Attitudes towards information and beliefs: how do biased perceptions influence individuals’ behavior? Francesca Bartoli∗ Università di Siena
Abstract Among others, Epstein [2005] suggests that psychological insights can be reconciled within a decision-theoretic framework not only by suitably modifying the utility function, but also by considering evolutions of beliefs that depart from the rational Bayesian information processing because of the subjective nature of the updating. We characterize non-Bayesian updating in psychological terms, focusing on two widespread phenomena that influence human judgments: the use of the representativeness heuristic and the conservatism bias. We show how overweighting (in the case of representativeness) or underweighting (in the case of conservatism) the relevance of information respectively leads to upward-biased and downward-biased posteriors, where the natural benchmark is the posterior computed by an agent who correctly perceives the signal and performs Bayesian updating. Given this framework, we then study how biased perceptions of information affect optimal decision making. The main results are that information processing biases reflect on the amount of evidence each type of agents needs before taking an action, but not on the confidence agents have on a certain state while taking the corresponding action.
∗ Correspondence address: Francesca Bartoli, Dipartimento di Economia Politica, Università di Siena, P.zza S. Francesco 7, 53100, Siena, email:
[email protected].
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1
Introduction
The concept of rational beliefs is based on the internal consistency constraint posed by Bayes’s rule, which formalizes the way in which new information should be incorporated into prior beliefs. In particular, the Bayesian paradigm states that an agent is said to be a rational decision maker under uncertainty, and may be viewed as having rational beliefs if, in the presence of new information, she updates her prior according to Bayes rule, obtaining a posterior belief that incorporates all the information observed. As Rabin and Schrag [1999] point out, modern information economics grounds on the assumption that individuals act as Bayesian statisticians, but this assumption suffers from several cognitive limitations. A growing body of research in psychology documents that human information processing systematically differs from what prescribed by Bayesian updating. For instance, it has been observed that people tend to misinterpret the evidence they face, and the flip side of this fact is that learning tends to exacerbate errors rather than correcting them. While psychologists have long recognized that individuals’ beliefs are biased by cognitive mistakes, economists have only recently started to incorporate such elements into the standard model of decision making under uncertainty. Among others, Epstein [2005] suggests that psychological insights can be reconciled within a decision-theoretic framework not only by suitably modifying the utility function, but also by considering evolutions of beliefs that departs from the rational Bayesian information processing because of the subjective nature of the updating. We find this intuition rather interesting, and indeed we interpret the author’s opinion while stating that the beliefs side could and should play a key role in behavioral economics modelling. Epstein explains updating distortions in terms of a conflict between commitment and urges in beliefs that arises after the realization of a signal. Loosely speaking, the negative scenario suggested by a bad signal may lead to a retroactive change of prior that takes place because of the presence of a tempting alternative. Although the author associates this panic to a tendency to interpret the bad signal as an even worse omen for the future than it could have been thought ex ante, no characterization of the psychological aspects that drive such changes is provided. In our opinion, the way in which a different prior looks like to emerge at the interim stage can be thought as a consequence of cognitive mistakes that lead to a (perhaps unconscious) choice about the reliability of the signal.
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That is, there are circumstances in which agents subjectively select the weight that should be put on data observed, and it is just this selection that leads to posteriors that contrast with the Bayesian ones, as if a new prior were chosen. Formally, the agent’s perception of the quality of information can be modelled through a probability measure θ, assuming that it is the subjective re-evaluation of θ that works as a choice over signals. We recognize that our characterization may seem trivial, but we think it allow us to picture information processing behaviors in a rather intuitive and realistic fashion. In exploring the reasons why biased attitudes to information could arise we focus on two widespread phenomena that characterize human judgements: the use of heuristics -and in particular, the use of the representativeness heuristic- and the conservatism bias. This matching is somewhat a standard of the literature. In their well-known model of investor sentiment, Barberis, Shleifer and Vishny [1998] state that representativeness is suggestive of stocks’ prices overreaction, whereas conservatism explains the underreaction evidence. In the same spirit, Griffin and Tversky [1992] attempt to reconcile representativeness and conservatism within a unique theoretical framework arguing that people update their beliefs according to the weight and the strength of new evidence, where strength refers to the salience and extremity of a news, and weight refers to statistical informativeness, such as sample size. Then, representativeness can be read as an excessive response to the strength of a particularly salient signal, in spite of a relatively low weight, whereas conservatism implies that people mildly react to a rather unimpressive signal, even though its weight calls for a larger response. Generally speaking, heuristics are simple, efficient rules of thumb that help people to deal with complex decision tasks or incomplete information by reducing the assessment of probabilities and the prediction of values to simpler judgmental operations. They are undoubtedly useful -for instance, the use of heuristics diminishes the time required to solve a problem by eliminating the need to consider unlikely possibilities or irrelevant statesbut at the same time quite dangerous in that not considering all the possible alternatives may lead to severe and systematic mistakes. In particular, a person who follows the representativeness heuristic tends to view events as typical or representative of some specific class and to ignore the law of probability in the process. Then, she evaluates an uncertain event by the degree to which (i) it is similar in its essential properties to its parent population, and (ii) it reflects the salient features of the process by which she 3
thinks it is generated. The first and still most relevant work that documents how people tend to evaluate situations using the representativeness heuristic is Tversky and Kahneman [1974]. The authors document that experimental subjects try to find a solution for the problem they are facing by seeking the closest match to past patterns, without paying attention to the primitive probability of matching the pattern. For example, if a detailed description of an individual’s personality matches up well with the subject’s experiences with people of a particular profession, the subject tends to significantly overestimate the actual probability that the given individual belongs to that profession. The key point is that in overweighting the representative description (i.e., the signal), she underweights the statistical base rate evidence of the small fraction of the population belonging to that profession. This tendency entails severe and systematic biases in the perception of the external environment, such as the fact that few impressive elements are enough to judge situations. Hence, we formalize the associated bias in perception in terms of a subjective re-evaluation of the reliability of information given by a probability measure θr stronger than θ. As noted before, representativeness has frequently been coupled with the conservatism bias in the behavioral economics literature, since their effects display in a rather symmetric fashion. Indeed, the conservatism bias -firstly studied by Edwards [1968]- entails a slow changing of beliefs in the face of new evidence. Edwards’ experiments show that people update their posteriors in the right direction, but by too little in magnitude relative to a rational Bayesian benchmark. We characterize an agent who suffers from the conservatism bias as interpreting new evidence in a way that conforms with her prior view of the world, but being rather cautious in her updating. Whatever is the signal that realizes, the agent prefers to mildly react to information, switching to a probability quality measure θw weaker than θ. We show how overweighting (in the case of representativeness) or underweighting (in the case of conservatism) the relevance of the signal leads respectively to upward-biased and downward-biased posteriors, where the natural benchmark is the posterior computed by a Bayesian agent who takes information at its face value. Given this framework, we then go a step further, questioning whether and how biased perceptions of information affect optimal decision making. We consider individuals with imperfect information about the true state who choose between two alternatives with 4
different risks (and different payoffs). Following Brocas and Carillo [2003], we endogenize the decision to collect more or less signals assuming that learning is feasible but costly. The novelty is that we first derive then compare the optimal stopping rules relative to three different types of agent: a Bayesian agent, an agent who uses the representativeness heuristic and an agent who displays conservative attitudes. Our first result is that qualitative differences in the way information is perceived lead to quantitative differences in the amount of evidence required before acting (proposition 3). Despite one could think that the presence or not of biases influence also the confidence agents have while undertaking the optimal action, this intuition is incorrect. Our second and more important result is that in equilibrium posterior beliefs do not depend on the way information is perceived (proposition 6). Perhaps surprising, this finding can be explained observing that the informational content of each signal affects the speed at which “bliss-point posteriors” are reached but not the relative probabilities of attaining them.1
1.1
Related literature
The treatment of information is a rather important modelling aspect, and information processing biases have been extensively analysed in the behavioral economics literature. Works on this field try to provide microeconomic foundations for several systematic biases observed in real-life situations, basing their arguments on hyperbolic discounting (Carillo and Mariotti, 2000, Bénabou and Tirole, 2002), regret and rejoicing (Loomes and Sudgen, 1982), anticipatory feelings (Caplin and Leahy, 2001, 2004) ego utility (K˝oszegy, 2000a, 2000b), and beliefs utility (Yariv, 2001, 2002). Specifically, Bénabou and Tirole [2002] and K˝oszegi [2000] assume an agent trying to assess her own ability either by actively choosing to selectively forget some negative past experiences (Bénabou and Tirole), or by stopping recording any further signals that may decrease judgments about the self (K˝oszegi). In both cases, information is processed in a purely Bayesian fashion: the agent is perfectly aware of the way she collects signals and she properly takes it into account when forming beliefs. As a result, both these forms of “confidence management” affect only the distribution over posterior beliefs, while beliefs are not biased on average. Unlike 1 With a slightly abuse of terminology, we refer to “bliss-point posteriors” as the posteriors the agents hold when they become reasonably confident about one hypothesis or the other. Throughout the paper we will sometimes replace it with the term “instrumental posteriors” to stress the fact that, having reached such posteriors, agents stop collecting information and take the optimal action.
