The following section discusses a simple framework for belief updating that augments the standard normative Bayesian benchmark to allow for several commonly hypothesized deviations from Bayes’ rule. This framework is borrowed from Möbius et al. (2014) and is commonly used for analyzing belief updating descriptively. The framework provides a basis for the empirical approach that we will use to test whether agents update their beliefs asymmetrically in response to ‘good-news’ and ‘bad-news’.
A simple model of belief formation
We consider a single agent who forms a belief over two states of the world, \(\omega \in \{A,B\}\), at each point in time, t. One of these states of the world is selected by nature as the ‘correct’ (or ‘realized’) state, where state \(\omega =A\) is chosen with prior probability \(\bar{p_{0}}\) (known to the agent). The agent’s belief at time t is denoted by \(\pi _{t}\in [0,1]\), where \(\pi _{t}\) is the agent’s belief regarding the likelihood that the state is \(\omega =A\) and \(1-\pi _{t}\) is the agent’s belief that the state is \(\omega =B\). In each period, the agent receives a signal, \(s_{t}\in \{a,b\}\), regarding the state of the world, which is correct with probability \(q\in (\frac{1}{2},1)\). In other words, \(p(a|A)=p(b|B)=q>\frac{1}{2}\). The history, \(H_{t}\), is defined as the sequence of signals received by the agent in periods \(1,\ldots ,t\), with \(H_{0}=\emptyset\). Therefore, the history at time t is given by \(H_{t}=(s_{1},...,s_{t})\).
To study how individuals update their beliefs, we follow Möbius et al. (2014) in considering the following model of augmented Bayesian updating:
$$\begin{aligned} {\text{ logit }}(\pi _{t+1})=\delta {\text{ logit }}(\pi _{t}) +\gamma _{a}\log \left( \frac{q}{1-q}\right) \cdot 1(s_{t+1}=a)-\gamma _{b} \log \left( \frac{q}{1-q}\right) \cdot 1(s_{t+1}=b) \end{aligned}$$
(1)
The parameters \(\delta\), \(\gamma _{a}\) and \(\gamma _{b}\) can be interpretted as follows. If \(\delta =\gamma _{a}=\gamma _{b}=1\) then the agent updates her beliefs according to Bayes’ rule. The \(\delta\) parameter captures the degree to which the agent’s prior affects her updating. For example, if \(\delta >1\) then the agent displays a confirmatory biasFootnote 3, whereby she is more responsive to information that supports her prior. In contrast, if \(\delta <1\) she is more responsive to information that contradicts her prior (i.e. base rate neglectFootnote 4). The former would predict that beliefs will polarize over time, while the latter would predict that over time beliefs remain too close to 0.5.
The parameters, \(\gamma _{a}\) and \(\gamma _{b}\) capture the agent’s responsiveness to information. If \(\gamma _{a}=\gamma _{b}<1\) then the agent is less responsive to information than a Bayesian. And if \(\gamma _{a}=\gamma _{b}>1\), then she is more responsive than a Bayesian. For example, if \(\gamma _{a}=2\), then whenever the agent receives a signal \(s_{t}=a\), she updates her belief exactly as much as a Bayesian would if he received two a signals, \(s_{t}=\{a,a\}\). The interpretation of the parameters is summarized in the first five rows of Table 1.
Table 1 Interpretation of parameters: a summary Affective states
In the preceding section, the affect or desirability of different states of the world played no role. However, in most situations in which individuals form beliefs, there are some states that yield an outcome that is preferred to the outcomes associated with other states—i.e. there are good and bad states of the world
To allow for the possibility that individuals update their beliefs differently in response to good-news in comparison to bad-news, we relax the assumption that belief updating is orthogonal to the affect of the information.Footnote 5 To do this, assume that each of the two states of the world is associated with a certain outcome—i.e. in state \(\omega =A\), the agent receives outcome \(x_{A}\), and in state \(\omega =B\), she receives \(x_{B}\). There are now two belief updating scenarios:
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Scenario 1 (symmetric): the agent is indifferent between outcomes (i.e. \(x_{A}\sim x_{B}\)); and
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Scenario 2 (asymmetric): the agent strictly prefers one of the two outcomes (i.e. \(x_{A}\succ x_{B}\)).
The question of interest is whether the agent will update her beliefs differently in the symmetric and asymmetric scenarios. Under the assumption that the agent’s behavior is consistent with the model described above in Eq. 1, this involves asking whether the parameters \(\delta\), \(\gamma _{a}\) and \(\gamma _{b}\), differ between the two contexts.
To guide our discussion, we consider the following two benchmarks. The first natural benchmark is Bayes’ rule, which prescribes that all three parameters equal 1 in both the symmetric and asymmetric contexts—statistically efficient updating of probabilities is unaffected by state-dependent rewards and punishments. According to Bayes’ rule, news is news, independent of its affective content.
Hypothesis 1
(Bayesian updating) Individuals update their beliefs according to Bayes’ rule. Therefore, \(\delta =1\), \(\gamma _{a}=1\) and \(\gamma _{b}=1\) in both symmetric and asymmetric scenarios.
The second benchmark that we consider is provided by the asymmetric updating hypothesis—that individuals respond more to ‘good-news’ than ‘bad-news’. Here, in our simple framework there are two ways to identify asymmetric updating.
