Berk–Nash Equilibrium: A Framework for Modeling Agents With Misspecified Models

We develop an equilibrium framework that relaxes the standard assumption that people have a correctly specified view of their environment. Each player is characterized by a (possibly misspecified) subjective model, which describes the set of feasible beliefs over payoff‐relevant consequences as a function of actions. We introduce the notion of a Berk–Nash equilibrium: Each player follows a strategy that is optimal given her belief, and her belief is restricted to be the best fit among the set of beliefs she considers possible. The notion of best fit is formalized in terms of minimizing the Kullback–Leibler divergence, which is endogenous and depends on the equilibrium strategy profile. Standard solution concepts such as Nash equilibrium and self‐confirming equilibrium constitute special cases where players have correctly specified models. We provide a learning foundation for Berk–Nash equilibrium by extending and combining results from the statistics literature on misspecified learning and the economics literature on learning in games.     

Utilitarian Preferences With Multiple Priors

This paper proposes a method for aggregating individual preferences in the context of uncertainty. Individuals are assumed to abide by Savage's model of Subjective Expected Utility, in which everyone has his/her own utility and subjective probability. Disagreement on probabilities among individuals gives rise to uncertainty at the societal level, and thus society may entertain a set of probabilities rather than only one. We assume that social preference admits a Maxmin Expected Utility representation. In this context, two Pareto‐type conditions are shown to be equivalent to social utility being a weighted average of individual utilities and the social set of priors containing only weighted averages of individual priors. Thus, society respects consensus among individuals' beliefs and does not add ambiguity beyond disagreement on beliefs. We also deal with the case in which society does not rule out any individual belief.     

EGO UTILITY,OVERCONFIDENCE, AND TASK CHOICE

This paper models behavior when a decision maker cares about and manages her self‐image. In addition to having preferences over material outcomes, the agent derives “ego utility” from positive views about her ability to do well in a skill‐sensitive, “ambitious,” task. Although she uses Bayes' rule to update beliefs, she tends to become overconfident regarding which task is appropriate for her. If tasks are equally informative about ability, her task choice is also overconfident. If the ambitious task is more informative about ability, she might initially display underconfidence in behavior, and, if she is disappointed by her performance, later become too ambitious. People with ego utility prefer to acquire free information in smaller pieces. Applications to employee motivation and other economic settings are discussed. (JEL: D83, D11)     

Evaluating the consequences of abandoning unprofitable customers: A comparison of direct and indirect abandonment strategies

Due to the increasing use of customer databases in many industries, companies nowadays have high visibility about the profitability of individual-level client relationships. This has led some researchers to recommend the implementation of value-based customer management strategies, including the abandonment of unprofitable customer relationships. In this context firms are faced with the choice between direct and indirect abandonment approaches. While direct abandonment strategies tend to be more efficient than indirect ones (i.e., immediate termination of the relationship vs. hoping that the unprofitable customer may leave on her own) they are also claimed to be associated with more severe negative reactions and higher abandonment cost. Based on two surveys among 965 US customers we provide evidence that, despite these beliefs, direct and indirect abandonment strategies do not differ in their negative impact on the abandoning firm. Given that (indirect) abandonment costs are therefore likely to be similar for both approaches, direct abandonment appears to be the recommendable approach for relationship dissolution.     