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this setting, where beliefs are a choice variable in the sense that the agent chooses when to stop receiving information, Yariv [2002] endogenizes the choice of beliefs, as well as the probability with which the agent misinterprets available information, making predictions concerning the quality of signals she will choose to obtain. The link between all these papers is that they conceive distortionary beliefs as being the outcome of a deliberate choice. The (more or less) implicit assumption is that individuals are able to rationally manage their own bias, but it can reasonably argued that psychological biases affect individuals’ perception in a way that is beyond their control. In this spirit, Rabin and Schrag [1999] present a model of confirmatory bias in which, with some exogenously given probability, individuals misread signals according to their first impressions, without realizing that these first impressions biased what they believed were “observations”. In this case not only the overconfidence that results in the long run is unintentional, but also it has no instrumental value. An interesting work that places between these two extreme is Compte and Postlewaite [2003]. The authors argue that confidence enhances performances, so that there are situations in which rational individuals prefer to have biased perceptions. Grounding on some related works in the psychology literature, they assume that agents’ probability of succeeding in an activity increases with the perceived probability of success. The interaction between the agent’s history of successes and failures and the probability of success in future attempts may induce biases in information processing, since the agent may selectively remember more positive past experiences that negative ones, which in turns leads to overconfidence. Note that in this model biased perceptions are not the outcome of an intentional choice but simply the product of recalled past experiences, which are themselves based on the types of self-attributions the agent makes. Even if there are circumstances in which having biased perceptions increases welfare, the authors themselves recognise that it may be hard for a rational agent to actively fool herself. In order to spoil the influences confidence has on performances, there may be a value to design activities in a way that facilitates biases in information processing, since to be passively fooled is rather more easy. The present work studies how information processing biases affect optimal decision making. As in Rabin and Schrag [1999], agent’s beliefs are a choice variable in the sense that the agent chooses when to stop collecting information, and the optimal stopping 6
rules depends on the way information is perceived. However, confidence distortions do not result in the short run, having essentially only temporary effects. Indeed, we will point out that information processing biases reflect on the amount of evidence the agent needs before taking an action, but not on the confidence the agent has while taking the action. One of the interesting implications is that in principal-agent contexts the principal could influence agent’s decisions simply managing the release of her own private information. In this sense, our model is close in spirit to that of Compte and Postlewaite [2003]: agents cannot rationally choose to bias their perceptions, but they cannot prevent others to profit from their subconscious processing biases. The paper is organised as follows. In section 2 we show how the presence of biased perceptions of information can be formalized in terms of non-Bayesian updating. We focus on conservatism and representativeness, and we rigorously characterize the posteriors associated with these two cognitive biases. In section 3 we present a simple model with three types of agents with imperfect information about the true state of the world who choose between two alternatives. The distinctive element is that two of the three types of agents suffer from cognitive biases that influence their perception of information, the third type of agent being a standard Bayesian. We describe the information acquisition process and we point out how different attitudes towards information influence the optimal stopping rule. We then characterize the beliefs associated with the cutoffs previously derived, and present the main result of the paper. Last, we offer some concluding remarks in section 4.
2
Information processing biases
To present our reasoning, we refer to financial investments. Stocks trading can be either a good or a bad strategy, depending on the underlying fundamental of the asset, as captured by a parameter ω ∈ Ω = {G, B} , where G and B respectively means good or bad. Assume agents have a prior probability about stocks being a profitable investment equal to: Pr(ω = G) = p >
1 2
Actions depend on the state of the world agents would perceive according to the
7
© ª realization of a signal sω ∈ S = sG , sB , ω = {G, B} , which is imperfectly correlated with the true state. Think of S as a space of perceived states of the world, while Ω is the
space of the true states. We argue that the different factors that drive performances on financial markets obscure the link between data observed and the unknown parameter, and we model the relation between them as follows: Pr[sG | G] = Pr[sB | B] = θ
Pr[sG | B] = Pr[sB | G] = (1 − θ)
and
where θ ∈ [ 12 , 1] is the probability of receiving a signal that corresponds to the true state of the world. The economic interpretation can be stated as follows. In a purely risky framework, the connection between states of the world and evidence collected is taken as objectively given. Whether this is undoubtedly true in controlled laboratory experiments, it seems less plausible for phenomena observed in real economic settings such as asset markets, where performances are influenced by a large number of factors that are difficult to disentangle and characterize in probability terms. Consider for instance an agent who tries to determine the profitability of an investment by collecting data such as stocks returns or market quotations. Because of the large number of exogenous factors that influence either returns or prices, she recognizes that the link between fundamentals and data observed is in a sense garbled. More formally, given a parameter space representing a feature of the environment the agent tries to learn (ω, in the present setting), a sequence of noisy signals is fully informative, since the distribution conditional on the parameter is known. On the other hand, when the role of the relevant parameter in determining data is uncertain (as it happens in financial markets), the meaning of the signals is obscure and the conditional distribution given the parameter is not unique. This can be captured by considering subjective evaluations of the reliability of the signal. Then, think of θ as representing the reliability of current data in reflecting the fundamental value of the stock, according to a subjective judgment made ex-ante. The re-evaluation of θ triggered by the observation of information will play a crucial role in characterizing non-Bayesian updating in a sense that will be specified later on. The timing of decisions works as follows. At time t = 0 the agent holds a subjective beliefs about ω summarized by p, that together with θ will give the probability distribu-
8
tion π defined over S × Ω. Think of π as the probability distribution that combines the agent’s prior view about the true state ω with beliefs about how ω is reflected in signals (sample evidence). At time t = 1 the agent observes the realization of the signal sω and updating is performed. Table 1 summarizes the time 0 priors about all the possible time 1 situations: TABLE 1
sG
sB
ω=G
pθ
p(1 − θ)
ω=B
(1 − p)(1 − θ)
(1 − p)θ
Table 2 shows how a signal of precision θ and a prior p over ω generate a probability distribution π(G | sω ) ≡ Pr(G | sω ) over the true state of the world being G. TABLE 2
sω
Pr(sω )
sG
pθ + (1 − p)(1 − θ)
sB
Pr(G | sω )
p(1 − θ) + (1 − p)θ
pθ pθ+(1−p)(1−θ) p(1−θ) p(1−θ)+(1−p)θ
Having observed the signal sω , agents update their beliefs about ω, then they choose an act from a signal-contingent menu of alternatives. Epstein [2005] argue that the perspective suggested by sω may induce agents to retroactively adopt a new prior q. While commitment asks for selecting an act that solves the maximization problem under (the Bayesian updating of) p, succumbing to temptation means that expected utility is computed by applying Bayes rule to q. In balancing these forces, agents behave as if they apply Bayes’ rule to a compromise measure p∗ that suitably mixes p and q. The novelty of the present study lies in the explicit characterization we provide for non-Bayesian updating through the probability measure θ. We deepen Epstein analysis by focusing on the way in which psychological biases influence the updating process, and in particular we argue that such biases affect agents’ perception of information, so that the posteriors that result differ from the ones we reasonably would have expected from the application of Bayes’ rule. The key element is the time 1 re-evaluation of θ triggered by the realization 9
of the signal, and it is exactly through the subjective perception of θ that we provide a possible justification for the apparent changing-priors attitude modelled by Epstein, which leads in turns to non-Bayesian updating. In order to better clarify the analysis, it is useful to show first what we mean by nonBayesian updating. Consider the posterior probabilities π(G | sω ) summarized in table 2,
and let µ(G | sω ) be the posterior agents would form if the observation of sω leads to a subjective re-evaluation of θ as ˆθ. Formally, this particular form of non-Bayesian updating
entails that posteriors are:
ω
µ(G | s ) =
½
pˆ θ ˆ pˆ θ+(1−p)(1−θ) ˆ p(1−θ) ˆ ˆ p(1−θ)+(1−p) θ
sω = sG (1) sω = sB
Note that µ(G | sω ) coincides with the correct Bayesian posterior π(G | sω ) induced by sω only if agents do not modify the first stage belief about the reliability of the signal. That is, if ˆθ = θ then µ(G | sω ) ≡ π(G | sω ) and the model reduces to the standard one. The subjective perception of θ is formalized in the simplest way. We simply distinguish between weak or strong re-evaluation of the reliability of the signal: ˆθ =
¾ ½ 1 θw , θr : ≤ θw ≤ θ ≤ θr ≤ 1 2
Then, we consider three different types of agent: a Bayesian agent who takes information at the face value, placing the “correct” weight θ to the signal; an agent who uses the representativeness heuristic and an agent who displays conservative attitudes. Representativeness characterizes an agent who puts too much emphasis on the most recent data, thinking that even short sequences suggest the underlying distribution. We associate the use of such an heuristic with the overweighting of θ as θr , whereas reconsidering θ as θw captures the updating attitudes displayed under conservatism. Obviously, as θw and θr respectively decreases or increases, the informational content accorded to signals varies as well. In particular, when θw = 1/2 the agent puts zero weight to data, and signals are considered as completely irrelevant. In fact, µw (G | sω ) ≡ p and updating takes no place at all. Conversely, θr = 1 means that the agent reads current information as fully representative of the underlying state of the world. In this case, µr (G | sG ) = 1 (just one
good signal is enough to reassure the agent about the good state), and µr (G | sB ) = 0 10
(just one bad signal is enough to let the agent completely change her mind about the underlying state of the world). Summing up, conditional to the realization of the signal sω two different attitudes towards information are possible. The use of the representativeness heuristic implies the overweighting of signals, whereas the conservatism bias leads to the mild evaluation of information (relative to the prior). Before switching to a t period setting, we present a formal characterization of these two updating biases.
