Modeling Trust and Liquidity Under Payment System Stress: A Multi-Agent Approach

Abstract: Operational disruptions in retail payments can induce behavioral responses that outlast technical recovery and may amplify liquidity stress. We propose a multi-agent model linking card payment outages to trust dynamics, channel avoidance, and threshold-gated withdrawals. Customers and merchants interact through repeated payment attempts, while customers additionally influence one another on a Watts-Strogatz small-world network. Customers update bounded memory variables capturing accumulated negative experience (scar) and perceived systemic risk (rumor), with merchants contributing persistent broadcast signals that may lag operational recovery. We prove that, under mild conditions on memory persistence and threshold gating, aggregate withdrawal pressure can peak strictly after the outage nadir, including during the recovery phase. Simulations reproduce behavioral hysteresis and confirm delayed peaks of outflows. We further study payment substitution via instant transfer: substitution consistently reduces peak avoidance, yet its effect on cumulative outflows is non-monotonic under realistic merchant broadcast persistence. Robustness experiments across random seeds show stable qualitative behavior. The model highlights why "status green" is not equivalent to risk resolution and motivates incident response strategies that address perception, merchant messaging, and post-recovery communication in addition to technical remediation.

• The paper develops a discrete-time multi-agent model linking payment failures, trust erosion, merchant messaging, social contagion, and threshold-gated withdrawals without requiring solvency shocks or strategic coordination.
• Simulations show that withdrawal pressure peaks after minimum payment reliability, often during technical recovery, because customer scar, rumor, and avoidance accumulate with bounded memory and sticky merchant broadcasts.
• The paper finds that instant-transfer substitution lowers peak avoidance by roughly 1–2 percentage points but has non-monotonic effects on cumulative outflows, making it a panic-dampening tool rather than a complete liquidity-risk control.

Overview

"Modeling Trust and Liquidity Under Payment System Stress: A Multi-Agent Approach" (2602.16186) develops a discrete-time multi-agent system (MAS) that links card payment outages to trust erosion, channel avoidance, and threshold-gated deposit withdrawals. The central claim is that technical recovery and behavioral recovery are systematically decoupled: under mild assumptions, aggregate withdrawal pressure peaks strictly after the outage nadir, often during the recovery phase. The paper formalizes this delayed-peak result analytically and corroborates it with agent-based simulations, and further shows that payment substitution via instant transfers reduces peak avoidance but has a non-monotonic effect on cumulative outflows.

Positioning in the literature

The paper sits at the intersection of four literatures that it argues have not been integrated end to end. Diamond–Dybvig-style bank run models formalize self-fulfilling withdrawals but abstract from transaction execution; Granovetter's threshold models and Watts–Strogatz network science explain cascades and diffusion topology; agent-based bank run models (e.g., dos Santos and Nakane) incorporate neighborhood effects but omit the payment experience layer; and operational risk studies of payment systems (Schmitz et al.; BCBS, IMF, Bank of England policy work) treat disruptions from an infrastructure and governance perspective, leaving end-user behavioral response outside the model. The contribution is to couple payment reliability directly to depositor behavior: runs emerge as a conditional, higher-order outcome of accumulated adverse micro-experiences, merchant messaging, and social contagion, rather than as an immediate reaction to balance sheet concerns or coordination failure alone.

Model structure

The system comprises three components: customer agents, merchant agents, and an exogenous payment infrastructure. Customers are embedded in a static Watts–Strogatz small-world network and hold state variables for trust $T_i(t)$, accumulated negative experience ("scar") $C_i(t)$, perceived systemic risk ("rumor") $R_i(t)$, a behavioral mode $$S_i(t) \in \{\text{OK}, \text{FRUSTRATED}, \text{AVOIDING}\}$$, and a deposit balance. Merchants maintain a private operational state and an externally visible broadcast state, which need not coincide: broadcast states carry a persistence timer $\tau_m$ that keeps "cash only"/degraded signage active for a minimum dwell time after operational recovery.

Payment attempts follow a stochastic process with individual baseline propensity, a common peak-demand multiplier $D(t)$, and a mode-dependent activity factor. Outcomes are SUCCESS, FAILURE, or UNKNOWN, with uncertain outcomes weighted at least as damaging as explicit failures ($\alpha_u \ge \alpha_f$), reflecting the empirical observation that timeouts are more corrosive to trust than rejections. Scar and rumor evolve as contractive bounded-memory processes; trust is a bounded moving average of normalized experience signals eroded by scar. Behavioral mode transitions are threshold rules on effective trust $T_i(t) - \kappa_C C_i(t)$, and rumor aggregates merchant broadcast severity (mapped to a 0/0.5/1 scale) with the fraction of avoiding neighbors.

Withdrawals are doubly gated: a customer must be in AVOIDING mode with scar and rumor above thresholds $\theta_C$ and $\theta_R$, and then withdraws with a sigmoid probability of $C_i(t)$0, removing a fraction of the remaining balance. Aggregate outflow $C_i(t)$1 feeds back into rumor formation, providing an additional endogenous amplification loop. The infrastructure itself is exogenous and does not adapt to agent behavior — a deliberate abstraction the authors flag as a limitation.

