Subjective State Space Research Articles

Indecisiveness, preference for flexibility, and a unique subjective state space

This paper unifies two seemingly different frameworks regarding uncertainty about tastes, namely preference for flexibility and indecisiveness. By identifying the set of lotteries as the domain of choice, and assuming preferences over alternatives and over menus as primitives, our main theorem axiomatizes a unique joint representation of expected multi-utility, as defined by Dubra et al. (2004), and ordinal expected utility, as defined by Dekel et al., (2001), wherein the set of expected utility functions in the former is equivalent to the subjective state space in the latter. This result indicates that preference for flexibility and indecisiveness both arise from an identical underlying uncertainty, albeit they may have different behavioral implications. Our key axiom is dominance consistency, which specifies that the addition of an alternative to a menu strictly improves the menu evaluation if and only if the alternative is undominated by the menu. Conversely, relaxing dominance consistency establishes a discrepancy between the subjective state spaces derived from the two frameworks, which can support “boundedly rational” behavioral implications, such as the illusion of control and projection bias.

Preference for flexibility and dynamic consistency with incomplete preferences

We generalize a previous result about dynamically consistent menu preferences to the case where preferences are not necessarily complete. We show that, as it is the case when preferences are complete, a subjective state space version of dynamic consistency is linked to a comparative theory of preference for flexibility. In words, an objective signal is interpreted as an event in the agent’s subjective state space and the agent acts in a dynamically consistent way after that if and only if we can attribute all the differences between the agent’s preferences before and after the signal to the fact that the agent values flexibility more before the signal than after.

Ambiguity and Awareness: A Coherent Multiple Priors Model

Abstract Ambiguity in the ordinary language sense means that available information is open to multiple interpretations. We model this by assuming that individuals are unaware of some possibilities relevant to the outcome of their decisions and that multiple probabilities may arise over an individual’s subjective state space depending on which of these possibilities are realized. We formalize a notion of coherent multiple priors and derive a representation result that with full awareness corresponds to the usual unique (Bayesian) prior but with less than full awareness generates multiple priors. When information is received with no change in awareness, each element of the set of priors is updated in the standard Bayesian fashion (that is, full Bayesian updating). An increase in awareness, however, leads to an expansion of the individual’s subjective state and (in general) a contraction in the set of priors under consideration.

Learning From Awareness

This paper introduces a novel channel of belief updating with awareness. Awareness is reflected by agents having state dependent subjective state spaces. An agent forms conjectures of his subjective state spaces in different states. He updates his belief over the true state by excluding states in which his conjectured subjective state spaces differ from his current state space. We restrict agents' capacities of belief updating through this channel by imposing axioms over agents' possible conjectures. We further use this framework to analyze belief updating when an agent's subjective state space expands. Importantly, in this framework the belief process is not necessarily a martingale. We use persuasion games as examples to illustrate this point. Since the receiver's belief process may not be a martingale, the sender might be better off sending an awareness signal to expand the receiver's state space than performing any feasible Bayesian experiment.

Subjective contingencies and limited Bayesian updating

We depart from Savage's (1954) common state space assumption and introduce a model that allows for a subjective understanding of uncertainty. Within the revealed preference paradigm, we take the analyst's perspective and uniquely identify the agent's subjective state space via her preferences conditional on incoming information. According to our representation, the agent's subjective contingencies correspond to sets of the analyst's states and, as such, are coarse. The agent uses an additively separable utility with respect to her set of contingencies; and she adopts an updating rule that follows the Bayesian spirit but is limited by her perception of uncertainty. We illustrate the relevance of our theory with applications to the confirmatory bias and correlation neglect, as well as to optimal contract design.

Revealed preference and the subjective state space hypothesis

This paper provides, for finite sets of choice data, revealed preference characterizations for the additive representations considered in Dekel et al. (2001). For a particular class of data sets, it is shown that the characterizing conditions can be reformulated as nonlinear systems of inequalities for which the existence of solutions can be verified using numerical methods.