2.1
Conservatism
We classify biased beliefs according to how they stand relative to the belief a Bayesian agent would adopt (i.e., relative to the “correct” Bayesian posterior). The agent is said to be suffering from conservatism if her beliefs are too moderate relative to the Bayesian posterior. Formally: Definition 1 (Conservatism) Conditional to all sω ∈ S, an agent holding a posterior
belief µw (G | sω ) is said to be suffering from conservatism if, relative to a Bayesian posterior π(G | sω ):
| µw (G | sω ) − p | 12 , new information can be so negatively impressive that the agent may end up to believe that the most likely true state is B. That is, sB by itself is so strong to induce a conditional belief π B < 12 , despite the prior p(G) is greater than 12 . Proposition 1 states that the conservative re-evaluation of the reliability of the signal implies that when sω = sB (i.e., when πB < p) posteriors are upwards biased so that µw B > π B . Now we point out that in the presence of bad news πB can be either ≷
1 2
depending on the
relative values taken by p and θ. Indeed: πB =
p(1 − θ) 1 > θ(1 − p) + (1 − θ)p 2
(6)
holds true if (1 − θ)p > θ(1 − p), i.e., if θ < p. Note that θ < p means either that the signal is actually not very salient, apart from any subjective reinterpretation made by the agent, or that the prior is particularly strong (p → 1). In both cases, even if beliefs are downwards adjusted, they still remain greater
1 w than 12 . Since µw B > π B , then also µB > 2 . This result is trivial and it does not enrich very
much the analysis. Rather, we think it is interesting to better characterize the situation in which θ > p. When information is actually salient, as well as when the agent is not very confident about the true state being G (i.e., when p → 12 ), a bad signal is enough to reverse beliefs, such that π B
π B , which holds true under the
assumptions that λ > 0 and θ > 12 . However, the more severe is the bias, the more the agent disregards the informative content of a bad signal, and it can be reasonably argued that there are situations in which conservatism may so deeply affect agents’ perception 13
that posterior beliefs will still favour the good state, despite a bad signal implying π B < 12 . Formally: Corollary 1 Assume sω = sB and θw < θ, so that µw B > π B . For every p > θ∈
( 12 , 1),
¯= such that θ > p, there exists a λ
¯ then π B < (i) for all λ > λ,
1 2
θ−p , θ− 12
¯> λ
1 2,
1 2
and
such that:
1 and µw B > 2;
¯ then µw = 1 . (ii) if λ = λ B 2
Proof. See the Appendix. The corollary shows that even in the presence of a strong signal for the bad state, the agent’s attitude towards information may be so conservative (λ >
1 2)
that she will still
perceive the good state as being the most probable one. Consider again the expression ¯ and note that (θ − 1 ) and (θ − p) proportionally increase as θ gets closer to 1. This of λ, 2 suggests that, for certain parameter values, conservatism implies stickiness in beliefs, regardless to the objective informational content of the signal. In section 3 we will show that such a stickiness in beliefs translates into procrastination of choices.
2.2
Representativeness
In the behavioral economics literature the term representativeness is used to characterize deviations from Bayesian updating in terms of the agent placing too much weight on observations relative to the weight placed on the prior. In general terms, this translates into the tendency to evaluate even a single piece of information as highly representative of the underlying state of the world. Given these premises, we model the behavior of an agent using the representativeness heuristic as being characterized by the ex-post re-evaluation of the reliability of information in terms of a quality measure θr greater than θ. Note that when representativeness is at issue, an external observer who assumes Bayesian behavior would interpret posteriors as indicating overreaction to signals. The following definition applies: Definition 2 (Representativeness) Conditional to all sω ∈ S, an agent holding a pos-
terior belief µr (G | sω ) is said to be suffering from representativeness if, relative to a
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Bayesian posterior π(G | sω ): | µr (G | sω ) − p |>| π(G | sω ) − p | In this case the bias entails that the difference between the posterior and the prior is bigger than the difference that would have been resulted under Bayesian updating. That is, the agent overvalues the reliability of new information, and she is rather prone to accordingly change her mind, whichever signal she observes.
r
ω
µ (G | s ) =
½
θr p θr p+(1−θr )(1−p) (1−θr )p θr (1−p)+(1−θr )p
sω = sG sω = sB
(7)
such that µr (G | sG ) ≥ π(G | sG ) and µr (G | sB ) ≤ π(G | sB ). That is, posteriors calculated using θr instead of θ display overreaction to new evidence. Following the notation introduced before, the severity of the representativeness bias is summarized by a parameter λ that takes value between 0 and 1, with the same meaning as in the case of conservatism. Then: θr = λ + (1 − λ)θ
(8)
where λ = 1 means now that the agent perceives the signal as fully representative of the underlying state ω. Equation (8) tells that if representativeness is extreme (λ = 1), then θr = 1. One signal perfectly informs the agent about the true state, so that µr (G | sG ) = 1 but
µr (G | sB ) = 0. On the other hand, Bayesian attitudes obtain when λ = 0, in which case θr = θ and µr (G | sω ) ≡ π(G | sω ). When λ lies between 0 and 1 then the biased posterior
µr (G | sω ) obtains as a convex combination between the Bayesian posterior π(G | sω ) and a prior view that features no uncertainty about the true state, i.e., p = 1. Formally: µr (G | sω ) = λ + (1 − λ)π(G | sω )
(9)
As in the conservatism case, π(G | sω ) attaches the “commitment” weight θ to the
signal and p = 1 is independent of sω . Thus, the measure µr (G | sω ) puts too much
emphasis to observation. Note that the way representativeness affects the perception of
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information entails that the resulting posterior can be thought as computed as if the agent is updating a new prior π r on S × Ω given by: π r = λ + (1 − λ)π
(10)
Proposition 2 formalizes the preceding.2 Proposition 2 (Representativeness) Conditional to all sω ∈ S, an agent who updates her beliefs according to the information quality measure θr displays representativeness attitudes with respect to an agent who performs Bayesian updating according to the measure θ.
Proof. See the Appendix. Note that the quality of information affects the updating of beliefs by itself. Given a certain degree to which the agent suffers from representativeness (i.e., the value of λ being equal), when information is rather impressive the difference between the biased posterior µrω and the Bayesian posterior πω is lower. Indeed, the difference between the biased and the Bayesian posterior is equal to λ(1 − p)(1 − θ) in case of a good signal, and equal to λ(p − θ − 1) in case of a bad signal, and it can be easily proven that both decrease in the value of θ. We think it is worthwhile to analyse what could happen when a bad signal realizes. Consider again λ(p − θ − 1). When θ > p the biased posterior remains closer to the correct Bayesian value, while if θ < p it is likely to become sensibly smaller. Assume that two alternatives a and b are at issue and let p =
1 2
define the pivot probability between
choosing a (if the state of the world is more likely to be G) or b (if the state of the world is more likely to be B). Assume further that p(G) > 12 . If the bad signal observed is particularly salient, beliefs are likely to be reversed, leading to a posterior π B
2 As
1 2
(see equation 6). Representativeness
in the case of conservatism, notation is simplified as follows: µrω = µr (G | sω ) and πω = π(G | sω ).
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implies downwards biased posteriors (µrB < π B ), and the more severe is the bias, the lower will be µrB . Since λ → 1 implies that µr (G | sB ) → 0, for certain parameter values representativeness may so severely influence the agent that the negative scenario suggested by a contrary signal suffices for a radical change of perspective. Formally: Corollary 2 Assume sω = sB and θr > θ so that µrB < π B . For every p > θ∈
( 12 , 1),
¯= such that θ < p, there exists a λ
¯ then π B > (i) for all λ > λ,
1 2
p−θ 1−θ ,
¯> λ
1 2,
1 2
and
such that:
and µrB < 12 ;
¯ then µw = 1 . (ii) if λ = λ B 2
Proof. See the Appendix. ¯ and note that (p − θ) and (1−θ) proportionally Consider the analytical expression of λ, 1 2.
increase as θ gets closer to
This suggests that representativeness may be so severe
(λ > 12 ) that the agent is prone to switch to the new perspective implied by a contrary signal, regardless to the objective informational content of the signal. We will better clarify this intuition the following section, while considering the influence information processing biases have on choices.
