Preference for Flexibility and Dynamic Consistency

Abstract

Dekel, Lipman, and Rustichini [3] characterize preferences over menus of lotteries that can be represented by the use of a unique subjective state space and a prior. We investigate what would be the appropriate version of Dynamic Consistency in such a setup. The condition we find, which we call Flexibility Consistency, is linked to a comparative theory of preference for flexibility. When the subjective state space is finite, we show that Flexibility Consistency is equivalent to a subjective version of Dynamic Consistency and that it implies that the decision maker is a subjective state space Bayesian updater. Later we characterize when a collection of signals can be interpreted as a partition of the subjective state space of the decision maker.

Introduction

The issue of understanding how agents react to new information is a topic extensively studied in individual decision theory. Within the realm of the Savagean theory of decision making, where the state space is regarded as exogenously given, a fully rational individual is usually associated with the property of Dynamic Consistency. When, however, the decision model at hand regards the state space as endogenous, as in the recently developed dynamic theory of choice over menus, developing a theory of how agents react to information becomes a much less transparent problem. In a nutshell, this paper is an attempt to provide such a theory by searching for the natural version of Dynamic Consistency in that environment.

Consider the following situation. At 11 AM Jane has to choose a place for her evening drink with her friends. Suppose different places differ only in the menu of drinks that they offer and Jane always has only one drink. Let X be the set of all possible drinks and $\mathcal{X}$ be the set of all conceivable drink menus, each menu being represented by a capital letter A, B, C, etc. As in Kreps [24], we assume that Jane has a well-defined preference relation ≿ on $\mathcal{X}$ and this relation exhibits preference for flexibility in the sense that, for any two menus A and B,

$$A\supseteq B\text{implies}A\succsim B.$$

Moreover, we assume that ≿ admits the following representation:

$$A\succsim B\text{iff}\sum _{s\in S}\pi (s)\underset{x\in A}{\max }U(x,s)⩾\sum _{s\in S}\pi (s)\underset{x\in B}{\max }U(x,s),$$

where S is a finite state space, π is a probability measure on S and U is a state-dependent utility function. The interpretation is that Jane is uncertain about the future and, in particular, she is not sure what kind of drink she will be in the mood for in the evening. The representation above thus says that she chooses a place that maximizes the expected utility she can get from the placeʼs drink menu, with respect to some prior π about her future tastes.

At lunch, Jane will meet with her friends and one of them is going to be selected as the designated driver for the evening. If Jane gets to be the designated driver, and only in that situation, she very much appreciates if the place they go to has orange juice, as the other drinks do no matter for her. We represent the “Jane being the designated driver situation” by the state $s^{⁎}\in S$. If we use j to represent the orange juice alternative, the discussion above can be formalized as

$$U(j,s^{⁎})>U(x,s^{⁎})\text{and}U(y,s^{⁎})=U(x,s^{⁎}),$$

for all $x,y\in X$ distinct from j, and

$$U(j,s)<U(x,s),$$

for all $x\in X$ and $s\in S$, distinct from j and $s^{⁎}$, respectively.

Suppose now that instead of choosing a place in the morning, Jane first goes for lunch with her friends where she is informed that she will not have to drive that evening. In terms of the representation above, this is equivalent to saying that she learns that the state $s^{⁎}$ will not happen. In the present paper we wonder how in the current setup, where Janeʼs state space is not observable, we can identify that this sort of situation is happening. More precisely, we investigate how we can learn from Janeʼs choices that the signal she received was interpreted as information about her state space and that upon learning this information she acted in a dynamically consistent way. Because the state space S is not observable, we cannot simply say that Jane satisfies the standard Dynamic Consistency condition. The main goal of the present paper is to find a subjective version of this condition that can be applied to preference relations over a space of menus.

Without an exogenously specified state space, we cannot write a condition that explicitly deals with the fact that Jane receives new information at lunch. Nonetheless, Janeʼs behavior still implies some consistency relating her preferences before and after lunch. Let ≿ represent her preference before lunch and ${\succsim }^{⁎}$ the one after lunch. Now suppose that A and B are two menus such that $A{\succ }^{⁎}B$, but $B\succsim A$. That is, before lunch she considers menu B at least as good as menu A, but after she learns that she will not have to drive in the evening, menu A becomes strictly more attractive than B. We note that, by (1) and by the assumption about how Jane updates her preferences, this can happen only if $j\in B$, but $j\notin A$. Intuitively, the only difference between Janeʼs preferences before and after lunch is that after lunch she no longer cares whether the place she goes has orange juice or not. So, if her before lunch preference relation values menu B more than her after lunch relation, it has to be because B offers exactly the alternative that loses its value once Jane learns she will not have to drive that evening.