Rural Nevada and Climate Change: Vulnerability,Beliefs, and Risk Perception

In this article, we present the results of a study investigating the influence of vulnerability to climate change as a function of physical vulnerability, sensitivity, and adaptive capacity on climate change risk perception. In 2008/2009, we surveyed Nevada ranchers and farmers to assess their climate change‐related beliefs, and risk perceptions, political orientations, and socioeconomic characteristics. Ranchers’ and farmers’ sensitivity to climate change was measured through estimating the proportion of their household income originating from highly scarce water‐dependent agriculture to the total income. Adaptive capacity was measured as a combination of the Social Status Index and the Poverty Index. Utilizing water availability and use, and population distribution GIS databases; we assessed water resource vulnerability in Nevada by zip code as an indicator of physical vulnerability to climate change. We performed correlation tests and multiple regression analyses to examine the impact of vulnerability and its three distinct components on risk perception. We find that vulnerability is not a significant determinant of risk perception. Physical vulnerability alone also does not impact risk perception. Both sensitivity and adaptive capacity increase risk perception. While age is not a significant determinant of it, gender plays an important role in shaping risk perception. Yet, general beliefs such as political orientations and climate change‐specific beliefs such as believing in the anthropogenic causes of climate change and connecting the locally observed impacts (in this case drought) to climate change are the most prominent determinants of risk perception.     

Identifying Promising Messages to Increase Hurricane Mitigation Among Coastal Homeowners in the United States

Hurricanes threaten the physical and financial well-being of coastal residents throughout the United States. Though hurricane-related losses are largely avoidable through property mitigation (e.g., structural modifications to existing homes), few homeowners invest in mitigation. Communication campaigns, which have influenced risk-related behaviors in other domains, hold promise for persuading coastal residents to engage in hurricane mitigation. The development of successful campaign messages relies, in part, on formative research to assess the potential influence of candidate message strategies. We present results from mixed-methods, theory-driven research to identify promising beliefs for persuading homeowners in coastal/coastal-adjacent regions of Alabama and Florida to install a high wind–resistant (HWR) roof. In Study 1, we elicited homeowners’ (n = 74) salient behavioral, normative, and control beliefs about installing an HWR roof. Using established procedures, we content analyzed open-ended responses and categorized them by thematic content. In Study 2, we surveyed another sample of homeowners (n = 533) to examine the extent to which salient beliefs/themes about installing an HWR roof (elicited in Study 1) are promising targets for a communication campaign, given their associations with homeowners’ intentions to retrofit. Results demonstrate that across elicited beliefs, common themes include the protection and property resilience reroofing affords, and anticipated expenses and financial barriers associated with reroofing. The most promising beliefs include behavioral beliefs that installing an HWR roof will protect oneself and one's family, and normative beliefs about the likelihood that one's family and community will install an HWR roof. We discuss the implications of findings for the development of hurricane mitigation messaging.     

No‐Betting‐Pareto Dominance

We argue that the notion of Pareto dominance is not as compelling in the presence of uncertainty as it is under certainty. In particular, voluntary trade based on differences in tastes is commonly accepted as desirable, because tastes cannot be wrong. By contrast, voluntary trade based on incompatible beliefs may indicate that at least one agent entertains mistaken beliefs. We propose and characterize a weaker, No‐Betting, notion of Pareto domination which requires, on top of unanimity of preference, the existence of shared beliefs that can rationalize such preference for each agent.     

Representing Preferences with a Unique Subjective State Space

We extend Kreps' (1979) analysis of preference for flexibility, reinterpreted by Kreps (1992) as a model of unforeseen contingencies. We enrich the choice set, consequently obtaining uniqueness results that were not possible in Kreps' model. We consider several representations and allow the agent to prefer commitment in some contingencies. In the representations, the agent acts as if she had coherent beliefs about a set of possible future (ex post) preferences, each of which is an expected‐utility preference. We show that this set of ex post preferences, called the subjective state space, is essentially unique given the restriction that all ex post preferences are expected‐utility preferences and is minimal even without this restriction. Because the subjective state space is identified, the way ex post utilities are aggregated into an ex ante ranking is also essentially unique. Hence when a representation that is additive across states exists, the additivity is meaningful in the sense that all representations are intrinsically additive. Uniqueness enables us to show that the size of the subjective state space provides a measure of the agent's uncertainty about future contingencies and that the way the states are aggregated indicates whether these contingencies lead to a desire for flexibility or commitment.     

