Channeled Attention and Stable Errors

Transcription

1 Channeled Attention and Stable Errors Tristan Gagnon-Bartsch Harvard Matthew Rabin Harvard Joshua Schwartzstein Harvard Business School Preliminary Extended Abstract Abstract A common critique of models with mistaken beliefs is that people should recognize their error after observations they thought were unlikely. This paper develops a framework for assessing when errors are likely to be discovered, in the sense that the error-maker will deem her mistaken theory implausible relative to a compelling alternative theory. The central premise of our approach is that people exhibit subjectively rational inattention, meaning a person may ignore or discard information her mistaken theory leads her to consider unimportant. We propose solution concepts embedding such channeled attention that predict when a mistaken theory will persist when a person attends to data if and only if it is perceived as valuable within her theory. We use our framework to study the attentional stability of common errors and psychological biases. While many costly errors are prone to persist, in some situations a person will recognize her mistakes via incidental learning : when the data she values given her mistaken theory happens to also tell her how unlikely her theory is. We investigate which combinations of errors, situations, and preferences tend to induce such incidental learning vs. factors that render erroneous beliefs stable. Applying the framework, we show, for example, how a person may never realize her self-control problems even when they lead to damaging behavior. And a person may never realize that he neglects correlation in others advice even when that neglect leads him to follow repetitive advice too much. Moreover, better feedback on the quality of others advice may make him less likely to realize his error. s: gagnonbartsch@fas.harvard.edu, matthewrabin@fas.harvard.edu, and jschwartzstein@hbs.edu. For helpful comments, we are grateful to Nava Ashraf, Max Bazerman, Erik the gorilla Eyster, Nicola Gennaioli, Botond Koszegi, George Loewenstein, Andrei Shleifer, and seminar participants at BEAM, CEU, Cornell, Dartmouth, Harvard, LSE, McGill, Princeton, and UC Berkeley.

2 1 Introduction Models formalizing specific ways that people misunderstand the world have become increasingly common in economics. These models span a wide range of errors from simple misconceptions about how the world works, to bad statistical reasoning, to faulty social inference, to distorted beliefs about one s own personal traits. A common critique of such models is that people should realize their errors after observing events they think are extremely unlikely or impossible. Someone who suffers from the gambler s fallacy, for instance, will see more streaks than she expects. A person who does not think through that others are following the crowd will see an unexpected degree of consensus. And a person who is naive about her procrastination problems will repeatedly pursue immediate gratification more than she predicts. It seems error-makers should often notice that something is amiss. Won t people get a clue? This paper clarifies and fleshes out the logic behind getting a clue. We develop solution concepts that provide guidance for when people notice their errors. These concepts rest on two central criteria. First, we clarify that unlikeness of what is observed shouldn t and wouldn t induce a person to abandon a theory, since any particular data is typically unlikely even for correct models. Rather, a person gets a clue only when noticed data is far more likely under a compelling alternative that a person considers in a moment of doubt. We do most of our analysis assuming that this light-bulb alternative the person might consider is the correct model of the world. Our primary emphasis is on a second criterion. Building on ideas introduced in Schwartzstein (2014), we examine the learning process of a person who may ignore or discard information that seems to her irrelevant given her beliefs. 1 We define solution concepts embedding such channeled attention that predict when a person s mistaken theory might survive after infinite data, where data is encoded and analyzed if and (in most of our analysis) only if it is perceived to be valuable within that theory. We then apply these principles to study the attentional stability of common errors and psychological biases. We show a large number of cases where a person is unlikely to learn of her errors. In other scenarios we predict a person will discover her mistakes via incidental learning: when data a person notices and remembers because her bad theory deems it useful also tells her as a by-product how unlikely that theory is. We investigate which combinations of errors and situations tend to provoke incidental learning, providing comparative statics on both preferences 1 The psychology literature confirms basic and intuitive features of attention important for our purposes: (1) attention is limited and often directed at what people think is relevant; (2) things not attended to initially are often not remembered later. One classic and salient demonstration of how directed our attention can be is Simons and Chabris (1999) finding if a person is told to count the number of completed passes during a short video clip of basketball, they commonly fail to notice someone dressed in a gorilla suit walking across the court. 1

