Learning Whom to Trust : Decision-Generated Credibility in Social Learning

Gabriel Bontemps; Abhishek Banerjee

cs.NEAug 25, 2026

Learning Whom to Trust : Decision-Generated Credibility in Social Learning

Authors: Gabriel Bontemps, Abhishek Banerjee

Organizations: GREDEG-CNRS, Université Côte d’Azur, Nice, France · Blizard Institute, Queen Mary University of London, London, United Kingdom of Great Britain and Northern Ireland (the) · Adaptive Decisions Laboratory, Department of Pharmacology, University of Oxford, Oxford, United Kingdom of Great Britain and Northern Ireland (the)

Abstract

Social interaction can improve collective learning but also amplify early mistakes. We study this tension when the credibility of social information is generated by the sender's own decision process rather than fixed ex ante. Reinforcement-learning agents make binary choices through a drift--diffusion process that jointly determines choice, decision time, and confidence; decision confidence then becomes social credibility by weighting anticipatory influence and retrospective social learning. Under balanced community exposure, the anticipatory field admits an exact quotient representation. Its local Jacobian is a scalar decision-sensitivity term multiplying the community-coupling matrix, which yields a common-mode amplification threshold and an analytical role for cross-community permeability in damping relative community differences. Monte Carlo experiments show the corresponding non-monotone performance pattern: moderate transmission accelerates correction, whereas strong transmission can lock populations into wrong consensus; low permeability instead sustains disagreement. Ablations reveal a dual role for confidence: credibility-sensitive transmission amplifies social error, while confidence-dependent private learning stabilises it. The model yields testable predictions linking sender confidence to receiver behaviour conditional on accuracy.

Figures & tables

Tradition	Typical mechanism	Exogenous element	What this paper changes
DeGroot / consensus	weighted opinion averaging	weights, signals	confidence generated by the choice process
Herding / cascades	action inference	reliability of social signal	credibility is decision-generated
Cooperative bandits	reward learning across agents	communication weight	two distinct social channels
Polarisation ABMs	homophily / social feedback	confidence usually absent	confidence-driven amplification
RLDDM (individual)	value-based diffusion choice	no social network	social transmission of credibility

Table 1: Positioning relative to adjacent model families.

Figure 1: Mechanism. Within a trial, latent values $Q_{i,t}$ are augmented by the anticipatory credibility field $S_{i,t}$ (top, blue) to form subjective values $V_{i,t}$ ; a drift–diffusion process turns the value contrast into an action $A_{i,t}$ , a decision time $\tau_{i,t}$ and process-generated confidence $C_{i,t}$ (red box). After the reward is observed, values are revised by a simultaneous update combining the private prediction error with the retrospective social prediction error (bottom, orange), whose strength is modulated by neighbours’ confidence. Confidence is the shared link between the two channels. The community coupling matrix $B$ governs how credibility circulates.

Figure 2: Confidence as a decision-process readout (all curves are the exact model maps, not fits). (a) Evidence accumulates to a boundary $\pm a$ at rate $v=\beta\Delta$ ; strong evidence yields fast, confident choices, weak evidence slow, uncertain ones. (b) Confidence $C=\Lambda(\kappa_{1}|v|/a-\kappa_{2}\log(1+\tau/\tau_{0}))$ rises with evidence strength and falls with decision time; the dashed ridge is the process’s own mean decision time $\mathbb{E}[\tau\mid v]$ . (c) Choice accuracy $P(+a)$ is monotone in the value contrast, but confidence is U-shaped in $\Delta$ — it tracks evidence magnitude , not correctness — so a strongly held wrong belief ( $\Delta<0$ , large) is transmitted with high credibility. This decoupling of confidence from accuracy is the decision-process basis for high-confidence error in the model.

Figure 3: Confidence is a valence-gated learning gain. Reading the $2\times 2$ matrix: the confidence-modulated learning rate ( 6 ) is largest ( $\alpha_{\max}$ ) for good news under low confidence and for bad news under high confidence, and smallest ( $\alpha_{\min}$ ) on the other diagonal. A low-confidence agent is therefore optimistic (soaks up success, brushes off failure); a high-confidence agent is revising (corrects on failure, discounts success) — a confidence-gated analogue of asymmetric reward-prediction-error learning. The same confidence $C$ plays a second role in social transmission. The normalised quantity $w_{i}=C_{i}/\sum_{j}C_{j}$ shown in the schematic summarises relative confidence within the local signal pool; the operative anticipatory exposure in the model remains $W_{ij}C_{j,t}$ as defined in Eq. ( 1 ). Confidence therefore acts both as an intra-agent learning gain and as a decision-generated multiplier on baseline social exposure. The exact curves are plotted in Appendix F (Figure 14 ).