Risk and Climate Change: Perceptions of Key Policy Actors in Canada

This article examines factors that predict perceptions of risk associated with global climate change. The research focuses on the perceptions of those associated with climate change policy making in the prairie region of Canada. The data are from an online survey ( n = 851) of those policy actors. The analysis integrates several dominant approaches to the study of risk perception: psychometric approaches that examine the effects of cognitive structure; demographic assessments that examine, for example, differences in perception based on gender or family status; and political approaches that suggest that one's position in the policy process may affect his or her perceived risk. Attitudes toward climate change are to a degree predicted by all of these factors, but only when indirect effects are observed. Sociodemographic characteristics have little direct effect on perceived risk, but do affect general beliefs that affect risk perceptions. Perceived risk is related more strongly to these general beliefs or world views than to more specific beliefs about the effects of climate change on weather patterns. Position within the policy process also contributes to our understanding of perceptions, with industry and governmental actors demonstrating similar attitudes, which contrast with environmental groups and university researchers.     

Can Relaxation of Beliefs Rationalize the Winner's Curse?: An Experimental Study

We use a second‐price common‐value auction, called the maximal game, to experimentally study whether the winner's curse (WC) can be explained by models which retain best‐response behavior but allow for inconsistent beliefs. We compare behavior in a regular version of the maximal game, where the WC can be explained by inconsistent beliefs, to behavior in versions where such explanations are less plausible. We find little evidence of differences in behavior. Overall, our study casts a serious doubt on theories that posit the WC is driven by beliefs.     

A Service Provider's Elicitation of Its Customers’ Demand Distributions by a Price Mechanism

In a service environment a service provider needs to determine the amount and kinds of capacity to meet customers’ needs over many periods. To make good decisions, she needs to know the probability distribution of her customers’ demand in each period. We study a situation in which customers’ demand for a given service is random in each period, but inelastic, or modeled well by this assumption, and cannot be delayed to the next period. This article presents a mechanism that allows a service provider to learn the distribution of a customer's demand by offering him a set of contracts through which he can partially prepay for future service for a reduced cost for units of service based on anticipated needs. We describe the form of a set of contracts that will cause the customer to reveal his demand distribution as he minimizes his expected costs. To justify the effort of organizing and offering contracts, we present an application that demonstrates the cost savings to the service provider with better capacity planning using the truthfully elicited distribution.     

If You're so Smart,why Aren't You Rich? Belief Selection in Complete and Incomplete Markets

This paper provides an analysis of the asymptotic properties of Pareto optimal consumption allocations in a stochastic general equilibrium model with heterogeneous consumers. In particular, we investigate the market selection hypothesis that markets favor traders with more accurate beliefs. We show that in any Pareto‐optimal allocation whether each consumer vanishes or survives is determined entirely by discount factors and beliefs. Whereas equilibrium allocations in economies with complete markets are Pareto optimal, our results characterize the limit behavior of these economies. We show that, all else equal, the market selects for consumers who use Bayesian learning with the truth in the support of their prior and selects among Bayesians according to the size of their parameter space. Finally, we show that in economies with incomplete markets, these conclusions may not hold. With incomplete markets, payoff functions can matter for long‐run survival, and the market selection hypothesis fails.     

A MODEL OF NONBELIEF IN THE LAW OF LARGE NUMBERS

People believe that, even in very large samples, proportions of binary signals might depart significantly from the population mean. We model this “nonbelief in the Law of Large Numbers” by assuming that a person believes that proportions in any given sample might be determined by a rate different than the true rate. In prediction, a nonbeliever expects the distribution of signals will have fat tails. In inference, a nonbeliever remains uncertain and influenced by priors even after observing an arbitrarily large sample. We explore implications for beliefs and behavior in a variety of economic settings.     

