# Theoretical Model for Analyzing the Links Between Nutritional and Organoleptic Standards, Food Adulteration Strategies, and Food Safety

Nacim Nait Mohand, Mohammed Said, Abdelhakim Hammoudi

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Nacim Nait Mohand, Mohammed Said, Abdelhakim Hammoudi. Theoretical Model for Analyzing the Links Between Nutritional and Organoleptic Standards, Food Adulteration Strategies, and Food Safety. 2025. hal-05301610

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# Theoretical Model for Analyzing the Links Between
Nutritional and Organoleptic Standards, Food Adulteration
Strategies, and Food Safety

Nacim NAIT MOHAND<fnref n="1" />, Mohammed Said RADJEF<fnref n="2" />, and Abdelhakim HAMMOUDI<fnref n="3" />

**Abstract.** This study analyzes the impact of food adulteration fraud strategies adopted by a company operating in a market subject to nutritional and organoleptic standards. The study is based on a mathematical model inspired by the 2009 milk crisis in China. This model examines the company's strategic decisions and highlights the relationships between food standards, fraud intensity, and health risk. We consider an agri-food company producing a good subject to a nutritional standard imposed by the public authority and an organoleptic standard dictated by consumer demand. The authority implements a control system based on tests designed to verify the product's compliance with the nutritional standard. However, these tests are imperfect and fail to detect certain substances that can mask fraud. The company may exploit this loophole by substituting a cheaper ingredient for the nutrient. The model is structured as a sequential game between two actors: the regulatory authority and the agri-food company. In the first period, the authority sets the nutritional standard based on public health objectives (non-linear bi-objective problems under constraint). In the second period, the company adjusts the fraud intensity while considering several factors: the prevailing nutritional standard, consumer sensitivity to variations in product composition, the cost of acquiring the substitute substance, and its impact on production capacity. The company seeks to maximize its expected profit (maximizing a nonlinear function under a linear constraint) while anticipating the risk of sanctions if fraud is detected through a health crisis. Solving the game using the backward induction technique reveals several results. Strengthening the nutritional standard may encourage the company to intensify fraud, thereby increasing health risks. However, this adverse effect can be mitigated by increasing financial sanctions. Additionally, better consumer awareness reduces the intensity of food fraud.

**Keywords:** Game theory · food adulteration · food standards · food safety.

## 1 Introduction

For centuries, operators in the agri-food sector have adopted fraudulent adulteration practices, deliberately altering the composition of their products through the integration, substitution or targeted adjustment of ingredients in order to optimise production processes (Hart 1952, Anagaw et al. 2024). These practices, developed strategically, take into account the organoleptic criteria specific to the target markets as well as the regulatory constraints in force (Manning 2016). Their main objective remains to increase production volumes and improve profitability. By optimising the cost/benefit ratio through the application of quantitative methods, they can fine-tune production strategies while minimising market risks, such as consumer boycotts, and the criminal risk if their practices are discovered.

<footnote n="1">¹ University of Bejaia, Faculty of Technology, LaMOS Research Unit, Algeria.</footnote>

<footnote n="2">² University of Bejaia, Faculty of Exact Sciences, LaMOS Research Unit, Algeria.</footnote>

<footnote n="3">³ National Research Institute for Agriculture, Food and the Environment (INRAE), AgroParisTech, Paris-Saclay Applied Economics, University of Paris-Saclay, 91120 Palaiseau, France.</footnote>
The analysis of issues relating to the causes, impacts and deterrent strategies of food adulteration practices has mobilised various disciplines, including the natural and life sciences, chemistry and law, among others. However, we will look at some of the work highlighting its economic dimensions, mathematical modelling and computer-based detection methods.

The studies by Zhang et al. (2023) and Lau et al. (2020) propose decision-making models based on game theory to analyze and reduce food adulteration. Based on these mathematical models, the authors demonstrate that the imposition of strict penalties and price adjustments is more effective than subsidies in deterring fraudulent practices. Their objective is to provide authorities with management tools to strengthen the integrity of supply chains and ensure sustainable development.

The works of Moore et al. (2012) and Goyal et al. (2022) propose approaches to counter food adulteration. Moore et al. created a database cataloging targeted products and known adulterants, providing an effective preventive tool. However, when unknown adulterants are used, isotope-based techniques, although efficient, remain costly and have limited preventive applications. Goyal et al. utilize artificial intelligence through machine learning and deep learning, augmented by IoT sensors, to improve fraud detection.