3 and information that make erroneous beliefs more stable or less stable. 2 Section 2 provides an extended example on incorrect medical beliefs that illustrating some basic components and principles of our framework. Section 3 then formalizes the baseline framework assuming full attention. Although the setup accommodates a broad class of errors, we primarily focus on simple quasi-bayesian models: agents are Bayesian updaters starting from a misspecified prior, π, over a set of parameters that either assigns positive probability to impossible parameters or zero probability to possible parameters. 3 We impose no further assumptions about which errors people are prone to make ex ante. Instead, our goal is to provide a general way of assessing when people are prone to discover such errors it should be applied once a misspecified model is input as a primitive. We formulate a criterion for when a person might find her misspecified prior inexplicable, meaning that she will eventually deem her model of the world an extremely unlikely explanation for what she observes. One possibility is that the person sees something that is impossible given her theory i.e., the theory is not measurable. 4 But we also label a theory inexplicable when observations are possible but extremely unlikely. Our framework makes clear that unlikely is necessarily defined as relative to a specified alternative model rather than in absolute terms. We therefore specify a light-bulb alternative, λ a model that economic actors might entertain when they doubt their conception of the world. We say that within a particular setting a misspecified model π is inexplicable with respect to a light-bulb alternative λ if (given the truth) what the person observes 2 Schwartzstein (2014) shows how a person can maintain a bad theory when he channels his attention based on a misunderstanding of which variables are important. His stylized setting does not permit the exploration of various themes emphasized below, and it does not allow for incidental learning, which looms as a central theme of this paper. 3 Many recent models, spanning a wide range of errors, are either quasi-bayesian or very close. Examples include Barberis, Shleifer, Vishny (1998) on stock-market misperceptions, Rabin (2002) and Rabin and Vayanos (2010) on the gambler s and hot-hand fallacies, Benjamin, Rabin, and Raymond (2016) on the non-belief in the law of large numbers, and Spiegler (2016) on biases in causal reasoning. Examples of biases about misreading information include Rabin and Schrag (1999) and Fryer, Harms, and Jackson (2015) on confirmation bias and Mullainathan (2002) on naivete about limited memory. Models of coarse or categorical thinking include Mullainathan (2000), Fryer and Jackson (2008), Jehiel (2005), Jehiel and Koessler (2008), Mullainathan, Schwartzstein, and Shleifer (2008), and Eyster and Piccione (2013). Relatedly, models that incorporate errors in reasoning about the informational content of others behavior include Eyster and Rabin (2005), Esponda (2008), and Madarász (2012). Such errors have been explored in sociallearning settings by DeMarzo, Vayanos, Zweibel (2003), Eyster and Rabin (2010), Eyster and Rabin (2014), Bohren (2016), and Gagnon-Bartsch and Rabin (2016). Models that assume false beliefs about others strategic reasoning or information include Camerer, Ho, and Chong (2004) and Crawford and Iriberri (2007). Misspecified models have also been considered in specific applications, such as firms learning about demand (Kirman 1975; Nyarko 1991) as well as macroeconomic forecasting (Sargent 1993; Evans and Honkapohja 2001). Farther from the quasi-bayesian approach, other models posit inconsistencies between a person s current and future beliefs. The model of naive selfcontrol we examine below posits that people believe they will have more self-control in the future than they truly will (e.g., O Donoghue and Rabin 1999). Models of projection bias, such as Loewenstein, O Donoghue and Rabin (2003), likewise posit that somebody may have systematically different beliefs about future tastes as a function of fluctuating contemporaneous tastes. 4 In fact, researchers formulating quasi-bayesian models typically (and openly) formulate them so that such observations cannot happen. This criterion is central to a methodological goal of such theories to pin down agents beliefs entirely through Bayesian updating. 2

