any fixed ranking over X. The reason this occurs is because there is still residual uncertainty in state s 3. In this state, the DM s ranking of sugar cookie vs. plain cookie is dependent on whether she simultaneously has a realized taste for tea. The role of the state space in this utility representation is to distinguish global uncertainty from local uncertainty. State s 3 corresponds to a global taste for a sweet craving, but this taste is coupled with a taste for tea which is only realizable when tea is on the menu. Thus, the label local refers to the fact that the residual uncertainty is indexed by the collection of menus. In contrast, the Kreps utility can be thought of as a model with only global uncertainty, so that once this uncertainty resolves the DM behaves as though she has a no uncertainty preference over menus (see 5

Definition 5 for a rigorously worded statement of this). It is interesting to compare the interpretation of the preference in the above example with the one given in Ergin (2003). Recall the following example taken from that paper: Example 2. (Ergin (2003)) Let X = {l, t, m}, where l = Lexus, t = Toyota, and m = Mercedes and consider the following violation of modularity {m, t} {t}, {l, t, m} {l, t} One can explain this violation as a consequence of costly contemplation. Imagine a prospective car buyer who must choose a car dealer to go to (this is the ex ante choice) with the understanding that she is committed to buying a car at the dealership she visits. The underlying state space here is the set of (ex post) rankings over the quality attributes of the cars. Once she chooses a dealership, she must go through the process of evaluating the available options. This is a costly process since she must weigh the relative benefit of buying a car with more features against the higher cost required to obtain these features. The preference {m, t} {t} then says that the buyer would rather be committed to buying a Toyota rather than first sort through the costly process of choosing between a Mercedes and Toyota when she anticipates that, at the end of the day, she will choose the Toyota anyways. The preference {l, t, m} {l, t} suggests that the more costly process of choosing between a Lexus, Toyota, and Mercedes is now potentially valuable since the added contemplation required to distinguish elements in this larger set may refine the buyers appraisal of the attributes of a Mercedes and lead her to conclude that the additional features are now worth the price. The decision model laid out in Ergin (2003) consists of the following ingredients. Let π = {E 1,..., E k } be a partition of an underlying state space S and let O be the collection of partitions of S. Also let µ be a distribution on S. For each event E S consider the conditional value function, V (A E) := max A s E µ(s E) u s(x). An Ergin utility, U ER, is defined as follows U ER (A) = max O ( E i π µ(e i ) V (A E i ) c(π)) where c : O R + is a cost function. Note that when c( ) 0 the optimal partition is the collection of atomic states and the function E i π µ(e i) V (A E i ) reduces to an ordinary Kreps utility. Any Kreps utility can be expressed in this form so that the Ergin model nests the Kreps model. Moreover, it does so in a completely different manner than the GPF utility. Returning to the GPF utility in Example 1 observe that the state space representation above is non-trivially a three-state representation. That is, each of the states is necessary for some menu, so that if we tried to collapse the states (e.g. by taking the sum of the state-dependent kernels) we would no longer be able to represent the given preference. We would like this to generally be true of any GPF representation. 6

Definition 4: Let C s (A) := {x A : (x, A) arg max A u s (x, A)} denote the set of ex-post commitments in state s and set U s (A) = max A u s (x, A). Let s be the menu preference induced by U s. Say that a GPF utility is fine if the following three properties hold, 1. S > 1. 2. s, s S. 3. There is a pair of states (s, s ), s s and a menu A such that C s (A) C s (A) =. The first restriction simply rules out single state representations. The second restriction rules out representations of where we simply add states where the kernel generates a cardinal representation of. Since these kernels are redundant in the representation they do not contribute to the non-triviality of the state space representation. Finally, the third restriction can be thought of as a weak independence condition on the state-dependent kernels. If this latter condition fails, then for any pair of states we can replace the kernels for these states with their sum and the corresponding GPF utility yields a representation of with a strictly smaller state space. Definition 5: P(X) is a no uncertainty preference if the following two properties hold, (i) B X, x B, B x, and (ii) if x B, x B, then whenever x A B, A B. Call any x B that satisfies (i) and (ii) a certainty equivalent for B. The following theorem is the main representation result in this paper. Note that the qualifier fine is important here since any monotonic preference is trivially representable by a GPF utility with a singleton state space. The content of the theorem is precisely that such a representation is possible with a non-singleton state space whenever the preference is non-trivially monotonic. Theorem 3. Let P(X). Then, satisfies Axiom 1 (Monotonicity) if and only if there is a pair (S, u(x, )) consisting of a subjective state space S and a state-dependent set monotonic utility kernel u s : X 2 X R + such that U GP F (A) := s S max x A u s (x, A) is a representation. Moreover, if is not a no uncertainty preference and the number of indifference classes of is at least 3, then the preference admits a fine GPF representation. Before turning to the proof of this result, we now discuss why the restrictions excluding no uncertainty preferences and preferences with less than 3 indifference classes are necessary for a fine representation. Lemma 1. If is a no-uncertainty preference that satisfies Axiom 1, then A sup(a) x, for any x sup(a). 7