McCallum et al. (2022) examine the impact of food fraud on consumer behavior and demonstrate that the perception of authenticity directly influences consumers' willingness to pay. These findings underscore the importance for public authorities to take into account consumer attitudes when designing strategies to combat food fraud.

Spink and Mayer (2011) evaluated the effectiveness of traditional analytical methods in detecting food fraud and stressed the need to reinforce existing control standards. They observed that these methods, designed to identify known contaminants, fail when faced with emerging adulterants. Their study highlights the health risks associated with adulteration, notably through the example of the milk adulteration crisis in China in 2009.

A striking example occurred during the melamine-contaminated milk crisis, when the company Sanlu adulterated infant milk by adding water and melamine to circumvent quality controls and distort protein levels measured by the nitrogen method. Under the pressure of rising international prices, the Chinese dairy sector, dominated by numerous small producers and a few large companies (Yili, Mengniu, Sanlu, Bright), saw some actors resort to this fraudulent dilution to increase their production volumes. In March 2008, initial suspicions emerged, followed in September by reports of severe illnesses and the death of six infants, with nearly 300,000 associated cases of illness (Keck 2009; Xiu and Klein 2010). Rapid intervention of the authorities led to massive recall of contaminated products, and in December 2008, the trial resulted in Sanlu bankruptcy. These events illustrate the magnitude of the health risks linked to food fraud and the inability of traditional controls to detect emerging adulterants.

In light of this major crisis and the current state of the literature on food fraud, it is clear that the extent of food adulteration cannot be reduced without an approach that integrates the economic and regulatory dynamics influencing market actors' decisions. On one hand, operators adjust their fraudulent strategies based on the acquisition costs of the substances used, the risks incurred, and consumer expectations. On the other hand, public policies impose nutritional standards and sanctions that modify these incentives. Therefore, to deter such practices, it is essential to understand how these factors interact to shape corporate behavior.

From this perspective, we propose a mathematical model integrating the rational and opportunistic behaviour of companies in order to identify the relationships between nutritional and
organoleptic standards, fines, the intensity of food fraud and the health risk. We consider a food company that is representative of the production fabric, operating in a market governed by two types of standards: a nutritional standard imposed by the public authority and an organoleptic standard reflecting consumer demand. The conformity control system is based on traditional tests to assess the dosage of the nutrient, but it is imperfect because it cannot detect certain substances used as substitutes, thus distorting the results. This flaw allows the company to adopt an adulteration strategy, with health risks for consumers and financial consequences for the company in the event of a health crisis.

We analyse this interaction between the authority's decisions and those of the company through a sequential game, highlighting the variations in equilibrium under different scenarios. Our results show that strengthening the nutritional standard can reduce the intensity of fraud, but does not systematically reduce the health risk, which can, in some cases, increase. In fact, in response to the reinforcement of the nutritional standard, the company may simultaneously slightly increase the proportion of its total offer that complies with the standard (reducing the intensity of fraud) and significantly increase the concentration of the substance used in the fraud, thereby increasing the health risk. To correct this perverse effect, fines need to be increased sufficiently. Another result, in line with the literature, indicates that improving consumer sensitivity to changes in the authentic composition of products helps to reduce both fraudulent practices and the health risk.

## 2 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.

In order to ensure the nutritional safety of products circulating on the market, public authorities implement two regulatory mechanisms:

1. A minimum nutrient content threshold $e$, (with $e \in [\underline{e}, \bar{e}] \subset [0, 1])^4$. 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 $\underline{e}$ and maximum $\bar{e}$ admissible levels to guarantee the nutritional quality of foods.

<footnote n="4">⁴ The 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>
2. An imperfect compliance control system, which is based on a conventional test<fnref n="5" /> for assessing the nutrient dosage in the product.

Let's consider a company F, representative of the productive fabric, with a production capacity Q, (Q ≥ 0) in product conforming to the standards: organoleptic $\bar{p}$, and nutritional e. Company 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 company 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.

## Genuine compliance strategy

If F decides to genuinely comply, then it offers a quantity Q on the market its profit π is given in this case by:

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

where: c, (c ≥ 0) is the unit cost of producing product of quality e, and ω, (ω ≥ c) is the unit selling price of the product on the market.