A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements

Rationalizability is a central solution concept of game theory. Economic models often have many rationalizable outcomes, motivating economists to use refinements of rationalizability, including equilibrium refinements. In this paper we try to achieve a general understanding of when this multiplicity occurs and how one should deal with it. Assuming that the set of possible payoff functions and belief structures is sufficiently rich, we establish a revealing structure of the correspondence of beliefs to sets of rationalizable outcomes. We show that, for any rationalizable action a of any type, we can perturb the beliefs of the type in such a way that a is uniquely rationalizable for the new type. This unique outcome will be robust to further small changes. When multiplicity occurs, then we are in a “knife‐edge” case, where the unique rationalizable outcome changes, sandwiched between open sets of types where each of the rationalizable actions is uniquely rationalizable. As an immediate application of this result, we characterize, for any refinement of rationalizability, the predictions that are robust to small misspecifications of interim beliefs. These are only those predictions that are true for all rationalizable strategies, that is, the predictions that could have been made without the refinement.     

Inductive Inference: An Axiomatic Approach

A predictor is asked to rank eventualities according to their plausibility, based on past cases. We assume that she can form a ranking given any memory that consists of finitely many past cases. Mild consistency requirements on these rankings imply that they have a numerical representation via a matrix assigning numbers to eventuality–case pairs, as follows. Given a memory, each eventuality is ranked according to the sum of the numbers in its row, over cases in memory. The number attached to an eventuality–case pair can be interpreted as the degree of support that the past case lends to the plausibility of the eventuality. Special instances of this result may be viewed as axiomatizing kernel methods for estimation of densities and for classification problems. Interpreting the same result for rankings of theories or hypotheses, rather than of specific eventualities, it is shown that one may ascribe to the predictor subjective conditional probabilities of cases given theories, such that her rankings of theories agree with rankings by the likelihood functions.     

Finite Order Implications of Common Priors

I characterize the implications of the common prior assumption for finite orders of beliefs about beliefs at a state and show that in finite models, the only such implications are those stemming from the weaker assumption of a common support. More precisely, given any finite N and any finite partitions model where priors have the same support, there is another finite partitions model with common priors that has the same nth order beliefs and knowledge for all n≤N.     

A Mechanism‐Design Approach to Speculative Trade

When two parties have different prior beliefs about some future event, they can realize gains through speculative trade. Can these gains be realized when the parties' prior beliefs are not common knowledge? We examine a simple example in which two parties having heterogeneous prior beliefs, independently drawn from some distribution, bet on what future action one of them will choose. We define a notion of “constrained interim‐efficient” best and ask whether they can be implemented in Bayesian equilibrium by some mechanism. Our main result establishes that as the costs of unilaterally manipulating the bet's outcome become more symmetric across states, implementation becomes easier. In particular, when these costs are equal in both states, implementation is possible for any distribution.     

投资者异质信念、预期演化与股票市场流动性

投资者异质信念不但是股票市场的现实，也会引起股票市场流动性的变化。本文建立从两期到多期的交易模型，利用马尔科夫决策过程模拟投资者的预期演化，旨在分析由于信息不对称和认知偏差导致的多种异质信念共存的情况下，投资者异质信念与股票市场流动性的内在关系。研究发现，投资者异质信念的波动程度、投资者风险厌恶程度和信息不对称程度均与股票市场流动性呈现负相关关系；而投资者规模与股票市场流动性正相关。利用MATLAB软件数值模拟多期交易过程，验证了在投资者预期演化的情况下，上述规律仍然成立。研究结论在一定程度上拓展了行为金融和市场微观结构的研究范畴，也对投资者优化决策行为及监管部门加强流动性管理有一定的参考价值。     

Measuring unfairness feeling in allocation problems

In this paper, we introduce a new measure of social fairness based on unfairness feelings of the players involved in an allocation problem, e.g., cake-cutting problem or shift scheduling. We only require that each player be described by a von Neumann–Morgenstern utility function. Next, we propose a social normalization of each player׳s utility function, based on how each player sees the other players׳ shares through her own utility function. Further, we extend this normalization idea to a setting where the players are represented by a weighted oriented graph, where the weights assess the relatedness of (or similarities between) the agents. Among other results, we establish some links between our measures of fairness and those classically used in the cake-cutting-problem literature.