Proof. Let x A be a certainty equivalent for A and let y sup(a). Then, x A A y, the latter relation following from monotonicity. Thus, since y x A we obtain that x A sup(a). Moreover, since sup(a) A, we obtain sup(a) x A A sup(a), which implies that A sup(a). Now assume that is a no-uncertainty preference that admits a representation by a GPF utility. Lemma 2. Let (S, u s (, )) be a GPF pair that represents a no-uncertainty preference. Then for any pair (s, s ) and any menu A, C s (A) C s (A). Proof. Let A := {x 1 x 2 x n } and let (S, u s (, )) be a GPF pair that generates. We claim that sup(a) C s (A), s S. Let S(x 1, A) := {s S : (x 1, A) arg max A u s (z, A)}. If there is some s S\S(x 1, A), then by set monotonicity max A\Cs(A) u s (z, A\C s (A)) < max A u s (z, A) so that A\C s (A) A - contradicting the preceding lemma. Thus, S(x 1, A) = S. The same argument applies to any x i A in the indifference class of x 1. It follows that C s (A) sup(a), s S. Note that the statedependent utilities u s : X 2 X R in this case are defined by u s (x, A) = u s (x, {x}), for any x sup(a). Thus, in order for a preference to admit a non-trivial state space GPF representation, it must not be a no uncertainty preference. The following example demonstrates that the lower bound on the number of indifference classes is tight, that is, monotonic preferences with no more than two indifference classes cannot generally be represented with fine GPF utilities. Example 3. Let X = {x, y, z, w} and consider the following monotonic preference, X A, A X, A B, A, B X Let (S, u s ( )) be a representing GPF pair. We claim that the representation cannot be fine. Note that for any x A X, we have u s (x, {x}) = u s (x, A). Putting U s (A) := max A u s (x, A) it follows that, for each s, U s (A) is constant on the class of sets {A : A X}. Since fineness requires that s, upwards set monotonicity then forces U s (X) = U s (A), A X. In other words, the kernels yield constant state-dependent menu utilities. Thus, the function U GP F (A) = s U s(a) cannot represent. 3 Proof of Theorem 3 Let U( ) be any cardinal representation of and define the kernel u s (x, A) U(A), x A. This yields a one state representation U GP F (A) = max A u s (x, A) = U(A). We will show that if is not a no uncertainty preference, then it admits a fine GPF representation. We will require the following lemma. 8

Lemma 3. Let P(X) be a monotonic preference. Let I(A 1 ), I(A 2 ),.., I(A k ) be an enumeration of the indifference classes of menus with respect to. Consider the sets Θ(i) := {A I(A i ) : B I(A i ), B A}. Then, is a no uncertainty preference if and only if for all i, Θ(i) contains only singleton menus. Proof. Assume Θ(i) contains only singletons for all i. Consider any non-singleton menu A and note that the minimal elements in the set {B : B A, B A} must be singleton by hypothesis. Pick x {B : B A, B A} and note that, by monotonicity, A C x, for any C A. Thus, x is a certainty equivalent for A. It follows that is a no uncertainty preference. Label the indifference classes I(A 1 ), I(A 2 ),.., I(A k ) such that A i A i+1 and pick any index i for which Θ(i) contains a non-singleton menu and let A Θ(i) be non-singleton. Guided by the lemma, we break the argument into two cases. In each of these cases, there are two issues to deal with: (i) representability, and (ii) fineness of the state space. Proof. Case 1: i such that Θ(i) contains a non-singleton menu A and such that I(A ) {A }. Choose a cardinal representation U( ) such that U(A j ) > 2 U(A j+1 ). Thus, if A A, then A A. Let A = D 1 D 2 be a partition of A and define kernels as follows. Sub-case (i): There is an i / {1, k} such that Θ(i) contains a non-singleton menu A and such that I(A ) {A }, and k 4. Define U = max{u(a) : A A i }. U(A), if A I(A j ), A j A i U, if A I(A j ), A j A i u s0 (x, A) = U(A), if A I(A i ), A A Pick another state s 1 with associated kernel, U(A i ), U(A), u s1 (x, A) = U(A), Define U(A), if x D 1, A = A U(A)/2, if x D 2, A = A. if A I(A j ), A j A i if A I(A j ), A j A i if A I(A i ), A A U(A)/2, if x D 1, A = A U(A), if x D 2, A = A. U GP F (A) := max A u s0 (x, A) + max A u s1 (x, A) and observe that (i) U GP F ( ) represents (by choice of U( )), and (ii) by choice of U( ) and minimality of A, the kernels u s0, u s1 are (upwards) set-monotonic. Note that this 9

