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# A Decision Support Tool for Food Fraud Regulation: Hybrid Framework Combining Game Theory, DCA Optimization, and Machine Learning

Nacim Nait-Mohand, Mohammed Said Radjef, Abdelhakim Hammoudi

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Nacim Nait-Mohand, Mohammed Said Radjef, Abdelhakim Hammoudi. A Decision Support Tool for Food Fraud Regulation: Hybrid Framework Combining Game Theory, DCA Optimization, and Machine Learning. 2025. hal-05316638

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# A Decision-Support Tool for Food Fraud Regulation: Hybrid Framework Combining Game Theory, DCA Optimization, and Machine Learning

Nacim NAIT MOHAND¹, Mohammed Said RADJEF², and Abdelhakim HAMMOUDI³

¹University of Bejaia, Faculty of Technology, LaMOS Research Unit, Algeria.,
nacim.naitmohand@univ-bejaia.dz

²University of Bejaia, Faculty of Exact Sciences, LaMOS Research Unit, Algeria.,
mohammedsaid.radjef@univ-bejaia.dz

³National Research Institute for Agriculture, Food and the Environment (INRAE),, AgroParisTech, Paris-Saclay Applied Economics, University of Paris-Saclay,, 91120 Palaiseau, France.,
abdelhakim.hammoudi@inrae.fr

## Abstract

This study presents a decision-support tool designed for the regulation of food fraud, based on the hybridization of four approaches: (i) an economic model inspired by the Chinese milk crisis 2008, opposing a food firm and a public authority. The model integrates the main hypotheses of food fraud mechanisms, considering organoleptic standards, production and compliance costs, the firm's ability to substitute the required nutrient with a less expensive chemical substance, as well as the cost of acquiring that substance. It also considers the probability of a health crisis occurring, the financial penalty stipulated by regulations, the overall health risk and the expected profits based on the firm's different strategic choices. (ii) A bilevel game where the authority first chooses the nutritional standard and the fine applicable in case of a health crisis to minimize health risk, and then the firm determines its optimal compliance level to maximize its expected profit. The game is solved by backward induction. (iii) The Difference of Convex functions Algorithm (DCA) method to locally solve step 2 of the game, and (iv) the ML model (CatBoost) to estimate the firm's best response function for solving step 1 of the game. The experimental results of the tool demonstrate that the organoleptic standard primarily drives the firm's compliance. Strict nutritional standards are effective at reducing health risk only when paired with high fines. Optimal regulation is adaptive: when production capacity, prices, or organoleptic standards are low, maximum standards and fines are required; otherwise, relaxing the standard best limits health risks.

**Keywords:** Food fraud; Game theory; DCA method; CatBoost; Decision-Support Tool.

## 1 Introduction

Economically motivated adulteration is a major public health issue. This form of fraud is mainly aimed at reducing production costs and increasing market share for the operators who engage in it. However, it compromises product quality and increases health risks, particularly
when the substances added are harmful [1, 2]. The decision to resort to these practices is often a deliberate strategy: fraudsters seek to preserve the organoleptic attributes of their products to avoid consumer boycotts and loss of market share. They also exploit loopholes in control systems to evade detection and reduce the risk of their products being rejected when they are placed on the market. Additionally, they aim to limit their exposure to financial penalties after a health crisis that could bring their practices to light [3, 4, 5].

The melamine-contaminated milk crisis in China is a striking example of the health risks associated with food adulteration. Faced with rising demand and prices, some companies deliberately adulterated infant milk by adding melamine [6]. This strategy allowed them to circumvent quality controls while artificially inflating the protein levels measured by conventional tests. The first suspicions arose in March 2008, culminating in September with reports of serious kidney disease: the final toll was six infant deaths and nearly 300 000 cases of illness linked to the contamination. The authorities' response led to a massive product recall and the bankruptcy of iconic companies such as the Sanlu Group [7, 8, 9]. This crisis highlighted the limitations of traditional controls in the face of sophisticated fraudulent strategies.

To address this issue, a large part of the literature has focused on improving control and detection mechanisms [10, 11]. These approaches, although necessary, remain reactive in nature and are consequently limited in the face of the adaptability and increasing sophistication of fraud strategies, confining them to a short-term struggle [12, 13]. A second branch of the literature focuses more on the development of regulatory incentive mechanisms to permanently alter the risk/reward equation of fraud [14, 15]. This work is part of a long-term approach, where prevention and strategic deterrence take precedence over simple ex post detection [16].

However, designing effective prevention tools requires in-depth mathematical economic modeling that incorporates multiple parameters and variables, the analytical resolution of which quickly becomes complex, if not impossible [17, 18]. The classic dilemma of sacrificing complex models for the sake of analytical tractability often leads to unreliable results, limiting the operational potential of policy recommendations. In this context, artificial intelligence tools appear to be a promising solution, capable of overcoming the computational limitations of traditional approaches, while preserving the richness and realism of economic models [19]. Using the power of learning algorithms, these tools offer the possibility of better anticipating, detecting, and preventing fraudulent behavior in agri-food chains.

The objective of this study is to develop a decision-support tool designed for a systemic and sustainable approach to combating food fraud. To this end, we propose a hybrid framework that integrates several complementary and interconnected modules.

The first module is a mathematical economic model that opposes a public authority to an agri-food firm. The authority determines, upstream, a nutritional standard that defines the minimum threshold of a nutrient required in each unit of the marketed product, as well as a financial penalty applicable in the event of fraud revealed during a public health crisis. The firm, downstream, adjusts its strategy to these regulatory decisions: it can either fully comply with the standard, bearing the associated compliance costs and obtaining a predictable profit, or engage in a fraudulent strategy. In this latter case, it maintains only a minimal level of compliance and substitutes the remaining proportion with a substance undetectable by standard controls but capable of simulating the required quality, allowing it to increase production volume and improve potential profits while incurring additional costs for acquiring the substance. To avoid arousing consumer suspicion, however, the firm must respect a critical compliance threshold that preserves the organoleptic attributes of the product, referred to as the organoleptic standard (the consumer's sensory sensitivity level to the product's perceived conformity and authenticity through its sensory attributes: taste, smell, texture, and appearance). If no public health crisis
occurs, fraud generates a higher profit than strict compliance; conversely, in the event of a crisis that exposes the fraud, the firm bears the fine imposed by the authority, which significantly reduces its expected gain.