4 is infinitely more likely under the light-bulb than her prior. Otherwise, π is explicable with respect to λ. Although we discuss examples where the light-bulb model is also misspecified, we primarily focus on the case where λ corresponds to the true model of the world. Section 3 concludes by establishing some baseline results assuming the decision maker attends to all available data. In Section 4, we turn to the primary focus of the paper: the role of channeled attention. Determining when a person will recognize her mistake relies on assessing which evidence she thinks she needs to pay attention to. We assume that the agent will rationally from the perspective of her wrong theory ignore data she deems irrelevant to her payoffs. This can be as straightforward as a person ignoring some features she thinks doesn t matter. Hanna, Mullainathan, and Schwartzstein (2014) provide evidence of noticing failures among Indonesian seaweed farmers. In truth it seems important that farmers use a particular size of pod (the strands of seaweed they attach to lines in the ocean to seed production), yet with the large variation in pods each farmer uses they are frequently not optimizing. Surveys indicate that the the vast majority appear to not know the size(s) they use. 5 To study the implications of channeled attention we define the notion of a sufficient attentional strategy (SAS), where a person attends to and optimally responds to information she thinks matters, but may neglect the rest. 6 This concept does not rule out agents attending to information they deem useless, but we often focus on minimal SAS s where the person notices no more than she finds useful. This framework can be thought of as one of subjectively rational inattention. Relative to other modern theories of rational inattention (e.g., Sims 2003, Woodford 2012, Gabaix 2014), we deemphasize attentional constraints and instead explore how mistaken theories distort the perceived benefits of attention. 7,8 We introduce two notions of SAS s that differ in their assumptions 5 There is an irony in farmers behavior that sharply illustrates the role of channeled attention. Their unawareness that pod size mattered seems to have induced haphazard choices that created the variation in the sizes they used. Their inattention was increasing the richness of the data from which they could learn were they paying attention. 6 Our notion of attending to information is unstructured in a way: rather than defining a list of variables a person might pay attention to, sufficient attention requires that any refining of all information she receives seems to the person superfluous. 7 Our concepts are a heuristic for (unmodeled) limit assumptions about attentional costs. Essentially, our model corresponds to the assumption that (a) these costs are very low but not zero and (b) people are patient and, as a result, they pay attention if and only if they perceive any benefit to doing so. Hence, our criterion is conservative: if a misspecified model is stable in our framework, then it would still be stable after incorporating realistic attentional costs. Furthermore, this heuristic allows us to highlight how the misguided allocation of attention itself (rather than non-trivial attentional costs per se) can play an important role in the persistence of mistaken beliefs. 8 Less closely related are models of paradigm shifts and testability (e.g., Hong, Stein, and Yu 2007; Ortoleva 2012; Al-Najjar and Shmaya 2014). While studying paradigm shifts has the flavor of analyzing when people have light-bulb moments, those papers do not study the interaction between light-bulb moments and inattention. The logic behind why channeled attention prevents light-bulb moments is similar to self-confirming equilibrium (Fudenberg and Levine 1993) and why people can maintain incorrect beliefs about options they rarely experiment with in bandit problems (Gittins 1979). However, beliefs are consistent with available data in those frameworks the friction is data collection while beliefs are only consistent with encoded data in ours the friction is data processing. Finally, we contribute to the growing literature on learning with misspecified models (e.g., Barberis, Shleifer, and Vishny 1998; Rabin 2002; Rabin and Vayanos 2010; Benjamin, Rabin, and Rabin 2015; Spiegler 2016; Esponda and Pouzo 2016; Bohren 2016; Heidhues, Kőszegi, and Strack 2016; Fudenberg, Romanyuk, and Strack 2016) by providing a 3