In this case, the producer's profit is the difference between the revenue generated by the sale of its quantities offered on the market and the costs associated with their production.

## Nutritional quality fraud strategy

If F decides to commit nutritional fraud, then it offers a quantity X (with X ≥ Q) produced on the market. To produce the quantity X, it decides to mix each quantity p, (with p ∈ [p̄, 1]) of the compliant product with a quantity α of a chemical substance to produce one unit of product. Consequently, the relationship between X, Q, and p is given by:

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

When company F decides to use a high proportion of the compliant product, its total quantity offered X approaches its production capacity Q. However, when it decides to use a proportion close to the standard $\bar{p}$, it can significantly increase its market supply.

The quantity α of the substance, necessary to produce one unit of the offer X(Q, p), is given by the expression<fnref n="6" />

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

<footnote n="5">⁵ One 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="6">⁶ The function α(e, p, m) satisfies two properties: 1) When p increases, α 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, α 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 *m* represents the quantity of substance necessary to achieve one unit of nutrient percentage in one unit of product offered<fnref n="7" />.

The total quantity *Q*<sub>*m*</sub> of the substance, necessary to produce the offer *X*(*Q*, *p*), is given by:

$$
Q_m(Q, p, e, m) = X(Q, p) \cdot \alpha(p, e, m). \tag{4}
$$

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

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

where $c_m$, ($c_m \ge 0$) *isthemarginalcostofthechemicalsubstance*.

The health risk $\phi$ associated with the strategy of fraud depends on: (i) the level of dangerousness $\mu$, ($\mu \in [0, 1]$) of the chemical substance used to produce ($Q, p$), and the quantity $me(1-p)$ of substance contained in each unit of $X(Q, p)$:

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

Relation (6) shows that the health risk $\phi(e, p, m, \mu)$ is an increasing function of the dangerousness of the substance ($\mu$) and of the amount of substance per unit of product $me(1-p)$. It can also be interpreted as the *probability that a health crisis occurs* in connection with $F$'s fraud strategy.

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:

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

Thus, the expected profit $\tilde{\pi}(p, e)$ of company $F$, when it chooses $p \in [\bar{p}, 1[$, is expressed as:

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

where $R$, ($R \ge 0$) is the fine (penal sanction) that the company must pay if its fraud is discovered.

Figure 1 below illustrates, in the form of a decision tree, the major assumptions of the model. Company *F* faces two choices: genuine compliance or adulteration (fraud), each leading to different scenarios depending on whether a health crisis occurs or not. The diagram thus highlights the trade-off between potential economic benefits and the risks related to public health, while taking into account imperfect public oversight and the sanction *R* in case the fraud is discovered.

<footnote n="7">⁷ In the case of the 2009 milk crisis in China, parameter *m* corresponds to the quantity of melamine needed to indicate 1% of protein in the composition of a one litre of milk when checking its compliance with the protein content standard.</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Decision tree of the company's compliance strategies and their health consequences.</figcaption></figure>

## 3 The Game

To highlight the strategic interaction between the decisions of public authorities regarding nutritional standards and those of firm F, which determines the nutritional quality of the product offered on the market, we model the situation as a sequential game with two steps.

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

**Step 2.** Company F chooses $p$ (with $p \in [\bar{p}, 1]$).

The decision criterion of company F is the maximization of its expected profit (8), taking into account production costs, potential savings associated with possible fraud, and the risk of sanctions, while the decision criterion of the public authority is bi-objective, aiming to minimize the health risk $\phi(e, p, m, \mu)$, deter fraudulent strategies, and improve the nutritional quality of marketed products (i.e., maximize $p$).

The solution of the game is achieved by the technique of backward induction.

**Solution of Step 2:** Given $e \in [e, \bar{e}]$), the rational behavior of F is to choose $p \in [\bar{p}, 1]$ that maximizes its expected profit $\tilde{\pi}(p, e)$ given by (8):

$$ \left\{ \begin{array}{l} \max_{p} \tilde{\pi}(p, e) \\ \text{subject to} \\ \bar{p} \le p \le 1. \end{array} \right. \iff \left\{ \begin{array}{l} \max_{p} \frac{Q(\omega p - c p^2) - e^2 m^2 (1-p)^2 (c_m Q^2 + p^2 R \mu)}{p^2} \\ \text{subject to} \\ \bar{p} \le p \le 1. \end{array} \right. \qquad (9) $$
In the rest of this work, we will set numerical values for all the parameters of the model to solve the optimisation problem (9), and determine the equilibrium of the game. This will allow us to examine the impact of these choices on the different outcomes of the game. More specifically, we will study how the variation in the threshold of the nutritional standard *e* influences the intensity 1 − *p* of the fraud practised by the company *F*, and consequently the effect on the evolution of the health risk *φ*. We will also analyse the effect of changes in the organoleptic standard *p* and the fine *R* on these indicators. This work is based on the following algorithm:

