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Stochastic compliance/evasion dynamics in tax models: a piecewise deterministic Markov process approach

This paper introduces a novel Piecewise Deterministic Markov Process (PDMP) framework that extends a deterministic tax evasion model by incorporating stochastic audit and imitation mechanisms, demonstrating that their interaction prevents extreme equilibria and instead generates persistent fluctuations around a stationary distribution that better reflects real-world compliance dynamics.

Original authors: Jonas Mayr, Amira Meddah, Irene Tubikanec

Published 2026-05-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jonas Mayr, Amira Meddah, Irene Tubikanec

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Tax Model That "Breathes"

Imagine a society as a giant, busy marketplace. In this market, people have different amounts of money (income classes) and they are constantly trading with one another. Some people pay their full share of taxes, while others try to hide their money to pay less (tax evasion).

For a long time, mathematicians tried to predict how this market would behave using a deterministic model. Think of this like a train on a fixed track. Once the train starts, you know exactly where it will go. In the old model, once a person decided to be a "tax evader" or a "tax payer," they stayed that way forever. They could move up or down in income (like changing train cars), but their attitude toward taxes never changed.

The authors of this paper say, "That's too rigid. Real people change their minds." They introduce a new model that acts less like a train and more like a game of tag with a referee. In this new game, people can suddenly switch teams (from evader to payer, or vice versa) based on random events. They call this a Piecewise Deterministic Markov Process (PDMP).

Here is how their new "game" works, broken down into three parts:

1. The Two Forces: The Referee and The Crowd

The paper identifies two main forces that cause people to switch their tax behavior. They modeled these as two separate "games" first, then combined them.

Force A: The Audit (The Referee)

The Metaphor: Imagine a referee blowing a whistle randomly. When the whistle blows, it's an audit.
How it works: If a person is cheating (evading taxes) and gets caught by the referee, they are forced to switch to the "honest" team.
The Result: If you only have the referee blowing whistles, eventually, everyone becomes honest. The "cheating" team disappears completely. The system settles into a state of 100% compliance.

Force B: Imitation (The Crowd)

The Metaphor: Imagine a group of friends hanging out. If one friend says, "Hey, I didn't pay my taxes and got away with it," the others might think, "That sounds smart. I'll do that too." This is imitation.
How it works: People watch their peers. If they see others successfully evading taxes, they are likely to copy them and switch to the "cheating" team.
The Result: If you only have this "copycat" effect, eventually, everyone becomes a cheater. The "honest" team disappears. The system settles into a state of 100% evasion.

2. The Real World: The Combined Game

The authors realized that in the real world, both the Referee (Audits) and the Crowd (Imitation) are happening at the same time. They built a third model where these two forces fight against each other.

The Analogy: Imagine a tug-of-war. On one side, the Referee is pulling people toward honesty. On the other side, the Crowd is pulling them toward cheating.
The Outcome: In the old "train" model, the system would eventually stop moving and settle on one side. But in this new "tug-of-war" model, the system never stops moving.
- Sometimes the Referee wins, and more people become honest.
- Then, the Crowd catches up, and more people start cheating.
- The numbers fluctuate up and down, but they tend to hover around a specific "middle ground."

This is the paper's main discovery: Real tax systems don't settle into a perfect, static state. Instead, they settle into a dynamic balance where there is always a mix of honest and dishonest people, constantly shifting but staying within a predictable range.

3. Why This Matters (According to the Paper)

The authors claim that their new model is more realistic because:

It respects the rules: Even though people are jumping back and forth between teams, the total amount of money and the total number of people in the society remain constant (just like in the old model).
It captures reality: It explains why we never see a society where everyone pays taxes perfectly, nor one where everyone cheats. We see a messy, fluctuating middle ground, and this model explains why that happens.
It predicts the "Stationary Distribution": Instead of predicting a single final number (e.g., "50% will cheat"), the model predicts a cloud of possibilities. It tells us that over a long time, the percentage of cheaters will bounce around a specific average, rather than freezing at one spot.

Summary

Old Model: People are stuck in their tax habits. The system is a straight line to a final destination.
New Model (PDMP): People are constantly switching habits due to Audits (pushing them to be honest) and Imitation (pushing them to cheat).
The Result: The system doesn't stop; it dances. It fluctuates around a stable average, creating a much more realistic picture of how tax compliance works in the real world.

The paper proves mathematically that this "dancing" behavior is stable and that the system will eventually settle into a pattern where the mix of honest and dishonest people remains consistent over time, even though individuals are constantly changing their minds.