argument presumes that there are at least two indifference classes below I(A i ). If instead there are at least two classes ranked above I(A i ), define kernels as follows: U(A), if A I(A 1 ) U(A 3 ), if A I(A j ), j = 2, 3 U(A), if A I(A j ), j > 3, j i u s0 (x, A) = U(A), if A I(A i ), A A U(A), if x D 1, A = A U(A)/2, if x D 2, A = A U(A 1 ), if A I(A j ), j = 1, 2 U(A), if A I(A j ), j 3, j i u s1 (x, A) = U(A), if A I(A i ), A A U(A)/2, if x D 1, A = A U(A), if x D 2, A = A Sub-case (ii): k 3 and A is not in the top indifference class. Note that, by definition, A cannot be the lowest indifference class which forces k = 3 in this sub-case. Define kernels as follows, U(A), if A I(A 1 ) u s0 (x, A) = U(A 2 ), if A I(A 2 ) U(A 2 ), if A I(A 3 ). Pick a state s 1 with kernel given by U(A 2 ), if A I(A 1 ) U(A), if A I(A 2 ), A A u s1 (x, A) = U(A), if x D 1, A = A U(A)/2, if x D 2, A = A U(A), if A I(A 3 ). Finally, pick a third state s 2 with kernel U(A 2 ), if A I(A 1 ) U(A), if A I(A 2 ), A A u s2 (x, A) = U(A)/2, if x D 1, A = A U(A), if x D 2, A = A U(A), if A I(A 3 ). Define U GP F (A) := max A u s0 (x, A) + max A u s1 (x, A) + max A u s2 (x, A) 10

and observe that (i) U GP F ( ) represents, and (ii) by choice of U( ) and minimality of A, the kernels u s0, u s1 are (upwards) set-monotonic. Note that this is non-trivially a three-state representation. It may be checked that we cannot collapse the state-dependent kernels and obtain a utility representation of with a smaller state space. Sub-case (iii): i = 1 and k 3 (the case where k 4 follows from an identical argument as in sub-case (i), in which a two-state representation may be constructed). For k = 3, let A be a minimal non-singleton menu in Θ 1 (A 1 ) and let A = D 1 D 2 as before. Define a three state representation as follows, U(A)/2, if A I(A 1 ), A A U(A)/2, if x D 1, A = A u s0 (x, A) = 0, if x D 2, A = A 0, if A I(A 2 ) 0, if A I(A 3 ). For the state s 1 define the kernel U(A)/2, if A I(A 1 ), A A 0, if x D 1, A = A u s1 (x, A) = U(A)/2, if x D 2, A = A 0, if A I(A 2 ) 0, if A I(A 3 ). Note that these kernels are set-monotonic by minimality of A. Pick a third state s 2 define the associated kernel, U(A 1 ), if A I(A 1 ) u s2 (x, A) = U(A 1 ), if A I(A 2 ) 0, if A I(A 3 ). Define U GP F (A) := 2 max A u si (x, A) i=0 and note that (i) U GP F ( ) represents, and (ii) the states are linearly independent, we cannot collapse the GPF representation onto a singleton state space by taking the sum of the kernels. Case 2: For each i, either (i) Θ(i) = {A i } = I(A i ) or (ii) Θ(i) contains only singletons. 3 3 Note that this restriction implies that the preference satisfies Axiom 2 (Commitment Monotonicity). Thus, if it also satisfies Axiom3 (Non-Emptiness) then it has an LPF representation, which by the arguments in case 2 imply that the state space representation is fine. We give an entirely different construction in this case precisely because we do not want to assume the Non-Emptiness axiom. 11