1) Initialization of parameters: Introduce $Q, \omega, c, c_m, \mu, c, m, \mu, \bar{p}, R$, and $e$.

2) Numerical Optimization: Determine $p^* = \arg\max_{p\in[\bar{p},1]} \tilde{\pi}(p, e)$.

3) Calculation of Indicators: Compute the fraud intensity $1-p^*$ and the health risk $\phi(e, p^*, m, \mu)$.

# 4 Results

In this section, we apply the algorithm presented previously to explore, via a numerical approach, the impact of regulatory decisions on the strategy of company F. In particular, we analyze how the strengthening of food standards and the deterrent increase of fines influence the intensity of fraudulent practices by adulteration and, by extension, the health risk.

## 4.1 Impact of strengthening the nutritional standard

The figure<fnref n="8" /> below illustrates how a change in the nutritional standard threshold simultaneously influences the intensity of fraud and the health risk.

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Variation in the intensity of fraud and health risk depending on the nutritional standard in place. From Figure 2, we observe that, as *e* increases, company *F* adjusts its compliance level *p*\* to maximize its expected profit.</figcaption></figure>

Faced with an increase in the minimum threshold *e* near the minimum recommended threshold *e*, company *F* strategically opts to maintain its compliance level at the minimum, i.e., *p*\* = *p*̄

<footnote n="8">⁸ This figure is constructed using the following parameter values: $Q = 1$, $e = 0.1$, $\bar{e} = 0.8$, $c = 2$, $m = 0.5$, $c_m = 5$, $\mu = 0.4$, $R = 10$, $\bar{p} = 0.2$, $\omega = 1$.</footnote>
(maximum fraud intensity). This choice is explained by the fact that, despite the increase in production costs, the costs of acquiring the chemical substance required for adulteration, and the potential sanctions associated with an increased health risk induced by the rise in threshold $e$, these additional costs remain relatively low compared to the gains generated by the expansion of the quantity produced and offered on the market. In fact, the net benefit obtained by expanding the supply more than compensates for the increased costs induced by the reinforcement of the nutritional standard. Thus, company $F$ favors a minimal compliance strategy to optimize its expected profit while remaining within an acceptable risk zone.

When the minimum threshold $e$ approaches the maximum recommended threshold $\bar{e}$, company $F$ is induced to increase its compliance level. This strategic choice is explained by the fact that, in this case, the potential gains from fraud remain relatively low compared to production costs, the costs of acquiring the chemical substance, as well as the costs related to health risk and the payment of the fine $R$. By raising its compliance level, company $F$ manages to mitigate the probability of a health crisis, even if this entails a less extensive production.

In the first situation, when $e$ increases near the minimum recommended threshold $\underline{e}$, company $F$ maintains a minimal compliance level ($p^* = \bar{p}$). To meet the reinforced nutritional requirements, $F$ decides to incorporate a larger amount of the substance, which directly increases the health risk.

In the second situation, when $e$ increases toward the maximum recommended threshold $\bar{e}$, company $F$ reduces its compliance level $p^*$. Here, the effect of increasing $e$ on the amount of substance required is attenuated by the reduction in fraud intensity, so that the health risk evolves in a concave manner: it initially increases with $e$, then tends to decrease when the increase in $e$ is offset by a stronger reduction in fraud intensity.