3
The model
In order to analyse how conservatism and representativeness bias the information acquisition process and the choice of action we place now into a T period setting. Assume agents can choose between stock trading (action a) and bonds (action b). Trading in the financial market is seen as a “risky” action that can generate either positive or negative payoffs depending on the (unknown) state of the world ω ∈ Ω = {G, B}. Bond investment does not entail any particular risk, and it generates non-stochastic earnings. Let us consider the following simplified utility representation:
u(a) =
½
h −h
if ω = G if ω = B
where 0 < l ¿ h.3
and
u(b) = l ∀ω ∈ Ω
3 Note that the excess returns of the risky action are centred around zero, so that price determination problems can be avoided.
17
Even if agents share a common prior beliefs about the true state being G equal to p > 1/2, they will choose between action a or b only after having observed a sufficient amount of confirmatory signals. Assume that at each date t one signal realizes, so that the passage of time can be measured through the number of signals agents collect before taking ω G a financial position. Given a sequence of sω and xB respectively 1 .....st signals, let x
denote the number of good and bad signals in the sequence, and let x = xG − xB ∈ Z denote the net number of good signals. The Bayesian posterior belief is given by: Pr (G
|
xG , xB ) =
Pr(xG , xB | G) Pr(G) = Pr(xG , xB | G) Pr(G) + Pr(xG , xB | B) Pr(B) G
G
= =
G
B
CxxG +xB θx (1 − θ)x p
B
CxxG +xB θx (1 − θ)xB p + CxxG +xB (1 − θ)xG θx (1 − p) G
(11)
B
=
θx p θ p + (1 − θ)x (1 − p) x
Note that the posterior depends only on x, the difference between the number of positive and negative signals, as if two different signals “cancel out”. Let Pr(G | xG , xB ) ≡ π(x), where π(x) is characterised by the following properties: (i) lim π(x) = 1 x→∞
(ii)
lim π(x) = 0
x→−∞
(iii) π(x + 1) > π(x). Following the notation introduced in the previous section, let µw (x) denote the conservatism posterior implied by θw , and µr (x) denote the representativeness posterior implied by θr . After t periods during which a net amount x of good signals is observed, the conditional probabilities about the true state being G are respectively given by:4 µrt (x) =
(θr
)x p
(θr )x p + (1 − θr )x (1 − p)
(12)
and µw t (x) = 4 Subscripts
(θw )x p (θw )x p + (1 − θw )x (1 − p)
here refer to time (and not to type of signal observed, as in the previous section).
18
(13)
Extending the results derived in proposition 1 and proposition 2, it is easy to check r that when x > 0 then µrt (x) > π t (x) > µw t (x); conversely, when x < 0 then µt (x)
12 ), the cost of waiting depends positively on x, and the expected payoff diminishes one period after the other by an amount proportional to [2π(x) − 1] . Since limx→−∞ π(x) = 0 and
limx→+∞ π(x) = 1, there should exist a number of good signals x defined by π(x) =
1 2
and
a cutoff x∗∗ > x such that when x = x∗∗ then π(x∗∗ ) > π(x) and the agent prefers to stop observing information and to take action a. Conversely, when π(x)
1 2
and to l(1 −δ) when π(x) < 12 ,
which means in turns that the optimal cutoffs x∗ and x∗∗ are both functions of h, l, δ.
Given the reduction in the expected payoff due to the delay, there exists a number of problems, and it does not affect the substance of the results.
20
either positive or negative signals that lead the agent to become reasonably confident about the true state, above which optimal actions are taken. New insights are provided by introducing into the standard model the assumption that individuals subjectively interpret the quality of information. Consider: Vtw (x) = max {l; h[2µw (x) − 1]; δ[ν w (x)Vt+1 (x + 1) + (1 − ν w (x))Vt+1 (x − 1)]} Vtr (x) = max {l; h[2µr (x) − 1]; δ[ν r (x)Vt+1 (x + 1) + (1 − ν r (x))Vt+1 (x − 1)]} where ν w (x) = µw (x)θw + (1 − µw (x))(1 − θw ) and ν r (x) = µr (x)θr + (1 − µr (x))(1 − θr ) First of all, note that the probabilities ν w (x) and ν r (x) of receiving another good signal
depend on θ, which implies that the specific way agents perceive information entails a different evaluation of the option value of waiting (exactly through ν w (x) and ν r (x)). As a consequence, the time devoted to learning should differ, which ultimately means that the optimal amount of information collected will depend on the reliability accorded to ∗∗ each signal, so that x∗t = x∗t (θ), and x∗∗ t = xt (θ).
Given the specific information processing behavior displayed, we show that agents would choose to observe a different number of positive signals before taking action a, despite holding a common prior probability in favour of state G. Relative to the cutoffs derived in Lemma 1, representativeness entails that the number of signals the agent needs to observe before taking the optimal action is smaller, whereas conservatism “damps the agent’s zeal”, increasing the time devoted to learning. The following Proposition applies: Proposition 3 Assume information processing biases, such that ˆθ = θw in the case of conservatism, and ˆθ = θr in the case of representativeness. Then, for all 0 < δ < 1, and 0 < l ¿ h, the optimal cutoffs implied by θw , θr are such that: (i) x∗t (θw ) < x∗t < x∗t (θr ); ∗∗ ∗∗ (ii) x∗∗ t (θ r ) < xt < xt (θ w ).
Proof. See Appendix. Summing up, lemma 1 states that agents trade-off the cost of delaying their choice between actions a or b with the benefit of acquiring a more consistent amount of information. Proposition 3 shows that when the perception of information is biased then the 21
number of signals that lead agents to stop collecting news and take the optimal action differs.
3.2
Biased perceptions and “bliss-point posterios”
Having shown that different attitudes towards information influence the timing of decisions, we consider now which are the consequences on the confidence agents have when choosing the optimal action. Given that conservatism and representativeness imply different posterior beliefs for any given signal, one could think that the three types of agents are not equally likely to invest in any of the two alternatives. However, by comparing the relative probabilities that each type of agents takes action a or action b, we will see that this intuition is incorrect. Let us consider first the beliefs held by the Bayesian agent (i.e, the agent whose perception of information is not biased). The most interesting property of the beliefs associated with the cutoffs derived in lemma 1 is that the confidence on the true state being G when the agent takes action a is smaller than her confidence on the true state being B when she takes action b. Roughly speaking, limited information suggesting favourable market movements is enough to convince the agent to invest in stocks, whereas only a large amount of negative signals induces her to choose bonds. Formally: Proposition 4 For all 0 < l ¿ h, there exist (x∗t , x∗∗ t ) at each date t such that: ∗ π(x∗∗ t ) + π(xt ) < 1.
Proof. Assume h = l. Then, for all t we have π(x∗t ) = 1 − π(x∗∗ t ). Since ∂x∗ t ∂h
∂x∗∗ t ∂h
< 0 and
< 0, then h > l implies that both cutoffs shift downward, so that x∗∗ − x ¯ Pr(τ = b | G) Proof. See Appendix. Basically, proposition 5 states that the probability of taking one action or the other is proportional to the distance between the prior and the beliefs held when the cutoffs are reached. As highlighted before, the greater is the payoff of action a under state G, the lower is the number of confirmatory signals above which the agent is willing to take such an action (see lemma 1). This means in turns that also the confidence on state G the agents requires for choosing a is smaller, so that, others things being equal, the agent is more likely to end up with a bliss-point belief that implies action a, and the opposite is true for action b. The most interesting implication is that, in equilibrium, the agent will invest on financial markets by mistake more often than she will opt for a safe strategy by mistake (part (iii)). As we highlighted before, this result merely depends on the specific payoffs structure we assumed and on the positive cost of learning. Differently put, an observed systematic bias towards certain actions is not sufficient to state that there exists an irrational bias in beliefs. As firstly pointed out by Brocas and Carillo [2002] in a different setting, the analysis of the probabilities derived in proposition 5 suggests another important finding: the agents’ confidence on a certain state while taking the corresponding optimal action does not depend on θ. This result may seem counter-intuitive, and in fact it can be reasonably
23
argued that, when learning is costly, agents’ willingness to keep on collecting information depends on the quality of each signal, summarized by θ. However, even if the agent could obtain a more accurate information for the same cost, this higher θ would not reflect on the relative probabilities of taking action a or b, but rather on the number of confirmatory signals that have to be observed before taking one action or the other. That is, the informational content of each signal reflects on the speed at which the bliss-point posteriors are reached but it does not affect the relative probabilities of attaining each of them, which means in turns that it does not affect their magnitude. The intriguing implication is that the relative probabilities of reaching one bliss point posterior or the other, as well as the magnitude of such bliss point posteriors, would still remain unaffected even in the presence of the information processing biases implied by conservatism and representativeness. Indeed, overweighting the relevance of information implies higher chances of becoming confident about the true state when signals are “correct” from the agent’s point of view (in the example, if sω = sG ). At the same time, an “incorrect” signal sB will move the posterior farther away in the opposite direction. These two effects perfectly compensate each other, and the intuition holds true also in the case of conservatism. In fact, less informative signals imply more smoothed movements in beliefs in both directions, so that even if a greater amount of confirmatory signals are required before taking the optimal action, contrary evidence will leave posteriors closer. To simplify notation, let µr∗∗ ≡ µr [x∗∗ (θr )], µr∗ ≡ µr [x∗ (θr )], µw∗∗ ≡ µw [x∗∗ (θw )],
and µw∗ ≡ µw [x∗ (θr )]. The following proposition formalizes the reasoning. Proposition 6 For all p ∈ ( 12 , π ∗∗ ), θ ∈ ( 12 , 1), 0 < δ < 1, and 0 < l ¿ h, we have that:
(i) Pr(τ = a | G), Pr(τ = a | B), Pr(τ = b | G), Pr(τ = b | B) do not depend on θ.