The second module formalizes the strategic interaction between the public authority's decisions and the firm's decisions as a bilevel game. To find the Nash equilibrium of the game, we will use the backward induction technique. This involves, in the first step, determining the firm's reaction function based on the public authority's decisions. In the second step, by anticipating the firm's reaction, the public authority determines its equilibrium strategy.

The third module employs the Difference of Convex Functions Algorithm (DCA), which is known for its efficiency in calculating the optimum of a function. This algorithm solves the firm's complex decision-making problem by determining the optimal level of compliance (or fraud) the firm should choose to maximize its expected profit, taking into account the regulatory decisions and the parameters of the economic model.

The fourth module is based on a supervised regression learning method, for which we selected the CatBoost model, applied to a large synthetic dataset. The explanatory variables (model parameters and public authority decisions) are generated using Latin Hypercube Sampling (LHS), while the response variable, corresponding to the firm's best response, is obtained by applying the DCA module.

The analysis of the numerical results from experiments conducted with our tool highlights several key insights for the regulation of food fraud.

The local analysis of the firm's strategic behavior, via the DCA module, demonstrated that strengthening the nutritional standard only reduces fraud and health risk when the standard is already strict. The combined effect of this standard and the fine reveals that these tools must be used in a complementary manner to be effective in reducing fraud and health risk. A moderate increase in the organoleptic standard may leave fraud and health risk unchanged. Conversely, more attractive market prices systematically encourage fraud, thereby amplifying the risk. Finally, expanding production capacity nonlinearly alters the fraud strategy and the evolution of health risk.

The global analysis conducted by the trained CatBoost module, using explainable AI tools, showed a remarkable convergence between different evaluation metrics (Feature Importance, Permutation Importance, and SHAP), confirming that the organoleptic standard is the predominant determinant of the firm's compliance level. SHAP analyses and Partial Dependence Plots (PDP) validated the direction of the effects observed locally, notably the positive impact of the nutritional standard, the fine, and the organoleptic standard on compliance, as well as the negative effect of market price and the non-linear effect of production capacity.

The integration of the DCA and CatBoost modules enabled the determination of optimal decisions for public authorities under different economic configurations. In situations of low production capacity or low market prices, the public authority should impose the maximum nutritional standard and the ceiling fine to maximize the deterrent effect. This combined strategy exploits the firm's sensitivity to financial penalties when the potential gains from fraud remain limited. The organoleptic standard also plays a crucial role: when it is high, the natural vigilance of consumers allows authorities to relax nutritional standards without increasing health risk. Conversely, when faced with high production capacity or attractive market prices, relaxing the nutritional standard becomes optimal to contain health risk at the source. The complementarity of the local (DCA) and global (CatBoost) approaches validates the robustness of these adaptive strategies, demonstrating that optimal regulation must constantly adjust to market conditions, productive capacities, and the consumers' organoleptic standard to effectively protect public health.
## 2 Related work

The fight against economically motivated adulteration in the agri-food sector is an inherently complex problem, long dealt essentially with by reactive approaches and by theoretical models whose tractability has often been obtained at the cost of strong simplifying hypotheses.

The historical foundations of adulteration are well documented[20], while recent studies have clarified its causes, risks, and regulatory challenges[16, 21]. In parallel, analyses of stakeholder behavior within agri-food chains have shown that fraud does not stem solely from economic incentives but also depends on the product's organoleptic attributes: the more these attributes allow imitation of authenticity, the greater the risk of fraud[2, 3, 22]. Consequently, the perception of authenticity and consumers' trust determine the effectiveness of public policies, which must therefore integrate this sensory dimension into the design of regulatory measures.

However, chemical and biological analytical methods, historical knowledge bases on adulterants[4], and the use of AI and IoT for anomaly detection[11, 23] validate compliance only ex post: they detect adulteration once harm has occurred and remain vulnerable to adaptive fraudulent strategies, as illustrated by the use of melamine to simulate protein content[6, 9].

Theoretically, game theory has represented a major advance: studies on honey adulteration[15] and on the evaluation of organic suppliers[14] show that policy effectiveness hinges on the regulator's ability to dynamically adjust oversight and penalties. However, these models often run up against the complexity of real-world parameters and may trade off realism for analytical tractability.

Our contribution is precisely aimed at filling this methodological gap. Unlike existing approaches, our hybrid framework integrates four complementary methodologies. The firm's non-convex optimization problem is solved using the DCA (Difference of Convex functions Algorithm), an efficient method for DC programs that converges to a local optimum by exploiting the decomposition of the objective function into the difference of two convex functions[24]. For the global estimation of the best response function, we rely on CatBoost, a gradient-boosting algorithm specialized in the native handling of categorical variables and noted for its robustness to overfitting thanks to ordered boosting[25]. The integration of explainable AI via SHAP[26] transforms predictions into quantified drivers, which are essential for justifying regulatory policies.

By bringing together game theory, nonconvex optimization via DCA, and interpretable supervised learning via CatBoost, our framework turns risk knowledge into adaptive regulatory prescriptions, anticipating operators' strategic responses and recommending optimized policies to minimize public-health risk.

## 3 Proposed tool

Our hybrid tool integrates four complementary methodologies into a unified framework for supporting regulatory decisions. The tool's architecture systematically combines economic modeling, game theory, optimization algorithms, and machine learning to analyze and predict food fraud behavior. The framework operates through four interconnected modules.

### 3.1 Economic model

Consider a market in which demand is expressed by consumers characterized by an organoleptic standard $\bar{p}$, ($\bar{p} \in [0, 1]$), which measures their level of stringency regarding the product's compliance with current nutritional standards. More precisely, if the proportion of
compliant product in each unit offered is below the threshold $\bar{p}$, consumers boycott the product; conversely, if this proportion exceeds $\bar{p}$, they agree to purchase and consume it. A value of $\bar{p} = 1$ indicates a high level of consumer sensitivity to variations in product composition, generally associated with consumers with a high level of food education, while a $\bar{p} = 0$ indicates an absence of discernment, a characteristic of emerging markets, or vulnerable consumers such as infants.