Following the insight provided by this example, we investigate in this paper what would be the natural translation of Dynamic Consistency when the state space of the model is subjective. We work in the setup of Dekel et al. [3]—henceforth DLR—and our main condition is a generalization of the idea discussed in the previous paragraph. In words, our condition says that if menus A and B are such that $A{\succ }^{⁎}B$, but $B\succsim A$, then it must be the case that ≿ sees some gain in flexibility when moving from menu A to menu $A\cup B$ that ${\succsim }^{⁎}$ does not see. Formally, it says that whenever $A{\succ }^{⁎}B$ and $B\succsim A$, there must exist a menu C such that $A\cup B\cup C{\sim }^{⁎}A\cup C$, but $A\cup B\cup C\succ A\cup C$.

The setup of lotteries of menus we use in this paper became popular after the works of DLR and Gul and Pesendorfer [18]. This two papers started a growing literature on preferences over menus. Several papers in this literature study variations of the self control representation in Gul and Pesendorfer [18]. (See Dekel, Lipman, and Rustichini [4], Kopylov [23], Noor [25], Noor and Takeoka [26] and Stovall [34].) Others derive particular cases of DLRʼs additive representation that correspond to specific psychological phenomena. (See Barbos [1], Dillenberger and Sadowski [7] and Sarver [32], for example.)

Recently, a few papers in this literature have considered the problem of subjective acquisition of information. For example, Ergin and Sarver [14] study an individual who is uncertain about her tastes, but can engage in costly contemplation before selecting an alternative from a menu. Their representation models contemplation strategies as subjective signals over a subjective state space. In a related paper, Ortoleva [28] models an individual who dislikes large choice sets because of the “cost of thinking” involved in choosing from them. In his most specialized representation, the individual thinks just enough to be able to make a choice.

Probably the two papers closest to this one, although still substantially different, are Dillenberger, Lleras, Sadowski, and Takeoka [6] and Dillenberger and Sadowski [8]. They work in a setup of menus of Anscombe–Aumann acts and model an individual who expects to receive a signal between the time of the choice of the menu and the time of the choice from the menu. The main differences between these two papers and the present one are that, first, in their case the individual expects to receive a signal after she has chosen a menu, but before she makes a choice from the menu. In our case, the signal arrives before the choice of the menu. Second, in their case the signal is subjective and appears only as part of the interpretation of the representation they derive. We work with objective signals, and what is subjective is how the individual interprets them.

Like this paper, Ozdenoren [29] and Sadowski [31] mix objective information with subjective states. Those two papers work with acts over an objective state space that return a different menu in each objective state of nature. The setup in this paper is slightly different, since we assume that each objective signal induces a different preference over menus. Of course there is a close relation between the two setups. In their case, it is as if the individual has to make a contingent plan for each possible objective signal she might receive.

In a setup with objective states, see Epstein and Le Breton [10], Ghirardato [15] and the references therein, for a discussion of Dynamic Consistency and Bayesian updating in the classic Savagean framework. For weakenings of Dynamic Consistency and a discussion of non-Bayesian updating rules, see Epstein [9], Epstein, Noor, and Sandroni [12] and Ortoleva [27]. Finally, there is also some literature on updating in the context of the multiple priors model of Gilboa and Schmeidler [16] (see Epstein and Schneider [13], Gilboa and Schmeidler [17], Hanany and Klibanoff [19] and Siniscalchi [33]).

The remainder of our paper is organized as follows. We discuss the primitives of the model in Section 2. In particular, we introduce the concept of a finite Positive Additive Expected Utility (PAEU) representation, axiomatized by DLR, Dekel, Lipman, Rustichini, and Sarver [5]—henceforth DLRS—and Dekel et al. [4]—henceforth DLR2. In Section 3, we present a comparative theory of preference for flexibility and relate it to Dynamic Consistency and Bayesian updating. In particular, we define the fundamental notion of Flexibility Consistency in Section 3.1, and we present our main result, relating it to a subjective version of Dynamic Consistency in Section 3.2. The analysis in Sections 3.1 and 3.2 focuses on a single pair of relations and, therefore, concentrates on a unique signal. In Section 3.3, we extend the analysis to incorporate the possibility of multiple signals and, in particular, we characterize when a collection of signals forms a partition of the state space. In Section 4, we briefly discuss some fundamental aspects of the analysis in this paper. Section 5 concludes. The proofs of the main results as well as a discussion about the testability of the main postulate investigated in this paper appear in Appendix A.