(ii) µr∗∗ = π ∗∗ = µw∗∗ and µr∗ = π∗ = µw∗ .
Proof. In order to prove part (i), consider for example Pr(τ = a | G) and assume that this bliss point posterior depends on θ. A higher θ implies higher confidence on the true state being G conditional to the signal sG , hence higher probabilities to take action a the next period, having observed sG . However, if sB obtains, the confidence on the true state being G decreases, and the higher is θ, the farther away posterior beliefs will be moved. Since these two effects are symmetric and perfectly compensate each other, Pr(τ = a | G) 24
does not depend on θ, and the same reasoning holds true for Pr(τ = a | B), Pr(τ = b | G), and Pr(τ = b | B). Part (ii) follows directly from part (i). Here there is the main result of the paper: psychological phenomena such as representativeness and conservatism bias the perception of information but they do not influence the degree of confidence individuals have while taking the optimal action. Even if the updating is biased, the posteriors associated with the optimal cutoffs coincide with the posteriors a Bayesian agent will hold in equilibrium, and the conclusion is that, from an instrumental point of view, the difference between Bayesian and biased attitudes towards information does not rely on “bliss-point” beliefs but rather on a different timing of decision. As proposition 3 shows, a biased perception of information affects the amount of evidence about the true state that agents require in order to become “reasonably confident” on that state. Relative to the Bayesian cutoffs, representativeness shrinks the “learning” interval, so that x∗∗ (θr ) < x∗∗ and x∗ (θr ) > x∗ , whereas conservatism widens it, leading to x∗∗ (θw ) > x∗∗ and x∗ (θw ) < x∗ . But these findings reveal the different speed at which the “reasonable” degree of confidence is reached, rather than implying a different degree of confidence agents need to achieve before acting. Obviously, during all the learning period, the posterior beliefs computed at a given date t differ, so that, the value of x r w being equal, then µrt (x) > π t (x) > µw t (x) if x > 0 and µt (x) < π t (x) < µt (x) if x < 0.
The conclusion is that, although information processing biases imply overconfidence or underconfidence in the short run, confidence distortions will not result in the long run. Summing up, agents who share a common prior belief about a certain state of the world, but display different attitudes towards information, will differ in the timing of decision but not in their confidence about the true state while taking the corresponding action.7 Even if downward biased posteriors (in the case of conservatism) or upward biased posteriors (in the case of representativeness) characterize the learning period, underconfidence or overconfidence have only temporary effects. The degree of confidence 7 The result strictly depends on the way we subjectively characterized the degree of informativeness of the signals. It can be reasonably argued that if the different values of θ are exogenously given and the acquisition of information has technical rather than psychological costs, then θ will influence the willingness to keep experimenting, since an higher θ would mean a more informative signal for the same cost. Nevertheless, the relative probabilities of reaching the upper-bound or the lower-bound posterior would still remain unaffected.
25
that characterizes the bliss-point posteriors coincides, and it is just the time at which such bliss-point posteriors are reached that differs. These results provide an alternative rationale to the anomalous markets movements that have been extensively analysed in the behavioral finance literature and that have often been explained in terms of investors’ states of mind. For example, anomalous trading volumes that follow announcements such as IPOs, mergers, and acquisitions, reveal investment decisions that are temporally staggered because of information processing biases; whereas the documented tendency of selling winning stocks or holding to much loosing ones can be respectively associated to impulsiveness and procrastination, rather than to pessimism and optimism, or to overconfidence.
4
Concluding remarks
This paper explores the consequences of systematically biased perceptions of information that arise because of the presence of the conservatism bias and the use of the representativeness heuristic. First of all, we model non-Bayesian updating in the simplest way, arguing that posteriors differ from the Bayesian ones because of the systematic overweighting or underwieghting of the quality of the signal. In that part we partially follow Epstein [2005]. However, while the author assumes changing priors and explains updating distortions in terms of a conflict between commitment and urges in beliefs, adopting the menus approach to model time consistent dynamic behaviors, our work lies a step behind in that we explicitly characterize the in way which information processing biases influence the updating process. Given this framework, we analyse whether and how information processing biases affect optimal decision making. Brocas and Carillo [2003] show that if learning is costly and the information acquisition process is endogenous, then individuals tend to favour the action that potentially generates the highest payoffs, although risking to obtain the lowest one. Even if this tendency represents a systematic bias in behavior, the stopping rule they derive is perfectly rational, given the payoffs structure considered and the costs of learning. The result merely depends on the option value of waiting, and holds true in standard settings. We consider how the analysis modifies when agents subjectively re-interpret the relevance of information, having associated the overweighting and underweighting of
26
signals with the use of the representativeness heuristics and with the conservatism bias, respectively. Two important results are at issue. First, we point out that these specific biases affect the timing of decisions: representativeness entails that little evidence in favour of a given hypothesis is enough in order to take the corresponding action, whereas conservatism increases the amount of confirmatory signals the agent requires before reaching a reasonable degree of confidence on the true state. This result is quite intuitive, as it confirms the behavioral characterizations we find in the psychology literature. The second and major result provides new interesting insights. Despite both conservatism and representativeness bias agents’ perceptions of the reliability of information, they do not have any influence on the relative probabilities of taking one action or the other. The conclusion is that biased perceptions of information affect the speed at which agents reach the level of confidence they need to achieve before acting, but not the magnitude of the level of confidence itself. We recognise that the specific way we model representativeness and conservatism naturally constraints the analysis, so that the findings cannot apply to all biases documented in the psychology literature. Nevertheless, we think that adding cognitive biases as an extra element to the discussion can be very useful not only to improve the understanding of the reasons that lie behind distorted behaviors, but also to identify situations where behaviors can be distorted along a desired direction through the intentional release of information.
27
References Barberis, N., A. Shleifer and R. Vishny (1998), “A model of Investor Sentiment”, Journal of Financial Economics 49, 307-343. Benabou, R. and J. Tirole (2002), “Self-confidence and Personal Motivation”, The Quarterly Journal of Economics 117, 871-915. Brocas, I. and J.D. Carillo (2002), “A Theory of Influence”, mimeo, USC. Brocas, I. and J.D. Carillo (2003), “Biases in Perceptions, Beliefs and Behaviors”, mimeo, USC. Camerer, C. (1995), “Individual Decision Making”, in: Handbook of Experimental Economics, J. Kagel and A. Roth (eds.), Princeton University Press. Caplin, A. and J. Leahy (2001), “Psychological Expected Utility and Anticipatory Feelings”, Quarterly Journal of Economics 116, 55-80. Caplin, A. and J. Leahy (2004), “The Supply of information by a Concerned Expert”, Economic Journal 114, 487-505. Carillo, J.D. and T. Mariotti (2000), “Strategic Ignorance as a Self-Discliplining Device”, Review of Economic Studies 67, 529-544. Compte, O. and A. Postlewaite (2003), “Confidence-Enhanced Performance”, PIER Working Paper 04-023. Edwards, W. (1968), “Conservatism in Human Information Processing”, in: B. Kleinmuntz (Ed.), Formal Representation of Human Judgment, New York: Wiley, (1968), 17-52. Edwards, W. (1971), “Bayesian and Regression Models of Human Informaiton Processing - A Myopic Perspective”, Organizational Behavior and Human Performance 6, 639-648. Epstein, L.G. (2005), “An Axiomatic Model of Non-Bayesian Updating”, mimeo, University of Rochester.
28
Epstein, L.G. and A.Sandroni (2004), “Non-Bayesian Updating: A Theoretical Framework”, mimeo, University of Rochester. Griffin, D. and A. Tversky (1992), “The Weighting of Evidence and the Determinants of Confidence”, Cognitive Psychology 24, 411-435. K˝oszegy, B. (2000a), “Ego Utility and Information Acquisition”, mimeo, MIT. K˝oszegy, B. (2000b), “Ego Utility, Overconfidence and Task Choice”, mimeo, MIT. Loomes, G. and R. Sugden (1982), “Regret Theory: An Alternative Theory of Rational Choice under Uncertainty”, Economic Journal 92, 805-24. Rabin, M. (1998), “Psychology and Economics”, Journal of Economic Literature 36(1), 11-46. Rabin, M. and J. Schragg (1999), “First Impression Matter: A Model of Confirmatory Bias”, Quarterly Journal of Economics, 37-82. Tversky, A. and D. Kahneman (1974), “Judgment under Uncertainty: Heuristics and Biases”, Science 185, 1124-31. Yariv, L. (2001), “Believe and Let Believe: Axiomatic Foundations for Belief Dependent Utility Functionals”, Cowles Foundation Discussion Paper Number 1344. Yariv, L. (2002), “I’ll See It when I Believe It - A Simple Model of Cognitive Consistency”, Cowles Foundation Discussion Paper Number 1352.