To ensure the safety of products circulating on the market, public authorities implement two regulatory mechanisms:

1. A minimum nutrient content threshold $e$, (with $e \in [\underline{e}, \bar{e}] \subset [0, 1]$)<fnref n="1" />. This regulatory requirement mandates that each unit of product offered contains at least a proportion $e$ of the specified nutrient. This threshold is set within a range defined by the Codex Alimentarius, which establishes recommendations for the minimum $e$ and maximum $\bar{e}$ admissible levels to guarantee the nutritional quality of foods.

2. An imperfect compliance control system, which is based on a conventional test<fnref n="2" /> for assessing the nutrient dosage in the product.

Let's consider a firm $F$, representative of the agri-food sector, with a production capacity $Q$, ($Q \ge 0$) in product conforming to the standards: organoleptic $\bar{p}$, and nutritional $e$. The firm $F$ sells its production on the market to meet consumer demand. However, given the failings of the control system and the sensitivity of consumers to variations in product composition, the firm may seek to increase the quantities offered on the market by resorting to fraud through adulteration. In this way, it can choose either to actually comply with food standards, or to commit fraud on the nutritional quality of the nutritional standard $e$, by incorporating a chemical substance that simulates the required nutrient content.

### 3.1.1 Nutritional quality compliance strategy

If $F$ decides to genuinely comply, it then offers a quantity $Q$ on the market. Its profit $\pi$ is given in this case by:

$$ \pi(Q, \omega, c) = \omega Q - c Q e^2, \quad (1) $$

where $c$, ($c \ge 0$) is the unit cost of producing a product of quality $e$, and $\omega$, ($\omega \ge 0$) is the unit selling price on the market. The quadratic term $e^2$ captures the convexity of marginal production costs: each increment in nutritional enrichment requires disproportionately larger investments in high-purity raw materials, precision formulation and encapsulation technologies, enhanced analytical testing and certification protocols, and strengthened quality-control and regulatory-compliance efforts<fnref n="3" />.

<footnote n="1">¹The Codex Alimentarius is the major international reference for the development of food standards. When defining their national standards, some countries take their inspiration directly from the Codex, while others adopt more strict or more flexible standards.</footnote>

<footnote n="2">²One example is tests based on analytical methods such as those of Kjeldahl and Dumas, which are used to measure the level of total nitrogen in a sample and estimate its protein content using a conventional multiplier coefficient. However, these methods do not distinguish between the origin of the nitrogen (protein or non-protein), which can distort the nutritional assessment. Similarly, the Rose-Gottlieb method, commonly used to determine lipids in milk and dairy products, is sensitive to sample variations, which can affect the accuracy of the results.</footnote>

<footnote n="3">³The literature on vertical differentiation classically models quality costs as convex, notably Mussa and Rosen (1978)[27].</footnote>
### 3.1.2 Nutritional quality fraud strategy

If $F$ decides to commit nutritional fraud, it then offers a quantity $X$ (with $X \ge Q$) produced on the market. To produce the quantity $X$, it decides to mix each quantity $p$, (with $p \in [\bar{p}, 1]$) of the compliant product with a quantity $\alpha$ of a chemical substance to produce one unit of product. Consequently, the relationship between $X$, $Q$, and $p$ is given by:

$$X(Q, p) = \frac{Q}{p}. \qquad (2)$$

The quantity $\alpha$, ($\alpha \in [0, 1]$) of the substance, necessary to produce one unit of the offer $X(Q, p)$, is given by the expression<fnref n="4" />

$$\alpha(p, e, m) = m e (1 - p), \qquad (3)$$

where $m \in [0, 1]$ measures the substance's inefficiency in achieving compliance with nutritional standard $e$: when $m$ is close to 0, the substance is highly efficient (a small quantity suffices), whereas when $m$ is close to 1, it is inefficient (a large quantity must be added).

The total quantity $Q_m$ of the substance, necessary to produce the offer $X(Q, p)$, is given by:

$$Q_m(Q, p, e, m) = X(Q, p) \alpha(p, e, m). \qquad (4)$$

The cost $C_m$ of acquiring the quantity $Q_m$ of the substance, necessary to produce the offer $X(Q, p)$, is given by:

$$C_m(Q, p, e, m, c_m) = c_m (Q_m(Q, p, e, m))^2, \qquad (5)$$

where $c_m$, ($c_m \ge 0$) is the marginal cost of the chemical substance.

The probability $\phi$ of a health crisis occurring due to the fraud strategy depends on: (i) the hazard level $\mu$ ($\mu \in [0, 1]$) of the chemical substance used to produce $X(Q, p)$, and (ii) the quantity $m e (1 - p)$ of the substance contained in each unit of $X(Q, p)$:

$$\phi(e, p, m, \mu) = \mu (m e (1 - p))^2. \qquad (6)$$

The potential profit $\hat{\pi}$ obtained by $F$, when it decides to commit fraud (with $p \in [\bar{p}, 1]$) and no health crisis occurs, is given by:

$$\hat{\pi}(p, e) = X(Q, p) \omega - cQ e^2 - C_m(Q, p, e, m, c_m). \qquad (7)$$

The expected profit $\tilde{\pi}$ of firm $F$, when it chooses $p \in [\bar{p}, 1]$, is expressed as:

$$\tilde{\pi}(p | (e, R)) = \phi(e, p, m, \mu) (\hat{\pi}(p, e) - R) + (1 - \phi(e, p, m, \mu)) \hat{\pi}(p, e), \qquad (8)$$

where $R$, ($R \in [0, \bar{R}]$) is the fine (penal sanction) that the firm must pay if its fraud is discovered.