The delayed-peak theorem

The main theoretical result, Theorem 1, states that under four assumptions — (A1) contractive bounded-memory dynamics for scar and rumor, (A2) threshold-gated avoidance and withdrawal eligibility, (A3) sticky merchant broadcasts, and (A4) monotone exogenous technical recovery — there exist outage and recovery trajectories such that $C_i(t)$2, where $C_i(t)$3 is the time of minimum payment success probability. The proof sketch proceeds in three steps: memory variables continue accumulating past the outage nadir due to persistence parameters; threshold crossings therefore occur during recovery; and broadcast persistence sustains rumor inputs beyond technical restoration. The result requires no strategic coordination, full information, or balance sheet deterioration — it arises purely from bounded memory, local information, and threshold rules. This is the paper's strongest claim, and its practical implication is direct: conventional operational metrics such as uptime, authorization rates, and mean time to recovery are insufficient indicators of systemic risk during the post-incident window. It should be noted that the theorem is existential — it establishes that delayed peaks can occur, not that they must under all parameterizations — and the proofs are presented as analytical sketches rather than full formal derivations.

Simulation findings

Simulations with $C_i(t)$4–$C_i(t)$5 customers and $C_i(t)$6–$C_i(t)$7 merchants, run across twelve random seeds with paired-seed comparisons for policy interventions, support three findings.

Behavioral hysteresis. Customer avoidance peaks after the outage nadir and decays slowly during recovery, while merchant broadcasts remain degraded well into the recovery phase. Perceived reliability recovers substantially more slowly than technical reliability.

Delayed withdrawal peaks. In all examined runs, maximum withdrawal pressure occurs strictly after minimum payment success probability, coinciding with the recovery phase. Withdrawals are negligible early in the outage because the compound eligibility conditions (sustained avoidance plus elevated scar and rumor) take time to accumulate. This is the paper's most operationally consequential result: liquidity risk can intensify precisely when systems are declared "status green."

Merchant broadcasts as a first-order amplifier. Simulations without broadcast stickiness show faster declines in avoidance and lower cumulative withdrawals, identifying merchant messaging — signage, cashier guidance, degraded acceptance — as an amplification channel distinct from both social contagion and infrastructure reliability. The authors argue this makes merchant communication a first-order policy lever rather than a secondary artifact.

The substitution paradox

The most nuanced result concerns instant transfer substitution. Substitution consistently reduces peak avoidance across all seeds (by roughly one to two percentage points in paired simulations) because successful alternative-channel transactions convert adverse experiences into reinforcing success signals at the height of the incident. However, its effect on cumulative outflows is non-monotonic under realistic broadcast persistence. The mechanism is that substitution keeps customers operationally engaged while behavioral risk remains elevated: customers who would otherwise disengage remain active and capable of executing withdrawals once eligibility thresholds are crossed. Usage of the substitution channel is intermittent rather than continuous, so the residual stream of card failures and timeouts continues to accumulate scar and sustain rumor. The operational implication is that substitution should be treated as a panic-dampening mechanism, not a complete risk control; it must be paired with interventions that actively accelerate rumor and scar decay. Robustness analysis confirms the qualitative patterns across seeds, though tail events in peak run pressure persist under substitution.

Limitations and open questions

The authors are explicit that the model is explanatory rather than predictive and is not calibrated to any institution or historical incident; parameters reflect plausible ranges, so numerical magnitudes should be read qualitatively. Several structural limitations bear directly on the results. The model omits bank balance sheets, interbank exposures, funding constraints, and central bank interventions, so it cannot speak to solvency outcomes or second-round systemic cascades. Customer decision-making is threshold-based rather than strategic or adaptive, abstracting from learning and heterogeneous information processing. Merchant behavior is stylized and excludes acquirer guidance and contractual incentives. The infrastructure is exogenous, precluding feedback loops between customer behavior (retries, load) and technical performance — a coupling that could plausibly strengthen or weaken the delayed-peak mechanism. Finally, only one substitution channel is modeled, and policy interventions such as withdrawal limits, deposit guarantees, and public communication are not endogenized. The most pressing open question the paper leaves is empirical: estimating memory decay rates, broadcast persistence times, and substitution usage patterns from incident logs and transaction-level data to validate whether the simulated delayed-peak magnitudes approximate real incidents.

Conclusion

This paper provides a coherent microfounded framework in which payment outages generate liquidity stress through bounded memory, threshold-gated behavior, sticky merchant messaging, and small-world social contagion, without requiring balance sheet deterioration or strategic coordination. Its formal and simulated results converge on the conclusion that peak run pressure can occur after technical recovery, that merchant broadcast persistence is a structural amplifier of behavioral risk, and that substitution channels mitigate peak panic while leaving cumulative outflow risk unresolved. The framework offers supervisors and incident responders a rationale for extending resilience metrics beyond technical indicators to behavioral and perceptual state variables, and a tractable base for future integration of balance sheet dynamics, endogenous infrastructure performance, and empirical calibration.