29
Appendix Proof of Proposition 1 First case: sω = sG → π G > p It should be proven that, in the presence of a confirmatory signal, the use of θw entails that µw G < π G . Let us verify that: µw G =
θw p θp < = πG θw p + (1 − θw )(1 − p) θp + (1 − θ)(1 − p)
where θw = λ 12 + (1 − λ)θ.
Assume λ > 0.8 Then, µw G < π G if and only if:
[λ 12
[λ 12 + (1 − λ)θ]p θp < θp + (1 − θ)(1 − p) + (1 − λ)θ]p + [1 − λ 12 − (1 − λ)θ](1 − p)
After some simple algebraic manipulations the inequality simplifies to: 1 λ(θ − )(p − 1) < 0 2 which always holds true since θ >
1 2
and p < 1.
Second case: sω = sB → π B < p
It should be proven that the use of θw entails that µw B > π B . Let us verify that: µw B =
(1 − θw )p (1 − θ)p > = πB θw (1 − p) + (1 − θw )p θ(1 − p) + (1 − θ)p
where θw = λ 12 + (1 − λ)θ.
Then, µw B > π B if and only if:
[λ 12
[1 − λ 12 − (1 − λ)θ]p (1 − θ)p > θ(1 − p) + (1 − θ)p + (1 − λ)θ](1 − p) + [1 − λ 12 − (1 − λ)θ]p
After some manipulations we obtain λ( 12 − θ) < 0. Hence, conservatism arises if θ > 12 , which holds true by hypothesis. 8 If
λ = 0 then θw = θ, and the model reduces to the standard one.
30
Proof of Corollary 1 By equation (6), if θ > p then πB < θw =
λ 12
1 2.
Recall that µw B =
(1−θw )p θw (1−p)+(1−θw )p
and
+ (1 − λ)θ, and consider: ¯ ¯ 1 − (1 − λ)θ] p[1 − λ 1 2 = µw = B 1 ¯ ¯ ¯ 1 − (1 − λ)θ]p ¯ 2 [λ 2 + (1 − λ)θ](1 − p) + [1 − λ 2
After some algebraic manipulations we get: ¯ − 1) = θ − p λ(θ 2 ¯= Then, λ
θ−p , θ− 12
¯ such that µw = which proves that there exist a λ = λ B
1 2
(ii). Since
¯ the l.h.s. is positive by definition and the r.h.s. is positive by assumption, for all λ > λ we have that µw B >
1 2
¯> (i). Note that λ
1 2
since (θ − 12 ) > θ − p holds true by assumption.
Proof of Proposition 2 First case: sω = sG → πG > p It should be proven that, in the presence of a confirmatory signal, the use of θr entails that µrG > π G , where µrG = µrG =
θr p θr p+(1−θr )(1−p)
and θr = λ + (1 − λ)θ. Let us verify that:
p[λ + (1 − λ)θ] pθ > = πG p[λ + (1 − λ)θ] + [1 − λ − (1 − λ)θ](1 − p) pθ + (1 − p)(1 − θ)
Assume λ > 0 (i.e., θr > θ). The inequality above simplifies to: λ(1 − p)(1 − θ) > 0 which holds true since p < 1 and θ < 1. Second case: sω = sB → πB < p. The use of θr implies that just one bad signal suggests that a negative scenario may be the actual one, leading to a posterior µrB < π B . To prove this, let us verify that: µrB =
p[1 − λ − (1 − λ)θ] p(1 − θ) < = µrB [λ + (1 − λ)θ](1 − p) + [1 − λ − (1 − λ)θ]p θ(1 − p) + (1 − θ)p 31
Solving the inequality we obtain: λ(p − θ − 1) < 0 which holds true since λ > 0 and p − θ < 1 (recall that p >
1 2
and θ ∈ ( 12 , 1)).
Proof of Corollary 2 By equation (6), if θ < p then π B > 12 . Recall that µrB =
(1−θr )p θr (1−p)+(1−θr )p
and θr =
λ + (1 − λ)θ. Consider then: ¯ − (1 − λ)θ] ¯ 1 p[1 − λ µrB = ¯ = ¯ ¯ − (1 − λ)θ]p ¯ 2 [λ + (1 − λ)θ](1 − p) + [1 − λ Simplifying and suitably rearranging the expression we get: ¯ − θ) p − θ = λ(1 ¯= Then, λ
p−θ 1−θ ,
¯ such that µw = which proves that there exist a λ = λ B
1 2
(ii). Since
¯ the l.h.s. is positive by assumption and the r.h.s. is positive by definition, for all λ > λ we have that µrB
(i). Note that λ
1 2
since p − θ < (1 − θ).
Proof of Lemma 1 (the proof suitably adapts that of Brocas and Carillo, 2003) TimeT. The value function the agent maximizes is: VT (x) = max {h(2π(x) − 1); l} Define: Yt (x) = Vt (x) − h(2π(x) − 1) Wt (x) = Vt (x) − l where Y (x) gives us the marginal gain of postponing the decision of investing in stocks one period ahead, and W (x) gives us the marginal gain of postponing the decision of investing in bonds one period ahead. 32
Then: YT (x) = max {0; l − h(2π(x) − 1)} WT (x) = max {0; h(2π(x) − 1) − l} By construction π(x) is increasing in x, which implies that YT (x) is non-increasing in x and WT (x) is non-decreasing in x. Since lim π(x) = 0 and lim π(x) = 1, there exists a threshold x ¯ defined by π(¯ x) = x→−∞
x→∞
1/2 such that for all x > x ¯ then τ T = a , and for all x < x ¯ then τ T = b . Time T − 1. Case 1: x ≥ x ¯ VT −1 (x) = max {h(2π(x) − 1); δ[ν(x)VT (x + 1) + (1 − ν(x))VT (x − 1)]} Since Yt (x) is defined on (¯ x, +∞) and Wt (x) is defined on (−∞, x ¯), we should consider only YT −1 (x) : YT −1 (x) = max {0; −(1 − δ)h(2π(x) − 1) + δν(x)YT (x + 1) + δ(1 − ν(x))YT (x − 1)} Since YT (x) is non-increasing in x, and π(x) is increasing in x, YT −1 (x) is decreasing in x. By hypothesis, x ≥ x ¯, where x ¯ is defined by π(¯ x) = 1/2. Therefore, there exists a
∗∗ cutoff x∗∗ T −1 such that for all x > xT −1 , the right hand side of YT −1 (x) becomes negative,
∗∗ then τ T −1 = a; and for all x ∈ [x, x∗∗ T −1 ), then τ T −1 = c. Note that the solution xT −1
should be such that x∗∗ ¯ and x∗∗ ¯, and so it solves: T −1 + 1 ≥ x T −1 − 1 < x
∗∗ ∗∗ ∗∗ ∗∗ 0 = −(1 − δ)h(2π(x∗∗ T −1 ) − 1) + δν(xT −1 )YT (xT −1 + 1) + δ(1 − ν(xT −1 ))YT (xT −1 − 1)
which can be simplified as: ∗∗ 0 = h · f (x∗∗ T −1 , δ) − l · g(xT −1 , δ)
33
where: ∗∗ ∗∗ ∗∗ f (x∗∗ T −1 , δ) ≡ (2π(xT −1 ) − 1) − δν(xT −1 )(2π(xT −1 + 1) − 1) ∗∗ g(x∗∗ T −1 , δ) ≡ δ(1 − ν(xT −1 ))
Differentiating with respect to h, l and δ we have:9 ∂x∗∗ T −1 ∂h ∂x∗∗ T −1 ∂l ∂x∗∗ T −1 ∂δ
£
£
£
∗∗ l · gx (x∗∗ T −1 , δ) − h · fx (xT −1 , δ) ∗∗ h · fx (x∗∗ T −1 , δ) − l · gx (xT −1 , δ) ∗∗ l · gx (x∗∗ T −1 , δ) − h · fx (xT −1 , δ)
¤
= f (x∗∗ T −1 , δ)
¤
∗∗ = h · fδ (x∗∗ T −1 , δ) − l · gδ (xT −1 , δ)
¤
= g(x∗∗ T −1 , δ)
£ ¤ ∗∗ ∗∗ ∗∗ Since f (x∗∗ T −1 , δ) > 0, g(xT −1 , δ) > 0, l · gx (xT −1 , δ) − h · fx (xT −1 , δ) < 0, h ·
∗∗ fδ (x∗∗ T −1 , δ) − l · gδ (xT −1 , δ) < 0, then we have:
∂x∗∗ ∂x∗∗ ∂x∗∗ T −1 T −1 T −1 < 0; > 0; >0 ∂h ∂l ∂δ Case 2: x < x ¯ The reasoning is fully symmetric. Consider: VT −1 (x) = max {l; δ[ν(x)VT (x + 1) + (1 − ν(x))VT (x − 1)]} WT −1 (x) = max {0; −(1 − δ)l + δν(x)WT (x + 1) + δ(1 − ν(x))WT (x − 1)} where WT −1 (x) is defined on (−∞, x ¯] . Since WT (x) is non-decreasing in x, and π(x) is increasing in x, WT −1 (x) is increasing in x. Therefore, there exists a cutoff x∗T −1 such
that for all x < x∗T −1 then τ T −1 = b, and for all x ∈ (x∗T −1 , x ¯], then τ T −1 = c. Note that
the solution x∗T −1 should be such that x∗T −1 + 1 > x ¯ and x∗T −1 − 1 ≤ x ¯, and so it solves: 0 = −(1 − δ)l + δν(x∗T −1 )WT (x∗T −1 + 1) + δ(1 − ν(x∗T −1 ))WT (x∗T −1 − 1) 9 The
subscripts x and δ denote a partial derivative with respect to that argument.