Assume that whenever Θ(i) contains non-singleton elements, then I(A i ) = {A i } = Θ(i) (note that this implies is not a no uncertainty preference). This implies the following dichotomy, either (i) I(A) = {A} or (ii) for A A i, Θ(i) contains only singleton menus. In the latter case, note that Θ(i) A is exactly the set of certainty equivalents for A. For this case, the constructions in the previous cases do not give a non-singleton state-space representation. We require an alternative construction. It suffices to prove the result for u s : X 2 X R (rather than R + ). We proceed by induction on the cardinality of the menu. Let Σ(n) = {A X A n} and put X(n) Σ(n) := {(x, A) : x A, A Σ(n)}. We inductively construct a sequence of states S(n) and state-dependent kernels u sn : X 2 Σ(n) R, s n S(n) such that the utility U n (A) := s n S(n) max x A u sn (x, A) represents on Σ(n). For Σ(1) let S(1) just be an index of the singleton menus. Let X = {x 1,..., x k } and let I 1,.., I r be a top down enumeration of the indifference classes of restricted to X. Choose a cardinal representation u( ) : X R such that u(x) > I k u(y), x I j, y I k, j < k. For s x S(1) define u sx (y) = u(x) if y x, and 0 elsewhere on Σ(1). Clearly, U 1 ({x}) = s S(1) u s({x}) represents on Σ(1). Let M 1 M 2... M k be an enumeration of the elements of 2 X of cardinality n+1. Assume that we have defined (S(n), u sn ) such that U n as defined above represents on Σ(n). We extend the representation to Σ(n) M 1 (the inductive argument that extends from Σ(n) j i 1 M j to Σ(n) j i M j is nearly identical and involves slightly more notation, hence is omitted). Let Σ M1 (n) denote the set of menus in Σ(n) contained in M 1. Put u x := max A ΣM1 (n) u s (x, A) and extend the kernels u s ( ) for each s S(n) as follows, { u u s (x, A) = x, if A = M 1 u s (x, A), otherwise. Clearly the extended kernel is (upwards) set monotonic. Put U (M 1 ) := s S(n) u s(x, A s ) and let UC n (M 1 ) := {A Σ(n) : A M 1 }, LC n (M i ) := {A Σ(n) : A M 1 }. Also define U = min A UCn (M 1 ) U n (A). The argument breaks into two cases: Case 1: U (M 1 ) > U. Sub-case (i): I(M 1 ) = {M 1 }. In this case, define M 1 new states s i M 1 with associated kernel as follows, (label M 1 = {x 1,.., x M1 }) (U U (M 1 ) ɛ)/ M 1, if x = x i, A = M 1 K, if x x i, A = M 1 u s i M1 (x, A) = 0, if A M 1, A UC n (M 1 ) K, if A M 1, A LC n (M 1 ). 12

where K < (U U (M 1 ))/ M 1 and ɛ > 0 is some small constant. Define an adjusted menu utility, Un ad (A) = max A u s i M1 (x, A) Note that (i) u s i M1 s S(n) max A u s (x, A) + i is upwards set monotonic, and (ii) U ad n represents on Σ(n) M 1. Sub-case (ii): M 1 A i, where Θ(i) consists of only singletons. This case itself breaks into two (possibly empty) sub-cases. To describe them consider the set of certainty equivalents for M 1, that is, CE(M 1 ) := {x M 1 : x C M 1, C M 1 }. We require the following facts; the proofs are straightforward and left to the reader. For all A M 1 such that A M 1, {s S(n) : CE(M 1 ) C s (A) } = S. For all x CE(M 1 ), s S(n), u s (x, A) = u s (x, {x}), A M 1, x A. Consider the maximal elements in the set I (M 1 ) := {A M 1 : A M 1 }. If U (M 1 ) > U, then either (a) There must be a pair of maximal elements in I (M 1 ), (A, A ), 4 and a corresponding pair of states, (s A, s A ) such that max A u sa (x, A) > max A u sa (x, A ), max A u sa (x, A ) > max A u sa (x, A) (b) CE(M 1 ) = {x} = C(M 1 ) and for any A M 1 with A M 1, C(A) = {x}. We claim that case (a) is empty. In this case, note that by the above facts there must be x A, x A A CE(M 1 ), A CE(M 1 ) with x A arg max A u sa (x, A), x A arg max A u sa (x, A ). Moreover, by the above facts u sa (x A, A) = u sa (x A, {x A }), u sa (x A, A ) = u sa (x A, {x A }) It follows that C({x A, x A }) = {x A, x A } - which is a contradiction since {x A } {x A, x A } x A. In the latter case (case (b)), consider the set of states S (n) defined as follows, (put u s (A) := max A u s (x, A)) S (n) := {s S(n) : (arg max ΣM1 (n) u s (A)) I(M 1 ) = } The argument given in case (a) implies that if U (M 1 ) > U, then S (n). Note that for s S (n), if A arg max ΣM1 (n) u s (B), then A M 1. Motivated by this observation, for each s S (n) we modify the kernel u s ( ) as follows { u K s + u s (x, A), if A LC n (M 1 ) s(x, A) = u s (x, A), if A UC n (M 1 ). 4 Note that if M 1 = {x, y} it cannot fall into this subcase since the extension procedure forces U (M 1 ) U n (x), for any x CE(M 1 ). 13

