Inferred Prior Beliefs of Other’s Information

In this section, we estimate the source beliefs (pA, pI), probabilities subjects consider the information comes from, which reflects how subjects attribute the information to the assigned and irrelevant urn. In our experiment, it is explicitly stated that the combination of source beliefs is (0.5, 0.5). We use the four posteriors elicited (first/second phase in the assigned/irrelevant urn) to estimate subjects’ (pA, pI) by conducting a maximum likelihood estimation.¹⁴ We follow a structural estimation method similar to that in Costa-Gomes and Crawford (2006) but impose a logit error structure instead of spike-logit because it is hard for subjects to exactly hit the Bayesian updating prediction given the complicated Bayesian calculation.

We allow for 21 possible types, ranging from pA = 0, 0.05..., to 1.¹⁵ We assume that each subject’s updating behavior is fixed across the 10 rounds. Formally, let k = 0, 5, ..., 100 (which stands for the source belief pA from 0%, 5%, ..., to 100%) index our types, R = 20 denote the total number of elicited probabilities (since each round consists of two updating decisions),¹⁶ and xⁱ_r denote subject i’s posteriors in choice r. Given subject’s type and information received, let t^i,k_r denote the predicted posterior for a type-k subject i in round r. In order to interpret the pattern of

14To properly investigate individual “updating” types, we use subjects’ first posteriors to cal-culate the target second posteriors, otherwise it could be problematic for those who deviate from the Bayesian posteriors in the first phase. For example, subject who report 60% as posteriors of the irrelevant urn and 38% as posteriors of the assigned urn in both phases is actually behaving as an “ignoring” type in the second phase. However, if we use the correct Bayesian posteriors in the first phase as benchmarks to calculate the second phase posteriors, we will mistakenly believe this subject is perfectly Bayesian.

15It is unnecessary to divide the types further since di↵erent pA would map into the same combination of balls. For example, suppose one subject has the balls 30 and 70 in the first and second phase, respectively. The Bayesian posteriors are 0.38 for the assigned urn and 0.61 for the irrelevant urn if pA= 0.5. If pA= 0.51, the corresponding posteriors hardly change, so we cannot distinguish the subject’s type.

16We assume that all posteriors are updated independently.

deviations from one’s updating, we specify a logit error structure in which, in every particular round, a subject updates to the exact predicted posterior of one’s type with highest probability, and the probability decreases as we move away from the predicted posterior. In particular, a type-k subject’s assigned urn posterior in round r satisfies the logit density function d^k_r(xⁱ_r, t^i,k_r , ) with precision parameter :

d^k_r(xⁱ_r, t^i,k_r , )⌘ exp [ E(xⁱ_r|t^i,kr )]

P

z_rⁱexp [ E(z_rⁱ|t^i,kr )]. (15)

where the expected payo↵ E(x|t^i,kr ) = x· t^i,kr + (1 x)· (x + 1)/2, the actual payo↵

subjects earn in the experiment. Therefore, the density of a type-k subject with updates xⁱ ⌘ (xⁱ1, ..., xⁱ_R) is

d^k(xⁱ, t^i,k, )⌘Y

r

d^k_r(xⁱ_r, t^i,k_r , ). (16)

Let p^k denote a subject’s prior probability of being type-k, with PK

k=1p^k = 1 and p⌘ (p¹, ..., p^K). By multiplying the right hand-side of (15) by p^k, summing over k and taking logarithms, the log-likelihood function for subject i becomes

ln L(p, ", s|xⁱ) = ln

Given the estimate of , it is clear from (17) that the maximum likelihood estimate of p sets p^k = 1 for the generically unique k that yields the highest d^k(xⁱ, t^i,k, ).

The maximum likelihood estimate of is the logistic scale parameter describing the spreading of subject’s updating.

Figure 7a shows that on average subjects assign di↵erent weights when facing conflicting and confirming information. The weight is p_A = 32% (median = 20%) when estimated using only rounds in which the information is conflicting, but it increases to pA= 44% (median = 45%) when using rounds in which information in confirming. The di↵erence of subject beliefs between confirming and conflicting is significant (44% 32%: t-test p < 0.001; Wilcoxon signed-rank test p = 0.003), suggesting the occurrence of an echo chamber e↵ect.

Figure 7: Models of Information Sources: (a) Two Urns (b) Three Urns.

The above model restricts the sum of pA and pI to necessarily equal to one, which implies the information must originate from either the assigned or irrelevant urn. This assumption adheres to our experimental design. However, people may underweight others’ information. Also, notice that subjects do not always update correctly compared to Figure 2a. Therefore, subjects may believe that the infor-mation received does not coincide with a ball drawn from one of the two urns. As a result, they might decide to discount or even ignore this information completely

when updating their beliefs in the second phase.