34
which can be written as: 0 = l · n(x∗T −1 , δ) − h · m(x∗T −1 , δ) where n(x∗T −1 , δ) ≡ [1 − δ(1 − ν(x∗T −1 ))] m(x∗T −1 , δ) ≡ δν(x∗T −1 )[2π(x∗T −1 + 1) − 1] Differentiating with respect to h, l and δ we have: ∂x∗T −1 ∂h ∂x∗T −1 ∂l ∂x∗T −1 ∂δ
£
£
£
l · nx (x∗T −1 , δ) − h · mx (x∗T −1 , δ) h · mx (x∗T −1 , δ) − l · nx (x∗T −1 , δ) l · nx (x∗T −1 , δ) − h · mx (x∗T −1 , δ)
¤
= m(x∗T −1 , δ)
¤
= h · mδ (x∗T −1 , δ) − l · nδ (x∗T −1 , δ)
¤
= n(x∗T −1 , δ)
£ ¤ Since n(x∗T −1 , δ) > 0, m(x∗T −1 , δ) > 0, l · nx (x∗T −1 , δ) − h · mx (x∗T −1 , δ) < 0, h ·
mδ (x∗T −1 , δ) − l · nδ (x∗T −1 , δ) > 0, then we have:
∂x∗T −1 ∂x∗T −1 ∂x∗T −1 < 0; > 0; xt+1 .
A3 Yt (x, h) ≤ Yt (x, h0 ) for all h0 < h. Hence, 35
∂x∗∗ t ∂h
< 0.
A4 Yt (x, l) ≥ Yt (x, l0 ) for all l0 < l. Hence,
∂x∗∗ t ∂l
A5 Yt (x, δ) ≥ Yt (x, δ 0 ) for all δ 0 < δ. Hence,
> 0.
∂x∗∗ t ∂δ
< 0.
Consider: Vt−1 (x) = max {h(2π(x) − 1); δ[ν(x)Vt (x + 1) + (1 − ν(x))Vt (x − 1)]} Yt (x) = max {0; −(1 − δ)h(2π(x) − 1) + δν(x)Yt+1 (x + 1) + δ(1 − ν(x))Yt+1 (x − 1)} Yt−1 (x) = max {0; −(1 − δ)h(2π(x) − 1) + δν(x)Yt (x + 1) + δ(1 − ν(x))Yt (x − 1)} Property (A1) states that Yt (x) is non-increasing in x, so the r.h.s. of Yt−1 (x) is ∗∗ decreasing in x, and therefore there exists a cutoff x∗∗ t−1 such that for all x > xt−1 then
τ t−1 = a, and for all x ∈ [¯ x, x∗∗ t−1 ), then τ t−1 = c. Given (A2), the r.h.s. of Yt−1 (x) is greater or equal than the r.h.s. of Yt (x). Hence, Yt−1 (x) ≥ Yt (x). As a result, both (A1)
∗∗ and (A2) hold at date t − 1, which guarantees that x∗∗ t−1 > xt .
To prove that properties (A3)-(A5) hold, consider a generic payoff h0 < h and compare:
Yt−1 (x, h) = max {0; −(1 − δ)h(2π(x) − 1) + δν(x)Yt (x + 1, h) + δ(1 − ν(x))Yt (x − 1, h)} Yt−1 (x, h0 ) = max {0; −(1 − δ)h0 (2π(x) − 1) + δν(x)Yt (x + 1, h0 ) + δ(1 − ν(x))Yt (x − 1, h0 )} Property (A3) entails that the r.h.s. of Yt−1 (x, h) is lower or equal than the r.h.s. of Yt−1 (x, h0 ). Hence, Yt−1 (x, h) ≤ Yt−1 (x, h0 ) and (A3) holds at date t − 1, which in turns ∂x∗∗ t−1 ∂h < ∂x∗∗ t−1 and ∂δ
means that
0. The same reasoning applies to properties (A4) and (A5), so that
∂x∗∗ t−1 ∂l
> 0.
>0
Case 2: x ≤ x ¯ The reasoning is fully symmetric. Assume: A1’ Wt (x) is non-decreasing in x and there exists a threshold x∗t such that if x ≤ x∗t then τ t = b and if x ∈ (x∗t , x ¯] then τ t = c.
A2’ Wt (x) ≥ Wt+1 (x) and therefore x∗t < x∗t+1 . A3’ Wt (x, h) ≥ Wt (x, h0 ) for all h0 < h. Hence, A4’ Wt (x, l) ≤ Wt (x, l0 ) for all l0 < l. Hence, 36
∂x∗ t ∂h
∂x∗ t ∂l
< 0.
> 0.
A5’ Wt (x, δ) ≥ Wt (x, δ 0 ) for all δ 0 < δ. Hence,
∂x∗ t ∂δ
< 0.
Consider: Vt−1 (x) = max {l; δ[ν(x)Vt (x + 1) + (1 − ν(x))Vt (x − 1)]} and: Wt (x) = max {0; −(1 − δ)l + δν(x)Wt+1 (x + 1) + δ(1 − ν(x))Wt+1 (x − 1)} Wt−1 (x) = max {0; −(1 − δ)l + δν(x)Wt (x + 1) + δ(1 − ν(x))Wt (x − 1)} Property (A1’) entails that the r.h.s. of Wt−1 (x) is increasing in x, and therefore there exists a cutoff x∗t−1 such that for all x < x∗t−1 then τ t−1 = b, and for all x ∈ (x∗t−1 , x ¯], then τ t−1 = c. Property (A2’) entails that the r.h.s. of Wt−1 (x) is greater or equal than the r.h.s. of Wt (x). Hence, Wt−1 (x) ≥ Wt (x). So, both (A1’) and (A2’) hold at date t − 1, and
x∗t−1 < x∗t .
Compare then: Wt−1 (x, h) = max {0; −(1 − δ)l + δν(x)Wt (x + 1, h) + δ(1 − ν(x))Wt (x − 1, h)} Wt−1 (x, h0 ) = max {0; −(1 − δ)l + δν(x)Wt (x + 1, h0 ) + δ(1 − ν(x))Wt (x − 1, h0 )} where h0 < h. Property (A3’) guarantees that the r.h.s. of Wt−1 (x, h) is greater or equal than the r.h.s. of Wt−1 (x, h0 ). Hence, Wt−1 (x, h) ≥ Wt−1 (x, h0 ) and (A3’) holds at date t − 1, which in turns means that (A4’) and (A5’) hold, so that
∂x∗ t−1 ∂h
∂x∗ t−1 ∂l
< 0. Following the same reasoning, also properties
> 0 and
∂x∗ t−1 ∂δ
< 0.
Proof of Proposition 3 Lemma 1 states that at each date t there exists a cutoff x∗∗ such that τ t = a if x ≥ x∗∗ and τ t = c if x ∈ [¯ x, x∗∗ ), and a cutoff x∗ such that τ t = b if x ≤ x∗ and τ t = c if x ∈ (x∗ , x ¯]. It should be proven now that x∗∗ is decreasing in θ, (formally, ∂x∗∗ /∂θ < 0),
and that x∗ is increasing in θ (formally, ∂x∗ /∂θ > 0). Case 1: x > x ¯ 37
Take θw , θr such that
1 2
≤ θw < θ < θr ≤ 1 and note that x > x ¯ entails xG > xB .
Hence, lim µ(x) = 1 and lim µ(x) = p, which means in turns that, given x, µ(x) is θ→1
θ→1/2
increasing in θ and µw (x) < µr (x). Furthermore, it is easy to check that ν(x) is increasing in θ. By lemma 1, the solution x∗∗ T −1 of the decision problem at date T − 1 solves: ∗∗ 0 = h · f (x∗∗ T −1 , δ) − l · g(xT −1 , δ)
where: ∗∗ ∗∗ ∗∗ f (x∗∗ T −1 , δ) ≡ (2µ(xT −1 ) − 1) − δν(xT −1 )(2µ(xT −1 + 1) − 1) ∗∗ g(x∗∗ T −1 , δ) ≡ δ(1 − ν(xT −1 ))
Differentiating with respect to θ : 10 ¤ ∂x∗∗ T −1 £ ∗∗ ∗∗ ∗∗ l · gx (x∗∗ T −1 , δ) − h · fx (xT −1 , δ) = h · fθ (xT −1 , δ) − l · gθ (xT −1 , δ) ∂θ
£ ¤ ∗∗ ∗∗ ∗∗ where l · gx (x∗∗ T −1 , δ) − h · fx (xT −1 , δ) < 0 and h·fθ (xT −1 , δ)−l·gθ (xT −1 , δ) > 0. Hence, ∂x∗∗ T −1 ∂θ
∗∗ < 0, which in turns means that x∗∗ T −1 (θ w ) > xT −1 (θ r ).