Technical Summary: Stochastic Compliance–Evasion Dynamics in Tax Models

Problem Statement
Tax evasion remains a persistent challenge with significant macroeconomic and social consequences, including reduced public revenue and increased income inequality. While existing research has utilized mathematical modelling to understand evasion, a specific limitation exists in the deterministic framework introduced by Bertotti and Modanese (2018). In that model, individual tax compliance behaviour is treated as static; agents are assumed to remain within their assigned evasion sectors indefinitely, with changes occurring only through income class transitions driven by economic interactions. This assumption ignores the dynamic nature of compliance, which in reality is shaped by institutional incentives (such as audit risks) and social factors (such as peer influence and imitation).

Methodology
To address this structural limitation, the authors extend the deterministic Ordinary Differential Equation (ODE) model into a stochastic framework using Piecewise Deterministic Markov Processes (PDMPs). This approach combines continuous deterministic evolution with discrete stochastic jumps.

The study formalizes three distinct PDMP models:

Audit PDMP: Models random enforcement events where audits induce transitions from non-compliant sectors to the fully compliant sector. The jump rate is state-dependent, scaling with the proportion of non-compliant individuals in the population.
Imitation PDMP: Models social influence where individuals adopt the evasion strategies of peers. This mechanism drives transitions from compliant sectors to higher evasion sectors. The jump rate is state-dependent, scaling with the proportion of compliant individuals (reflecting the idea that a larger compliant population offers more opportunities for observing and imitating evasion).
Combined PDMP: Integrates both mechanisms, allowing audits and imitation to act simultaneously. The total jump rate is the sum of the individual rates, and the specific event type (audit vs. imitation) is determined probabilistically at each jump time.

For each model, the authors define the three characteristic components of a PDMP: the deterministic flow (governed by the original ODE), the jump rate function, and the transition kernel (the map describing state changes at jump times). They rigorously verify that these stochastic extensions preserve the fundamental conservation laws of the original system: total population mass and global income.

Numerical simulations are conducted using a modified Euler method (to ensure population conservation) combined with a thinning procedure for simulating the jump times of the PDMP.

Key Contributions and Results

Opposing Long-Term Dynamics: The study demonstrates that the two stochastic mechanisms produce fundamentally opposing long-term outcomes.
- The Audit PDMP drives the system toward a state of full compliance. Repeated audits systematically reduce the non-compliant population, leading to a stationary distribution concentrated at a fully compliant equilibrium.
- The Imitation PDMP drives the system toward full non-compliance. Imitation events systematically shift the population toward higher evasion sectors, resulting in a stationary distribution concentrated at a state of maximal evasion.
The Combined Model: The central contribution is the formulation of a combined PDMP where these opposing forces interact. Unlike the individual mechanisms which drive the system to extreme equilibria, the combined model does not converge to a single deterministic point. Instead, numerical evidence suggests the system converges to a stationary distribution that fluctuates around the deterministic trajectory of the original ODE model.
Parameter Sensitivity: The simulations reveal that the system's behavior is sensitive to four key parameters: audit intensity ( $\gamma$ ), audit effectiveness ( $\delta$ ), imitation frequency ( $\eta$ ), and imitation strength ( $\epsilon$ ). While intensity parameters ( $\gamma, \eta$ ) control the frequency of events, strength parameters ( $\delta, \epsilon$ ) control the magnitude of population transfers during those events.
Stationary Distribution Characteristics: In the combined model, the stationary distribution is centered near the deterministic equilibrium but exhibits systematic deviations. These deviations are attributed to the interaction between the transition kernels and the nonlinear, class-dependent economic structure (specifically, the boundary conditions where the poorest never pay and the richest never receive).

Significance and Claims
The paper claims to present the first application of PDMPs to model behavioural dynamics and audit effects in tax evasion. The primary significance of this work lies in its ability to incorporate behavioural stochasticity into socio-economic models, moving beyond the static treatment of compliance found in previous deterministic frameworks.

The authors argue that the combined PDMP offers a more realistic representation of tax compliance dynamics observed in real economies, where populations do not settle into extreme states of total compliance or total evasion but rather exhibit persistent fluctuations around a partial compliance equilibrium. The proposed framework is presented as a versatile tool for integrating stochastic behavioural mechanisms into complex systems, with potential for future extensions involving refined behavioural sectors, alternative audit policies, or learning mechanisms. The work establishes a mathematical foundation for studying how policy parameters and social interactions co-evolve to create macroscopic distributions of compliance and income.