MOZAIC: A mathematical approach to formalizing behavioral theories – applied to an agent-based model of mobility

1 Introduction

Global environmental change represents one of the most pressing challenges now and in the near future, leading to problems such as temperature rise, increased occurrence of extreme weather events, air and water pollution, loss of biodiversity, etc. (IPCC, 2023). Human activities (e.g., burning fossil fuels, industrial processes, and transportation) are the primary driver of global environmental change in the current epoch, referred to as the Anthropocene. Human behavior acts as both a significant destructive force and a lever for potential solutions. However, understanding and changing human behavior is complex, not least because of the micro-macro feedback within social systems. It requires a deeper understanding of the behavior itself and the resulting dynamics at the societal level (Schill et al., 2019; Steffen et al., 2008; Constantino et al., 2021). An approach that fosters such understanding is social simulation; it provides computational models that simulate human behavior and the resulting dynamics over time. Different scenarios with varying conditions can be explored and analyzed by adjusting the relevant parameters and thereby the rules of the model (Jager, 2017; Macal, 2016).

Agent-based modeling (ABM) is a particularly promising methodology, which has the advantage of implementing autonomous agents (e.g., humans, households, companies, etc.) as individual entities with varying decision-making processes, allowing each entity to make decisions based on their own circumstances (Macal, 2016). ABM enables modeling heterogeneous agents and their behavior (An et al., 2021) to understand and explore the emergence of macro-level phenomena from micro-level interactions (Macal and North, 2010). Viewed “with both enthusiasm and optimism due to its potential to create a “revolution” among the social, ecological, behavioral, and complexity sciences” (An et al., 2023), ABM has experienced broad popularity and a wide distribution (An et al., 2021). At the same time, ABM comes with unsolved challenges, including calibration and validation procedures, computational efficiency, transparency, and reusability, and, most importantly in the given context, adequate representation of human behavior (Huang et al., 2022; An et al., 2021; Balke and Gilbert, 2014). However, while some of these challenges are specific to ABM, many are generally present in complex systems modeling (An et al., 2021).

Although ABMs reflect great potential for valuable insights and to support decision- or policy-making that concerns human-environment interactions, many ABMs are implemented using oversimplified approaches to modeling human behavior (Huang et al., 2022; Mehdizadeh et al., 2022). In addition, modeling human behavior is a challenge itself, as human behavior is context-dependent, multifaceted, and complex (Schwarz et al., 2020; Schlüter et al., 2017). Although there are various ways to derive decision rules for a decision model, a major differentiation can be made between ad-hoc and more sophisticated theory or data-driven approaches (Mehdizadeh et al., 2020; Schlüter et al., 2017).

In this paper, we focus on theory-driven approaches that allow modelers to integrate the knowledge on human behavior accumulated in the social sciences over centuries with an abundance of respective theories explaining human behavior. Theory-driven approaches constrain how agent behavior is implemented, reducing the danger of ad-hoc modeling decisions (Antosz et al., 2023). Modelers are often unaware of this abundance of available theories and may also lack the competence to put them into code adequately (Schlüter et al., 2017). Navigating theories outside one's own discipline is challenging, as a shared “lingua franca” for ABMs and decision models is still evolving within interdisciplinary research (Grimm et al., 2020). Terms and theories often differ or overlap subtly and not necessarily consistently in various fields, creating difficulty in integrating theory-driven approaches effectively (Grimm et al., 2020). Even for a selected theory at hand, choosing the most suitable formalization is usually difficult, leading to different implementations of a single theory (Polhill and Gotts, 2017; Muelder and Filatova, 2018).

Overall, two main challenges in implementing theory-driven approaches are (i) the identification of relevant theories, and (ii) the lack of formalization (Jager, 2017; Schlüter et al., 2017; Schwarz et al., 2020; Scholz et al., 2023). Although challenge (i) has not yet been solved to a sufficient degree (Wijermans et al., 2023a), there is a growing number of publications and resources that provide some guidance to find and select a suitable theory for a model (see, e.g., Balke and Gilbert, 2014; Schlüter et al., 2017; Constantino et al., 2021; Groeneveld et al., 2017; Mehdizadeh et al., 2020; Mooij et al., 2023; Klabunde et al., 2015; Wijermans et al., 2023a; Antosz et al., 2023; Rietkerk-van der Wijngaart et al., 2025). In this paper, we focus on challenge (ii): the formalization of theory, which has received comparatively less attention, with fewer papers highlighting the problem (see, e.g. Jager et al., 2000; Schlüter et al., 2017; Scholz et al., 2023) and providing common best practices (Schwarz et al., 2020). Existing efforts toward standardized model descriptions, such as the ODD protocol and its extensions, improve transparency and communication but do not address the formalization of the underlying theories themselves (Muller et al., 2014). As social science theories are formulated primarily in natural language, they necessarily remain ambiguous. This leaves room for interpretation when a social science theory is formalized for the use in a computational model and there are little to no best practices or standards for doing so. Even the term “formalization” can be interpreted in several ways, referring, e.g., to graphical representations, such as decision trees or flow charts (e.g., Schwarz et al., 2020), ontological systems that decompose theories into labeled constructs and typed relationships (West et al., 2019), specific cognitive architectures (Epstein, 2014), mathematical formulae, and pseudocode or code (e.g., Bronson and Jacobson, 1990; Schwarz et al., 2020; Schlüter et al., 2017; Muelder and Filatova, 2018). The different approaches can be grouped into more computational science-oriented (e.g., using UML or object-oriented design) and more formal mathematical ones. We see value in different toolkits to cater to the needs of a diverse community of social simulation modelers, and identify a need for frameworks on how to formalize.

In this paper, we showcase the potential of mathematical formalization and how it can help reduce ambiguity. Muller et al. (2014) argue for the development of a common standard beyond source code and natural-language descriptions; we propose mathematical formalization as one concrete realization of such a standard to reap benefits described by Suppes (1968), such as explicit and self-contained assumptions, generality, and standardization. Using mathematical formalization to describe specific social simulation models is not new, as mathematical formulae that provide formal descriptions for agents' decision making can be found in various examples, such as the well-know Sugarscape (Epstein and Axtell, 1996) or a host of opinion dynamics models since the voter model (Clifford and Sudbury, 1973). Epstein (2014) suggested and mathematically formalized a specific cognitive architecture building upon a neurocognitive model of decision making. Compared to that, what we propose in this article is theory-agnostic: we provide a process for formalizing diverse social science theories rather than suggesting a particular agent design.

The MOZAIC framework for Mathematical fOrmaliZAtIon for deCision models that we describe in this paper is a systematic method for mathematically formalizing social science theory into a decision model. It aims at achieving less ambiguous interpretations of theories that are reusable, transferable, and hence comparable between models. The purpose of the present paper is twofold: first, we suggest a list of steps that modelers can use as guidance to arrive at formal decision models for their ABM. Secondly, we apply the MOZAIC framework for an example case, a simple ABM on mobility tool ownership, and for five example theories identified by Schlüter et al. (2017) as commonly used in social simulation models: rational choice theory, bounded rationality, habitual decision making, prospect theory, and descriptive norms. As a result, we provide mathematical definitions of these five theories that serve as mathematical building blocks (MBB). How these blocks relate to and may complement the initiative on reusable building blocks (RBB) and AGENTBLOCKS (Berger et al., 2024; Filatova et al., 2025), which provides atomic blocks of code of specific mechanisms or processes relevant to many ABMs in a given domain, will be discussed in Section 2.