Assume that the inequality holds at an unspecified date t ∈ {1, ..., T − 1}: A6 Yt (x, θw ) > Yt (x, θr ), hence
∂x∗∗ t ∂θ
< 0.
If property (A6) holds at date t − 1, then it holds for all t ∈ {0, ..., T } . Compare then: Yt−1 (x, θw ) = max {0; −(1 − δ)h(2µw (x) − 1) + δν w (x)Y t (x + 1, θw )+ w
+δ(1 − ν (x))Y t (x − 1, θw )} Yt−1 (x, θr ) = max {0; −(1 − δ)h(2µr (x) − 1) + δν r (x)Y t (x + 1, θr )+ +δ(1 − ν r (x))Y t (x − 1, θr )} Since the r.h.s. of Yt−1 (x, θw ) is greater or equal than the r.h.s.of Yt−1 (x, θr ), then property (A6) holds at date t − 1, which means that it holds for all t ∈ {0, ..., T } . That is,
∂x∗∗ t ∂θ
1 0 The
∗∗ ∗∗ < 0, hence x∗∗ t (θ r ) < xt < xt (θ w ).
subscripts x and θ denote a partial derivative with respect to that argument.
38
Case 2: x < x ¯ Note that x < x ¯ means that xG < xB . Then, given x, µ(x) is decreasing in θ, since lim µ(x) = 0 and lim µ(x) = p. Furthermore, also ν(x) is decreasing in θ, since ν θ (x) < 0
θ→1
θ→1/2
for µ(x) < 12 . Consider the decision problem at date T − 1 (see the proof of Lemma 1). The solution
x∗T −1
solves: 0 = l · n(x∗T −1 , δ) − h · m(x∗T −1 , δ)
where: n(x∗T −1 , δ) ≡ [1 − δ(1 − ν(x∗T −1 ))] m(x∗T −1 , δ) ≡ δν(x∗T −1 )[2µ(x∗T −1 + 1) − 1] Differentiating with respect to θ : ¤ ∂x∗T −1 £ l · nx (x∗T −1 , δ) − h · mx (x∗T −1 , δ) = h · mθ (x∗T −1 , δ) − l · nθ (x∗T −1 , δ) ∂θ
£ ¤ Since l · nx (x∗T −1 , δ) − h · mx (x∗T −1 , δ) < 0 and h·mθ (x∗T −1 , δ)−l ·nθ (x∗T −1 , δ) < 0, then ∂x∗ T −1 ∂θ
> 0, which in turns means that x∗T −1 (θw ) < x∗T −1 (θr ).
Assume that the inequality hold at an unspecified date t ∈ {1, ..., T − 1}: A6’ Wt (x, θw ) < Wt (x, θr ), hence
∂x∗ t ∂θ
> 0.
If property (A6’) holds at date t − 1, then it holds for all t ∈ {0, ..., T } . Compare then: w
w
Wt−1 (x, θw ) = max {0; −(1 − δ)l + δν (x)Y t (x + 1, θw ) + δ(1 − ν (x))Y t (x − 1, θw )} r
r
Wt−1 (x, θr ) = max {0; −(1 − δ)l + δν (x)Y t (x + 1, θr ) + δ(1 − ν (x))Y t (x − 1, θr )} Since the r.h.s. of Wt−1 (x, θw ) is lower or equal than the r.h.s.of Wt−1 (x, θr ), then property (A6) holds at date t − 1, which means that it holds for all t ∈ {0, ..., T } . That is,
∂x∗ t ∂θ
> 0, hence x∗t (θw ) < x∗t < x∗t (θr ).
39
Proof of Proposition 5 (the proof suitably adapts that of Brocas and Carillo, 2002) Part (i). Let us consider first the probabilities of undertaking action a. Denote by φω (x) the probability of reaching an unspecified difference of confirmatory signals equal to x before reaching an unspecified (negative) difference of confirmatory signals equal to x when the current difference of confirmatory signals is x and the true state is ω. φG (x) = θφG (x + 1) + (1 − θ)φG (x − 1)
∀ x ∈ {x + 1, ..., x − 1}
φB (x) = (1 − θ)φB (x + 1) + θφB (x − 1)
∀ x ∈ {x + 1, ..., x − 1}
Consider φG (x). We have that: 1 1−θ G φG (x + 1) − φG (x) + φ (x − 1) = 0 θ θ which generic solution is of the form: φG (x) = k1 · z1x + k2 · z2x where (k1 , k2 ) are constants and (z1 , z2 ) are the roots of: 1 1−θ z2 − z + =0 θ θ Hence, z1 =
1−θ θ
and z2 = 1. Since by definition φG (x) = 0 and φG (x) = 1, then: k1 k1
µ µ
1−θ θ 1−θ θ
¶x ¶x
+ k2
= 0
+ k2
= 1
Simple calculations yeld: k1 k2
1 Θx − Θx Θx = − x Θ − Θx =
40
where Θ =
1−θ θ .
The general solution of the second-order difference equation is therefore: φG (x) =
1 − Θx−x 1 − Θx−x
∀ x ∈ {−β + 1, ..., x − 1}
Since the case ω = B is symmetric, it is easy to check that the solution of φB (x) can be obtained simply by switching θ and (1 − θ) : φB (x) =
Θx−x − Θx−x 1 − Θx−x
∀ x ∈ {−β + 1, ..., x − 1}
Recall that Pr(ω = G) = p ≡ Pr(G | 0). Denote by π x the posterior belief that the true state is G when the difference of signals is x, and by π x the posterior belief that the true state is G when the difference of signals is x. Applying the definition of Pr(G | x) we have: πx πx
= 1+ =
1+
Hence: φG (0) =
1 ¡ 1−θ ¢x θ
1 ¡ 1−θ ¢x θ
1−p p 1−p p
⇔ Θx =
p 1 − πx 1 − p πx
⇔ Θx =
p 1 − πx 1 − p πx
p − πx πx p − πx 1 − πx and φB (0) = πx − πx p πx − πx 1 − p
Assume now that x ≡ x∗∗ and x ≡ x∗ , hence π x = π∗∗ and πx = π ∗ . The expressions above can be rewritten as: φG (0) = φB (0) =
p − π ∗ π∗∗ = Pr(τ = a | G) π∗∗ − π∗ p p − π ∗ 1 − π∗∗ = Pr(τ = a | B) π∗∗ − π∗ 1 − p
Part (ii) Since the problem is fully symmetric, the probabilities of taking action b can be easily calculated following the same steps. Denote by ϕω (x) the probability of reaching an unspecified difference of confirmatory signals equal to x (where x is negative) before reaching an unspecified difference of signals equal to x. The reasoning is analogous, except that ϕG (x) = 1 and ϕG (x) = 0, which implies that the general solution when the true
41
state is G is: ϕG (x) =
1 − Θx−x 1 − Θ−x+x
∀ x ∈ {x + 1, ..., x − 1}
By switching θ and (1 − θ) we find the general solution when the true state is B : ϕB (x) =
Θ−x+x − Θ−x+x 1 − Θ−x+x
∀ x ∈ {x + 1, ..., x − 1}
Then: ϕG (0) = ϕB (0) =
p − πx πx p − π ∗∗ π ∗ ⇒ Pr(τ = b | G) = ∗ πx − πx p π − π ∗∗ p p − πx 1 − πx p − π ∗∗ 1 − π∗ ⇒ Pr(τ = b | B) = ∗ πx − πx 1 − p π − π∗∗ 1 − p
Part (iii) Consider the expressions derived above. We have that: Pr(τ = a | B) =
p − π ∗ 1 − π∗∗ p − π ∗∗ π∗ > ∗ Pr(τ = b | G) ∗∗ ∗ π −π 1−p π − π∗∗ p
By proposition 2, 1 − π∗∗ > π ∗ , then
1−π ∗∗ 1−p
>
π∗ p
(note that p >
1 2
by assumption).
Since the distance between the prior and the lower-bound belief is greater than the distance between the prior and the upper-bound belief, then also that, in absolute value, π
∗∗
negative, hence the fraction
∗
∗
∗∗
p−π ∗ π ∗∗ −π ∗
− π = π − π . Moreover, both p − π p−π ∗∗ π∗ −π∗∗
becomes positive).
42
∗∗
>
p−π ∗∗ π ∗ −π ∗∗ ∗
and π − π
(note ∗∗
are