Figure 4: Illustrative baseline regimes. Mass on the optimal arm $m_{c,t}(a^{\star})$ in each community (mean and $10$ – $90$ percentile band over $R=300$ Monte Carlo replications; $N=400$ agents, horizon $T=300$ ). The same mechanism produces efficient consensus, wrong consensus and polarisation under the scenario parameters of Table 5 .

Regime	$R$	$\lambda$	$P(\mathrm{regime})$ [Wilson $95\%$ LB]	$\mathcal{R}_{T}$ (SE)
Efficient	300	0.40	1.00 [0.987]	450 (1)
Wrong	300	0.90	1.00 [0.987]	11,988 (0)
Polarised	300	0.90	1.00 [0.987]	5,998 (0)

Table 2: Illustrative baseline regimes. Each scenario is run for

R

Monte Carlo replications; the empirical frequency of the designated regime is

1.00

in every case, reported with its Wilson

95\%

lower bound.

\mathcal{R}_{T}

is mean group regret with Monte Carlo standard error. Scenario parameters are in Table 5 .

Figure 5: Social-transmission phase diagram. Probability of efficient consensus, wrong consensus and polarisation, and the dominant regime, over the anticipatory social-transmission intensity $\lambda$ and cross-community permeability $\Gamma=1-B_{cc}$ . A corrective band at moderate $\lambda$ separates an unresolved region ( $\lambda\to 0$ ) from a distortive region (large $\lambda$ ); within the distortive region, low permeability yields polarisation and high permeability yields wrong consensus. Grid $13\times 13$ , 120 replications per cell, regime threshold $0.9$ , horizon $T=240$ ; the first three panels show regime probabilities and the fourth the dominant (modal) regime. The Wilson half-width at 120 replications is at most $0.09$ .

Figure 6: Amplification bifurcation of the representative-community reduction. The early-lead order parameter $z^{\star}$ is zero (stable neutral state) below $\lambda^{\star}$ and splits into a correct ( $+z^{\star}$ ) and a wrong ( $-z^{\star}$ ) amplified branch above it. The transition is mildly subcritical (a bistable window precedes $\lambda^{\star}$ ), so a sufficiently large early lead can lock in just below threshold. Small reward gap $|\Delta\mu|=0.06$ , $B_{cc}=0.85$ .

Figure 7: How early community leads evolve as social influence strengthens. Streamlines show the direction of motion, colour indicates its speed, and dots mark fixed points. (a) Under weak influence, shared and opposing leads both decay toward neutrality. (b) At intermediate influence, shared leads grow while opposing leads decay, producing consensus. (c) Under strong influence, both modes grow, permitting corrective $(+,+)$ , distortive $(-,-)$ and polarised $(\pm,\mp)$ endpoints. The three panels therefore isolate the modal information not contained in the scalar bifurcation diagram. Frozen-value reduction, $|\Delta\mu|=0.06$ , $B_{cc}=0.85$ .

Figure 8: Mechanism isolation under common random numbers ( $R=300$ ). (a) Regime composition for the full model and five ablations at the contested operating point (Table 5); (b) mean group regret with $95\%$ bootstrap intervals; (c) paired structural contrast in wrong-consensus incidence, $\Delta P(\mathrm{wrong})=P(\mathrm{wrong}\mid\mathrm{full})-P(\mathrm{wrong}\mid\mathrm{ablation})$ , with paired-bootstrap $95\%$ intervals. Positive contrasts indicate that removing the component lowers wrong-consensus incidence; negative contrasts indicate that removing it raises wrong-consensus incidence. At this operating point, removing the anticipatory channel, the retrospective channel, or the decision-time confidence map eliminates wrong consensus; removing credibility weighting weakens but does not eliminate it; constant learning rates intensify it.

Variant	$P(\mathrm{wrong})$ [ $95\%$ ]	$P(\mathrm{efficient})$ [ $95\%$ ]	$\mathcal{R}_{T}$ [ $95\%$ ]	$\Delta P(\mathrm{wrong})$ [ $95\%$ ]
Full	0.83 [0.78, 0.87]	0.17 [0.13, 0.22]	8,571 [8,148, 8,962]	—
No anticipatory	0.00 [0.00, 0.01]	0.00 [0.00, 0.01]	3,913 [3,908, 3,919]	+0.83 [+0.79, +0.87]
No retrospective	0.00 [0.00, 0.01]	1.00 [0.99, 1.00]	438 [433, 442]	+0.83 [+0.79, +0.87]
No credibility weighting	0.73 [0.67, 0.77]	0.27 [0.23, 0.33]	7,501 [7,002, 7,996]	+0.10 [+0.04, +0.16]
Constant learning rates	0.99 [0.98, 1.00]	0.01 [0.00, 0.02]	10,239 [10,142, 10,303]	$-0.16$ [ $-0.21$ , $-0.12$ ]
Alternative confidence	0.00 [0.00, 0.01]	1.00 [0.99, 1.00]	911 [882, 941]	+0.83 [+0.79, +0.87]

Table 3: Mechanism isolation at the contested operating point (Table 5 ),

R=300

replications under common random numbers.