The health risk $\Phi$ is obtained by multiplying the probability of a crisis by the quantity supplied $X(Q, p)$ (number of consumers exposed):

$$\Phi(e, R | p) = \phi(e, p) X(Q, p), \qquad (9)$$

<footnote n="4">⁴The function $\alpha(e, p, m)$ satisfies two properties: 1) When $p$ increases, $\alpha$ decreases, i.e., $\frac{\partial\alpha(p, e, m)}{\partial p} \le 0$, which means that the higher the proportion of compliant product, the less substance is required for compliance. 2) When $e$ increases, $\alpha$ increases, i.e., $\frac{\partial\alpha(p, e, m)}{\partial e} \ge 0$, indicating that a stricter standard necessitates the use of more substance to simulate compliance.</footnote>
where $R$ affects $\Phi$ only through $p$.

<figure><img src="image_3.png" /><figcaption>Figure 1: Decision tree of the firm's compliance strategies and their health consequences.</figcaption></figure>

## 3.2 Bilevel game

To highlight the strategic interaction between, on the one hand, the decisions of the public authority regarding nutritional standards and the fine applicable in the event of a food safety crisis, and, on the other hand, those of firm $F$, which determines the nutritional quality of the product offered on the market, we model the situation as a two-player sequential game in two steps.

**Step 1.** The public authority sets the nutritional standard $e$ (with $e \in [e, \bar{e}]$) and the fine $R \in [0, \bar{R}]$ applicable in the event of a health crisis in order to minimize health risk $\Phi$ (9).

**Step 2.** The firm $F$ chooses $p$ (with $p \in [\bar{p}, 1]$) which maximizes its expected profit $\tilde{\pi}$ (8).

The game solution is based on the backward induction technique, formalized in **Algorithm 1**. In **Step 2**, firm $F$ choose the compliance level $p^*(e, R)$ that maximizes its expected profit (8) given the public authority's decisions $(e, R)$. In **Step 1**, the public authority anticipates firm $F$'s response $p^*(e, R)$ and chooses $(e^*, R^*)$ to minimize the health risk (9). The final equilibrium strategy $p^*(e^*, R^*)$ is then determined. [H] Game resolution procedure [1] **Initialization of parameters**: Introduce: $Q, \omega, c, c_m, \mu, m$, and $\bar{p}$. **Step 2 – Firm's best response**: Compute $p^*(e, R)$: $p^*(e, R) = \arg\max_{p \in [\bar{p}, 1]} \tilde{\pi}(p | (e, R))$. **Step 1 – Public authority's optimization**: Determine: $(e^*, R^*) = \arg\min_{e \in [e, \bar{e}], R \in [0, \bar{R}]} \Phi((e, R) | p^*(e, R))$. **Final step – Firm's final decision**: Recompute: $p^* = p^*(e^*, R^*)$
## 3.3 DCA for the Resolution of Step 2 of the Game

Given $e \in [e, \bar{e}]$) and the fine $R \in [0, \bar{R}]$, the rational behavior of the firm $F$ is to choose $p \in [\bar{p}, 1]$ that maximizes its expected profit $\tilde{\pi}(p | (e, R))$ given by (8):

$$ \max_{p \in [\bar{p}, 1]} \tilde{\pi}(p | (e, R)) \iff \max_{p \in [\bar{p}, 1]} \frac{p Q \omega - e^2 (c p^2 Q + (1-p)^2 m^2 (c_m Q^2 + p^2 R \mu))}{p^2} \quad (10) $$

The analytical resolution of problem (10) proves to be complex. Therefore, in the remainder of this work, we will adopt a numerical resolution of the firm's decision problem, considering different parameter values as well as various choices of $e$ and $R$ made by the public authority. The resolution of the problem (10) is equivalent to solving the following minimization problem: $\min_{p \in [\bar{p}, 1]} -\tilde{\pi}(p | (e, R))$. The function $\tilde{\pi}(p) = -\tilde{\pi}(p | (e, R))$ can be decomposed as a difference of convex (DC) functions: $\tilde{\pi}(p) = h(p) - g(p)$, with $h(p) = a_0 p^{-2} + a_2 - a_3 p + a_4 p^2$ and $g(p) = a_1 p^{-1}$, and $a_i \ge 0$ for all $i \in \{0, 1, 2, 3, 4\}^5$. To solve this equivalent problem, we apply the following Difference of Convex functions Algorithm (DCA)[24]: [H] Difference of convex functions algorithm (DCA) [1] Initialization: Introduce: $Q, \omega, c, c_m, \mu, m, \bar{p}, R$, and $e$. Set initial point $p_0 = \bar{p}/2$, iteration counter $k \leftarrow 0$, and desired precision $\varepsilon > 0$. Repeat Compute $y_k \leftarrow g'(p_k)$ Update $p_{k+1} \leftarrow \arg \min_{p \in [\bar{p}, 1]} (h(p) - y_k p)$ Increment iteration counter: $k \leftarrow k + 1$ Until $|p_k - p_{k-1}| \le \varepsilon$ Return $p_k$.

## 3.4 Machine learning estimation of best response

This section presents a machine learning regression approach to estimate the firm's best response function $p^*(e, R)$. Estimating this function enables the analysis of the effects of public policy decisions and economic variables on the firm's strategic choices, while also serving as a complementary tool to DCA for solving Step 1 of the bilevel game.

### 3.4.1 Dataset construction

We constructed a sample of $N = 5 \times 10^5$ sets of inputs $(Q, c, m, c_m, \mu, \bar{p}, \omega, e, R)$ using Latin Hypercube Sampling (LHS), a method that ensures stratified coverage of the input feature space while maintaining good statistical efficiency $^6$.

For each set of parameters $\theta^{(i)} = \{Q^{(i)}, c^{(i)}, m^{(i)}, c_m^{(i)}, \mu^{(i)}, \bar{p}^{(i)}, \omega^{(i)}, e^{(i)}, R^{(i)}\}_{i=1}^N$, the best response $p^{*(i)}(e, R)$ was determined numerically using algorithm 3.3 (DCA).

The resulting dataset$^7$ is defined by $\mathcal{D} = \{(\theta^{(i)}, p^{*(i)})\}_{i=1}^N$. It consists of $5 \times 10^5$ observations of 10-dimensional vectors of a relationship between the input variables $(Q, c, m, c_m, \mu, \bar{p}, \omega, e, R)$ and the output variable $p^*(e, R)$.

