1 Introduction

Throughout our lives, we are constantly receiving new information about ourselves and our environment. The way that we filter, summarize and store this new information is of critical importance for the quality of our decision making. Theories of human behavior under dynamic uncertainty are therefore enriched by an understanding of how individuals process new information. Economists typically write down models where information is summarized in the form of probabilistic ‘beliefs’ over states of the world, and updated upon receipt of new information according to Bayes’ rule. However, the assumption that individuals process information in a statistically accurate way has increasingly been called into question, with many studies documenting systematic deviations.Footnote ¹ One important strand of this literature examines whether belief formation and updating is influenced by the affective content of the new information, i.e. whether individuals update their beliefs symmetrically in response to ‘good-news’ and ‘bad-news’ (see, for example, Eil and Rao Reference Eil and Rao2011; Ertac Reference Ertac2011; Möbius et al. Reference Möbius, Niederle, Niehaus and Rosenblat2014; Coutts Reference Coutts2019).

This literature tests an implicit assumption of Bayesian updating, namely that the only object that is relevant for predicting an individual’s belief is her information set—her beliefs are completely unaffected by the prizes and punishments she would receive in different states of the world. This fundamental assumption—that people update their beliefs symmetrically—is important because it underpins the entire orthodox approach to modelling uncertainty.

In this paper, we test whether individuals update their beliefs symmetrically in response to ‘good-news’ and ‘bad-news’ when states differ only in the financial rewards they offer. To do this, we conduct a laboratory experiment in which subjects complete a series of belief updating tasks. In each task, there are two possible states of the world. Subjects are told the prior probabilities of each state, and then receive a sequence of partially informative binary signals. We elicit subjects’ beliefs after each signal. The financial rewards associated with the two states are either identical (symmetric), or different (asymmetric). This experimental design allows us to compare posterior beliefs in situations where the entire information set is held constant, but the rewards associated with the states of the world are varied. For example, we can compare how two groups of individuals revise their beliefs when both groups share the same prior belief and receive an equally informative signal, but for one group of individuals the signal is ‘good news’ and for the other the signal constitutes ‘bad news’. We can also conduct a similar exercise for a single individual, comparing contexts where she has identical priors and signals, but the signal constitutes ‘good news’ in one context, and ‘bad news’ in the other. These comparisons provide a clean test of the asymmetric updating hypothesis.

The experiment considers belief updating in two contexts. In the symmetric treatment, subjects have an equal stake in each of the underlying states. In the two asymmetric treatments, a larger bonus payment is paid in one of the two states of the world. This design permits two separate tests of the asymmetric updating hypothesis. First, we can compare how the same individual responds to ‘good-news’ and ‘bad-news’ within the asymmetric treatments. Second, we can conduct a between-subject comparison of belief updating in the symmetric treatment and asymmetric treatments. Each individual in our experiment faces only one incentive environment. However, since we exogenously endow participants with a prior over the states of the world, we are able to repeat the exercise several times for each individual and study how they update from each of five different priors, $p_0$ , chosen from the set $\{\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6}\}$ .

The experimental design and analysis aim to address several challenges that are present when studying belief updating in the presence of state-dependent stakes. First, we use exogenous variation in the priors to ensure that the estimates are robust to the econometric issues that arise when a right-hand side variable (i.e. the prior) is a lagged version of the dependent variable (i.e. the posterior). Second, we avoid a second type of endogeneity issue, which arises when the underlying states are defined as a function of some personal characteristic of the individual (e.g. her relative IQ) that might also be related to how she updates (see Online Appendix C for further details). Third, we measure the influence that hedging has on belief elicitation when there are state-dependent stakes. Furthermore, we conduct several exercises to correct our estimates for this hedging influence—both experimentally, and econometrically. Fourth, our experimental design allows us to study belief updating from priors spanning much of the unit interval. Importantly, averaging across all subjects, the design generates a balanced distribution of ‘good’ and ‘bad’ signals.

