Decomposing Differences in Learning from Private and Social Information

For experimental details, see the Experimental Design (Public) section.

2026-08-25

2026-10-31

Posterior beliefs: the reported probability (0–100%) assigned to the state favoured by the information received.

The objects of interest are the three adjacent treatment comparisons, each measuring one mechanism's contribution to the overall difference in response to private and social information.

I analyse posterior beliefs in two ways. First, I regress beliefs on treatment indicators, controlling for signal precision. Differences between adjacent treatment coefficients measure each mechanism's contribution in percentage points of reported probability. Second, I estimate a Grether-style regression of the log-odds of reported beliefs on the log-likelihood ratio of the information interacted with treatment indicators. The interaction coefficients represent the weights placed on each information type in belief updating. Adjacent differences between these coefficients provide the corresponding decomposition.

For both analyses, belief reports are winsorised at the 5th and 95th percentiles separately within each signal precision level. Both specifications control for individual, part-order, round-order, and signal-presentation-version fixed effects. Standard errors are clustered at the individual level.

The secondary outcomes are willingness-to-pay (WTP), confidence, and usefulness rankings for the four information types.

Beyond belief updating, I examine additional aspects of participants' responses to private and social information: their valuation of each information type, their confidence in beliefs formed from it, and their subjective rankings of its usefulness. I also compare WTP, confidence, and usefulness rankings across the four information conditions to explore the potential mechanisms behind these differences.

I also consider several exploratory analyses:

1. Individual-level associations across outcomes. I test whether participants' responses to different information types show consistent or different patterns across beliefs, WTP and confidence.
2. Heterogeneity by signal precision. I test whether each mechanism’s contribution varies across the five signal precision levels.
3. Heterogeneity in participant response types. I test whether participants can be grouped according to their response patterns across the four information conditions. For example, one mechanism may account for most of the overall difference for some participants, whereas others may respond similarly to all four information types.
4. Beliefs about others' rationality. I test how much this belief can account for the difference in behaviour between the A-bot and A-other conditions.

For participants with a unique switching point, WTP for an information type is measured as the midpoint value between the highest amount at which they choose the information and the lowest amount at which they choose the money. WTP is coded as the boundary value for participants who always choose the information or always choose the money.

Participants whose multiple-price-list choices exhibit wrong-direction switching (switch from money back to information as the monetary amount increases) will be excluded from the WTP analyses. If more than 20% of WTP observations are not well-defined, I will also use an alternative WTP measure: the number of rows in which the participant chooses the information.

The main specification regresses WTP on information-type indicators, controlling for participant and price list order fixed effects. Standard errors are clustered at the participant level.

Confidence is measured on a 0–100% scale and captures how certain participants are about their reported belief. The main specification regresses confidence on information-type indicators, controlling for signal precision, participant, part-order, and signal-presentation-version fixed effects. Standard errors are clustered at the participant level.

Usefulness rankings are coded from first (most useful) to fourth (least useful). I examine how the distribution of these rankings differs across the four information types.

This experiment is conducted online on Prolific. Participants complete multiple incentivised rounds of a standard balls-and-urns belief updating task. In each round, there are two boxes of balls, Box A and Box B, each containing a mixture of purple and green balls in symmetric compositions (Box B reverses the colour composition of Box A). The computer randomly selects one of the two boxes as the “True Box”, each with equal chance. Participants receive one piece of information about the True Box and then report the probability that the True Box is Box A versus Box B.

The experiment has 5 parts. The first 4 parts are the main within-subject treatments, with 5 rounds each. These parts vary the type of information participants receive and are presented in randomised order:
1. S-own: the participant draws one ball from the True Box by clicking a button.
2. S-other: the participant sees one ball drawn from the True Box for another participant.
3. A-bot: the participant sees the guess of a robot that sees one ball drawn from the True Box and follows the optimal rule.
4. A-other: the participant sees the guess of another participant who sees one ball drawn from the True Box.

In the A-bot and A-other parts, guesses are displayed as binary symbols indicating which box the actor considered more likely; participants never see the ball behind a guess. The guesses shown in A-other are matched to real choices made by other participants in the corresponding S-own rounds. If an A-other round is selected for bonus payment, the matched participant’s True Box determines the true state.

The four treatments are constructed so that the three adjacent comparisons isolate the mechanisms of interest: S-own vs. S-other identifies the preference for own information; S-other vs. A-bot identifies inferential complexity; and A-bot vs. A-other identifies beliefs about others' rationality.

Within each part, the signal precision (the share of the dominant colour in the box) takes five values: 5/7, 3/4, 4/5, 5/6, and 7/8. Each precision level appears exactly once per part, in random order. Box compositions are displayed in one of four presentation versions: basic (Box A has more purple balls), colour-swapped (Box B has more purple balls), scaled to larger counts at identical ratios, or scaled and colour-swapped. The four presentation versions are randomly assigned across parts without replacement. This ensures that no two parts show exactly the same compositions.

After the last round of each part, participants report their confidence: how certain they are (0-100%) that their belief report, within plus and minus 5 percentage points, is their best choice. At the end of the A-other part, participants additionally report how likely they think it is that the other participant’s guess was based on seeing a purple versus a green ball.

After completing all four parts, participants rank the four information types from most to least useful and briefly explain their ranking in free text.

In the fifth part, I elicit participants' willingness-to-pay for each information type using multiple price lists. The final round differs from the earlier rounds in that, instead of reporting a probability, participants make a binary guess about the True Box and earn a fixed bonus if their guess is correct and nothing otherwise. The final round uses a fixed composition with precision 11/15, which participants see before making their price-list choices. For each information type, participants choose in each of ten rows between seeing information of that type in the final round and receiving a fixed amount of extra money, ranging from $0 to $0.9 in $0.1 increments. The order of the four price lists is randomised. The computer then randomly selects one information type and one row and implements the participant's choice in the final round. Participants see the information if they choose it. If they choose the money, they do not see the information and receive the corresponding amount of extra money if this round is selected for payment. In both cases, they then guess the True Box and report their confidence in that guess.

Finally, participants answer a set of demographic and feedback questions.

The study also includes two attention checks. Participants who fail either attention check or receive a low-authenticity rating on either of Prolific's LLM or bot authenticity checks will be excluded from the main analyses.

Not available

All randomisation done by a computer.

The four information treatments vary within participant, and the order is randomised at the participant level.

No

350 participants

7000 posterior belief reports: 350 participants * 4 information types * 5 rounds 1400 confidence reports: 350 participants * 4 information types 1400 willingness-to-pay observations: 350 participants * 4 information types

All 350 participants complete all four within-subject treatments.