By detailing all steps in the suggested formalization process, we hope that this work might serve as an orientation, inspiration, and transparent resource for other models. The resulting mathematical formalizations aspire to be reusable in case-specific implementations of social science theories in ABMs in other contexts. Particularly for those who prefer formulae over pseudocode, they may provide a valuable starting point to formalize psychological constructs enabling modelers to choose and combine theories suited to their application. The remainder of the paper is organized as follows: Section 2 discusses current practices in formalization in the field of ABM and how mathematical formulae might be beneficial; it then introduces the four steps of the MOZAIC framework. Section 3 describes the application of the MOZAIC framework to formalize social science theories for a Mobility Transition Model. A discussion and conclusion including an outlook for further research are presented in Section 4.

2 Mathematical formalization of social science theories for ABM

A systematic approach of how to formalize would reflect great potential to support modelers in their process of formalizing social science theories, aiding a discussion toward community standards and best practices. Formalization generally refers to the translation of concepts, statements, or a theory that is expressed in a natural language into a formal language. As mentioned, there are various options of which formal language to use; we focus on mathematical language that includes formulae, but also natural language descriptions of what each piece of notation represents and which characteristics it is assumed to fulfill. One can consider mathematics itself a language, as, e.g., Pólya (2004), who characterizes mathematical notation as a “langue bien faite”—a language well adapted to its purpose, concise and precise, with rules which, unlike the rules of ordinary grammar, suffer no exception. [p. 135], making mathematical formalization a specified and unambiguous translation of concepts, statements, or theories to mathematics. Where their use is feasible and suitable, mathematical definitions are, on the one hand, precise, and on the other hand flexible in their level of contextualization or generality: filling in all the details, they can be as precise as computer code, whereas when stating more abstract properties, they can remain at a more general level, like graphical representations, but may avoid ambiguities. With the help of mathematical definitions of concepts, one can capture general structures and precisely formulate requirements or restrictions, enabling the reuse of formalized theory across various application cases and models.

As we are interested in the formalization of social science theories in the context of modeling, briefly distinguishing formalization from modeling seems helpful: mathematical modeling refers to the process of simplifying a real world situation while preserving its key characteristics, translating the real world problem into a mathematical representation, deriving results within mathematics, and then using these results to gain insight into the real world situation or, if the model does not yet fit, to refine and improve the mathematical representation (Blum and Leiß, 2007). While both activities include a translation to mathematics, modeling thus mostly refers to a concrete real-world situation, whereas the term formalization is rather used for concepts or theories at a more general level. The mathematical representations of concepts or theories can then be further specified for use in a model of a concrete situation (see Figure 1).

However, the distinction between formalization and modeling is not clear cut. In fact, when translating a social science theory into mathematical language, verbal descriptions always leave room for interpretation, requiring further specification from a modeler (Schlüter et al., 2017; Schwarz et al., 2020). Such a specification is often impossible to obtain at a general level: in practice, when moving from an abstract description of a theory to a specific code implementation in a model and for a given system under study, making a natural language definition more precise and specifying it to a concrete case are two sides of the same coin. Both require subjective decisions by a modeler—creativity even (Muelder and Filatova, 2018)—leading to a multitude of varying and case-specific uses of a single social science theory in models (Scholz et al., 2023).

Previous works on formalizing social science theories defined elements within decision models that can be explained by social science theory (e.g., Schlüter et al., 2017) or abstracted back from details given by case-specific ABM codes, finding commonalities but also differences in structures, a paramount example is the work by Muelder and Filatova (2018) for the theory of planned behavior. What is rarely done in ABM is to provide a formal definition of the social science theory used as an intermediate step between the theory and the ABM code, a step more precise than the natural language definition, but more general than the code. A noteable exception is the recent work on reusable building blocks (Filatova et al., 2025) that proposes exactly such intermediate level elements for reuse in different ABM. These blocks come with a main description and one or several implementations that avoid repeating the reinventing of how to formalize agent-behavior (or other common ABM parts) over and over. The authors compare RBB to software engineering and its prime concepts of abstraction and modularity, focusing on the latter with modular blocks of code. The approach we propose in this paper—mathematically formalizing theory into mathematical building blocks—is less specific (case-/code-wise) while similarly reducing ambiguity and allowing the application and contextualization for different models¹. Figure 2 illustrates where we collocate MBB on the spectra from generality to specificity and from natural to formal language in comparison with the starting point, social science theories, other types of formalization (e.g., UML diagrams or RBB), and the outcome: a case-specific ABM.

Note that with the MOZAIC framework we do not suggest rephrasing a whole ABM in mathematical language—ABMs are used precisely when a description of some dynamics in terms of a few equations is not possible (Secchi et al., 2024)—but instead want to formalize small elements in an ABM to make these unambiguous by stating assumptions explicitly. For using MOZAIC, no mathematical knowledge beyond high school is needed, as we do not employ mathematical results but only use basic concepts (such as functions) to describe important parts of a model accurately.

To clarify where the MOZAIC framework can be used, it helps to position it within the modeling cycle (see Figure 3). While various versions of a modeling cycle exist, we set our focus on the key steps in “the modeling world”: problem definition, conceptual model, formal model (the focus of this paper and not explicitly part of every modeling attempt), computational model, simulation, and analysis. Hence, the MOZAIC framework is located between developing a conceptual model (oneself or using an already existing one) and implementing the resulting formalizations in a computational model. The conceptual model might or might not already specify a social science theory (if not, this needs to be decided during the first step of the formalization). However, what should be decided beforehand is the general setting and conceptualization.

To structure the formalization process, the MOZAIC framework suggests a stepwise approach; its four steps are detailed in Sections 2.1–2.4 and consider a conceptual model to be given as a starting point.

2.1 Step 1: behavioral context and theory selection

As a preparatory step, the context of the behavior of interest should be understood and described. Clarifying the type of phenomenon in scope and the context in which it takes place allows one to identify relevant aspects of the behavior of interest and, based upon this, to select a suitable theory. In addition to a growing number of publications from the modeling community², behavioral science, particularly psychology, offers guidance on how to analyze behavior and select a suitable theory. Although such knowledge is not yet commonly integrated into the development of ABM, attention is growing and Rietkerk-van der Wijngaart et al. (2025) describe advantages and a structured way of conducting behavioral analysis. They propose four steps: identifying the type of behavior (or behavioral change), identifying influencing factors, finding relevant theories, and finding data/specifying data requirements. Those steps can be a point of reference for laying the ground for the formalization process described in the following steps. Note that in the example case in Section 3, we select not one theory but five examples. This is due to the fact that this paper aims to explore the proposed method rather than to define an ABM's decision model in the first place.