Our results show no evidence in favor of asymmetric updating in response to ‘good-news’ in comparison to ‘bad-news’ in the domain of financial outcomes. Several robustness exercises are carried out in support of this conclusion. Furthermore, we find that average updating behavior is well approximated by Bayes’ rule.Footnote ² This average behavior masks substantial heterogeneity in updating behavior at the individual level, but we find no evidence in support of a sizeable subgroup of asymmetric updators.

The evidence reported here contributes to the recent literature studying the asymmetric updating hypothesis across different contexts (e.g. Eil and Rao Reference Eil and Rao2011; Ertac Reference Ertac2011; Grossman and Owens Reference Grossman and Owens2012; Mayraz Reference Mayraz2013; Möbius et al. Reference Möbius, Niederle, Niehaus and Rosenblat2014; Kuhnen Reference Kuhnen2015; Schwardmann and Van der Weele Reference Schwardmann and Van der Weele2019; Gotthard-Real Reference Gotthard-Real2017; Charness and Dave Reference Charness and Dave2017; Heger and Papageorge Reference Heger and Papageorge2018; Buser et al. Reference Buser, Gerhards and Van Der Weele2018; Coutts Reference Coutts2019). The results in this literature thus far are highly heterogeneous, with some papers finding a greater responsiveness to good-news, some a greater responsiveness to bad-news, and some no evidence of an asymmetry. The objective of the experiment discussed in this paper is not to isolate the contextual features that activate and deactivate asymmetric updating; rather the objective is to provide a clean test of the asymmetric updating hypothesis for contexts in which states differ only in their material rewards and have no direct ego-relevance. The reason for focusing on this context (where states yield different financial outcomes) is that it characterizes a large class of economically important decision problems under uncertainty—most economic models with uncertainty fit this description. This paper does not disentangle the reasons for why updating is asymmetric in some settings but not in others. Instead, by providing a clean test of the asymmetric updating hypothesis for one specific domain, the paper contributes to the growing collective body of evidence pertaining to asymmetric updating across a range of contexts. There are several candidate contextual and experimental design factors that could be generating the heterogeneous results observed in the literature as a whole. Section 7 below offers a discussion of some of these candidate explanations, and asks whether the existing body of evidence can help us to detect a systematic pattern that organizes the results.

The remainder of the paper proceeds as follows. Section 2 outlines the theoretical framework, Sect. 3 details the experimental design, Sect. 4 provides some descriptive statistics, Sect. 5 presents the empirical specification, Sect. 6 discusses the related literature and Sect. 7 concludes.

2 Theoretical framework and hypotheses

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. (Reference Möbius, Niederle, Niehaus and Rosenblat2014) 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’.

2.1 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 $\overline{p_0}$ (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, $p\left(a\right|A)=p\left(b\right|B)=q>\frac{1}{2}$ . The history, $H_t$ , is defined as the sequence of signals received by the agent in periods $1,\dots ,t$ , with $H_0=\varnothing$ . 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. (Reference Möbius, Niederle, Niehaus and Rosenblat2014) in considering the following model of augmented Bayesian updating:

(1) $\begin{array}{c}\text{logit}\left({\pi }_{t+1}\right)=\delta \text{logit}\left({\pi }_t\right)+{\gamma }_{a}log\left(\frac{q}{1-q}\right)·1(s_{t+1}=a)-{\gamma }_{b}log\left(\frac{q}{1-q}\right)·1(s_{t+1}=b)\end{array}$

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 bias Footnote 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 neglect Footnote 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

Belief updating distortion	Parameter values
Bayesian updating	$\delta =1$ , ${\gamma }_a=1$ and ${\gamma }_b=1$
Confirmatory bias	$\delta >1$
Base rate neglect	$\delta <1$
Conservatism	${\gamma }_j<1$ for $\forall j\in \{a,b\}$
Overresponsiveness	${\gamma }_j>1$ for $\forall j\in \{a,b\}$
Optimistic updating (in ASYMMETRIC)	${\gamma }_a>{\gamma }_b$
Pessimistic updating (in ASYMMETRIC)	${\gamma }_a<{\gamma }_b$

2.1.1 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:

• • Scenario 1 (symmetric): the agent is indifferent between outcomes (i.e. $x_A\sim x_B$ ); and

• • 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: ${\gamma }_a^A>{\gamma }^S>{\gamma }_b^A$ .