where K s is chosen such that max A ( K s + u s (x, A)) = max A u s (x, {x}), x CE(M 1 ) and A arg max ΣM1 (n) U s (A). Note that (i) the modified kernel is set-monotonic, (ii) the modified GPF pair (S(n), {u s ( )}) represents on Σ(n) if the original pair did, and (iii) the modified pair (S(n), {u s ( )}) is fine. Now start over the process of extending these kernels to M 1. For each state s S(n) put u s (x, M 1 ) = max A M1 u s (x, A) and let U (M 1 ) = s S(n) max M 1 u s (x, M 1 ). Putting U = min A UCn (M 1 ) U n (A) as before, the above argument then shows that we must have U (M 1 ) U. Case 2: U (M 1 ) U. 5 If equality holds and I n (M 1 ) =, then add a state s with kernel defined as follows, { K, if A UC n (M 1 ) u s (x, A) = 0, otherwise. Note that (by monotonicity) the kernel is set-monotonic on the extended domain X(n) Σ(n) x M1 (x, M 1 ). Moreover, putting Un ad (A) := max A u s (x, A) + max A u s (x, A) s S(n) we get an extension of to Σ(n) M 1. If I n (M 1 ), then we already have an extension and no new states need to be introduced. If strict inequality holds and I n (M 1 ), then add a new state s with kernel defined as follows { U U (M 1 ), if A = M 1 u s (x, A) = 0, otherwise. Again, the kernel is set-monotonic on the extended domain. Moreover, extending the utility Un ad ( ) to include s as above, we get an extension to Σ(n) M 1. Finally, if strict inequality holds and I n (M 1 ) =, then if U (M 1 ) > U n (A), A LC n (M 1 ) we already have an extension. Otherwise, since U > max{u n (A) : A LC n (M 1 )} (since U n ( ) represents on Σ(n)) we add a new state s with kernel defined as follows { K, if A = M 1 u s (x, A) = 0, otherwise. where K (max{u n (A) : A LC n (M 1 )}, U ). Defining the extension Un ad as above we get an extended representation to Σ(n) M 1. This completes the induction step, and hence the proof of the Theorem. One of the reasons we consider the GPF utility an important model of preference for flexibility is that it properly nests both the Kreps model for preference for flexibility and 5 We include this as the default case in the inductive algorithm when UC n (M 1 ) =. 14

the LPF utility model from our companion paper. Note that a Kreps utility is just a special case of a GPF utility where the GPF kernels are not menu-dependent, that is, u(x, A) = u(x, B), A, B. It is slightly more subtle to show that the LPF utility is a specialization of the GPF utility. To make the argument completely transparent assume that the commitment operator C( ) is strictly monotonic. 6 Let (U LP F, u LP F (, )) be an LPF pair and define a state space S as follows. For each x X, put s x := {(x, A) x C(A)}. Thus, the state s x is the set of ex-post decisions (x, A) where x is a commitment element of A. Put S := x X s x and define state-dependent utility kernels as { u LP F (x, A), if y = (x, A) s x u sx (y) = 0, otherwise. Note that u sx ( ) is a monotonic kernel since C(A) C(B) whenever A B. 7 Moreover, U(A) = max z A u sx (z, A)) = u LP F (x, A) = U LP F (A) s x S {s x: x C(A)} Note that if Σ LP F has monotonic commitment correspondence, then it is a no uncertainty preference. Moreover, for a pair {x, y} note that C({x, y}) = {x, y} so that u sx (x, {x, y})) = u LP F (x, {x, y}) > 0 = u sx (y, {x, y})) and similarly, u sy (y, {x, y})) > u sy (x, {x, y})) so that the above GPF representation is fine. 4 Refining the State Space Since all monotonic preferences admit a single state GPF representation, it is particularly important that we have some means of testing whether a candidate multiple state GPF representation is truly a multiple state representation or merely a single state representation in the guise of a non-singleton representation. As an illustration of the latter, consider the following example. Example 4. Let S = {s 1,.., s n }. Define u si (x, A) U(A)/i and U GP F (A) = i max A u si (x, A) Note that this fails the definition of fineness on two of three counts, (i) s S such that s =, and (ii) there is no pair (s, s ) along with a menu A such that C s (A) C s (A) =. 6 Without this assumption, one needs to refine the definition of the LPF utility to show that it is a special case of a GPF utility. It follows from our proof of Theorem 2.2 that any preference in Σ LP F admits such a refined representation, so that the class of refined LPF utilities captures the entire set Σ LP F. Details are in the appendix. 7 This is where the (strict) monotonicity of the commitment map is used. 15

Our definition of fineness easily rules out multi-state representations that are transparently disguised single state representations like the one given above. However, there is a sense in which our definition of fineness of the state space is still too broad. Recall the following example from Case 1 of the proof of the Theorem. Example 5. Let have three indifference classes and recall the following three state representation, U(A), if A I(A 1 ) u s0 (x, A) = U(A 2 ), if A I(A 2 ) U(A 2 ), if A I(A 3 ). The first state distinguishes the top indifference class from the others, the latter two states differentiate the middle class from the bottom class and assures the definition of fineness is met. U(A 2 ), if A I(A 1 ) U(A), if A I(A 2 ), A A u s1 (x, A) = U(A), if x D 1, A = A where U GP F (A) := 2 i=0 max A u si (x, A). U(A)/2, if x D 2, A = A U(A), if A I(A 3 ). U(A 2 ), if A I(A 1 ) U(A), if A I(A 2 ), A A u s2 (x, A) = U(A)/2, if x D 1, A = A U(A), if x D 2, A = A U(A), if A I(A 3 ). Note that if we simply drop the state s 0 from the representation, then the utility 2 U (A) = max A u si (x, A) i=1 also represents. Moreover, the latter representation is also fine. However, it is not a genuine two-state representation since U (A) = 2 i=1 max A u si (x, A) = max A ( 2 i=1 u s i (x, A)). This brings us to the following refinement of the notion of fineness. Definition 6: Let (S, u s ) be a GPF representation of. Say that each state s S is relevant if the following property holds: For each s S, there is a pair (A, B) such that A B, but B S\s A, where S\s denotes the menu preference generated by the utility, U (A) = s S\s max A u s (x, A). 8 We say that the repre- Definition 7: Let (S, u s ( )) be a GPF representation of. sentation is non-collapsible if the following conditions hold: 8 This definition is taken from DLR (2001). 16

