Trust evolves according to three components: i) A baseline contribution to trust (e.g., from institutional or reputational factors). ii) Erosion from dishonesty. iii) Natural decay of trust in the absence of reinforcement. Formally, trust follows:

where U ( t ) ∈ [ 0 , 1 ] , c is the baseline trust contribution, α > 0 captures the marginal erosion of trust due to lying, and δ > 0 is the natural decay rate.

At each instant, the advisor derives utility from two sources: i) The value of trust itself, reflecting the long-term benefit of credibility. ii) Immediate gains from deception, which increase in trust—because lies are more effective when credibility is high—but grow concavely with the level of deception.

The payoff structure therefore captures the dual role of credibility: it provides ongoing value and at the same time amplifies the benefit of lying.

The advisor selects a deception strategy U(t) over the horizon [0, T] to maximize the present value of discounted utility:

where the integrand consists of two parts: i) Trust value S ( t ) , reflecting the long-term benefit of credibility. ii) Deception gain γ S ( t ) f ( U ( t ) ) , capturing the immediate benefit of lying, which scales with trust and the incentive parameter γ , f ( U ) is concave and increasing, so that lying produces diminishing returns as intensity rises. In the baseline case, f ( U ) = U , yielding diminishing returns from lying intensity.

Equation (1) and Equation (2) define a nonlinear optimal control problem. Applying Pontryagin’s Maximum Principle reveals the potential for singular controls, where marginal costs and benefits of lying balance, complicating closed-form characterization. To address this, we adopt a direct transcription approach:

a) The horizon [0, T] is discretized into n intervals of length Δt = T/n.

b) Trust dynamics are imposed as difference equations.

c) The continuous-time problem is reformulated as a nonlinear program that can be solved numerically using modern interior-point algorithms.

This approach ensures tractability and provides accurate approximations to the optimal deception path.

The baseline solutions in vary smoothly, but in practice deception is often episodic. To capture this, we introduce three refinements:

1)Effort budget constraint. Advisors face a finite deception capacity B, limiting cumulative deception across the horizon. This discourages continuous moderate lying and pushes effort into bursts when trust is high: ∑ t = 0 n − 1 U t Δ t ≤ B .

2)Anti-middle penalty. Behavioral evidence suggests lying is deployed in an all-or-nothing fashion. We add a convex penalty term, λ 01 U t ( 1 − U t ) which vanishes at U t = 0 and U t = 1 but penalizes intermediate deception, reinforcing episodic bursts.

This “anti-middle” penalty is supported by extensive evidence in behavioral economics and psychology showing that individuals often avoid moderate levels of dishonesty and instead gravitate toward either full honesty or more substantial misreporting. [11] documents that even small lies generate moral and psychological costs, creating a threshold effect in lying behavior. Complementing this, [3] show that people manage their self-concept by maintaining a balance between economic gain and moral identity, making partial dishonesty especially aversive. Once individuals cross that moral boundary, however, larger lies can become psychologically easier—a mechanism further reinforced by [2], who demonstrate that moral disengagement reduces incremental psychological costs at higher deception levels. Together, these findings provide strong empirical grounding for our modeling assumption that intermediate deception levels carry disproportionately high subjective costs, naturally producing the episodic all-or-nothing patterns predicted by the model.

3)Alternative gain specification.In the baseline, returns are concave, γ S ( t ) U ( t ) , encouraging interior solutions. To amplify episodicity, we also consider linear returns γ S ( t ) U ( t ) , which favor concentrated effort. Under these refinements, deception emerges in bursts: lying switches on sharply when trust is high, persists briefly, and shuts down as trust erodes. Trust erosion acts as a natural cap on sustained dishonesty.

illustrates these episodic dynamics. Panel A shows U(t) spiking at discrete intervals; Panel B shows jagged declines in S(t) aligned with those bursts. Compared to , deception is no longer smooth but clustered when trust is abundant.

Figure 2. Episodic optimal deception and trust trajectories. (A). Optimal lying effort U(t)—episodic extension. (B). Trust S(t)—episodic extension.

Panels A and B plot optimal deception effort U(t) and trust S(t) under the episodic refinements, for γ = 0 (red), 0.50 (blue), 1 (green), 2 (purple), and 3 (magenta).

- Low incentives (γ = 0, 0.5). Deception is negligible, and trust remains high.

- Moderate incentives (γ = 1). Lying appears in short bursts early, depleting trust modestly before stabilizing.

- High incentives (γ = 2). Deception spikes more sharply, causing rapid trust decline followed by stabilization at a low level.

- Very high incentives (γ = 3). Lying is highly opportunistic: bursts occur abruptly and intensely, then vanish. Trust collapses quickly and stabilizes at a minimal level.

Together, and show episodic dishonesty as the natural outcome of constrained deception capacity and strong incentives. Bursts exploit high trust while it is available, and erosion then shuts down further lying.

We define an episode as a maximal consecutive set of periods with Ut > 0. shows how episode duration varies with the incentive intensity γ. Average (and max) episode length increases monotonically with γ: for γ = 0 episodes are absent; as γ rises, lying consolidates into fewer, longer bursts. In our episodic capacity setup, total time spent lying is constrained by the effort budget B; stronger incentives mainly reallocate that fixed capacity into longer, more concentrated episodes.

