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Article Excerpt
Summary. In the literature on choice under unforeseen contingencies, the decision maker behaves as if she aggregates possible instances of future rankings indexed by a set S. The set S is interpreted as a subjective state space even though subsequent rankings need not conform to any one of the aggregated utilities. This paper proposes a definition for a subjective state space under unforeseen contingencies that is topologically unique, derives its existence from preference primitives as opposed to the representation of preferences, and does not commit to an interpretation in which states correspond to future realized rankings. The definition topologically concurs with and extends the identification of the 'essentially unique' subjective state space due to Dekel, Lipman and Rustichini [4].
Keywords and Phrases: Endogenous state space, Unforeseen contingencies, Preference for flexibility, Incomplete preference relations, Partial orders.
JEL Classification Numbers: D11, D81, D91.
1 Introduction
In a seminal paper, Kreps [8] showed that a preference for flexibility implies that a decision maker acts as if she possesses an endogenous state space. For example, if a menu from which the decision maker will later consume, {a, b}, is strictly preferred to both the menu {a} and the menu {b}, then the decision maker cannot both (i) know how she will compare a and b when it is time to consume, and (ii) be consequentialist. If, for instance, she 'knows' that she will prefer a over b later, then it is pointless to pay a premium to retain the added flexibility of having both a and b available for future choice; it cannot be that {a, b} is strictly preferred to a, unless the decision maker cares about attributes of choice that are not related to final outcomes or final consumption (i.e., the decision maker likes freedom of choice for its own sake and is therefore not a consequentialist). Assuming consequentialism, a preference for flexibility at least intuitively implies that the decision maker is uncertain of her future rankings. Kreps [9] later notes that this is a reasonable way to behave when the decision maker suspects that she cannot account for all future uncertainty in a 'Savagian' framework. Since she is unable to list all welfare relevant events, the decision maker does not commit to any particular ex-post ranking when selecting menus. Another complementary interpretation is that the decision maker does know her future rankings but they happen to be incomplete. (1)
Kreps demonstrates, under parsimonious assumptions on preference over choice sets, that the decision maker has a utility representation that suggests an endogenous state space of possible future tastes. Specifically, Kreps derives a representation for preference over subsets of a finite space of prospects, D, that has the following structure:
x [>] y [right arrow] [summation over (s[member of]S)] [max.[d[member of]x]] U (s, d) > [summations over (s[member of]S)] [max.[d[member of]y]] U (s, d)
x and y are menus (subsets of D), S is an index set derived endogenously, and for each s [member of] S, U (s, d) is a real-valued function over D. The existence of the index set, S (i.e., the set of utility functions), is interpreted as an endogenous state space of future rankings that the decision maker might use subsequently or ex-post to select a single item from the chosen menu. This space, however, is not unique and the formulation is not normative in the sense that there is no guarantee that the decision maker will in fact resort to using one of the rankings implied by the representation in her actual ex-post choice. There are many equivalent utility representations that specify different index sets. These, in general, take the form:
U (x) = u ([max.[d[member of]x]] U (1, d), [max.[d[member of]x]] U (2, d),...)
where u is non-decreasing in all its arguments (it can be viewed as an aggregator of possible future indirect utility functions). To obtain an intuition for this result note that the U (i, d)'s in the representation imply a Pareto frontier for the menu x, and thus the current utility of the menu x can be equivalently written as depending on the Pareto frontier of x. Moreover, any set of utility functions that produce the same Pareto frontier can be used to construct an equivalent representation.
Dekel, Lipman and Rustichini (DLR) [4] show that the representation of such preferences and S can be essentially pinned down by allowing D to be a lottery space over a finite set of outcomes and insisting that a decision maker is indifferent between a menu and its convex hull (generated by all probabilistic mixtures of the menu's elements). The latter assumption leads to aggregated U (s, d)'s (or 'ex-post utilities') that are expected utility functionals:
U(x) = u([max.[d[member of]x]] [E.sub.d][[[psi].sup.1]], [max.[d[member of]x]] [E.sub.d][[[psi].sup.2]],..., [max.[d[member of]x]] [E.sub.d][[[Psi].sup.[alpha]]],...) (1.1)
where the d's are distributions over some finite set of outcomes. Although alternative representations exist, a state space of future expected utility tastes has a weakly smaller cardinality than any other representation whenever u(x) is monotonically increasing in the expected utility functionals. (2)
Both Kreps and DLR describe a setting where choice is essentially static. The decision maker chooses among menus, uncertainty over her subjective states is assumed to resolve and then the decision maker selects from the menu. However, there is no explicit modeling of ex-post choice and no role for consistency between realized tastes and tastes inferred from ex-ante preferences. Ideally, one would like to establish that the decision maker's ex-post ranking is actually one of the U (s, d)'s. A 'simple' way to achieve this is to explicitly impose an axiom stating that the decision maker's ex-post ranking is one of the utility functions that appears in the ex-ante representation. This, however, is completely unsatisfactory: axioms must be imposed directly on decision makers' choice behavior and not on the mathematical representation of their behavior. Moreover, such an assumption runs contrary to the spirit of modeling decision making under unforeseen contingencies since it rules out the possibility that the decision maker's ex-post ranking will correspond to a non-expected utility function that is consistent with the Pareto ordering implied by the set of expected utility functions used in the representation. One possible rebuttal of this criticism is to claim that if all menus are convex in terms of probability mixtures, an optimal choice for any non-expected utility functional will always be consistent with an optimal choice for some expected utility functional. (3) This 'as if' argument, while somewhat compelling, can also be applied to describe the static choice behavior of anyone who violates the expected utility axioms even in the absence of changing tastes or unforeseen contingencies; i.e., she can be indifferent (or averse) to mixing over alternatives and when selecting from any choice set, her choice can be rationalized as the optimal choice of some expected utility maximizer. If the 'as if' argument is not satisfactory in discussing deviations from expected utility in the classical static setting, then perhaps it ought not...
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