### 3.4.2 Regression model

Various supervised regression techniques were evaluated on the $\mathcal{D}$ data to model the relationship between the explanatory variables $(Q, c, m, c_m, \mu, \bar{p}, \omega, e, R)$ and the target variable $p^*(e, R)$, with model performance being compared using three standard metrics: coefficient of

<footnote>$^5$The coefficients are defined as: $a_0 = c_m e^2 m^2 Q^2$; $a_1 = Q(2 c_m e^2 m^2 Q + \omega)$; $a_2 = e^2(cQ + m^2(c_m Q^2 + R\mu))$; $a_3 = 2e^2 m^2 R\mu$; $a_4 = e^2 m^2 R\mu$;</footnote>

<footnote>$^6$The features are sampled using LHS over the following ranges: $Q \in [0, 50]$, $c \in [0, 4]$, $c_m \in [0, 2]$, $\omega \in [4, 20]$, $R \in [0, 5000]$, $(m, \mu, \bar{p}, e) \in [0, 1]^4$.</footnote>

<footnote>$^7$The synthetic dataset used is available at: https://bit.ly/3IYFiA6</footnote>
determination $R^2$, mean absolute error *MAE*, and mean square error *MSE*. Five-fold cross-validation (5-fold CV) was applied to ensure the robustness of the results and avoid overfitting.

<table><caption>Table 1: Comparative performance of regression algorithms</caption><thead><tr><th>Algorithm</th><th>R<sup>2</sup></th><th>MAE</th><th>MSE</th><th>R<sup>2</sup> (CV 5-fold)</th><th>Std. dev.</th></tr></thead><tbody><tr><td>Simple Linear Regression</td><td>0.752</td><td>0.104</td><td>0.0197</td><td>0.754</td><td>0.0012</td></tr><tr><td>Polynomial Regression (n = 2)</td><td>0.890</td><td>0.065</td><td>0.0087</td><td>0.891</td><td>0.0008</td></tr><tr><td>XGBoost</td><td>0.961</td><td>0.030</td><td>0.0030</td><td>0.962</td><td>0.0010</td></tr><tr><td>LightGBM</td><td>0.958</td><td>0.029</td><td>0.0030</td><td>0.958</td><td>0.0000</td></tr><tr><td><strong>CatBoost</strong></td><td><strong>0.968</strong></td><td><strong>0.027</strong></td><td><strong>0.0030</strong></td><td><strong>0.968</strong></td><td><strong>0.0000</strong></td></tr></tbody></table>

Among models tested with their default parameters, gradient boosting (XGBoost, Light-GBM, and CatBoost) methods notably outperform linear and polynomial methods. In particular, CatBoost achieves the best performance and thus was chosen for the rest of the analysis. To further boost its accuracy and generalization capacity, its hyperparameters were optimized by a Bayesian approach<fnref n="8" />. This phase allowed for obtaining significantly better results with the following metrics: $R^2$ = 0.978, *MAE* = 0.02, *MSE* = 0.002, and $R^2$ (CV 5-fold)=0.978 (±0.0000).

# 4 Experimental Results

In this section, we conduct numerical experiments to validate the proposed tool, examining (i) the *local effects* of authority decisions and market parameters on firm behavior via the DCA algorithm, (ii) the *global analysis* of compliance determinants through explainable AI applied to CatBoost, and (iii) the resulting *optimal regulatory* strategies.

## 4.1 Local analysis of firm behavior

We apply algorithm 3.3 to study how public authority decisions and market variables shape firm *F*'s best response $p^*(e, R)$ and the resulting health risk $\Phi(e, R | p^*(e, R))$.

### 4.1.1 Strengthening nutritional standard

The figure<fnref n="9" /> shows how changes in the nutritional standard threshold affect fraud intensity and health risk.

<footnote n="8">⁸Optimized CatBoost hyperparameters: depth = 10; learning rate = 0.3; 12 leaf reg 8.3; random strength 1.3; bagging temperature = 0.0.</footnote>

<footnote n="9">⁹Constructed with baseline parameter values: $Q = 2, c = 1, c_m = 2, \omega = 2, R = 10, \bar{R} = 200, (m, \mu) = (0.5, 0.5), (e, \bar{e}) = (0.1, 0.8), \bar{p} = 0.2$.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 2: Variation in fraud intensity and health risk under nutritional standard.</figcaption></figure>

From Figure 2, we observe that, as $e$ increases, firm $F$ adjusts its compliance level $p^*(e, R)$ to maximize its expected profit. Near the minimum threshold $\underline{e}$, the firm maintains minimal compliance $p^*(e, R) = \bar{p}$ (maximum fraud intensity). By contrast, when $e$ approaches the upper threshold $\bar{e}$, $F$ increases $p^*(e, R)$.

In response to a tightening of the nutritional standard $e$ near the minimum threshold $\underline{e}$, firm $F$ chooses to maintain minimal compliance, $p^*(e, R) = \bar{p}$ (maximum fraud intensity). Although stricter standards raise the costs of compliant production, the purchase of the adulterating substance, and the risk of crisis and sanctions, these remain small compared to the gains from expanding output and sales. The firm therefore adopts a maximum fraud intensity strategy to maximize expected profit. However, this approach requires using more of the adulterating substance per unit, which directly increases the health risk. By contrast, when $e$ approaches the upper threshold $\bar{e}$, firm $F$ chooses to increase its compliance level $p^*(e, R)$ to mitigate the probability of a health crisis and reduce associated costs (fine $R$). In this case, the effect of $F$'s increased compliance (reduction in quantity offered) outweighs the effect of the stricter standard (more substance required), leading to an overall reduction in health risk.

### 4.1.2 Increasing fine

The figure<fnref n="10" /> shows the joint impact of tightening the nutritional standard and increasing the fine on the firm's strategy and the resulting health risk.

<footnote n="10">¹⁰Constructed with baseline parameter values: $Q = 2$, $c = 1$, $c_m = 2$, $\omega = 2$, $R = 10$, $\bar{R} = 200$, $(m, \mu) = (0.5, 0.5)$, $(\underline{e}, \bar{e}) = (0.1, 0.8)$, $\bar{p} = 0.2$.</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 3: Variation in fraud intensity and health risk under nutritional standard and fine.</figcaption></figure>

From Figure 3, we can see that increasing the fine $R$ encourages firm $F$ to reduce its fraud intensity $1 - p^*(e, R)$. This result can be explained by the fact that a higher fine increases the financial risk in the event of fraudulent practices being detected following a health crisis. To mitigate the probability of such a crisis, firm $F$ therefore chooses to increase its compliance level $p^*(e, R)$, which leads to a reduction in the health risk.

