When a Bayesian learns new information and changes her beliefs, she must on average become concomitantly more certain about the state of the world. Consequently, it is rare for a Bayesian to frequently shift beliefs substantially while remaining relatively uncertain, or, conversely, become very confident with relatively little belief movement. We formalize this intuition by developing specific measures of movement and uncertainty reduction given a Bayesian’s changing beliefs over time, showing that these measures are equal in expectation and creating consequent statistical tests for Bayesianess. We then show connections between these two core concepts and four common psychological biases, suggesting that the test might be particularly good at detecting these biases. We provide support for this conclusion by simulating the performance of our test and other martingale tests. Finally, we apply our test to data sets of individual, algorithmic, and market beliefs.

Rationality leads people to imitate those with similar tastes but different information. But people who imitate common sources develop correlated beliefs, and rationality demands that later social learners take this correlation into account. This implies severe limits to rational imitation. We show that (i) in most natural observation structures besides the canonical single-file case, full rationality dictates that people must “anti-imitate” some of those they observe; and (ii) in every observation structure full rationality dictates that people imitate, on net, at most one person and are imitated by, on net, at most one person, over any set of interconnected players. We also show that in a very broad class of settings, any learning rule in which people regularly do imitate more than one person without anti-imitating others will lead to a positive probability of people converging to confident and wrong long-run beliefs.

We develop a model that fleshes out, extends, and modifies existing models of referencedependent preferences and loss aversion while accomodating most of the evidence motivating these models. Our approach makes reference-dependent theory more broadly applicable by avoiding some of the ways that prevailing models—if applied literally and without ancillary assumptions—make variously weak and incorrect predictions. Our model combines the reference-dependent gain-loss utility with standard economic “consumption utility ” and clarifies the relationship between the two. Most importantly, we posit that a person’s reference point is her recent expectations about outcomes (rather than the status quo), and assume that behavior accords to a personal equilibrium: The person maximizes utility given her rational expectations about outcomes, where these expectations depend on her own anticipated behavior. We apply our theory to consumer behavior, and emphasize that a consumer’s willingness to pay for a good is endogenously determined by the market distribution of prices and how she expects to respond to these prices. Because a buyer’s willingness to buy depends on whether she anticipates buying the good, for a range of market prices there are multiple personal equilibria. This multiplicity disappears when the consumer is sufficiently uncertain about the price she will face. Because paying more than she anticipated induces a sense of loss in the buyer, the lower the prices at which she expects to buy the lower will be her willingness to pay. In some situations, a known stochastic decrease in prices can even lower the quantity demanded.