2.2 Step 2: agents, environment, and interaction structures

In the second step, we suggest to define agents, environment, interaction structure, and, in particular, the state variables of all of the aforementioned that are relevant to decision making. This step can be considered as a mathematical formalization of the section entities, state variables, and scales of the ODD protocol's overview section (Grimm et al., 2020). As we are interested in human decision making, we consider a set H={h1,…,hH} of H ∈ ℕ human agents, where individuals will be referred to as h_*i*. Throughout, i and j will be used as indices. Mathematically defining the agents' state variables means choosing a notation and explicating assumptions with the help of statements. For example, one would specify the domain of definition, i.e., the set that a variable is part of: e.g., whether a state variable is a number in ℕ or ℝ or one of a finite number of categories such as an age group. Schlüter et al. (2017), Wijermans et al. (2023a), and Müller et al. (2013) identify relevant elements in the agents' states, such as goals, values, knowledge, and assets—if such elements are to be used in an ABM, they need to be mathematically defined in this step. A model context is needed to further carry out a mathematical formalization, as we do for an example in Section 3.

Other types of agent beyond humans are not the focus in this framework, but may enter the formalization process in so far as their state variables might also influence the humans' decisions. In this case, the state variables of the other agent types have to be defined, just as those of the human agents. The same is true for other entities: state variables of, e.g., the environment should be defined wherever relevant to agents' decision making.

The interaction structures specify who can interact with whom. Such structures can have various forms such as networks, where agents are nodes and interaction possibilities are indicated by the edges between nodes; they can be defined through geographical vicinity, or even randomly (which then needs to be described by the underlying probability measure). Again, in full generality, no useful mathematical formalization is possible.

2.3 Step 3: decision model elements

In the third step, the decision model elements are defined. The fundamental starting point in considering decision making seems to be the question of Müller et al. (2013) about subjects and objects of decision making. The subjects, human agents, have been discussed in the previous step, while objects will vary with the model context. We suggest defining a notation for the set of decision options that an agent has and will use O in the following. We consider a finite set of K ∈ ℕ options. An infinite set of options might result, e.g., when an agent can allocate a fraction of a resource to some activity, so that O=[0,1]. However, this case is not relevant here³.

Another kind of decision model element arises from the fact that human decision making is considered a multi-stage process in much of the literature (see, e.g., Muelder and Filatova, 2018). Different definitions of stages exist, and stages with the same label can be interpreted differently. Schlüter et al. (2017) present an ABM-specific starting point by listing perception, evaluation, selection, and behavior as processes in decision making. Depending on the theory selected, all or some of the named processes may be relevant and therefore need to be formalized. Here, we focus on the distinction between evaluation and selection that is widely observed in the literature (see, e.g., Schwarz et al., 2020; Wijermans et al., 2023a; Schlüter et al., 2017). As the term evaluation suggests, in this step of decision making, different alternatives are equipped with some kind of value. Note that the set of alternatives under consideration can be the set of decision options just defined, but this need not always be the case. In economics, a prominent example is a utility function mapping each alternative to a number, enabling a direct comparison. We will consider utility in the context of the example model in Section 3.3.2. For ABM, Schlüter et al. (2017) describe evaluation as the “process by which an individual determines the significance, worth, or condition of the perceived state of the social and bio-physical environment” and refer to returns from past decisions, fairness or equity of distribution of benefits within a society, and compliance with social norms as examples. Although the example of returns can be mapped to a number, this might need further operationalization for concepts such as fairness or norm compliance. Selection concepts abound in the literature, including random choice, optimization, imitation of other agents, and satisficing; again, with different variants of many of these mechanisms. For most selection processes, no standard formalization is available; examples will be defined in Section 3.4.

2.4 Step 4: mathematical formalization of theory with decision model elements

In this final step, the entities and decision model elements defined above need to be combined into formulae that represent the selected theory in the model. We suggest beginning with a short natural language sketch of the theory and references to the relevant literature.

Given formal definitions of agents and their states as well as of decision model elements, as obtained from the two previous steps, the formalization of a social science theory then corresponds to defining a decision-making mechanism according to the theory. Ideally speaking, this requires a formula that represents how some inputs from a previously defined set X∈X, which will generally depend on the agent taking the decision, translate into a choice between the available options, in other words, a function ϕ:X(hi)→O. Realistically speaking, this function will generally not be available in closed form but be built from several parts or in several steps, e.g., evaluation and selection steps. If these two steps can be formalized as two functions, decision making may in some cases be defined as function composition. Also, the input to decision making will generally depend on more than just the agent itself, involving information from the environment, other agents, etc. Hence, discussing this step makes most sense in an example context and for example theories, as will be done in Section 3.4.

2.5 Formalization results: mathematical building blocks (MBB)

Going through the steps described above, one arrives at the formalization of a social science theory in the context of a specific model. Such formalizations consist of mathematical definitions together with statements about requirements or conditions, while leaving open some details that have to be filled in case-by-case. For example, one might define a decision-making mechanism as a function and specify what kind of input this function should take to produce what kind of outputs, and whether it should fulfill further conditions, such as monotonicity. The concrete function to be used then needs to be defined for the code implementation, and ideally, another modeler could take the same general formalization and fill in another concrete function for use in their model. In this way, MBB are created as a form of RBB, which specify constraints and assumptions made. Being mathematically specific about assumptions provides the ground for a sound discussion about assumptions and theory development.

To demonstrate usefulness and applicability, we apply the MOZAIC framework to a specific case in the next section. In doing so, we also point to the need for contextualization, which is a permanent necessity when developing a formal model of a real-world case. The application has a further function by giving particular consideration to Step 4: the mathematical formalization of the theory and the decision model, which is at the heart of MOZAIC. By applying this step to five different theories, we showcase different features, and, at the same time, produce a set of MBBs of social science theories that can be used and contextualized by modelers for their models.

Although Step 4 might also serve as a stand-alone method, the information from the previous steps is required anyhow, so that we recommend following all steps in the given order to ease the process of specifying a well-defined contextualized mathematical formalization. Note that, as is probably always the case in modeling, the distinction between the steps is somewhat artificial. In practice, rather than completing steps in a row, there will be a back-and-forth between steps, as one might notice only in a later step that some aspect or element was missing or should be revised.

3 Applying the MOZAIC framework to formalize social science theories for a mobility transition model

To apply the framework to a simplified example model, we briefly introduce this model's purpose and patterns along the lines of the ODD protocol's overview section (Grimm et al., 2020). The Mobility Transition Model (MoTMo) was developed to model mobility behavior in Germany, spanning multiple decades, to enhance the understanding of daily human mobility in the long term and under various policy settings. The purpose of MoTMo is to provide information on long term dynamics to support informed decision making by policymakers, stakeholders, and citizens (Wolf et al., 2023). The patterns that the model should reproduce to fulfill this purpose are the past evolution of the number of privately owned vehicles in the study regions, as provided by KBA (2025), as well as insights and attributes about the population's mobility choices as obtained from the “Mobilit't in Deutschland” (MiD) survey studies (BMDV, 2018). The present work arises from the development of a new version of the model for an urban and a rural region that puts a strong focus on theory-driven decision models. In this setting, we provide theory-driven decision models for five selected theories.