Figure 3 also highlights the combined impact of strengthening nutritional standards and increasing penalties in the event of a crisis on the resulting health risk. It shows that the transition from a less stringent threshold $e_0$ to a more demanding threshold $e_1$ must be accompanied by a significant increase in the fine $R$ in order to achieve a reduction in health risk (see, for example, the change in health risk when moving from $e_0 = 0.4$ to $e_1 = 0.7$ for values $R \le 50$ and $R = 100$). The shift from $e_0$ to $e_1$, while keeping fines low, may induce firm $F$ to maintain its compliance at the minimum level ($p^*(e, R) = \bar{p}$) (see Figure 2). In this case, the firm increases the concentration of the fraudulent substance in its supply, kept constant otherwise, in order to simulate compliance with $e_1$, which leads to a higher health risk. By contrast, if the shift from $e_0$ to $e_1$ is accompanied by a substantial increase in the fine, firm $F$ is incentivized to reduce its fraud $1 - p^*(e, R)$, thereby lowering the health risk.

### 4.1.3 Raising organoleptic standard

The figure<fnref n="11" /> shows how changes in the organoleptic standard $\bar{p}$ affect fraud intensity and health risk.

<footnote n="11">¹¹Constructed with baseline parameter values: $Q = 2$, $c = 1$, $c_m = 2$, $\omega = 2$, $R = 100$, $\bar{R} = 200$, $(m, \mu) = (0.5, 0.5)$, $e = 0.7$, $(\underline{e}, \bar{e}) = (0.1, 0.8)$.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 4: Variation in fraud intensity and health risk under the organoleptic standard.</figcaption></figure>

Figure 4 shows that an increase in the organoleptic standard $\bar{p}$ has a nonlinear impact on fraud intensity and health risk.

When the organoleptic standard $\bar{p}$ increases within the lower range, the firm chooses to maintain a constant compliance level $p^*(e, R)$ that exceeds this standard. This deliberate strategy allows the firm to minimize the risk of sanctions in the event of a health crisis. Thus, within this interval, an increase in the organoleptic standard has no effect on either fraud intensity $(1 - p^*(e, R))$ or the overall health risk $\Phi$. Beyond a critical threshold of $\bar{p}$, however, the firm switches to a minimum compliance strategy by setting $p^*(e, R) = \bar{p}$. Under this regime, any further increase in the standard directly compels the firm to reduce its fraud intensity $(1 - p^*(e, R))$ in order to avoid an immediate consumer boycott. As a result, the health risk $\Phi$ mechanically decreases.

### 4.1.4 Rising Prices

the figure<fnref n="12" /> shows how variations in the market price affect fraud intensity and health risk.

<footnote n="12">¹²Constructed with baseline parameter values: $Q = 2$, $c = 1$, $c_m = 2$, $R = 100$, $(m, \mu) = (0.5, 0.5)$, $e = 0.7$, $\bar{p} = 0.2$.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 5: Variation in fraud intensity and health risk under the market price.</figcaption></figure>

From Figure 5, it can be observed that an increase in the market price $\omega$ incentivizes the firm to lower its compliance level $p^*(e, R)$, thereby intensifying its fraudulent activities (1 – $p^*(e, R)$). This strategic adjustment leads to several consequences: higher acquisition costs for the chemical substance due to increased quantities required, an elevated probability of a health crisis, and a greater risk of financial penalties. Nevertheless, the firm anticipates that the substantial growth in sales revenue from the expansion of its supply will more than compensate for these additional costs and risks. From a public health perspective, the company's decision to increase fraud in response to rising prices has a double aggravating effect: it increases both the concentration of the adulterated substance per unit and the total volume distributed, resulting in a cumulative increase in the overall health risk $\Phi$.

### 4.1.5 Expansion of compliant production capacity

The figure<fnref n="13" /> below illustrates how variations in the production capacity $Q$ affect the intensity of fraud and the associated health risk.

<footnote n="13">¹³This figure is generated using the following parameter values: $\omega = 2$, $e = 0.7$, $R = 100$, $m = 0.5$, $c = 1$, $c_m = 2$, $\mu = 0.5$, $\bar{p} = 0.2$, with the production capacity $Q$ varying from 1 to 5.</footnote>
<figure><img src="image_8.png" /><figcaption>Figure 6: Variation in fraud intensity and health risk under production capacity.</figcaption></figure>

Figure 6 shows that when the *Q* compliant production capacity of firm *F* increases from a low level to a moderate level, the firm chooses to intensify its fraudulent $1-p^*(e, R)$. This strategy allows it to increase its supply and, consequently, its sales revenue, as the profits generated remain higher than the costs of acquiring the adulterant and the increased risks of a health crisis and regulatory sanctions. However, beyond this moderate level, the firm's decision-making is reversed: the marginal costs associated with fraud and the threat of sanctions then exceed the potential additional revenues. In this new equilibrium, the firm reduces the intensity of its fraud, which leads to a decrease in health risks.

## 4.2 Global analysis of firm behavior using CatBoost model

In this section, we complement the local analysis of Section 4.1 by examining the determinants of compliance across the entire parameter domain defined by the dataset $\mathcal{D}$ (Section 3.4.1). Unlike the previous approach, which focused on specific scenarios, we analyze here the global trends and the relative importance of all parameters simultaneously.

Using explainable AI tools applied to the CatBoost model, we identify the factors that exert the most significant influence on the compliance level $p^*$ and assess the robustness of the relationships observed locally. Partial dependence plots allow us to visualize the average marginal effects of each variable, thus providing a comprehensive view of the determinants of compliance across the whole parameter landscape.

The Table 2 reports a comparative overview of feature relevance, combining three complementary tools: Feature Importance, Permutation Importance, and SHAP values.