P(\mathrm{wrong})

and

P(\mathrm{efficient})

carry Wilson

95\%

intervals; regret carries a

95\%

bootstrap interval. The last column reports the paired structural contrast

\Delta=P(\mathrm{wrong}\mid\mathrm{full})-P(\mathrm{wrong}\mid\mathrm{ablation})

, with a paired-bootstrap

95\%

interval. A positive

\Delta

indicates that removing the component reduces wrong-consensus incidence, whereas a negative

\Delta

indicates that removing it increases wrong-consensus incidence. A component is described as necessary at this operating point only when its removal reduces

P(\mathrm{wrong})

to approximately zero.

Figure 9: Decision-process confidence and early confident error. Distribution of confidence $C$ for correct and wrong choices, early ( $t<T/2$ ) and late ( $t\geq T/2$ ), in a neutral contested scenario; densities are normalised to unit area and dashed lines mark group means. Early decisions number $\approx 1.8\times 10^{6}$ (correct) and $\approx 6\times 10^{4}$ (wrong). Early wrong choices carry a wide range of confidence, including high values (mean $0.73$ versus $0.98$ for correct choices): confidence attaches to error, which is what lets credibility-weighted transmission amplify it. The right panel shows that confidence concentrates near one late in learning as communities reach consensus.

Figure 10: Validity of the community-level approximation. (a) Mean absolute endpoint error over the $11\times 11$ $(\lambda,\Gamma)$ grid, with contours delimiting one-, five- and ten-percentage-point tolerances. (b) Empirical share of parameter cells meeting each tolerance, stratified by distance from an agent-model regime boundary. All 36 interior cells are within one percentage point; errors are concentrated among the 85 boundary or adjacent cells. (c) Mean absolute micro–meso error over time for efficient consensus, wrong consensus and a four-community stress test; the logarithmic scale distinguishes small persistent discrepancies from large but transient ones. The dotted guides in panels (b) and (c) mark a five-percentage-point tolerance.

Figure 11: Computational-to-neural correspondence. Each model quantity (left) implements a decision computation (centre) with a candidate neural correlate (right). A functional analogy: it motivates the measurements and manipulations of Figure 12 ; the model commits to the centre column.

Figure 12: A model-based experimental design. Inducing confidence orthogonally to accuracy realises the confident-error cell (upper right), where the model predicts distortive transmission. The model supplies the regressors and falsifiable signatures (right), turning predictions (i)–(iv) into a testable protocol for dyadic or networked group-decision experiments.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 13: Trajectory-level comparison of the agent model and its community approximation. Solid lines report the agent-ensemble mean and dashed lines the deterministic quotient; vertical position is the probability of choosing the better option, displayed with log-odds spacing so differences near zero and one remain visible. (a) In efficient consensus, both models select the better arm but the quotient converges faster. (b) In wrong consensus, both select the worse arm and their aggregate paths remain close. (c) In the four-community stress test, colours identify communities and line style identifies model level; both models preserve direction and community ordering, with a persistent difference in tail probabilities.

Symbol	Default	Role	Config key
$\beta$	6.0	value sensitivity (drift scaling)	beta
$\sigma$	1.0	internal decision noise	sigma
$a$	1.0	speed–accuracy threshold	a_thr
$\kappa_{1}$	3.0	confidence: evidence weight	kappa1
$\kappa_{2}$	1.0	confidence: decision-time weight	kappa2
$\tau_{0}$	0.5	reaction-time scale	tau0

Appendix

Table 4: Global default parameters ( code/credibility/config.py ). Confidence map is decision (Eq. 5 ); values are projected to

[0,1]

each trial; reaction times use the inverse-Gaussian approximation.

Scenario	$N$	$T$	$R$	$\Delta\mu$	$\lambda$	$\eta$	$\Gamma$	init. lead
Fig. 2 efficient	400	300	300	0.20	0.40	0.30	0.30	neutral
Fig. 2 wrong	400	300	300	0.10	0.90	0.20	0.30	asym.
Fig. 2 polarised	400	300	300	0.10	0.90	0.25	0.02	asym.
Fig. 3 phase ( $\lambda,\Gamma$ swept)	160	240	120	0.10	[0,1.6]	0.25	[.01,.5]	asym.
Fig. 4 ablation point	400	260	300	0.10	0.60	0.30	0.15	asym.
Fig. 5 confidence	300	260	60	0.10	0.50	0.25	0.15	neutral

Appendix

Table 5: Scenario parameters for all reported figures (standard scale).