3.1 Step 1: behavioral context and theory selection

Mobility decisions and the resulting behavior impact sustainability through global factors, such as CO₂ emissions and energy consumption, as well as local factors, including space consumption, air pollution, social equity and accessibility. Here, we focus on global factors and people's daily mobility. Air travel and long-distance public transportation (PT) depend on a very different behavioral context and are therefore beyond the scope of this work. The emissions resulting from human mobility behavior depend on the mode of transport. In Table 1, the emissions by mode of transport for Germany in 2017 are listed as published by the German Federal Environment Agency (Allekotte et al., 2021), where the passengers' kilometers account for the capacity and occupancy of a transport vehicle.

Table 1. Emissions and modal shares by transport mode in Germany in 2017.

For more detailed information, please refer to Tables 2, 3 in Allekotte et al. (2021).

Motorized individual transport has the highest emission rates in road traffic and, combined with its popularity, 75.3% of passenger transport emissions are caused by motorized individual transport. Walking and cycling have very little emissions. Due to the important contribution of motorized individual transport to overall emissions, this mode is the main focus of the decision model here. The most significant factor in using motorized individual transport is availability (Schmid et al., 2023). For motorized individual transport, the availability is defined by the car ownership status. Car ownership in turn leads to a car-centered mobility style, as thoroughly investigated by Haustein and Hunecke (2007, 2013), and Haustein and Kroesen (2022). Also, due to a car's lifetime, the attribute car ownership has to be analyzed in the long term, which aligns with the scope of MoTMo. We therefore concentrate on the decision-making process for car ownership and have thus identified the type of behavior and/or behavior change that we want to focus on, completing Step 1 of Rietkerk-van der Wijngaart et al. (2025))'s Behavioral Analysis. In ongoing work, MoTMo is combined with the Multi-Agent Transport Simulation (MATSim) model (Horni et al., 2016), another large-scale ABM, which additionally models the daily mobility trajectories of all individuals in a given region, thereby providing detailed information about decision making on mode usage.

Decisions about car ownership are influenced by many different factors; an overview is given in Table 2, categorized, and sorted in preparation for the formalization process, as described in Step 2 of Rietkerk-van der Wijngaart et al. (2025)'s Behavioral Analysis. Evidently, the large number of factors is challenging. In addition, many factors are difficult to quantify or measure, and some are interdependent or based on multiple sub-factors themselves. To avoid these complications here, we focus on geographical and financial aspects for the following reasons: first, these factors are among the most significant (Haustein and Kroesen, 2022; Gu et al., 2021; Schmid et al., 2023). Second, a minimal selection of clearly distinguishable factors is made to avoid interdependence between the factors used. For example, social factors such as status or prestige might depend on financial factors such as socio-economic status. Third, we keep the decision model as simple as possible to clearly illustrate similarities and differences between the social science theories considered.

Table 2. Overview of relevant decision factors for car ownership (based on Haustein and Kroesen, 2022; Gu et al., 2021; Schmid et al., 2023).

In fact, departing from Step 3 in the Behavioral Analysis by Rietkerk-van der Wijngaart et al. (2025) to some extent, we explore a broader spectrum of five theories: rational actor, bounded rationality, habitual decision making, prospect theory, and descriptive norms theory—rather than aiming to identify the most relevant ones at this point. The selection draws on Schlüter et al. (2017), who base their selected six theories—of which we drop the theory of planned behavior⁴—from a pool of about 30 representing a broad spectrum of both human behavioral drivers and usage in different scientific disciplines.

3.2 Step 2: agents, environment, and interaction structures

Given the above context, we define entities and their state variables as follows:

• As part of the environment in the model, the available transport modes are considered entities. Although they do not have agency, transport modes differ among each other and might undergo changes throughout a model simulation. An example attribute (which can vary, e.g., through policy interventions) is their cost, denoted c ∈ ℝ₊. To start with the simplest possible model, we consider just two modes, car and other, where the latter summarizes public transport, cycling and walking. For notational convenience, below we define the set of modes to be M={car, other} and refer to the modes as m∈M.
• Another kind of entity in the model is its spatial units that we refer to as locations l∈L={l1,…,lL}, where L denotes the number of locations in a given context. We consider locations that are cells in a given grid. In other models, one might also use administrative units (quarters in a city, etc.) or any kind of polygon adapted to a specific context. For simplicity, the only attribute of locations is a spatial type r∈R={r1,r2,…,r7} defined by the regional statistical spatial typology of BMDV (2021). We consider seven different spatial types (four in urban and three in rural areas, where r₁ indicates a metropolitan area and r₇ the most rural kind of area) and the spatial type of a specific location l_*j* is denoted r(l_*j*).
• Finally, the model contains one type of agent: a set H={h1,…,hH} of H ∈ ℕ humans or private persons. Although households play a role in MoTMo, we ignore them here for simplicity. The attributes of the agents' include their mobility toolbox b ∈ {0, 1}, where 1 indicates car ownership and 0 indicates that the agent does not own a car. We abstract from the availability of other modes—for simplicity, we consider these available for all agents—and note that we also disregard the possibility of owning more than one car per agent. Other attributes of the agents are the location of the home l∈L and the socio-economic status s∈S={s1,s2,…,s5}, where we refer to the five status groups defined by MiD (BMDV, 2018) from very low to very high. The attributes of an agent h_*i* will be denoted b(h_*i*), l(h_*i*), or s(h_*i*), respectively.

The ODD protocol (Grimm et al., 2020) further asks for specifying the model's scales, referring to its temporal and spatial resolution and extent. For the example model, the geographical resolution and extent are given by the data underlying the locations, while the temporal extent we are interested in is several years to decades. Focusing on decision making about car-ownership, we consider a quarterly time step. These scales will not explicitly occur in the MBBs introduced below. However, in some contexts, scales may play a role for whether or not an MBB is suitable, e.g., if different factors influence some behavior on shorter or longer time scales, so defining scales as required by the ODD protocol is useful also at this step in our framework.

Given the entities' state variables, it becomes obvious that model specification is an iterative process. The reason for choosing the state variables described here will only be fully apparent when defining, e.g., a utility function below.

3.3 Step 3: decision model elements

As decision model elements to formalize several theories, we will need decision options (Section 3.3.1) and an evaluation concept that we define via utility (Section 3.3.2).

3.3.1 Decision options

As described above, among the two kinds of mobility decisions—relating to ownership/availability and usage of transport modes—we focus on the former and assume, as mentioned, that except for cars all other transport modes are always available. Hence, it is sufficient to consider the agents' decision between the two alternatives of owning a car and not owning a car. The object of an agent's decision making in our case is thus one of its own state variables (as car ownership is given by its mobility toolbox b).

For a given agent, the set of options to decide between can be written in two ways. Focusing on the resulting car ownership status, the options are the following.

