An agent employs a stochastic automaton to search for the best alternative drawn randomly with replacement from a menu of unknown composition. In each state, the automaton inquires about some attribute of the currently drawn alternative. I study how the minimum complexity of near-optimal automata depends on the *language*—collection of binary attributes used to describe alternatives. I show that the tight lower bound on transitional complexity among all languages is 3·⌈log_{2 }(m)⌉, where *m* is the number of alternatives valued distinctly. I also provide a linear upper bound.
If a language facilitates additive utility representation with the smallest possible number of attributes, it reaches the lower complexity bound. This result is if and only if when
(3/4)·2^{n} < m ≤ 2^{n} for a natural *n*. Finally, I drop the assumption that each attribute has 2 values and characterize the lower complexity bound for general languages.

We develop and analyze a model of framing under ambiguity. Frames are circumstances, unobservable to the analyst, that shape the agent’s perception of the relevant ambiguity. The analyst observes a choice correspondence that represents the set of possible choices under the various decision frames. We provide axioms that allow us to operationalize each frame as a set of priors, while the agent’s utility index remains fixed. We show that the analyst can identify the unique minimum set of decision frames. One agent is more consistent than another if the former has a unique choice whenever the latter does. We characterize comparative consistency in terms of the model parameters and apply this result to characterize the aggregation of preferences that satisfy the Unanimity criterion. Finally, we characterize the behavior of agents who recognize that they are subject to different frames and learn by combining their frames into a single model.

[Online Appendix]

Goods and services—public housing, medical appointments, schools—are often allocated to individuals who rank them similarly but differ in their preference intensities. We characterize optimal allocation rules in such settings, considering both the case in which individual preferences are known and ones in which they need to be elicited. Several insights emerge. First-best allocations may involve allocating to some agents lotteries between very high-ranked and very low-ranked goods. When preference intensities are private information, second-best allocations always involve such lotteries and, crucially, may coincide with first-best allocations. Second-best allocations may also entail disposal of services. We also discuss a market-based alternative approach and show how it differs.

An expected utility maximizer learns various aspects of her preferences in different decision frames; each frame is defined as a collection of observable features of the choice environment. We interpret each feature as a Blackwell experiment (signal structure) associated with a state space that describes uncertainty of the agent's preferences, and consider an agent who updates her beliefs using the Bayes rule. An analyst observes the resulting stochastic choice of a population of agents with heterogeneous state-dependent utility functions. We show that almost any stochastic choice that admits a random utility representation within each frame is consistent with such model. However, when the state space has limited cardinality, the resulting stochastic choice should satisfy additional constraints on the sums of Block-Marschak polynomials constructed from choice frequencies under each decision frame.

An agent with a privately known continuous type applies for approval. While the agent always prefers approval over rejection, approving an agent with low type has social costs. The agent sends a report about her type that she can inflate by engaging in signaling costs that have the single-crossing property. We study approval mechanisms without transfers that maximize a social welfare function that takes into consideration both the approval decision and the signaling costs of the agent. We show that threshold approval rules, which are widespread in society, are never socially optimal. By introducing some randomness in the rule, we can reduce the signaling costs without substantially changing the screening of the rule. If we further assume that the marginal cost is strictly log-supermodular, an assumption satisfied by the quadratic-loss function, we show that the optimal approval mechanism induces an approval probability that is continuous in the agent's type. We provide necessary first-order conditions for the optimal rule, and illustrate them with an analysis of the case of quadratic-loss function and uniform distribution of the agents' types.