Preference Discovery and Experimentation

I provide axiomatic foundations for a model of taste uncertainty with endogenous learning through consumption. In this setting, uncertainty is over an unobservable, subjective state space. Preference over lottery-menu pairs is sufficient to identify the state space and the learning process. In this model, the agent is viewed as if he learns the utility of an object upon its consumption. This information is used to improve choice from the follow-on menu. This implies a trade-off between consumption value and information leading to experimentation. I provide a behavioral definition of experimentation. While the literature focuses on identifying subjective states through a demand for flexibility, I show that experimentation also (partially) identifies taste uncertainty. Furthermore, only choice data on a small set of binary menus (analogous to financial options) is required to uniquely identify model parameters. An application of the model shows that the compromise effect can often be understood as rational experimentation.

Preference discovery and experimentation

I provide axiomatic foundations for a model of taste uncertainty with endogenous learning through consumption. In this setting, uncertainty is over an unobservable, subjective state space. Preference over lottery–menu pairs is sufficient to identify the state space and the learning process. In this model, the agent is viewed as if he learns the utility of an object upon its consumption. This information is used to improve choice from the follow-on menu. This implies a trade-off between consumption value and information leading to experimentation. I provide a behavioral definition of experimentation. While the literature focuses on identifying subjective states through a demand for flexibility, I show that experimentation also (partially) identifies taste uncertainty.

Active learning with a misspecified prior

We study learning and information acquisition by a Bayesian agent whose prior belief is misspecified in the sense that it assigns probability zero to the true state of the world. At each instant, the agent takes an action and observes the corresponding payoff, which is the sum of a fixed but unknown function of the action and an additive error term. We provide a complete characterization of asymptotic actions and beliefs when the agent's subjective state space is a doubleton. A simple example with three actions shows that in a misspecified environment a myopic agent's beliefs converge while a sufficiently patient agent's beliefs do not. This illustrates a novel interaction between misspecification and the agent's subjective discount rate.

Subjective Contingencies and Limited Bayesian Updating

We depart from Savage’s (1954) common state space assumption and introduce a model that allows for a subjective understanding of uncertainty. Within the revealed preference paradigm, we uniquely identify the agent’s subjective state space via her preferences conditional on incoming information. According to our representation, the agent’s subjective contingencies are coarser than the analyst’s states; she uses an additively separable utility with respect to her set of contingencies; and she adopts an updating rule that follows the Bayesian spirit but is limited by her perception of uncertainty. We illustrate our theory with an application to the Confirmatory Bias.

Self-consistency, consistency and cycles in non-partitional knowledge models

This paper expands the correspondence model of knowledge to a framework where there is an objective and many subjective state spaces, one for each player. At every objective state, each player lists the subset of her subjective states that she considers possible. In this context, the question of “self-consistency” arises: given beliefs of a player, is it possible that all of these beliefs come from a common prior via Bayesian updating? This paper provides necessary and sufficient conditions for self-consistency. If all players are self-consistent, the question of consistency arises: is there a common prior that rationalizes the beliefs of all players? Is it possible that self-consistency or consistency holds regardless of the numerical values of beliefs of the players? This paper provides necessary and sufficient conditions for consistency of beliefs.

Learning the Krepsian state: Exploration through consumption

We take the Krepsian approach to provide a behavioral foundation for responsive subjective learning processes. In contrast to the standard subjective state space models, the resolution of uncertainty regarding the true state is endogenous and depends on the decision maker's actions. There need not be full resolution of uncertainty between periods. When the decision maker chooses what to consume, she also chooses the information structure to which she will be exposed. When she consumes outcomes, she learns her relative preference between them; after each consumption history, the decision maker's information structure is a refinement of the previous information structure. We provide the behavioral restrictions corresponding to a recursive representation exhibiting such a learning process. Through the incorporation of dynamics we are able to identify the set of preferences the decision maker believes possible after each history of consumption, without appealing to an environment with risk.