• o₁: the agent decides to own a car, i.e., the mobility toolbox is maintained or changed to b = 1
• o₂: the agent decides not to own a car, i.e., the mobility toolbox is maintained or changed to b = 0

Focusing, on the contrary, on whether the decision means a change to the agent's mobility toolbox or not, which may be the more relevant perspective, one can alternatively consider the options “buying a car,” “selling a car,” “keeping a car” or “remaining without car.” The definition of these options is a bit more complicated, as they depend on the current mobility toolbox of the agent. As soon as this is the case, time needs to be made explicit: we denote it by an index t ∈ ℕ. It is customary to denote by time indices t, t+1, etc. the time steps of the model. As we here do not specify the complete time step of the model, but the sub-step of a single agent's decision making, we will denote the value of the agent's state variable “pre-decision” b_*t*, and its “post-decision” value b_*t*+Δ. In a full model step, the value of Δ will depend on the number of agents and the number of further substeps included. Then, the options can be written as:

• o₃: the agent decides to stick to the current car ownership status, i.e., the mobility toolbox is maintained, b_*t*+Δ = b_*t*
• o₄: the agent decides to sell the car, i.e., if b_*t* = 1, then b_*t*+Δ = 0 (otherwise, this option is not applicable)
• o₅: the agent decides to buy a car, i.e., if b_*t* = 0, then b_*t*+Δ = 1 (otherwise, this option is not applicable)

For reasons that will become clear later, we will consider the set O={o1,o2,o3} as the available options for an agent's decision, although this mixes the two perspectives outlined. The process by which an option is selected will be defined on the basis of the respective theories below.

3.3.2 Evaluation through utility

The next element of the decision model concerns an evaluation step. As we will use a utility function, we first have a few general remarks on this concept. The concept of utility evaluates alternatives by mapping them to abstract utility values, usually real numbers, which do not necessarily say much in themselves but are comparable in that the terms smaller, larger, equal are well-defined. Utility originates in economics as a general formalization of preferences over some set of alternatives A, where preferences are denoted, e.g., ≼, that is, a_*i*≼a_*j* means that the agent considers an alternative a_*j* at least as desirable as another one, a_*i*. Note in particular, that to use utility, the preferences need to be given; a utility function is merely one concise way of representing them. This representation is possible only if the preference ordering fulfills the conditions of reflexivity, completeness, and transitivity⁵, that is, only in this case can the preferences be represented by a utility function u:A→ℝ that fulfills the following:

For any ai,aj∈A:u(ai)≤u(aj)⇔ai≼aj.

In other words, a utility function represents the ordering by assigning real numbers to each element in a set, where a higher number expresses a higher preference (Debreu, 1954). Since utility functions are easier to work with than preference orderings, they are widely used, sometimes without questioning the assumptions that this entails.

Real-world decision making is often based on evaluations corresponding to preferences that are, e.g., incomplete or not transitive (see, e.g., Kahneman and Tversky, 1979, for the fact that most humans are not rational in the sense of rational preference orderings). Decisions may also be based on evaluations that are difficult to match directly with preference orderings at all, for example, if they are situated and depend on social structures represented within the individual, such as the social identity approach (Wijermans et al., 2023b; Tajfel and Turner, 1979). In such cases, assuming the existence of a utility function would disregard these aspects. Whether or not using a utility function is justified will then depend on a model's context and purpose.

In the context of consumption, a prominent topic in economics, utility functions are often used to express preferences over bundles of goods, assuming that these goods are infinitely divisible, i.e., their amount can be any real number. We denote the amount of each good in a bundle as a vector y ∈ ℝ^*n* for the case of n goods. While the general assumption on goods is that more is better, it is not a priori clear how to compare two bundles where one has more of a first good, but the other one contains more of a second good. The evaluation depends on the preferences of an agent that are expressed again via a utility function. In this case, the utility function provides an aggregation of the amounts of all goods into one value, u:ℝ^*n* → ℝ. The idea of utility as an aggregation can be transferred to mapping several criteria into a single value. Two very common examples of utility functions that provide such aggregations are (i) a weighted sum of criteria and the corresponding multiplicative formula, which is (ii) a weighted product of criteria, where weighting is obtained using exponents. The latter is known as the Cobb-Douglas utility function. Many other functions can be used, expressing various assumptions about preferences. In addition, in settings with uncertainty, the concept is usually extended to the expected utility. We will not go further into these topics here but return to our example.

Having decided to use a utility function for aggregating several criteria, we have to define both the function and the criteria. As probably in most models, rather than first defining preferences and then representing them by a utility function, we directly define the function and assume the given corresponding preferences. In our context, rather than evaluating the decision options just defined, we consider the fact that agents decide about a means to an end, i.e., owning a car or not enables them to use different transport modes. For the evaluation step in the decision models defined below, we therefore consider as alternatives the transport modes, so that A=M≠O. Criteria for evaluating a transport mode may include time needed, costs,⁶ convenience, etc. To keep the utility function simple, we focus on just two criteria that relate to the geographical and financial decision factors discussed above: convenience and cost of the modes in M. We hence need to define what is measured for these criteria. And while decision making is considered from the perspective of an individual agent, this measurement may well derive from other model entities and their attributes, as long as these can be mapped to the agent. In particular, geographically anchored factors—as convenience in our case—will depend on a spatial location that an agent inherits from its home location.

• In view of the convenience criterion, a spatially explicit infrastructure quality is defined for each transport mode, where, for simplicity, we abstract from single locations, but consider the infrastructure quality as depending on the regional statistic type of locations (see Section 3.2). That is, we define a function q:R×M→Q, where Q=[0,1] and a larger number indicates a higher quality. For agent h_*i*, the relevant infrastructure quality values for the two modes are those of the region type of its home location: q(r(l(h_*i*)), m). As this becomes difficult to read, we abbreviate: q(h_*i*, m): = q(r(l(h_*i*)), m). In MoTMo, the definition of this function is derived from survey data, and we will not consider its details here, as this would go too deeply into the definition of the regional statistical spatial typology used, e.g., to decide how a specific urban type relates to a specific rural one. Roughly speaking, for other, the quality decreases when going from more urban to more rural spatial types, and vice versa for car. Comparing the two modes for a fixed spatial type, in the more rural cases, the quality of car should exceed that of other, and vice versa for very urban cases, e.g., for an agent h_*i* with r(l(h_*i*)) = r₁, we get that q(h_*i*, car) < q(h_*i*, other), and for h_*j* with r(l(h_*j*)) = r₇, the converse is true q(h_*j*, car) > q(h_*j*, other). Where exactly this quality comparison flips again depends on data and will not be further discussed here.
• Similarly, the financial load of each transport mode experienced by an agent will depend not only on the cost of the transport mode c(m) ∈ ℝ₊ but also on the socio-economic status of the agent. We define the financial load f as a function of the socio-economic status and the mode f:S×ℝ+→F with F=[0,1]. In the implementation of the model, its values for the 10 possible combinations (of two modes and five status groups) are determined from data. At an abstract level, we require the function to be increasing in cost and decreasing in the social status component, while a concrete definition of the function will have to be tested and possibly changed in the model validation step. From the agent's perspective, the relevant values are those corresponding to the agent's socio-economic status for both modes, f(s(h_*i*), c(m)), which we analogously abbreviate to f(h_*i*, m): = f(s(h_*i*), c(m)).

