Invariant Causal Routing for Governing Social Norms in Online Market Economies

Xiangning Yu; Qirui Mi; Xiao Xue; Haoxuan Li; Yiwei Shi; Xiaowei Liu; Mengyue Yang

cs.LGMar 4, 2026

Invariant Causal Routing for Governing Social Norms in Online Market Economies

Authors: Xiangning Yu, Qirui Mi, Xiao Xue, Haoxuan Li, Yiwei Shi, Xiaowei Liu, Mengyue Yang

Organizations: Tianjin University, Tianjin, China · Institute of Automation, Chinese Academy of Sciences, Beijing, China · Peking University, Beijing, China · University of Bristol, Bristol, United Kingdom

Abstract

Social norms are stable behavioral patterns that emerge endogenously within economic systems through repeated interactions among agents. In online market economies, such norms -- like fair exposure, sustained participation, and balanced reinvestment -- are critical for long-term stability. We aim to understand the causal mechanisms driving these emergent norms and to design principled interventions that can steer them toward desired outcomes. This is challenging because norms arise from countless micro-level interactions that aggregate into macro-level regularities, making causal attribution and policy transferability difficult. To address this, we propose Invariant Causal Routing (ICR), a causal governance framework that identifies policy-norm relations stable across heterogeneous environments. ICR integrates counterfactual reasoning with invariant causal discovery to separate genuine causal effects from spurious correlations and to construct interpretable, auditable policy rules that remain effective under distribution shift. In heterogeneous agent simulations calibrated with real data, ICR yields more stable norms, smaller generalization gaps, and more concise rules than correlation or coverage baselines, demonstrating that causal invariance offers a principled and interpretable foundation for governance.

Figures & tables

Figure 1. Social norms in onlineeconomies.

Figure 2. Three-stage framework for discovering causal strategies in social norm formation. Stage I tests causal effects by identifying implicit contracts $(\theta,g\mid\psi)$ where policy $\theta$ enables group $g$ to reach the norm under context $\psi$ . Stage II learns a compact rule router $S^{\star}$ selecting effective strategies for robust norm attainment. Stage III explains success by attributing norms to key factors where user responses differ from the baseline $\theta_{0}$ .

Figure 3. Last-50-year trajectories of

\phi_{1}

(ST) and

\phi_{2}

(RI) with norm bands (per objective

\times

group). Shaded regions of the same color indicate the band within which individuals of the corresponding group are expected to remain under the associated social norm. When all trajectories lie within their respective same-colored shaded regions, the social norm is fully attained.

Norm	Baseline	Current	Group	PNS
RI-1	NI	FAI	ALL	0.966 [0.904, 0.993]
RI-1	GMV	FAI	ALL	0.981 [0.899, 1.000]
RI-1	BAL	FAI	ALL	0.962 [0.893, 0.992]
ST-1	BAL	GMV	ALL	0.857 [0.722, 0.933]
ST-1	FAI	GMV	ALL	0.887 [0.774, 0.947]
ST-1	UW	GMV	ALL	0.905 [0.779, 0.962]

Table 1. Representative PNS-significant causal routes selected by Stage I and Stage II. Entries report PNS with 95% binomial confidence intervals as value [lower, upper] .

Method	PNS ${}_{\text{train}}$ $\uparrow$	Cov ${}_{\text{train}}$ $\uparrow$	PNS ${}_{\text{test}}$ $\uparrow$	Cov ${}_{\text{test}}$ $\uparrow$	Rules $\downarrow$	Rules ${}_{\text{norm}}$ $\downarrow$	Gap $\downarrow$	Perf. $\uparrow$
PNS+Greedy	0.989	0.990	0.953	0.966	24	0.300	0.036	0.862
PNS+Greedy (pruned)	0.972	0.981	0.931	0.938	12	0.150	0.041	0.883
Corr+Greedy (Pearson)	0.805	0.958	0.741	0.931	46	0.575	0.064	0.600
Corr+Greedy (Pearson, pruned)	0.742	0.840	0.677	0.796	18	0.225	0.065	0.627
Coverage-Driven	0.622	0.944	0.565	0.915	48	0.600	0.057	0.449
Coverage+Corr Hybrid	0.416	0.564	0.384	0.550	80	1.000	0.032	0.114

Table 2. Ablation results on disjoint train/test seeds (Perf recomputed with

K{=}80

). Higher

\uparrow

is better; lower

\downarrow

is better.

Figure 4. Key factors attribution for invariant causal routes.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

IC	CRRA	IFE	$e_{p}$	$e_{q}$	$\rho_{e}$	$\sigma_{e}$	$super_{e}$
IC1	1.0	2.0	$2.2\!\times\!10^{-6}$	0.990	0.982	0.20	504.3
IC2	1.5	1.8	$5.0\!\times\!10^{-6}$	0.985	0.975	0.18	400
IC3	0.7	2.5	$8.0\!\times\!10^{-6}$	0.985	0.990	0.15	350
IC4	1.2	2.2	$1.0\!\times\!10^{-5}$	0.920	0.950	0.22	450
IC5	2.0	1.5	$3.0\!\times\!10^{-6}$	0.995	0.990	0.25	600

Appendix

Table 3. Initial-condition (IC) configurations used in all experiments.

Item	Value
Population	$n_{\text{households}}=100$
Episode length (env)	300 steps (episode termination)
Model / Buffer	hidden size $=256$ ; replay buffer $=10^{6}$
Optimization	Adam; $lr=3\times 10^{-4}$ ; batch $=128$ ; $\gamma_{\text{RL}}=0.975$ ; $\tau=5\times 10^{-3}$
Schedule	$1500$ epochs; epoch length $=500$ env steps (across episodes); update freq.=2; eval $=100$ episodes
PPO (aux)	$\gamma_{\text{PPO}}=0.99$ ; $\tau=0.95$ ; clip $=0.1$ ; $v_{\text{loss}}=0.5$ ; ent $=0.01$

Appendix

Table 4. Training parameters. One step = one year; evaluation window

T{=}50

(post burn-in).

IC	Train seeds	Test seeds
IC1	1101, 1103, 1105, 1107	1102, 1104, 1106, 1108
IC2	1201, 1203, 1205, 1207	1202, 1204, 1206, 1208
IC3	1301, 1303, 1305, 1307	1302, 1304, 1306, 1308
IC4	1401, 1403, 1405, 1407	1402, 1404, 1406, 1408
IC5	1501, 1503, 1505, 1507	1502, 1504, 1506, 1508

Appendix

Table 5. Seeds for ablation experiments. Train uses odd seeds; Test uses even seeds within each IC.

Stage	Time	Space
Stage I (PNS)	$\mathcal{O}(C\,M\,G)$	$\mathcal{O}(1)$
Stage II (routing)	$\mathcal{O}\big(CG^{2}M+RC+R\log R+LRBU+L^{2}BU\big)$	$\mathcal{O}(RBU)$
Stage III (attribution)	$\mathcal{O}(LU)$	$\mathcal{O}(U)$

Appendix

Table 6. Asymptotic complexity (Big-

O

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