As shown in Table 2, the three methods agree in ranking the organoleptic standard $\bar{p}$ as by far the most influential characteristic. The next most important variables are the substance's effectiveness *m* and nutritional standard *e*, while characteristics such as marginal production cost *c* have a negligible impact. The consistency between the measurements reinforces confidence in the model's interpretation. To visualize the influence of each characteristic on individual predictions, we use a SHAP beeswarm plot (Figure 7), which illustrates both the magnitude and direction of each variable's effect on the model's output.
<table><caption>Table 2: Comparison of feature importance metrics (CatBoost: Feature Importance (FeatImp), Permutation Importance (PerImp), and SHAP values (ShapV)) for the best response estimation model of *F*.</caption><thead><tr><th>Feature</th><th>FeatImp</th><th>PerImp</th><th>ShapV</th></tr></thead><tbody><tr><td>p&#773;</td><td>36.77</td><td>36.63</td><td>0.192</td></tr><tr><td>m</td><td>17.07</td><td>17.08</td><td>0.060</td></tr><tr><td>e</td><td>18.36</td><td>18.33</td><td>0.059</td></tr><tr><td>&omega;</td><td>5.69</td><td>5.71</td><td>0.022</td></tr><tr><td>c<sub>m</sub></td><td>6.86</td><td>6.94</td><td>0.022</td></tr><tr><td>Q</td><td>7.33</td><td>7.40</td><td>0.017</td></tr><tr><td>R</td><td>3.90</td><td>3.90</td><td>0.014</td></tr><tr><td>&mu;</td><td>3.79</td><td>3.78</td><td>0.013</td></tr><tr><td>c</td><td>0.23</td><td>0.22</td><td>0.001</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>Figure 7: SHAP feature importance for firm *F*'s best response.</figcaption></figure>

From the figure 7, it appears that higher values of *e*, *R*, and *p&#773;* are associated with positive SHAP values, indicating that an increase in these factors has a positive effect on the best response *p*\*(*e*, *R*) of firm *F* (consistent with Figures 2, 3 and 4). For the price *ω*, most SHAP values are negative, which is consistent with its negative effect on the best response of firm *F* (as also illustrated in Figure 5). Production capacity *Q* has a variable effect, with SHAP contributions that are sometimes positive and sometimes negative (consistent with the result illustrated in Figure 6).

To consolidate these observations, the following partial dependence plots (PDPs) illustrate the average marginal effect of *e*, *R*, *p&#773;*, *ω* and *Q* on the best response of firm *F*.
<figure><img src="image_10.png" /><figcaption>Figure 8: Partial dependence plots for the best response prediction of firm F.</figcaption></figure>

## 4.3 Optimal regulatory strategies

This section is devoted to the numerical solution of step 1 of the game, during which the public authority must jointly establish the nutritional standard $e \in [e, \bar{e}]$ and the fine $R \in [0, \bar{R}]$ to minimize the health risk of the market (9).

Our methodology is based on anticipating the best response $p^*(e, R)$ of firm F, a behavior that we have modeled using the DCA algorithm (see 3.3) and CatBoost algorithm (see Section 3.4.2). The public authority's optimization problem is then addressed by exhaustively exploring a discretized parameter grid: e is explored over the interval [0.2, 0.8] and R over [0, 2000].

A sensitivity analysis of the optimal solution is then conducted by varying certain key exogenous factors individually (in particular, production capacity $Q$, market price $\omega$, and organoleptic standard $\bar{p}$), with the other parameters held constant. This approach makes it possible to identify the specific influence of each factor on the determination of the optimal choices ($e^*, R^*$).

## 4.4 Optimal regulatory strategy based on production capacity

The following figures<fnref n="14" /> show the variation in the optimal strategy of the public authority, the health risk, the best response of F as a function of the compliant production capacity of firm F.

<footnote n="14">¹⁴These figures are plotted with the following fixed parameters: $\omega = 12$, $\bar{p} = 0.5$, $m = 0.5$, $c_m = 1$, $\mu = 0.5$, $c = 2$. The compliant production capacity $Q$ is explored over [0.5, 20].</footnote>
<figure><img src="image_11.png" /><figcaption>Figure 9: Variation $e^*, R^*, \Phi(e^*, R^*)$ and $p^*(e^*, R^*)$ as a function of $Q$.</figcaption></figure>

From Figures (9), we can see that the two approaches used (DCA and CatBoost) to anticipate the best response of $F$ converge on the same results: for a low $Q$, the public authority must impose the strictest nutritional standard ($e^* = \bar{e}$) and set a fine $R^*$ at its maximum level. For a high capacity $Q$, the public authority must impose the minimum nutritional standard ($e^* = \underline{e}$) and set the fine $R^*$ at the minimum threshold.

When the production capacity $Q$ of firm $F$ is low, the authorities have an incentive to set the nutritional standard and the fine at their maximum levels. This strategy fully exploits the deterrent effect of regulation on the intensity of fraud ($1 - p^*$): the stringency of the standard and the severity of the sanction encourage the firm to choose a high level of compliance $p^*(e, R)$ (see Figures ??). However, as $Q$ increases within a moderate range, the firm has an incentive to slightly increase the intensity of its fraud ($1 - p^*$) while still maintaining $p^* > \bar{p}$. In this configuration of moderate production capacity and limited expansion, the strategy $(e, R) = (0.8, 2000)$ remains optimal for reducing health risk, as it helps contain the increase in fraud while limiting the amount of harmful substance per unit produced.

When the production capacity $Q$ becomes very high, the firm reaches a level of maximum fraud ($p^*(e, R) = \bar{p}$), since the potential gains from fraud become predominant. In this situation, the public authority has an incentive to relax the nutritional standard ($e^* = 0.2$) in order to reduce the amount of harmful substance per unit (relation 3), which mitigates the health risk despite the high level of fraud. In this regime, the fine $R$ has no effect on the firm's behavior, so its level is irrelevant for the authority.
## 4.5 Optimal regulatory strategy based on market price

The following figures show the impact of market price variations on optimal regulation decisions<fnref n="15" />.