3 Experimental design

The experiment is designed to test the asymmetric updating hypothesis using both within-subject and between-subjects comparisons of updating behavior. The experiment consists of three treatment groups. The treatment T1:Symmetric corresponds to Scenario 1 (symmetric: no exogenous state-contingent stakes) and the other two treatment groups, T2: Combined and T3: Separate, correspond to Scenario 2 (asymmetric: state-contingent stakes) discussed above.

The two asymmetric treatments, T2: Combined and T3: Separate, have identical financial incentives. The rationale for running two treatments with identical incentives was to conduct additional checks to ensure that our results were not driven by the influence of a hedging motive. To do this, we vary only the way that the incentive environment is described to participants (i.e. we only vary the framing of the incentives).

The experiment proceeds in three stages. The first stage comprises the core belief updating task in which we elicit a sequence of reported beliefs from subjects as they received a sequence of noisy signals. In the second stage we collect the reported probabilities associated with known objective probabilities on the interval [0, 1] required for the Offerman et al. (Reference Offerman, Sonnemans, van de Kuilen and Wakker2009) correction approach, as well as data on risk preferences. In the third stage, we obtain data on several demographic characteristics as well as some further non-incentivized measures. In each of the first two stages, one of the subject’s choices is chosen at random and paid out. In addition the participants receive a fixed fee of £5 [€5] for completing Stage 3, as well as a show-up fee of £5 [€5].Footnote ⁸

3.1 The belief updating task (Stage 1)

The Belief Updating Task is the primary task of the experiment, and is summarized in Fig. 1. The task consists of five rounds. In each round, participants are presented with a pair of computerized ‘urns’ containing blue and red colored balls. Each of these two urns represents one of the two states of the world. The composition of the two urns is always constant, with state $\omega =A$ represented by the urn containing more blue balls (5 blue and 3 red), and state $\omega =B$ represented by the urn containing more red balls (5 red and 3 blue).

3.1.1 Priors

The five rounds differ from one another only in the exogenous prior probability that $\omega =A$ is the true state, with this prior, $p_0$ , chosen from the set $\{\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6}\}$ . In each round, this prior is known to the participant. The order of these rounds is randomly chosen for each individual. Conditional on the prior, $p_0$ , one of the two urns is then chosen through the throw of a virtual die, independently for each individual in each round.

3.1.2 Belief updating

In each round, after being informed of this prior probability, $p_0$ , the participant receives a sequence of five partially informative signals, $s_t$ , for $t=\{1,2,3,4,5\}$ . These signals consist of draws, with replacement, from the urn chosen for that round. Therefore, if the state of the world in a specific round is $\omega =A$ then the chance of drawing a red ball is $\frac{3}{8}$ and the chance of drawing a blue ball is $\frac{5}{8}$ for each of the draws in that round (see Fig. 1).

In each round, we elicit the participant’s reported belief, $r_t$ , about the likelihood that state $\omega =A$ is the correct state of the world six times (i.e. for $t=\{0,1,2,3,4,5\}$ ). We first elicit her reported belief, $r_0$ , directly after she is informed of the exogenous prior probability, $p_0$ , and then after she receives each of her five signals we elicit $r_t$ for $t=\{1,2,3,4,5\}$ . Overall, we therefore elicit 30 reported beliefs in Stage 1 from each individual (6 reported beliefs in each of 5 rounds).Footnote ⁹

3.1.3 Incentives and treatment groups

The Belief Updating Task is identical across treatment groups, with the exception of the incentives faced by participants. In each treatment, a participant’s payment consists of two components: (1) an exogenous state-contingent payment, and (2) an accuracy payment that depends on both their stated belief and the true state.