Active Learning with Misspecified Beliefs

We study learning and information acquisition by a Bayesian agent who is misspecified in the sense that his prior belief assigns probability zero to the true state of the world. In our model, at each instant the agent takes an action and observes the corresponding payoff, which is the sum of the payoff generated by a fixed but unknown function and an additive error term. We provide a complete characterization of asymptotic actions and beliefs when the agent's subjective state space is a doubleton. A simple example with three actions shows that in a misspecified environment a myopic agent's beliefs converge while a sufficiently patient agent's beliefs do not. This shows that examples of myopic agents with non-converging beliefs in the prior literature require all myopically optimal actions to be informative, and illustrates a novel interaction between misspecification and the agent's subjective interest rate.

Identifying subjective beliefs in subjective state space models

This paper takes the Dekel, Lipman, and Rustichini (2001) framework, in which subjective beliefs over subjective states cannot be identified, and proves a conjecture made in their paper: if the Bernoulli utility functions are additively separable and one of the terms is state-independent, then beliefs are uniquely identified. The main departure from existing literature is that beliefs are identified without imposing extra objective elements into the model.

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.

How Is Non-knowledge Represented in Economic Theory?

In this article, we address the question of how non-knowledge about future events that influence economic agents’ decisions in choice settings has been formally represented in economic theory up to date. To position our discussion within the ongoing debate on uncertainty, we provide a?brief review of historical developments in economic theory and decision theory. Uncertainty is thereby understood as either based on decision-making in the context of a?state space representing the exogenous world, as in Savage’s axiomatisation and some successor concepts (ambiguity as situations with unknown probabilities), or, based on decision-making over a?set of menus of potential future opportunities, providingthe possibility of derivation of agents’ subjective state spaces (unawareness as situation with imperfect subjective knowledge of all future events possible).

Contingent Preference for Flexibility: Eliciting Beliefs from Behavior

Following Kreps (1979), I consider a decision maker who is uncertain about her future taste. This uncertainty leaves the decision maker with a preference for flexibility: When choosing among menus containing alternatives for future choice, she weakly prefers menus with additional alternatives. Standard representations accommodating this choice pattern cannot distinguish tastes (indexed by a subjective state space) and beliefs (a probability measure over the subjective states) as different concepts. I allow choice between menus to depend on objective states. My axioms provide a representation that uniquely identifies beliefs, provided objective states are sufficiently relevant for choice. I suggest this result as a choice theoretic foundation for the assumption, commonly made in the (incomplete) contracting literature, that contracting parties who know each others ranking of contracts, also share beliefs about each others future tastes in the face of unforeseen contingencies.

Contingent preference for flexibility: Eliciting beliefs from behavior

Following Kreps (1979), I consider a decision maker who is uncertain about her future taste. This uncertainty leaves the decision maker with a preference for ‡exibility: When choosing among menus containing alternatives for future choice, she weakly prefers menus with additional alternatives. Standard representations accommodating this choice pattern cannot distinguish tastes (indexed by a subjective state space) and

Preference for Flexibility and Random Choice

We study a two-stage model where the agent has preferences over menus as in Dekel, Lipman, and Rustichini (2001) in the first period and then makes random choices from menus as in Gul and Pesendorfer (2006) in the second period. Both preference for flexibility in the first period and strictly random choices in the second period can be, respectively, rationalized by subjective state spaces. Our main result characterizes the representation where the two state spaces align, so the agent correctly anticipates her future choices. The joint representation uniquely identifies probabilities over subjective states and magnitudes of utilities across states. We also characterize when the agent completely overlooks some subjective states that realize at the point of choice.

Equivalence of the information structure with unawareness to the logic of awareness

This paper proves the Li (2009) [13] unawareness structure equivalent to the single-agent propositionally generated logic of awareness of Fagin and Halpern (1988) [4]. For any model of one type one can construct a model of the other type describing the same belief and awareness. Li starts from an agent unable to perceive aspects of the world and distinguish states, modeled with subjective state spaces coarser than the objective state space. Fagin and Halpern limit the agentʼs language or cognitive ability to reasoning only about a subset of the primitive propositions describing the world. Equivalence of these approaches suggests they capture a natural notion of unawareness in a minimal way.