1. The representation is fine. 2. Let the collapsed kernel be denoted by u s (x, A) := s u s(x, A). Then, the utility defined by U (A) := max A u s (x, A) does not represent. The following proposition extends Theorem 3 to the domain of non-collapsible GPF utilities. Proposition 1. Let P(X) be a monotone preference (with at least 2 indifference classes) that is not a no-uncertainty preference. If X is strict, 9 then it admits a non-collapsible GPF representation where every state is relevant. Proof. Using the 2 case breakdown as in the proof of Theorem 3, we check in each case that the given construction is either (i) already non-collapsible, or (ii) can be easily amended to assure non-collapsibility. Case 1: i such that Θ(i) contains a non-singleton menu, A, and I(A i ) {A i }. We leave to the reader the task of constructing a non-collapsible GPF representation when X = {x 1, x 2 }. When X 3, recall that the construction in the previous proof gives the following two state representation, U(A), if A I(A j ), A j A i U, if A I(A j ), A j A i u s0 (x, A) = U(A), if A I(A i ), A A U(A), if x D 1, A = A U(A)/2, if x D 2, A = A. where U = max{u(a) : A A i }. Pick another state s 1 with associated kernel, U(A i ), if A I(A j ), A j A i U(A), if A I(A j ), A j A i u s1 (x, A) = U(A), if A I(A i ), A A U(A)/2, if x D 1, A = A U(A), if x D 2, A = A. Clearly, the representation is fine and each state is relevant. We check that the kernels are not collapsible. Actually, we check the stronger claim that there isn t any linear combination of the kernels that yields a one-state representation. Towards contradiction, let u s (x, A) = α u s0 (x, A) + β u s1 (x, A). We claim that the utility U (A) := max A u s (x, A) 9 Due to this restriction, we do not get a full extension of Theorem 3 to non-collapsible utilities. There are various relaxations of this assumptions that yield the same result, however the statement we have given is the cleanest to state among them. The search for a necessary and sufficient condition for non-collapsibility is the subject of ongoing research. 17

cannot represent. Note that this requires U (A ) = U(A), A I(A ), A A which implies either (i) α U(A) + β U(A)/2 = (α + β)u(a) or (ii) α U(A)/2 + β U(A) = (α + β)u(a). In the former case, we must have α = 1, β = 0 which contradicts relevance of state s 1. Similarly, we cannot have α = 0, β = 1. Case 2: Either Θ(i) consists of only singletons or (ii) I(A i ) = {A i }. We verify that the state space obtained at the end of the inductive procedure must be non-collapsible. Choose the largest, highest ranked menu A such that I(A ) = {A }. We will require the following facts, whose proofs are straightforward and left to the reader. Let X = {x 1,..., x m }, where x 1 x 2 x m. Then, there is an l such that A = {x l, x l+1,..., x m }. {B : B A } = {B : B {x 1,..., x l 1 } = }. These facts (and the maximality of A with respect to the property that I(A) = A) imply that if B is such that B > A, then (i) B A, and (ii) CE(B) (that is, B has certainty equivalents). Let N denote the stage in the induction argument reached before requiring an extension of the utility to A, with (S(N), u si ( )) the terms in the GPF utility constructed up through stage N. Let Σ(N) = {B : either (i) B < A, or (ii) B = A B A }, Σ A (N) = {B : B A }. Put u x := max A ΣA (N) u s (x, A) and for each s S(N) define, { u u s (x, A) = x, if A = A u s (x, A), otherwise. Putting U (A ) := s S(N) u s(x, A s ) and UC N (A ) = {B Σ(N) : B A } we get two cases for the extension, (i) U (A ) min{u N (B) : B UC N (A )} and (ii) U (A ) < min{u N (B) : B UC N (A )}. In either of these cases, we first want to verify that if the state space S(N) is not collapsible, then the extended space S(N +1) obtained after extending the representation to A is still not collapsible. Sub-case (i): U (A ) min{u N (B) : B UC N (A )}. Assume that S(N) is not collapsible and let u s c(x, A) = s S(N) u s(x, A), U c (A) = max A u s c(x, A). By non-collapsibility there are menus (A, B) Σ(N) (for later reference, call this is a pivotal pair) such that U N (A) U N (B), but U c (A) > U c (B) (or such that U N (A) < U N (B) and U c (A) = U c (B)). Since U c ( ) U N ( ) this is only possible if U c (B) < U N (B). Since {B : B A } = {B : B {x 1,.., x l 1 } = }, by maximality of A, if B A, then CE(B) and the elements of Θ(B) are singletons. Thus, B x, for some x CE(B). Since U N ( ) represents on Σ(N) we obtain U c (x) = U N (x) = U N (B). This implies U N (B) U c (B) U c (x) = U N (B) 18