In treatments T2: Combined and T3: Separate, the state-contingent payment is £10 in state $\omega =A$ in comparison to £0.10 in state $\omega =B$ , making $\omega =A$ the more attractive state of the world. In T1: Symmetric, participants receive an equal state-contingent payment of £0.10 in both states.

Fig. 1

Overview of experimental design

In all three treatments, participants receive nearly identical detailed written instructions describing the belief updating task as well as the two payment components. With the aim of ensuring that participants understand the incentive environment, the instructions present the QSR as a choice from a list of lotteries (this approach is also used, for example, by Armantier and Treich (Reference Armantier and Treich2013) and Offerman et al. (Reference Offerman, Sonnemans, van de Kuilen and Wakker2009)). To this effect, subjects are presented with payment tables, which inform them of the precise prospect they would face for each choice of $r_t$ , in increments of 0.01. An abbreviated version of the three payment tables associated with each of the three treatment groups is presented in Table 2.Footnote ¹⁰ In order to represent all payments as integers in the instructions and payment tables, we adopt the approach of using experimental points. The exchange rate is 6000 points = $\pounds$ 1.

Table 2

Comparison of incentive summary tables between treatment groups

T1: SYMMETRIC			T2: COMBINED			T3: SEPARATE
Probability judgment for Urn A	True Urn		Probability judgment for Urn A	True Urn		Probability judgment for Urn A	True Urn
	Urn A	Urn B		Urn A	Urn B		Urn A	Urn B
100	12,600	600	100	72,000	600	100	12,000	0
95	12,570	1770	95	71,970	1770	95	11,970	1170
90	12,480	2880	90	71,880	2880	90	11,880	2280
85	12,330	3930	85	71,730	3930	85	11,730	3330
80	12,120	4920	80	71,520	4920	80	11,520	4320
75	11,850	5850	75	71,250	5850	75	11,250	5250
70	11,520	6720	70	70,920	6720	70	10,920	6120
65	11,130	7530	65	70,530	7530	65	10,530	6930
60	10,680	8280	60	70,080	8280	60	10,080	7680
55	10,170	8970	55	69,570	8970	55	9570	8370
50	9600	9600	50	69,000	9600	50	9000	9000
45	8970	10,170	45	68,370	10,170	45	8370	9570
40	8280	10,680	40	67,680	10,680	40	7680	10,080
35	7530	11,130	35	66,120	11,130	35	6930	10,530
30	6720	11,520	30	66,120	11,520	30	6120	10,920
25	5850	11,850	25	65,250	11,850	25	5250	11,250
20	4920	12,120	20	64,320	12,120	20	4320	11,520
15	3930	12,330	15	63,330	12,330	15	3330	11,730
10	2880	12,480	10	62,280	12,480	10	2280	11,880
5	1770	12,570	5	61,170	12,570	5	1170	11,970
0	600	12,600	0	60,000	12,600	0	0	12,000

Table 2 shows the difference between treatments T2: Combined and T3: Separate. While participants in these two treatments face precisely the same incentives, the treatments differ in the salience of the hedging motive. The only difference between the two treatments is the way in which the payment information is summarised in the payment table. In T2: Combined, the payment table shows the combined payment from both (1) the exogenous state-contingent payment, and (2) the accuracy payment. Therefore, it summarizes the reduced form prospect associated with each reported probability choice ( $r_t$ ) for subjects. In T3: Separate, the payment table shows only the accuracy payment, so subjects need to integrate the two payments themselves to spot the hedging opportunity.

