The Echo Chamber

In principle, subjects should update their beliefs of both urns regardless of the information received in the second phase because there is always a chance the new information could be from either urn. However, the irrelevant urn has the natural

8We test the coefficient 3 from the model: Beliefs 0 + 1Bayesian + 2Confirming +

3Interaction + ✏, where the dummy variable Confirming indicates the new information is con-firming (=1) or not (=0), Interaction is the interaction term of Bayesian and Concon-firming.

advantage that one should only update it according to the new information regarding the ball of the second phase, since the first ball only carries information about the assigned urn. Therefore, we can easily infer how subjects attribute new information to each urn in the second phase from their updating behavior.

Figure4 plots elicited probabilities against second-phase information.⁹ The red dots are elicited beliefs around 0.5, adhering to the Bayesian prediction of the first phase, indicating “fully dissociate” subjects who do not update irrelevant urn beliefs at all (and should completely attribute the new information to the assigned urn). On the other hand, the blue crosses along the 45-degree line indicate “fully attribute”

types who completely ignore the fact that there is some probability that the new information is from their assigned urn.¹⁰ These two types are strongly biased since they put extreme weight on the new information when updating the irrelevant urn.

However, they account for 76.7% of the choices when we allow 5 percentage points of error. The intermediate types with more reasonable weights are shown as green triangles in Figure4, but consist only 18.7% of the choices. This includes those who follow Bayesian updating. Lastly, the remaining 4.6% of choices in black are difficult to rationalize, and might reflect confusion or some other information processing method. We summarize the updating behavior in the Table2.

In Figure 5, we separate second-phase information into confirming and conflict-ing information as defined in section 3.1.2. To compare the di↵erence in behavior between receiving confirming and conflicting information, we use a dummy

indi-9We drop the choices if their first phase beliefs of the irrelevant urn are out of the range, [0.45, 0.55]. The remaining choices plotted in the Figure4contain 74% of the data.

10The purple dot-cross symbols are overlapping area of the two types, in which we cannot distinguish their types.

Figure 4: Types of Behavior (Irrelevant Urn)

Table 2: Types of Behavior (Irrelevant Urn)

Types of Choices Definition Percentage

Either Either fully dissociate or fully attribute type. 16.3 % Fully Dissociate Other subject’s information comes from the assigned urn. 25.4 % Fully Attribute Other subject’s information comes from the irrelevant urn. 35 % Intermediate Put reasonable weights on other subject’s information 18.7 % Others Choices cannot be classified into above four types. 4.6 %

cating confirming information to predict the occurrence of two distinct types of behavior, completely attribute the information to the assigned urn (Fully Disso-ciate) and the irrelevant urn (Fully Attribute). Table 3 report fixed-e↵ect panel regression results clustered at the subject level, predicting whether the inferred prior belief fully attributes the new information to the irrelevant urn using whether information is confirming or not. For confirming information, 33.7% of the choices completely attribute the new information to the assigned urn, while 31.1% of the choices completely attribute the new information to the irrelevant urn. However, when subjects receive conflicting information, only 16.5% of the choices attribute new information to the assigned urn, significantly lower than that under confirming information. Moreover, 39% of the choices completely attribute new information to the irrelevant urn, significantly higher than that under confirming information. This results demonstrates a confirmation bias where subjects overweight (underweight) the possibility that new information came from the assigned urn when it confirms (refutes) their prior.

Table 3: Attribution of the Information

(1) (2)

Fully Attribute to Assigned Urn Irrelevant Urn Confirming Information 0.165^⇤⇤⇤ -0.079^⇤⇤⇤

(0.022) (0.025)

Constant 0.172^⇤⇤⇤ 0.390^⇤⇤⇤

(0.017) (0.019)

N 914 914

Note: Standard errors in parentheses,^⇤ p < 0.05,^⇤⇤ p < 0.01,^⇤⇤⇤ p < 0.001

Among those who completely attribute the new information to the irrelevant urn (Fully Attribute), their updated beliefs of the assigned urn should remain unchanged

Figure 5: Elicited Beliefs of the Irrelevant Urn: (a) Confirming, and (b) Conflicting Information.

because they believe the information is coming solely from the irrelevant urn. Indeed, the posteriors of the assigned urn show that 75% do not update the assigned urn beliefs much.¹¹ The remaining 25% also changes their beliefs regarding the assigned urn, overreacting the new information.

In contrast, among those who completely attribute the new information to the assigned urn (Fully Dissociate), beliefs of the assigned urn should be updated as if they have two balls from that urn, resulting in a Bayesian updating process similar to equation (3) in section 2.3.2.¹² Unexpectedly, 54% of these choices stick to their first-phase posteriors of the assigned urn. This implies at least 25.4%⇥ 54% = 13.7% of all choices completely ignore the new information and update neither urn.¹³ Figure 6 plots the remaining choices after excluding those which completely ignore the

11This number is calculated by allowing 5% error. In fact, 63% have the exact same first and second posterior beliefs.

12The Bayesian prediction of having two balls from the same urn is: Pr(✓max|s¹, s2) = Pr(s2|✓^max)· Pr(✓^max|s¹)/ [Pr(s2|✓^max)· Pr(✓^max|s¹) + Pr(s2|✓^min)· Pr(✓^min|s¹))].

1313.7% is the lower bound since 25.4% excludes choices when second phase information are close to 50 that could be either Fully Dissociate or Fully Attribute.

new information. Figure 6a compares the elicited probabilities of fully dissociate types and the Bayesian posterior assuming that both balls came from the same urn.

Even though subjects fully dissociate the information from the irrelevant urn, the updating behavior systematically under-weights the new information from the other subject, resulting in a slope of 0.67 that is significantly lower than 1 (p < 0.001).

In fact, the elicited probabilities are closer to the Bayesian probability prediction derived in section 2.3.2 (Figure 6b), although the slope (0.78) is still lower than 1 (p < 0.001).

Figure 6: Fully Dissociate: (a) Two Balls from Assigned Urn. (b) Correct Bayesian.