so that U N (B) = U c (B). Hence, in the pivotal pair (A, B) we must have A, B A. Recall that the states that we added to S(N) in this case had kernels defined as, (U U (A ))/ A, if x = x i, A = A K, if x x i, A = A u s i M1 (x, A) = 0, if A A, A UC n (A ) K, if A A, A LC n (A ). where K < (U U (A ))/ A. Thus, calling the extended representation U N+1, we find that U N+1 (A) = K + U N (A), U N+1 (B) = K + U N (B) and the pair (A, B) is still pivotal. Thus, the extended state space S(N + 1) is not collapsible. Sub-case (ii): U (A ) < min{u N (B) : B UC N (A )}. As in the previous sub-case we know that the only pivotal pairs could be in {B Σ(N) : B A }. Recall that in this case we extended by adding one more state and with kernel defined by, { K, if A = A u s (x, A) = 0, otherwise. where K is some small positive constant. Clearly, the pair (A, B) remains pivotal under the extended representation U N+1 in this case as well. We now verify that if the space S(N) is collapsible, then we can assure an extension S(N + 1) that is not collapsible. This is where the strictness of the singleton ranking is invoked. Consider the set of menus {B : B = A \x, x A }. By the strictness of the singleton ranking, not all of these menus are in the same indifference class. Thus, let B, B respectively denote selections from this set in (resp.) the highest and second-highest indifference class. Since the representation at stage N is collapsible, let (s c, u s c) denote the collapsed representation on Σ(N). Let x A \B. Find 0 < ɛ < U c (B ) U c (B ) and define a new state s with kernel as follows, { ɛ, if x = x, A = A u s (x, A) = 0, otherwise. Let S(N + 1) = {s c, s } and consider the GPF utility given by (S(N + 1), u s c, u s ). By choice of ɛ and A, this is a non-collapsible representation on Σ(N) A. This argument works when there is a unique highest ranked element in the set {B : B = A \x, x A }. If there are ties at the top, then it must be the case that the top ranked sets have certainty equivalent equal to x l, the top-ranked element in A. Now consider the shifted kernel, u sc(x, A) = u s c(x, A) K, u s c(x, A) K, u s c(x l, A), u s c(x, A), if A LC N (A ), x l / A if A LC N (A ), x l A, x x l if A LC N (A ), x l A 19 otherwise.

where K is some large constant. It follows that with u s ( ) as defined above, the space S(N + 1) = {s, s c } with kernels (u s c, u s ) is non-collapsible. Thus, in either case of the inductive extension we can assure an extension that is non-collapsible. To complete the argument we want to show that the non-collapsibility property is preserved as we extend the utility up to the remaining menus. Note that by choice of A, every menu to which we still have to extend the representation is strictly preferred to A, and moreover, all these menus have certainty equivalents. Assume we have a noncollapsible representation up through stage M (for some M > N) and let S(M) be the non-collapsible state-space. Let D 2 X \Σ(M) be the menu to which we want to extend the representation. Since A D, there is a pivotal pair (A, B) (the same pair from stage N in the lower contour set of A ) for the representation (S(M), u s ( )) on the space Σ(M). Moreover, any additional kernels added to the representation shift all elements of the lower contour set of A by a uniform constant, so that pivotal-ness of the pair (A, B) is preserved. 10 5 Discussion The theorem shows that if we allow for monotonic preferences to exhibit either menudependence or menu-independence, then we exhaust the full set of monotonic preferences. Thus, both utility forms representing Σ LP F and Σ KP F are special cases of the utility given in Theorem 3. This observation, in and of itself, doesn t tell us why the two utility forms appear as special cases of the function in the Theorem. To better understand the connection think of the state space as consisting of two components, a global component and a local one. Put S := A X {S S(A)}. Consider an element (s, s A ) of the union. The first component represents a global ex-post preference, one that is not induced or dependent on the menu A under consideration. The second component is a local, or menu-dependent, ex-post taste. To make matters concrete consider the issue of college choice, taken from our previous paper. A high-school senior s possible tastes for schools may break along two dimensions. The first dimension is global in the sense that it is not-determined by the set of schools the senior is accepted to. The second dimension is local in the sense that it is entirely dependent on the set of available options. For example, the global dimension could be a possible preference for school location {East Coast, West Coast}. Here global means that the set of locations that form the support of the preferences does not change depending on the menu under consideration. We have already discussed examples of local, or menudependent, state spaces. Assume A is some set of liberal arts schools, distributed across 10 We can assure that all states are relevant in the above construction as follows. Take u s c to be a single state representation on Σ N (which is always possible by monotonicity). Extend to A as above with the kernel u s and note that the pair (A, B ) assures relevance of the pair of states {s c, s }. Moreover, since any kernels added to the representation shift the utility of A and B by the same constant, these states remain relevant as we extend the representation to the full domain of menus. 20