Why do we have two treatments with identical incentives, differing only in presentation? The T2: Combined treatment presents participants with incentives faced in the simplest and clearest way. This is desirable as it maximizes subject understanding of the incentives. In contrast, T3: Separate lowers the salience of the hedging opportunity with the aim of inducing more accurate belief reporting. Implementing both treatments serves multiple purposes: (1) it allows us to evaluate the influence of the presentation of incentives on reported beliefs (i.e. whether a hedging motive affected reporting), (2) it provides a way to test the internal validity of the correction mechanism we use, and (3) it provides us with a form of internal replication of our results, since we can test each hypothesis more than once.

4 Data and descriptive evidence

The experiment was conducted at the UCL-ELSE experimental laboratory in London as well as at the WZB-TU laboratory in Berlin. There were two sessions for each of the three Treatment groups at each location, making twelve sessions and 222 participants in total. At each location, participants were solicited through an online database using ORSEE (Greiner Reference Greiner2015) and the experiment was run using the experimental software, z-Tree (Fischbacher Reference Fischbacher2007). On average, sessions lasted approximately 1.5 h and the average participant earned £19.7 in London and €20.3 in Berlin. Realized payments ranged between $\pounds$ 11 [€11] and $\pounds$ 34 [€34].

One challenge faced by belief updating studies is ensuring that subjects understand and engage with the task. In order to facilitate this, we were careful to ensure that the instructions received were as clear and simple as possible. Nonetheless, there remained a non-trivial fraction of participants who took decisions to update in the ‘incorrect’ direction upon receiving new information. In order to ensure that the behavior we are studying is reflective of actual updating behavior of individuals who understood and engaged with the task, we restrict our sample for our main analysis by removing rounds where an individual updates in the incorrect direction.Footnote ¹² We also estimate all the main results on the full sample, and the general patterns of behavior are similar.

While randomization to treatment group should ensure that the samples are balanced on observable and unobservable characteristics, Table 9 in the Appendices provides a check that the selection of our preferred sample has not substantially biased our treatments groups. Overall, the treatments appear to be balanced, with the exception that individuals in T3: Separate are more likely to speak English at home than individuals in T2: Combined.

5 Empirical specification

In this section, we first discuss the calibration exercise used to correct the reported beliefs. Second, we describe the core estimation equations used in our analysis. These build on the work by Möbius et al. (Reference Möbius, Niederle, Niehaus and Rosenblat2014) as their data has a similar structure to ours. One key difference in our data is the exogenous assignment of the participants’ entire information set. We exploit this feature of our data to address endogeneity issues that can arise when studying belief updating.

5.1 The belief correction procedure

The belief correction procedure that we adopt involves assuming a flexible parametric form for the participants’ utility and probability weighting functions in order to estimate their personal belief-distortion function (this is the R function discussed in Equation 8 in Online Appendix B.2.1). We estimate this function for each individual separately in order to correct the reported beliefs at the individual level. Essentially, we are simply fitting a curve through each subject’s belief elicitation distortion, for the relevant incentive environment. Online Appendix B.2.4 contains a more detailed discussion of the estimation of the belief correction function, and illustrates the main ideas through Figures 5 and 6 (for further details about this procedure, see Offerman et al. Reference Offerman, Sonnemans, van de Kuilen and Wakker2009 and Kothiyal et al. Reference Kothiyal, Spinu and Wakker2011).

In Fig. 2 we plot the CDFs of the uncorrected reported beliefs as well as the reported beliefs that have been corrected at the individual level. The left panel displays the distribution of reported beliefs, prior to correction, in each of the three treatments. One interesting feature of this figure is that, in spite of the fact that subjects in T2: Combined and T3: Separate are offered precisely the same incentives, the distributions of reported beliefs of these two groups are significantly different from one another (Mann–Whitney rank-sum test, $p<0.01$ ). The larger mass of reported beliefs to the left of 50 in T2: Combined suggests that individuals are more likely to respond to the hedging opportunity when it is more salient.

