Harstad and Selten (this forum) raise interesting questions about the relative promise of optimization models and bounded-rationality models in making progress in economics. This article builds from their analysis by indicating the potential for using neoclassical (broadly defined) optimization models to integrate insights from psychology on the limits to rationality into economics. I lay out an approach to making (imperfect and incremental) improvements over previous economic theory by incorporating greater realism while attempting to maintain the breadth of application, the precision of predictions, and the insights of neoclassical theory. I then discuss how many human limits to full rationality are, in fact, well understood in terms of optimization.

USING EXPECTED-UTILITY THEORY, economists model risk aversion as arising solely because the utility function over wealth is concave.This diminishing-marginal-utility-ofwealth theory of risk aversion is psychologically intuitive, and surely helps explain some of our aversion to large-scale risk: We dislike vast uncertainty in lifetime wealth because a dollar that helps us avoid poverty is more valuable than a dollar that helps us become very rich.Yet this theory also implies that people are approximately risk neutral when stakes are small.Arrow (1971, p. 100) shows that an expected-utility maximizer with a differentiable utility function will always want to take a sufficiently small stake in any positiveexpected-value bet.That is, expected-utility maximizers are (almost everywhere) arbitrarily close to risk neutral when stakes are arbitrarily small.While most economists understand this formal limit result, fewer appreciate that the approximate risk-neutrality prediction holds not just for negligible stakes, but for quite sizable and economically important stakes.Economists often invoke expected-utility theory to explain substantial (observed or posited) risk aversion over stakes where the theory actually predicts virtual risk neutrality.While not broadly appreciated, the inability of expected-utility theory to provide a plausible account of risk aversion over modest stakes has become oral tradition among some subsets of researchers, and has been illustrated in writing in a variety of different contexts using standard utility functions.2In this paper, I reinforce this previous research by presenting a theorem that calibrates a relationship between risk attitudes over small and large stakes.The theorem shows that, within the expected-utility model, anything but virtual risk neutrality over modest stakes implies manifestly unrealistic risk aversion over 1 Many people, including

This article proposes an approach to improving the psychological realism of economics while maintaining its conventional techniques and goals--formal theoretical and empirical analysis using tractable models, with a focus on prediction and estimation. Besides tolerating the imperfections that come with precision, models should aim for two crucial criteria: power and scope. The approach advocated is to develop portable extensions of existing models that embed preexisting theories as parameter values, while introducing the new psychological assumptions as alternative parameter values, and make the model portable by defining it in all cases where existing models make predictions.

People like to help those who are helping them, and to hurt those who are hurting them. Outcomes reflecting such motivations are called fairness equilibria. Outcomes are mutual-max when each person maximizes the other's material payoffs, and mutual-min when each person minimizes the other's payoffs. It is shown that every mutual-max or mutual-min Nash equilibrium is a fairness equilibrium. If payoffs are small, fairness equilibria are roughly the set of mutual-max and mutual-min outcomes; if payoffs are large, fairness equilibria are roughly the set of Nash equilibria. Several economic examples are considered, and possible welfare implications of fairness are explored.

We experimentally investigate the time-inconsistent taste for immediate gratification and future-preference misprediction. Across 7 weeks, 100 participants choose the number of unpleasant transcription tasks given various wages to complete immediately and at different future dates. Participants preferred 10–12% fewer tasks in the present compared to any future date, leading to an estimated $β $ of 0.83. Comparing predictions with actual immediate-work choices provides evidence against substantial sophistication, with estimates implying that participants understand no more than 24% of their present bias. Finally, we find evidence of “projection bias”: participants wished to complete 4–12% fewer tasks when decisions were elicited right after completing tasks rather than before.

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.

There is evidence that people do not fully take into account how other people's actions depend on these other people's information. This paper defines and applies a new equilibrium concept in games with private information, cursed equilibrium, which assumes that each player correctly predicts the distribution of other players' actions, but underestimates the degree to which these actions are correlated with other players' information. We apply the concept to common-values auctions, where cursed equilibrium captures the widely observed phenomenon of the winner's curse, and to bilateral trade, where cursedness predicts trade in adverse-selections settings for which conventional analysis predicts no trade. We also apply cursed equilibrium to voting and signalling models. We test a single-parameter variant of our model that embeds Bayesian Nash equilibrium as a special case and find that parameter values that correspond to cursedness fit a broad range of experimental datasets better than the parameter value that corresponds to Bayesian Nash equilibrium.

We develop a rational dynamic model in which people are loss averse over changes in beliefs about present and future consumption. Because changes in wealth are news about future consumption, preferences over money are reference-dependent. If news resonates more when about imminent consumption than when about future consumption, a decision maker might (to generate pleasant surprises) overconsume early relative to the optimal committed plan, increase immediate consumption following surprise wealth increases, and delay decreasing consumption following surprise losses. Since higher wealth mitigates the effect of bad news, people exhibit an unambiguous first-order precautionary-savings motive.