This specification of decision factors illustrates how mathematical formalization helps make explicit assumptions at a more general level than through a concrete code implementation. Functional forms, i.e., properties that the functions should have in order to be useful representations of infrastructure quality or financial load could be specified and/or discussed at an abstract level. Thus clarifying structures at this abstract level constitutes an intermediate step between a concept description and an example implementation in a model, and this is one of the benefits of mathematical formalization.

Using all of the above, we define an example utility function, with which each agent hi∈H can evaluate the individual utility of the transport modes m∈M={car, other}:

uhi(m)=α1q(hi,m)+α2f(hi,m),(1)

where weights α₁ > 0 and α₂ < 0 reflect our assumptions that the utility of a mode increases with a higher quality of infrastructure and decreases with the associated financial load. The explicit values of these weights need to be defined in the implementation of the model, obtained via calibration to real-world data. Note that the specification of the utility function explains some of the choices made in Step 2 above, illustrating the close connection between the steps that often requires some iteration in model development. However, clarifying structures at this abstract level constitutes an intermediate step between a concept description and an example implementation in a model, and this is one of the benefits of mathematical formalization.

In the implementation of the large-scale mobility ABM MoTMo, the utility calculation will be extended with further criteria, respectively, decision factors to increase realism. A relatively straightforward extension might be to vary the values of the weights between agents to represent that people might weight the criteria differently. This could be based on data for groups of agents (e.g., if data suggest that agents who have to transport kids might put a larger weight on convenience or similar) and/or by random perturbations to summarily represent heterogeneity in agents' preferences. However, to minimize complexity in this work, the simple form in Equation 1 is used in the following. To sum up the decision model elements, Table 3 lists all relevant inputs to the decision making of an agent for use in the following formalization of theories.

Table 3. Overview of an agent's decision factors and necessary related information from the model.

Elements are used implicitly to determine others and are hence not explicitly used in the formalizations below.

3.4 Step 4: mathematical formalization of theory with decision model elements

As discussed, we describe this step for not one but five theories for the purpose of illustrating the MOZAIC framework. Four out of these theories include the evaluation and selection steps, so that their common structure can be summarized in our context by Figure 4.

3.4.1 Rational actor or homo economicus

The main characteristics of rational actor theory are the assumptions that an agent is perfectly informed, acts in their own self-interest, and makes consistent and logically reasoned decisions based on well-defined preferences, as described in Section3.3.2. This is formalized (in economic and other previous literature) in making decisions maximizing utility. For a more in-depth discussion of the theory, refer to Simon (1955), Frank (1987), Monroe (2001), and Schlüter et al. (2017). In cases where utility is defined over O, one can simply define the chosen option as:

o*=argmaxo∈O(uhi(o)),(2)

where argmax denotes the set of values for which a function is maximal. Thus, in this simple case, utility plays the role of evaluating alternatives and maximization that of selecting one of them. However, as mentioned in Section 2.3, the stages of a decision process with the same label can be interpreted differently. In Schlüter et al. (2017)'s formalization of the rational agent, the goal utility maximization is part of the agent's state. No evaluation step is carried out, representing the fact that the rational agent is omniscient and does not need to evaluate the incoming information. The same could be observed for bounded rationality below.

Note that even in the above simple setting, if the set of options with maximal utility value contains more than one of the options, one has to define how one of them is then selected. In addition, the utility function need not be defined for the options directly, and then the connection between the decision to be made and the utilities evaluated must be made explicit. Both these tasks are impossible at this level of generality and can hence only be carried out on examples.

In a decision model based on the rational actor theory for MoTMo, utilities are defined for transport modes, but the decision concerns the agents' availability of the mode car. To define the link between evaluation and selection, we assume that agents' decisions are aimed at always having the transport mode with the highest utility available, leading to the following formalization with M={car,other} and utilities u_{h_*i*}(m) as defined in Equation 1. We consider the mode(s) with the maximum utility:

m*=argmaxm∈M(uhi(m))=argmaxm∈M(α1q(hi,m)+α2f(hi,m)).(3)

If this contains only the mode car, an agent selects the decision option o₁ and owns a car. If m^* contains only other, an agent selects the decision option o₂ and does not own a car. If the utility of both modes is equal, i.e., m*=M, maximization alone leaves the selection step undetermined. Given the long-term nature of car ownership, we define that an agent selects the decision option o₃ and adheres to the current car ownership status. In mathematical formulae, with o^* denoting the result of the decision, we get:

o={o1,IFm={car}o2,IFm={other}o3,IFm=M.(4)

Other options for the case that m*=M could be to randomly draw the choice, where the probabilities would have to be defined, to imitate another agent, etc.

3.4.2 Bounded rationality

The bounded rationality theory can be seen as a modification or extension of the rational actor theory. The main characteristics of bounded rationality are that an agent still acts in their own self-interest, but due to limitations, e.g., in cognitive capacity, information availability, or time, utility is not necessarily maximized. These different examples indicate that the ideas behind bounding rationality are manifold, and, in fact, there are numerous variants of the theory. The interpretation that is adequate in a given context is determined by the specific case and the parameters used or available. For a more in-depth discussion of the theory, refer to Simon (1978), Gigerenzer and Selten (2001), and Schlüter et al. (2017).

To illustrate, we consider two possible formalizations of bounded rationality. They are based on key aspects of the theory, as emphasized by Schlüter et al. (2017). In the first case, agents strive for satisfaction or gratification rather than maximization of their utilities. For formalization, we suggest introducing a gratification threshold GT, so that if a utility exceeds GT, the agent does not gain additional value, in other words, instead of the original utility u(a), consider min(u(a), GT) for any alternative under consideration. The explicit value of such a threshold needs to be evaluated via validation in alignment with real-world data.

For our example, with u_{h_*i*}(m) as defined in Equation 1, bounded rationality can be defined by truncating utility to the value of the gratification threshold. This means that maximization is applied to min(u_{h_*i*}(m), GT) for each mode, i.e.,

m*=argmaxm∈M(min(uhi(m),GT)),(5)

while the selection step remains as above in Equation 4. As a result, the formalization is quite analogous to the rational actor approach (See section 3.4.1): the same evaluation of utilities and the same definition of the final choice apply, while the only difference is in the input used in the maximization to determine m^*, thus changing Equation 5 with respect to Equation 3.

In the second formalization, we focus on the idea of limits in information availability and make a change in the utility evaluation step. As this is specific to the utility function used, it is again impossible to usefully define this step in full generality. Instead, its illustration for the example model is the following.

Removing the information on the financial decision factor in Equation 1 represents the idea that agents ignore the financial load of owning a car in their decision-making process. Although this might not be completely realistic, there is evidence, however, that people often underestimate this burden (Haustein and Hunecke, 2007). Formally, this can be represented by redefining the utility function so that it forgets about financial information, i.e., an agent uses its value of the infrastructure quality q(h_*i*, m) but instead of its financial load f(h_*i*, m) considers 0 (the neutral element in the additive utility function) as input to the utility function. The resulting new utility function, say ũ:M→ℝ, based on the limited available information is:

ũhi(m)=u(q(hi,m),0)=α1q(hi,m).(6)

Then, in complete analogy to Equation 3 of the rational agent, this function is used to determine m^* via

m*=argmaxm∈M(ũhi(m)),(7)

and the decision is taken again according to Equation 4.