Fig. 2

CDFs of corrected and uncorrected reported beliefs

The right panel shows the corrected beliefs. It suggests that the belief correction approach was successful in removing the strong hedging influence in T2: Combined. After correction, the beliefs in the two treatments with identical incentives, T2: Combined and T3: Separate, are similar. This evidence underscores the usefulness of the belief correction mechanism in reducing bias in reported beliefs when there is a salient hedging opportunity.

With the corrected beliefs in hand, we proceed to the main analysis. The analysis is done using both the corrected and uncorrected beliefs.Footnote ¹³

5.2 Core estimation specifications

Our core estimation equations aim to test for systematic patterns in updating behavior, within the framework of Eq. 1. First, we examine whether there are systematic deviations from Bayes’ rule in updating behavior, independently of having a stake in one of the two states of the world. Second, we assess the influence that having a stake in one of the two states of the world has by (1) testing for an asymmetry in updating within treatments where there is a state-contingent stake (i.e. T2: Combined and T3: Separate); and (2) testing whether there are differences in updating behavior between the treatments with and without a state-contingent stake.

The first estimation equation follows directly from Eq. 1, allowing us to test the asymmetric updating hypothesis, and also to test for other common deviations from Bayes’ rule, such as a confirmatory bias or base rate neglect:

(2) $\begin{array}{c}{\tilde{\pi }}_{i,j,t+1}=\delta {\tilde{\pi }}_{i,j,t}+{\gamma }_a\tilde{q}·1(s_{i,j,t+1}=a)-{\gamma }_b\tilde{q}·1(s_{i,j,t+1}=b)+{ϵ}_{i,j,t+1}\end{array}$

where ${\tilde{\pi }}_{i,j,t}=\text{logit}\left({\pi }_{i,j,t}\right)$ and $\tilde{q}=log\left(\frac{q}{1-q}\right)$ ; j refers to a round of decisions; and the errors, ${ϵ}_{\mathrm{ijt}}$ , are clustered at the individual, i, level.

To determine the belief updating pattern within each incentive environment, we estimate this equation separately for each treatment. Then to test for significant differences between the coefficients in different incentive environments, we pool our sample and interact the treatment variable with all three of the coefficients in this equation (i.e. $\delta$ , ${\gamma }_a$ , and ${\gamma }_b$ ). This provides us with a test of whether the parameters differ between either of the two asymmetric treatments and the symmetric treatment.

5.2.1 Endogeneity of the Lagged Belief

One potential concern with the identification of the parameters of Eq. 2 is that the right hand side contains lagged versions of the dependent variable. This implies that there is a possible endogeneity of the lagged beliefs, ${\pi }_{i,j,t}$ , if they are correlated with the error term (i.e. if $E\left\{{\tilde{\pi }}_{i,j,t}{ϵ}_{i,j,t+1}\right\}\ne 0$ ).Footnote ¹⁴ If this is the case, it can result in biased and inconsistent estimators for the parameters of the regression. Our experiment was designed to avoid this issue by virtue of exogenously assigning the subjects’ entire information set. This allows us to use the exogenously assigned prior probability of state $\omega =A$ being the true state, $p_{i,j,t=0}$ , as well as the sequence of signals observed, $s_t$ , to construct an instrument for the lagged belief, ${\pi }_{i,j,t}$ , in Specification 2. We do this by calculating the objective Bayesian posterior, given the agent’s information set at time t, and using this as an instrument for her belief, ${\pi }_{i,j,t}$ .

The approach used here also avoids a second type of endogeneity issue that can arise when studying belief updating when the states of the world are a functions of personal characteristics (e.g. when examining beliefs regarding individual attributes, such as one’s own skills, IQ, or beauty) or personal choices. When this is the case, the conditional probability of observing a specific signal depends on the state of the world, and therefore can be correlated with personal characteristics (see Online Appendix C and earlier versions of this paper for further discussion of this issue).