N

is total agents,

T

horizon,

R

replications,

\Delta\mu=\mu_{1}-\mu_{2}

\lambda

anticipatory weight,

\eta

retrospective weight,

\Gamma=1-B_{cc}

permeability. Regime threshold

0.9

throughout. Configuration: code/credibility/scenarios.py .

Model element	Equation	Code	Test
Anticipatory signal	Eq. 1	model.py: anticipatory_field_block	test_quotient_field_exact
Augmented value	Eq. 2	model.py: simulate_blocks	test_block_dense_*
DDM choice	Eq. 4	ddm.py: choice_prob_up	test_ddm_choice_*
Reaction time	§ 3	ddm.py: sample_decision_time	test_ddm_mean_fpt_*
Confidence	Eq. 5	confidence.py	test_confidence_bounded
Private learning	Eq. 6	model.py: _private_lr	test_simultaneous_*

Appendix

Table 6: Equation-to-code-to-test map. Code paths are under code/credibility/ ; tests under code/tests/test_credibility.py .

Figure 14: Quantitative companion to Figure 3 . (a) the exact confidence-gated learning rates for good news ( $\alpha_{\max}-(\alpha_{\max}-\alpha_{\min})C$ ) and bad news ( $\alpha_{\min}+(\alpha_{\max}-\alpha_{\min})C$ ), crossing at $C=\tfrac{1}{2}$ . (b) An illustrative normalised relative-confidence index $C/(C+\bar{C})$ and the social learning rate $\gamma C^{\omega}$ ( $\omega=1,2$ ) versus sender confidence. The operative anticipatory exposure in the model is $W_{ij}C_{j,t}$ as defined in Eq. ( 1 ).

Check	Variation	Result
R1 threshold	$0.80/0.90/0.95$	wrong-area $=0.53/0.53/0.53$ (invariant)
R2 horizon	$T=200\!-\!1000$	near-zero stays unresolved; corrective/distortive stable
R3 initial lead	$\delta=0\!-\!0.16$	P(wrong) $=0.12\!\to\!1.00$ (small lead suffices)
R4 reward gap	$\|\Delta\mu\|=0.02\!-\!0.20$	P(wrong) $=1.00\!-\!1.00$ (persists)
R5 random graphs	balanced/SBM/noisy	distortive P(wrong) $=1.00/1.00/1.00$
R6 grid resolution	$7^{2}/9^{2}/13^{2}$	wrong-area $=0.49/0.53/0.51$

Appendix

Table 7: Robustness summary (Appendix G ). Entries are the dominant-regime area fractions of the

(\lambda,\Gamma)

plane, or P(wrong) at the operating point, as indicated. Full tables: paper/tables/robustness_*.csv .

Figure 15: Robustness of the phase structure. (a) regime area fractions are invariant to the terminal threshold; (b) P(wrong) versus horizon for three operating points; (c) P(wrong) and P(efficient) versus early-lead strength $\delta$ ; (d) P(wrong) versus reward gap with Wilson intervals; (e) P(wrong) under balanced, weighted-SBM and noisy graphs; (f) regime area fractions versus phase-grid resolution. Full tables in Table 7 .

Figure 16: Stochastically generated early lead (neutral initialisation, $R=200$ per $\lambda$ ). (a) P(wrong) is $\approx 0$ below the local amplification threshold $\lambda^{\star}\approx 0.67$ and rises above it, mirrored by a fall in P(efficient). (b) the frequency of early suboptimal leads rises with $\lambda$ and they are locked in with fidelity $\approx 1$ . Data: tables/endogenous_early_lead.csv .

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Social Learning is a decentralized decision-making paradigm in which spatially dispersed agents collect streaming observations regulated by one of a finite number of models (the hypotheses). The agents are interested in assigning probability scores (the beliefs) to the possible hypotheses. To this end, the agents exchange their beliefs according to a certain communication graph. It has been shown that, under reasonable conditions on the identifiability of the decision model and the network connectivity, each agent ultimately places all the belief mass on the true hypothesis governing the data. However, several questions remain unanswered regarding the evaluation of the social learning performance. One recently adopted performance metric is the rejection rate, i.e., the rate at which the beliefs about the erroneous hypotheses vanish. One contribution of this work is to establish that the rejection rate leads to several paradoxes, which make it unsuitable as a valid performance measure. We then focus on studying the error probability measure. For a binary Gaussian problem, we derive an analytical formula characterizing the ratio between the individual agents' probabilities and the optimal Bayesian probability. The formula shows that this ratio is expressed by the product of two terms quantifying the effect of the network connectivity and the role of the prior information. As a result, an irreducible gap emerges between the decentralized and the centralized error probabilities, which is agent-dependent and does not disappear asymptotically.

Abstract

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