These two examples were chosen not only to illustrate different interpretations of bounded rationality but also to show that there is no unique way of separating between the two steps of evaluation and selection, nor necessarily a unique formal description of a specific procedure. For example, the definition of the new utility function in the second example was useful but not necessary. Instead, one could have redefined the agents' decision factors to remove the information on f and replace it by 0, and then used u instead of ũ in the following.

3.4.3 Habitual decision making

Habitual decision-making theory highlights the role of habits in human behavior, suggesting that individuals rely on established patterns of behavior rather than actively evaluating choices. This theory is also referred to with the psychological concept of reinforcement as reinforcement learning. The focus then is on the feedback mechanism, which is driven by agents' past experiences and the outcomes of their behavior over time. In a first instance of decision making, a behavior is considered intentional and goal-directed. This behavior becomes more and more likely if it is reinforced by a positive outcome or becomes less likely if it is discouraged by a negative outcome, leading to habit formation. Habitual decision making means that individual maintains a given behavior as long as the needs are satisfied. If needs are no longer satisfied, the behavior might persist for some time, but change eventually. For a more in-depth discussion of the theory, refer to Skinner (2019), Pavlov (1927), Graybiel (2008), and Schlüter et al. (2017).

To describe habitual decision making at a general level, consider a decision taken at time t and denote the result ot+Δ. For habits to be considered, a previous decision ot must be known. The general pattern will then be the following.

ot+Δ={ô,IF something happensotelse,(8)

where something has to be defined and ô is a placeholder for the outcome of a decision mechanism that also has to be defined. This can, but does not need to, be done through one of the examples above.

For our model, we also include the current ownership of mobility tools in the definition of option o₃. We consider an initial decision, and hence the current car ownership status, i.e., the agent's mobility toolbox b_*t*(h_*i*), given. An agent's first decision about owning a car might, e.g., have been taken when a driving license was obtained, and in the model, an initial state of each agent's mobility toolbox needs to be set. We further suggest allowing an agent to enter the decision-making process only if at least one of the following circumstances applies.

• The needs of an agent are no longer satisfied, i.e., the utility of the agent's current car ownership status is below a gratification threshold GT as defined in the bounded rationality approach (see Section 3.4.2).
• An agent experiences a mobility milestone. A mobility milestone is a life event that is explicitly relevant to mobility behavior, such as obtaining a driving license, a traffic accident, a change of residence, the birth of a child, etc. For more details on the concept of mobility milestones, see Rau and Manton (2016). From the perspective of the decision model, these milestones are exogenously given, e.g., obtained from a demographic model's dynamics or simply probabilistically for our purposes⁷.

If at least one of the two conditions is satisfied, the agent enters the decision-making process, which then has to be defined explicitly. Otherwise, no decision is taken, which we define formally as ot+Δ*=o3.

ot+Δ={ô,IFuhi(ot)<GT OR a mobility milestone occurso3else,(9)

and ô could be defined as one of the examples above. Adapting Figure 4 to capture the habitual approach leads to Figure 5. While this MBB formalizes habitual decision making, the formation of the habit here occurs indirectly, through sticking to the previous decision if nothing happens. Without going into formulae, an example MBB in a probabilistic setting could represent the idea of discouraging a behavior if outcomes are not good by having an agent enter the decision making loop randomly, where the probability for doing so increases whenever observed utility is low, but decreases with high utility.

3.4.4 Prospect theory

Prospect theory, developed by Kahneman and Tversky (1979), uses rational actor theory as a starting point and adapts the assumptions to reflect the cognitive biases that the authors systematically discovered and which contradict consistent rationality in people's behavior. As a first concept, the theory highlights an irrational willingness to seek or avoid risk. “Losses are more important than gains” (Kahneman and Tversky, 1979) is a key conclusion that explains how people evaluate potential gains and losses relative to a reference point rather than universally, and how losses are evaluated more negatively than equivalent gains are evaluated positively. As a result, individuals confronted with potential gains are rather risk-seeking and individuals confronted with potential losses are rather risk-averse. In addition, the theory highlights how individuals overweight unlikely events and underweight highly probable ones. The resulting main characteristics of prospect theory are different systematic cognitive biases (loss aversion, certainty effect, isolation effect, and overweighting of low probabilities, reflection effect, etc.) that influence an individual's decision making, often leading to irrational and inconsistent choices. Note that different cognitive biases are systematic modifications of the way in which the rationality of an actor might be bounded, which closely links it to the approaches based on the rational actor and the theory of bounded rationality. For a more in-depth discussion of the theory, refer to Kahneman and Tversky (1979) and, Schlüter et al. (2017).

To formalize risk aversion, we would need a model that uses probabilities. However, even in the simple deterministic example model, a cognitive bias can be represented. The decision options o₁ and o₂ are not symmetric in that owning a car allows to use other modes, but not owning a car does not allow to use one. Hence, the action of selling a car can be interpreted as a loss. In the respective decision model, we thus focus on the cognitive bias of loss aversion as an example. For formalization, we suggest introducing a loss aversion barrier LB and besides that proceeding as with rational actor theory. LB affects the decision-making process only for agents who currently own a car, i.e., b_*t* = 1. If in the utility maximization, m^* as defined in Equation 3 equals {other}, which would lead a rational agent to the decision option o₂ of not owning a car after the decision, in the case of prospect theory, the agent will only decide for this option if the utility of other exceeds that of car by at least the amount of LB, formally:

ot+Δ={o2,IF bt=1 AND m=other AND uhi(other)−uhi(car)>LBo1, IF bt=1 AND m=car OR (m=other AND uhi(other) −uhi(car)≤LB)o^ IF bt=0,(10)

where ô is defined by Equation 4.

3.4.5 Descriptive norm

Descriptive norm theory, as originally described by Cialdini et al. (1990), suggests that individuals tend to act according to their observation of established behavior patterns of others in a specific social context. They do this rather than actively evaluating each choice and especially when they perceive this behavior as common or typical. The theory highlights that social norms, in particular, those based on observed behavior, influence behavior by creating a sense of what is acceptable in a group or a society. For a more in-depth discussion of the theory, refer to Schlüter et al. (2017), Cialdini et al. (1990), and Kashima et al. (2014).

In the decision model based on descriptive norm theory, the formalization is fundamentally different from the other approaches in that the utility of transport modes does not play a role. An agent decides on the basis of observable behavior in the population, which means that it has to be defined which behavior an agent is able to observe. For formalization, let Ni⊆H be the set of agents that agent h_*i* is able to observe. This set has to be specified for each agent and can be given by a fixed or evolving network between agents, through spatial proximity, or can simply be the set H if an agent can observe all agents. As an example, we assume that the behavior of agents in the immediate surroundings has the greatest impact and introduce a network of neighbors based on geospatial proximity, e.g., allowing an agent to observe others with home locations within a given radius ρ from its own home location, i.e., Ni={hj∈H|d(l(hi),l(hj))≤ρ}, where d is a distance measure between locations. The definition of N and parameters such as ρ should be validated and adapted according to real-world data for any specific model.