We can modify our model to accommodate the possibility of under-weighting information. Subjects may view the information as useless for making any inference, and thus ignore and attribute it to a “useless urn” added to our model to deal with such situations. If the information comes from the useless urn, each ball is drawn with equal probability. In other words, this information is completely random and not helpful to update any posteriors at all. The theoretical predictions of Pr(s₂|s1, ✓_max) derived in equation (5) becomes¹⁷

Figure 7b shows that subjects are still significantly prone to attributing infor-mation to the irrelevant urn when it is conflicting (59% 45%: t-test: p = 0.001;

Wilcoxon signed-rank test: p = 0.002). However, this e↵ect disappears for the as-signed urn—subject beliefs of the information source are not significantly di↵erent between conflicting and confirming information (25% ⇠ 21%: t-test and Wilcoxon signed-rank test: p > 0.1). Instead, the e↵ect is entirely on the useless urn, showing

17Equation 18 demonstrates how to break down the probability Pr(s2|s¹, ✓max) to three urns.

We can also apply the same method to the remaining three required probabilities, Pr(s2|s¹, ✓min), Pr(s2|s¹, !max), and Pr(s2|s¹, !min).

Figure 8: Information Sources Distributions: (a) Two Urns (b) Three Urns.

that subjects tend to ignore the information when it is confirming (33% 16%:

t-test: p < 0.001; Wilcoxon signed-rank test: p < 0.001). The distributions of sub-jects in the two models are shown in Figure 8, and individual beliefs of the source are listed in Table 6.

To illustrate the di↵erential processing of confirming and conflicting information, we consider three representative types: Subjects who attribute the information com-pletely to the assigned urn (pA = 1), completely to the irrelevant urn (pA= 0), and those close to Bayesian (pA= 0.5). Applying the same maximum likelihood estima-tion with these 3 types (p_A= 0, 0.5, 1) instead of 21 types (p_A = 0, 0.05,· · · , 1), we estimate individual types and classify subjects accordingly. The results shown in Ta-ble4indicate that 24.4% more subjects attribute the information completely to the assigned urn when it is confirming. In contrast, 10.6% more subjects attribute the information completely to the irrelevant urn when it is conflicting. Table 4 uncov-ers this alternation at the individual level. Subjects along the diagonal (49.6%) are consistent under both information. Importantly, the upper triangle subjects (37.4%, underlined) put more weight on the assigned urn when moving to confirming

infor-mation (from conflicting inforinfor-mation). In other words, these subjects exhibit an

“echo chamber e↵ect,” since they are more likely to believe that confirming infor-mation comes from their assigned urn and vise versa.

Table 4: Individual Type Transition: Conflicting vs. Confirming (%) Confirming pA

Conflicting pA 0 0.5 1 Total 0 21.1 21.1 12.2 54.4 0.5 6.5 25.2 4.1 35.8

1 1.6 4.9 3.3 9.8

Total 29.3 51.2 19.5 100.0

It is apparent that subjects are not necessary consistent between belief-updating of the assigned urn and the irrelevant urn. This may be caused by the inability to properly assign probabilities between the two urns. In particular, subjects could update the two urns independently, instead of comprehensively evaluate the infor-mation and simultaneously update their beliefs about the assigned and irrelevant urn. Hence, they utilize the information and assess the probability for it to come from each urn separately. If they deem the information irrelevant, it is attributed to a useless urn, in which each ball (1 to 100) is drawn with equal chance, instead of the other urn. Therefore, subjects assign underlying beliefs (pA, p_U) and (p_I, p_U) when assessing the assigned and irrelevant urn, respectively.

We compare underlying beliefs pAand pI when receiving confirming and conflict-ing information. Specifically, we predict underlyconflict-ing beliefs with a constant and the dummy for Confirming information to predict pA in each round, and cluster stan-dard errors at the subject level to control for repeated observations. We exclude choices which could only be rationalized with impossible beliefs that are not in the interval [0, 1], which happens more often for the irrelevant urn. This leaves us with

846 observations for the assigned urn, in contrast to 775 observations for the irrele-vant urn. Table5column (1) and (2) show that the directions of coefficients confirm the asymmetric updating. When the information is aligned with their priors, sub-jects put insignificantly more weight (2.4%) on the assigned urn, but significantly less (-18.7%, p < 0.001) weight on the irrelevant urn. However, notice that some information are more confirming or conflicting than others. For instance, when in-formation is 51, one can hardly infers anything. Similarly, the inin-formation may not really be confirming or conflicting for subjects where the initial draws are close to 50. Thus, we regard information as strongly confirming or conflicting when neither the initial draw nor the new information are between 40 and 60. The results shown in column (3) and (4) indicated that the e↵ects are even larger at 5.6% (p < 0.05) and -27.3% (p < 0.001) for the assigned and irrelevant urn, respectively.

Table 5: Independent Source Beliefs

Source Beliefs: (1)

Stronger Confirming/Conflicting 7 7 3 3

N 846 775 555 518

Table 6: Individual Source Beliefs

Two Urns Three Urns Two Urns Three Urns

Conflicting Confirming Conflicting Confirming Conflicting Confirming Conflicting Confirming

ID p_A p_A p_A p_I p_A p_I ID p_A p_A p_A p_I p_A p_I