Line plot of average and maximum episode length against γ ∈ { 0 , 0.5 , 1 , 2 , 3 } .

We next introduce a two-state environment (Low/High). High states make deception more tempting; persistence is parameterized by P(H→H) and P(L→L).

(grid of 3D surfaces) introduces environmental persistence. When high-demand states are persistent, episodes lengthen dramatically under strong incentives. Frequent switching shortens deception. Persistence and incentives interact: γ magnifies the impact of stability, producing sustained lying when opportunities remain stable.

Figure 3. (A) Average lying episode length vs. incentive intensity; (B) Lying episode lengths vs transition probabilities.

Summary.Across , three insights emerge:

1) Baseline dishonesty is smooth and front-loaded ().

2) Structural refinements generate bursty episodes that mirror real dishonesty ().

3) Episodes lengthen with incentives and persistence, showing how external environments amplify internal motives to deceive ().

These findings highlight trust erosion as a natural brake but also reveal vulnerabilities: when incentives are high and environments stable, dishonesty becomes prolonged and opportunistic.

In the baseline model (), deception appears as smooth, front-loaded effort. However, when capacity limits and behavioral penalties are introduced (), lying emerges in distinct bursts. Advisors exploit high trust briefly and then stop once credibility erodes. This episodic pattern is consistent with empirical evidence that deception often occurs in concentrated episodes rather than gradual drifts. For managers, this suggests that dishonesty should not be expected to rise smoothly with incentives—it will instead cluster in critical moments when trust is abundant. Monitoring systems must therefore be sensitive to bursts of activity rather than average levels.

Stronger incentives make lying more attractive but also prolong deceptive episodes. demonstrates that average episode length increases monotonically with incentive intensity γ. In financial advising, tying compensation too strongly to short-term performance may inadvertently encourage sustained misrepresentation. In supply chains, performance-based contracts may incentivize exaggeration of demand forecasts. The challenge is to calibrate incentives so that they motivate effort without creating extended opportunities for dishonesty.

Environmental conditions interact with incentives to shape deception dynamics. As shows, when high-demand states are persistent, deceptive episodes become substantially longer, especially under strong incentives. In volatile environments, dishonesty is self-limiting because trust erodes quickly and opportunities dissipate. But in stable settings, dishonesty can persist and accumulate. This implies that context matters as much as incentives: preventive mechanisms such as auditing, advisor rotation, or third-party verification are especially critical in stable environments.

Each proposed managerial intervention directly targets a specific parameter or dynamic in the model. Auditing increases the effective psychological or economic cost of deception, corresponding to a higher marginal cost of U(t) or a steeper penalty function f(U); this reduces the incentive for episodic lying by making high-intensity deception less attractive. Advisor rotation works through a different channel: it effectively resets or boosts the trust state S(t) by introducing a new advisor who is not burdened by prior erosion, thereby counteracting the model’s natural decay process governed by δ. Rotation also weakens the perceived persistence of the environment (captured in the discounting of future trust gains), reducing incentives to cluster deception when trust is abundant. Transparency mechanisms, such as shared forecasting dashboards, reduce the advisor’s relative influence on S(t) by lowering the marginal gain from deception α, thereby flattening the reward function associated with dishonest behavior. Together, these interventions show how managerial levers can be mapped directly onto core structural elements of the model, providing a practical toolkit for mitigating episodic opportunism.

Our results also reinforce the well-known asymmetry in trust: dishonesty erodes credibility rapidly, while recovery is slow and partial. Even a brief episode of deception leaves lasting scars, with trust stabilizing at lower levels. This fragility suggests that organizations should prioritize proactive trust-building measures (e.g., transparency, verification systems) rather than relying on post-crisis repair.

Our findings connect directly to research on information sharing in supply chains. [17] show that trust in forecast accuracy affects coordination and efficiency. However, most supply chain models assume static conditions or repeated interactions with fixed strategies. By contrast, our dynamic framework reveals how episodic dishonesty can emerge endogenously and persist when incentives and environmental stability align. For example, in a supply chain contract, a supplier may exaggerate forecasts during stable demand periods to secure favorable terms, only to revert to honesty once credibility is depleted. This episodic perspective highlights the need for dynamic monitoring and contract designs that adapt to changing credibility levels over time.

Although our model is stylized, its predictions resonate with behavioral research: episodic lying, incentive-driven dishonesty, and asymmetric trust dynamics are all observed empirically. Embedding these behavioral patterns in an optimal control framework provides a rigorous lens to derive comparative statics and managerial insights. The integration of behavioral evidence with dynamic optimization thus advances both theory and practice, offering new ways to design incentives, monitor dishonesty, and preserve trust.

A limitation of the present framework is its deterministic treatment of trust erosion, which does not capture stochastic discovery of deception or sudden, catastrophic trust collapse—both of which may arise in real advisory relationships.

To guide future empirical work, the model yields several testable predictions. First, the duration and intensity of deceptive episodes should increase when the advisor’s marginal incentive for misrepresentation is higher (i.e., higher α), particularly when trust S(t) is abundant. Second, environments with stronger “anti-middle” psychological penalties should exhibit a more bimodal distribution of deception levels—characterized by long periods of full honesty punctuated by concentrated bursts of high-intensity lying. These hypotheses offer a structured path for behavioral and field experiments to empirically validate the model’s key mechanisms.