<figure><img src="image_12.png" /><figcaption>Figure 10: Variation $e^*, R^*, \Phi(e^*, R^*)$ and $p^*(e^*, R^*)$ as a function of $\omega$.</figcaption></figure>

We can see from Figure (10) that the market price $\omega$ directly determines the optimal regulation strategy. When the price is low, the authority must impose the maximum nutritional standard ($e^*=e$) and set the fine at the highest level $R=\bar{R}$). In this case, the potential gains from fraud remain limited, and the combination of strict standards and maximum penalties is sufficient to deter the firm, thus ensuring a high level of compliance and reduced health risk.

When the price rises sharply, the profits from fraud become too attractive. It is then in the authority's interest to impose the minimum nutritional standard ($e^*=e$) as a high threshold would encourage the firm to resort to widespread adulteration to maintain its margins, which would increase the health risk. In this case, the level of the fine becomes irrelevant: regardless of its severity, it no longer offsets the expected gains from fraud. The public authority must therefore adjust its policy by relaxing the standard when prices rise, in order to limit the amount of substance per unit produced and thus contain the health risk.

## 4.6 Optimal regulatory strategy based on organoleptic standard

The figures<fnref n="16" /> below show the evolution of the optimal strategy of the public authority, the health risk, and the best response of firm *F*, by varying the organoleptic standard $\bar{p}$.

<footnote n="15">¹⁵Fixed parameters used: $Q = 5$, $\bar{p} = 0.5$, $m = 0.5$, $c_m = 1$, $\mu = 0.5$, and $c = 2$. The market price $\omega$ varies over [4, 20].</footnote>

<footnote n="16">¹⁶The parameters used for this analysis are as follows: $\omega = 5$, $Q = 5$, $m = 0.5$, $c_m = 1$, $\mu = 0.5$, $c = 2$, and $\bar{p} \in [0.1, 0.9]$.</footnote>
<figure><img src="image_13.png" /><figcaption>Figure 11: Variation $e^*$, $R^*$, $\Phi(e^*, R^*)$ and $p^*(e^*, R^*)$ as a function of $\bar{p}$.</figcaption></figure>

From Figure (11), the optimal regulatory strategy depends on the stringency of the organoleptic standard. When this standard is low, the authority should impose the maximal nutritional standard $e^* = \bar{e}$ and set the fine at its highest level $R^* = \bar{R}$. With a low sensory requirement, the firm voluntarily chooses a higher compliance level to minimize expected financial sanctions; in this setting, tightening the standard and maximizing the penalty effectively reduces the fraud. Conversely, when the organoleptic standard is *high*, the firm aligns its compliance exactly with ($p^* = \bar{p}$). The authority should then set the nutritional standard at the minimum $e^* = \underline{e}$ in order to limit the quantity of the substance per unit produced and thereby contain the health risk.

# 5 Conclusion

In this paper, we propose a decision-support tool designed to deter food adulteration. Grounded in a realistic economic representation of supply chains, it models the strategic interaction between a regulatory authority and a firm using a bilevel game. The firm's non-convex optimization problem is solved using the DCA (Difference of Convex functions Algorithm), while CatBoost estimates the global best response. The integration of these components enables the identification of the game equilibrium and the derivation of optimal regulatory rules aimed at reducing food fraud-related health risks.

Testing our tool has yielded operational recommendations and optimal strategies for public authorities to deter fraud and mitigate associated risks.

First, the DCA algorithm identifies several key findings: (i) strengthening nutritional standards must be systematically accompanied by increased penalties, as these two levers act synergistically for effective deterrence; (ii) rising market prices significantly increase the economic appeal of fraud, leading to reduced compliance levels and amplified health risks; (iii) raising
the organoleptic standard (e.g., through consumer sensory education policies) proves to be an
effective mechanism for limiting fraudulent practices; (iv) production capacity exhibits a dual
effect: at low levels, the pursuit of revenue expansion may incentivize increased fraud despite
sanction risks, whereas at high levels, the same fraud intensity exposes a larger volume of units,
multiplying aggregate risk and penalty exposure.

Second, analysis using the optimized CatBoost model establishes a precise hierarchy of food fraud determinants. Through three complementary metrics: Feature Importance, Permutation Importance, and SHAP values. The approach reveals the predominance of the organoleptic standard, which alone explains nearly 37% of compliance level variance. This is followed by the efficiency of the adulterating substance in simulating compliance and the nutritional standard, confirming their decisive influence. Partial Dependence Plot (PDP) analysis and SHAP values validate the robustness and consistency of the relationships observed locally by DCA across the entire parameter space, while quantifying the marginal effect of each variable on fraud decisions.

Third, the methodological integration of local DCA and global CatBoost approaches enables the definition of adaptive and optimal regulatory strategies. This synergy reveals that authorities should adjust their regulatory policies (nutritional standards, penalties) according to economic conditions: when facing low production capacity, low organoleptic standards, or low market prices, the optimal strategy combines maximum nutritional standards and ceiling penalties. Conversely, under conditions of high production capacity, high organoleptic standards, or high prices, relaxing nutritional standards becomes optimal to contain health risks.

After calibration to the local context, our tool applies to both developing and developed markets. In developing markets, where consumers' organoleptic benchmarks are lower, the optimal strategy (in the sense of reducing public-health risk and fraud) combines stricter nutritional standards, higher penalties, and targeted sensory education; this package raises the expected cost of fraud and reduces its profitability. In developed markets, where organoleptic benchmarks are higher, policy should maintain or raise these sensory standards while finely adjusting nutritional standards. When strong economic incentives make eradication of fraud difficult even under high penalties, a risk-reduction strategy, selectively relaxing nutritional standards to lower the hazard per unit, can be more effective than further tightening penalties.

This study presents a first version of a decision-making tool for the preventive fight against food fraud, where the current results constitute a proof of concept intended to be further developed. Each module could be expanded: (i) the economic model could integrate multiple competing firms, endogenous price formation, and a multi-criteria objective function; (ii) the game module could introduce temporal dynamics, and adaptive learning mechanisms; (iii) the resolution methods could be enhanced through multi-objective Bayesian optimization approaches while maintaining full traceability via explainable AI tools. These potential extensions affirm the evolving nature of this methodological framework.

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