the East and West coast (she only applies to schools in these locations). Then, the set S(A) could be the set of majors/activities available at these schools. We may think of the LPF utility kernel as being computed by first grouping global states together by the ex-post decision (x, A) implemented in those states, and then taking the LPF kernel value of the decision (x, A) to be the average value of u(s, (x, A)) over these global states. Roughly speaking, the LPF utility averages out the effects of the global state space by first conditioning on the local state space (e.g. fixing a menu A) and then taking an expectation over global states. A Kreps utility is obtained by conditioning in exactly the opposite direction. That is, for a fixed global ex-post state s, the Kreps kernel value u s (x) is obtained by averaging over the relative value of x in menu A, u s (x, A), taken across all menus A. 21

6 Appendix In this appendix, we refine the class of LPF utilities and show that any such utility is a specialization of a GPF utility with a non-collapsible state space. For any preference Σ LP F recall that we have the following partition structure. Put {B 1,..., B n } an enumeration of the maximal commitment sets and let Φ i = {A : C(A) B i }. Let (U LP F, u LP F ) be an LPF pair with the following property, u LP F (x, A) n U LP F (B) > U LP F (D), whenever A Φ i, B, D j>i Φ j Note that from the proof of Theorem 2 (see Chandrasekher (2007)) we know that we can always choose such an LPF representation. We now verify that any LPF utility that satisfies this additional property is a specialization of a GPF utility with non-collapsible state space. Let Ui follows. = max Φi U LP F (B) and define n states s 1,..., s n with associated kernels as U LP F (A) (n i) Ui+1, if A Φ i u si (x, A) = Uj+1, if A Φ j, j < i 0, if A Φ j, j > i. Consider the utility function given by U GP F (A) = n max A u si (x, A) i=1 Observe that U GP F (A) = U LP F (A) so that the utility represents. We claim that the kernels u si (x, A) are upwards set monotonic. Let A B and consider the pair (u si (x, A), u si (x, B)) for any fixed i. If A, B Φ l, then clearly u si (x, B) u si (x, A). If A Φ l, B Φ k, then if k = i we require U LP F (B) (n i) Ui+1 = u si (x, B) u si (x, A). Since U LP F (B) (n i) Ui+1 > Ui+1 by construction and Ui+1 u si (x, A) (as A Φ l and l > i) we obtain u si (x, B) > u si (x, A). The remaining case similarly follows. We verify that the GPF representation described above is non-collapsible. To check that the representation is fine first note that none of the state-dependent preferences si equals. Second, since not all the B i are singletons (else the preference is a no uncertainty preference) we can choose a non-singleton commitment subset Bi B i and appropriately modify the kernels as in the proof of Case 1 of Theorem 3 to find a pair of states (s i, s j ) such that C si (Bi ) C sj (Bi ) =. This same construction also implies that the state space is not collapsible. Finally, observe that each state i is relevant (assuming that each of the sets B i are non-singleton) since the ith kernel is required to represent the menus in Φ i. To extend to the case where some B i are singleton recall that the collection {B 1,..., B n } partitions X into order intervals. In the event that some of the 22

B i are singleton we inductively define states s i with kernels u si as follows. If B 1 is nonsingleton, let (s 1, u s1 ( )) be defined as before. If B 1 is singleton, let B j(1) be the first non-singleton B j after B 1. Define u s1 as follows { U LP F (A) (n 1) Uj(1) u s1 (x, A) =, if A Φ j, 1 j < j(1) 0, else. where n ( n) is a constant to be determined. Let u s2 be as defined in the non-singleton case and similarly define u s3 if B j(1)+1 is non-singleton. If B j(1)+1 is singleton, let B j(2) denote the first non-singleton B j after B j(1)+1. Put U LP F (A) (n 3) Uj(2), if A Φ j, j(1) + 1 j < j(2) Uj(1)+1 u s3 (x, A) =, if A Φ j(1) Uj(1), if A Φ j, j < j(1) 0, if j j(2). U LP F (A) (n 4) Uj(2)+1, if A Φ j(2) Uj(2), if A Φ j, j(1) + 1 j < j(2) u s4 (x, A) = Uj(1)+1, if A Φ j(1) Uj(1), if A Φ j, j < j(1) 0, else. Continue this construction until we exhaust the set {B 1,..., B n } and retroactively set n equal to the number of states obtained at the end of this construction. Note that the same argument as before shows that the state space is fine and cannot be collapsed. Moreover, each state is clearly relevant by construction. 23