## 4.2 Impact of the deterrent increase in fines

The following figure<fnref n="9" /> shows the joint impact of a strengthening of the nutritional standard and an increase in the fine on: Company $F$'s strategy and the health risk.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Variation in the intensity of the fraud and the health risk as a function of the nutritional standard in force and the fine.</figcaption></figure>

<footnote n="9">⁹ This figure is constructed using the following parameter values: $Q = 1$, $\underline{e} = 0.1$, $\bar{e} = 0.8$, $c = 2$, $m = 0.5$, $c_m = 5$, $\mu = 0.4$, $\bar{p} = 0.2$, $\omega = 1$.</footnote>
Faced with the increase in fine $R$, company $F$ is confronted with an elevated financial risk in the event that its fraudulent practices are detected. To mitigate the probability of a health crisis, company $F$ chooses to increase its compliance level $p^*$. Consequently, the intensity of fraud $1-p^*$, decreases, which leads to a reduction in the associated health risk. Although this strategic response by $F$ results in a decrease in the quantities produced, it significantly reduces the risk of criminal sanctions.

From Figure 3, several key lessons emerge regarding the combined impact of strengthening nutritional standards and increasing financial penalties in the event of a public health crisis:

When the fine remains moderate, a tightening of the nutritional standard (i.e., the shift from a less demanding standard to a stricter one) is not sufficient to reduce the health risk. In this case, company $F$ only slightly increases its compliance level and compensates for the reduction in compliant production by increasing the concentration of the substance in the product. Thus, despite the reinforcement of the standard, the expected fine remains too low to effectively deter fraud, and the health risk increases.

On the other hand, when the reinforcement of the standard $e$ is accompanied by a significant increase in the fine, company $F$ is encouraged to substantially raise its compliance level. The fear of more severe criminal sanctions then leads the company to reduce the concentration of the substance, which in turn decreases the health risk.

## 4.3 Impact of consumer sensitivity to quality variation

The figure<fnref n="10" /> below illustrates the impact of changes in the organoleptic standard on the intensity of fraud and the health risk.

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Variation in the intensity of fraud and health risk as a function of organoleptic standards.</figcaption></figure>

<footnote n="10">¹⁰ This figure is constructed using the following parameter values: $Q = 1$, $e = 0.1$, $\bar{e} = 0.8$, $c = 2$, $m = 0.1$, $c_m = 5$, $\mu = 0.4$, $R = 10$, $\omega = 1$, $e = 0.5$.</footnote>
It follows from Figure 4 that an increase in the sensitivity level of consumers to variations in product composition (i.e., an increase in $\bar{p}$) leads to a decrease in the intensity of fraud $1-p^*$, and in the associated health risk. This result remains valid even when nutritional standards are set at low levels. Mathematically, this is a rather intuitive observation: increasing the lower bound of the domain over which the maximal argument of a function is determined naturally leads to an increase in that maximal argument. Economically, this result underscores the major role that heightened consumer awareness plays in deterring food fraud and in protecting consumer health.

Another intuitive result emerging from Figure 4, concerns the increase in the marginal cost of the substance, used in the practice of adulteration. Indeed, when this cost rises, the acquisition costs of the substance become higher, which incentivizes the company $F$ to reduce the intensity of its adulteration practice. Consequently, the intensity of fraud decreases, leading to a reduction in the associated health risk.

## 5 Conclusion

Food adulteration represents a significant and persistent challenge to food safety, jeopardizing not only the nutritional quality of products but also, collaterally, public health. This practice, driven by opportunistic economic motivations, highlights the limitations of traditional control mechanisms and underscores the necessity for more effective regulation. In response to these challenges, public authorities must adopt a proactive approach, combining incentives and sanctions, to deter such practices and ensure that consumers receive products that comply with nutritional and sanitary requirements.

In this context, this study proposes a public decision-making tool intended to inform regulatory choices. We have developed a mathematical model that simulates the technical mechanisms of food adulteration by quantifying the substances used in adulteration, the costs associated with the production of compliant and adulterated products, and the economic gains resulting from increased production volumes. The model facilitates the analysis of firms' trade-offs in the face of regulatory demands, the risk of sanctions, and consumer reactions. By employing tools from game theory, we highlight the strategic levers that can guide public decisions towards more effective policies, capable of reducing the intensity of fraud and limiting its impact on public health.

The results presented in this work reveal several interactions between nutritional standards, organoleptic standards, and food safety risks. These linkages emerge from the strategic behavior of companies that possess discretionary power to circumvent public controls on compliance with nutritional quality standards. We have demonstrated that investment in consumer awareness and education constitutes an essential lever in the fight against food fraud. Our findings also underscore the need for a combined approach, articulating both the reinforcement of nutritional standards and an increase in fines, to maximize the deterrent effect on fraud and mitigate health risks.
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