First, if we only consider the behavior of individuals within the asymmetric scenario, we can ask whether there is an asymmetry in updating after signals that favor the more desirable state \(\omega =A\) (‘good-news’), relative to signals that favor the less desirable state \(\omega =B\) (‘bad-news’). For example, if \(\gamma _{a}>\gamma _{b}\), this would indicate that the agent updates more in response to ‘good-news’. We refer to such an agent as an optimistic updater. Conversely, if we have \(\gamma _{a}<\gamma _{b}\) then the agent updates more in response to ‘bad-news’. We refer to such an agent as a pessimistic updater.
Second, if we compare behavior between the symmetric and asymmetric scenarios, we can ask whether the parameters of Eq. 1 differ according to the scenario. We use the postscript \(c\in \{A,S\}\) to distinguish the parameters in the two scenarios—i.e. \(\delta ^{S}\), \(\gamma _{a}^{S}\) and \(\gamma _{b}^{S}\) in symmetric and \(\delta ^{A}\), \(\gamma _{a}^{A}\) and \(\gamma _{b}^{A}\) in asymmetric. In the symmetric treatment, where the agent is completely indifferent between the two states, there is no reason to expect her updating to be asymmetric. Therefore, we assume that \(\gamma _{a}^{S}=\gamma _{b}^{S}=\gamma ^{S}\). Thus, the difference \(\gamma _{a}^{A}-\gamma _{a}^{S}\) reflects a measure of the increase in the agent’s responsiveness when information is desirable, relative to when information is neutral in terms of its affect. Similarly, \(\gamma _{b}^{A}-\gamma _{b}^{S}\) is a measure of the increase in the agent’s responsiveness when information is undesirable, relative to the case in which information is neutral in affect.
Hypothesis 2
(Asymmetric updating) Individuals update their beliefs asymmetrically, responding more to good than bad news. Therefore, within the asymmetric scenario, we will observe \(\gamma _{a}^{A}>\gamma _{b}^{A}\). And in a comparison between the symmetric and asymmetric scenarios, we will observe \(\gamma _{a}^{A}-\gamma _{a}^{S}>0\) and \(\gamma _{b}^{A}-\gamma _{b}^{S}<0\). Together, we can summarize the asymmetric updating hypothesis parameter predictions as follows: \(\gamma _{a}^{A}>\gamma ^{S}>\gamma _{b}^{A}\).
Belief elicitation and incentives
To empirically test the hypotheses above using an experiment, we would like to be able to elicit our participants’ true beliefs. However, eliciting beliefs when studying the relationship between preferences and beliefs presents additional challenges. In particular, one needs to account for the inherent hedging motive faced by participants who have a stake in one state of the world (see Karni and Safra 1995 for a discussion). To obtain unbiased reported beliefs, we adopt the approach developed by Offerman et al. (2009), and extended to accommodate state-dependent stakes as in Kothiyal et al. (2011).
The central idea behind this approach is to acknowledge that the incentive environment within which we elicit beliefs in the laboratory may exert a distortionary influence on reported beliefs. We therefore measure this distortionary influence of the incentive environment in a separate part of the experiment. Once we have constructed a mapping from true beliefs to reported beliefs within the relevant incentive environment, we can invert this function to recover the participant’s true beliefs from her reported beliefs. Our objective, therefore, is to recover the function that each individual uses to map her true beliefs to the beliefs that she reports within the given incentive environment.
The incentive environment that we use in our experiment to elicit beliefs is the quadratic scoring rule (QSR).Footnote 6 Online Appendix B.2 provides a detailed discussion of the way in which reported beliefs might be distorted under the quadratic scoring rule. In the absence of state-dependent stakes, it is well documented that under the QSR a risk averse agent should distort their reported belief towards 50%. With state-dependant stakes, a risk averse EU maximizer will face two distortionary motives in reporting her belief: (1) she will face the motive to distort her belief towards 50% as discussed above; and (2) in addition, there is a hedging motive, which will compel a risk averse individual to lower her reported belief, \(r_{t}\), towards 0% as the size of the exogenous stake increases.Footnote 7
If the participants in our experiment are risk neutral expected utility maximizers, the reported beliefs, \(r_{t}\), that we elicit under the QSR will coincide with their true beliefs, \(\pi _{t}\), and no belief correction is necessary. However, to account for a hedging motive (e.g. due to risk aversion), we measure the size of the distortionary influence of the elicitation incentives at an individual level and correct the beliefs accordingly. The following section provides the intuition for how this correction works.
A Non-EU ‘truth serum’
The Offerman et al. (2009) approach proposes correcting reported beliefs for the reporting bias generated by risk aversion or non-linear probability weighting. This approach involves eliciting subjects’ reported beliefs, r, corresponding to a set of risky events where both the participant and the analyst know the objective probabilities, p (known probability). This is done under precisely the same QSR incentive environment in which the subjects’ subjective beliefs, \(\pi\), regarding the events of interest are elicited (unknown probability). If a subject’s reported beliefs, r, differ from the known objective probabilities, p, this indicates that the subject is distorting her beliefs due to the incentive environment (e.g. due to risk aversion). The objective of the correction mechanism is therefore to construct a map, R, from the objective probabilities, \(p\in [0,1]\), to the reported beliefs, r, for each individual under the relevant incentive environment. Given this map, R, we can invert the function and recover the subject’s true beliefs from her reported beliefs about events with unknown probabilities, \(\pi\).
In Online Appendix B.2, we offer a detailed discussion of how the Offerman et al. (2009) method operates, describes the underlying assumptions, and demonstrate how it can be augmented (as proposed in Kothiyal et al. 2011) to allow for the scenario where there are state-contingent stakes (i.e. \(x\ne 0\)).