5 pertaining to memory: volitional recall corresponds to cases where a person need only remember the data her theory deems useful, while automatic recall insists that a person remember anything she noticed in the past. It is direct that when the person follows a minimal SAS (with or without automatic recall), her model will be attentionally measurable: she never notices events her theory says will not happen. While channeled attention renders all theories measurable, it does not always render them explicable. We say (roughly) that π is attentionally inexplicable relative to a λ and a given SAS if the noticed data is infinitely more likely under λ than under π. Otherwise, the theory is attentionally explicable relative to λ and the SAS. When a given misspecified model is attentionally explicable relative to the truth, we call the SAS used an attentionally stable equilibrium given π. Often there exist attentionally stable equilibria for models that are inexplicable with full attention. But in other instances, an inattentive person may still discover the implausibility of her theory as a by-product of attending to the information she deems valuable. This can even happen when a person s mistaken beliefs lead her to pay attention to things a rationally inattentive person would not. For instance, a person who pays close attention to patterns because she believes in the gambler s fallacy may incidentally collect the data necessary to discover other mistaken aspects of her model. Before drawing out some general results, we first examine applications of the framework in Section 5. We first explore how channeled attention can lead a person to persistently underestimate her self-control problems. While some researchers (e.g., Ali 2011) argue that rational learning should correct such overestimation of self control, we demonstrate that the path to sophistication faces many obstacles. For a person who each day decides whether to take an action (e.g., exercise) that has an immediate cost and delayed reward, overestimation of self control is part of an attentionally stable equilibrium. Intuitively, the person sees no reason to carefully track her behavior and thus never confronts convincing evidence that she takes the action less often than expected. If instead the person considers making an up-front payment to reduce these costs (e.g., buying a gym membership), then the degree to which she tracks her behavior depends on how uncertain she is about the value of up-front payment. If she is sure it is worthwhile, then overestimation of self control is still part of an attentionally stable equilibrium. She may, for example, persistently pay not to go to the gym (DellaVigna and Malmendier 2006). Alternatively, if she faces uncertainty in the benefits of up-front payments because she is uncertain about her behavior, then she may correctly learn her behavior without learning her lack of self control: she may misattribute her failure to take the action to unexpectedly high costs. In such equilibria, she may persistently fail to make worthwhile up-front investments because she neglects their commitment value. We then analyze the implications of channeled attention for a person who neglects the correlated framework to assess when misspecified models are stable. 4

6 nature of others advice (as in DeMarzo, Vayanos, and Zwiebel 2003; Eyster and Rabin 2010, 2014; Enke and Zimmermann 2017). Will she notice this correlation over time? Imagine, for instance, a person in a new job who seeks advice from her colleagues and needs to learn the quality of their advice. If her bad theory is that others advice is conditionally independent (not thinking through that colleagues also talk with each other), she thinks she can learn about the quality of a colleague s advice simply by comparing his recommendation with the eventual outcome. She thus won t attend to the correlation across colleagues, will not discover it is a mistake to neglect this correlation, and, as a result, will persistently overreact to consensus advice. But what if she sometimes doesn t observe the outcome associated with her colleagues recommendations? She would then rationally update her beliefs about a colleague by comparing his or her advice to that of other colleagues she has come to believe are smart not because she thinks the advice is correlated conditional on the truth, but as a proxy for guessing how correlated each colleague s advice is with the truth. In such a case with limited information, she is forced to notice that the high correlation in others advice is inconsistent with her original model: the person incidentally learns that it is a mistake to treat their advice as (conditionally) independent. In this setting, then, more information can make the error more stable. In Section 6 we turn to establishing some more general implications, with both simple formal results and broad principles. The simplest principle bears repetition: If a person thinks she has nothing to learn, she thinks she does not need to pay attention. Indeed, in an i.i.d. world where a person is certain about everything that might affect her preferred behavior, there is always a minimal SAS under which a person simply does not learn if her beliefs are wrong. So, for instance, if people are armed with the theory that riskier, managed funds must outperform index funds, and she does not mind the risk, she may simply invest in these funds and never notice that this is not true. A second principle is also apparent here: such errors may occur no matter the scale. We also show in Section 6 some principles about how preferences affect whether a person figures out her error. In the case of managed funds, for instance, a person who was more risk averse and therefore keen to find out if managed funds are worth the risk may pay enough attention to learn that they are dominated by index funds. While this illustrates how preferences can matter for noticing errors, we also show that, fixing the information structure, certain types of mistakes are attentionally stable no matter the person s preferences. The simplest form of a misspecified model that is preference-independent attentionally explicable (under both automatic and volitional recall) is one where a person simply does not think some distinctions or variables are relevant to outcomes at all. 5

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