In this case, since we need to consider the decisions of many agents at the same time, we include an agent index after the time index in the decision, e.g., o_{*t, i*} denotes the decision of agent h_*i* at time t, dropping the h to avoid cluttering the notation. An agent h_*i*'s decision for t + Δ can then be formalized depending on the current decisions of the agents in Ni. For simplicity, we assume that Ni contains N agents and label these 1, …, N. The decision is then represented as:

ot+Δ,i*=g(ot,1,…,ot,N),(11)

where g is a function that conveys the idea of following its input. For example, it could provide the decision option that the majority of the observed group is using. In a probabilistic case, the requirement that g should fulfill would be that the probability of choosing an option should increase with the number of agents in the group that use that option. Adapting Figure 4 to capture the descriptive norm approach leads to Figure 6.

4 Discussion and conclusions

This paper presented the MOZAIC framework, offering a systematic process for formalizing various social-science theories into modular, reusable mathematical representations, and applied it to an ABM on long-term mobility decisions. To this end, we selected and formalized five examples of social science theories in MBBs. As the focus of this paper is on addressing the gap between the social science theories described in natural language and their implementations as code, the ABM was presented in a simplified way to enable high-level discussions about the mathematical formalizations rather than getting lost in details.

The MOZAIC framework guides the formalization process in four steps. In most ABM development activities, these steps will be embedded in an iterative process consisting of definition, validation, updating the definition, further validation, etc., as sketched in the modeling cycle in Figure 3.

• Step 1: overview Behavioral context and theory selection: first, the model's context must be delineated. This step requires gathering empirical information and finding relevant scientific studies related to human behavior in the system under study, leading to the selection of a theory to be formalized.
• Step 2: definition of elementary model components Agents, environment, and interaction structures: based on the results of step 1, elementary model components, such as agents and their environment, are specified in mathematical language.
• Step 3: definition of decision model components Decision model elements: possible decision options, a set of decision criteria, and how to evaluate them, are specified in mathematical language.
• Step 4: decision model Mathematical formalization of theory with decision model elements: finally, the social science theory is formalized by defining how model components defined in the previous steps are assembled to form the decision model, delivering a decision-making process for agents.

Several steps as a guide to formalizing social theories have also been proposed by Schwarz et al. (2020), but systematically translating to mathematical definitions is a new component of the MOZAIC framework. Applying this framework's steps to the example ABM and theories, we gathered the following insights on the theories themselves, partially confirming findings from previous works:

• The rational actor theory, widely criticized due to unrealistic assumptions, is easy to formalize in the sense that there already exist mathematical descriptions extensively elaborated in economics, leading to “fewer assumptions on missing elements” (Schlüter et al., 2017). This makes its widespread use understandable. However, the formalization also showed that and where assumptions have to be made (e.g., to define the decision model elements).
• The bounded rationality theory comes with more realistic ideas on how human decision-making processes are constrained by limitations in information, cognitive capacity, and time. However, its broad definition allows and at the same time requires significant subjective interpretation leading to various routes taken under this theoretical heading. To illustrate, we provided two formalizations for this theory. This limits comparability and highlights the need to accurately state the assumptions made.
• Prospect theory shares its core idea with bounded rationality theory, but adds detail by identifying specific cognitive biases, such as loss aversion. It hence seems promising in guiding how to bound rationality based on empirical findings.
• The habitual theory emphasizes habituation and reinforcement of behavior through positive or negative experiences (relating to the status quo bias in prospect theory). Although this theory determines how habits lead to maintaining a given situation, it does not specify how a new decision is taken when habits are interrupted. In practice, this means that a combination with a different decision mechanism—either based on a different theory or chosen ad hoc— is needed.
• The descriptive norm theory focuses on the role of observed behavior of others. As it skips the step of evaluating the available options from an agent's perspective, it structurally differs from the other theories considered.

The resulting MBB for the five theories are complementary to work on reusable building blocks (Berger et al., 2024) and AGENTBLOCKS (Filatova et al., 2025). Agreeing with Filatova et al. (2025) that “a catalog of theory-grounded RBBs for key social mechanisms and decisions (possibly based on competing theories) will create a breakthrough and enable collaborative knowledge sharing with non-programming domain experts,” we propose mathematical definitions as formal descriptions that are independent of programming languages. Using an MBB in an ABM demands further work for its implementation as code; here the implemented RBBs that are entailed in the AGENTBLOCKS library may be used more immediately. However, at a more abstract level, MBB allow illustrating and discussing intermediate steps of the formalization process. Using mathematical language may provide the following benefits.

• Formal decision models can be compared. For the above examples, similarities arise in the common use of elements for more than one theory (e.g., the basic utility function). In addition, the same selection formula could be used for several theories.
• A formalization need not go all the way to a model implementation, but may remain more general. This resonates with statements from the literature about how theories do not provide the complete set of details needed to build a model (Schwarz et al., 2020; Schlüter et al., 2017). Formalizations may merely make the underlying structures explicit, so that a modeler can see precisely which elements have to be specified to implement the theory.
• Properties that, e.g., a function should have, can be discussed at an abstract level rather than considering potential example functions one by one. As pointed out by Schlüter et al. (2017), “ambiguities [...] on how variables relate to each other make it very important to be explicit about assumptions made.” Mathematical definitions help making assumptions explicit that may go unnoticed in a concrete model implementation. And a mathematical definition may carry more precise information than, e.g., an arrow in a diagram indicating an influence of a variable on another (Schlüter et al., 2017): it may directly define this influence. A discussion of properties of elements in a theory at a somewhat general level may also be beneficial in the step of theory selection, allowing modelers to judge whether or not some of these properties are adapted for a given model context.

We see mathematical definitions as a stepping stone between social science theories (that are formulated in natural language and general) and model code (formal language and case-specific). Ideally, as illustrated in Figure 1, they are in fact located between the two, closer to theories on a scale from general to specific (the vertical axis in the figure) and closer to model code on the other scale, as they also use a formal language. In addition, mathematical definitions are more specific and applicable than graphical illustrations (see Figure 2) and more general and transferable than implemented code. They enable a modeler to determine and specify elements that represent structural characteristics in a theory and to only outline those elements that must be filled with model-specific content. Leveraging the potential of mathematical formalization could contribute to ameliorating the difficulty in finding “a common language that would be interpreted in the same way by different modelers when specifying their [decision model] characteristics,” which Wijermans et al. (2023a) report.

Of course, there are limitations to mathematical formalization and points that have not been covered in this paper: first, the formalizations presented are based on a simplified model. When used in a complex ABM, they are supposed to support the formalization process, but their implementation will require considerably more effort. Second, there are important aspects of ABMs which are not addressed in the simple model, such as temporal changes, particularly significant in relation to learning or habituation, changes in the model environment, that impact the behavior of an agent, and interactions between agents. Third, we use a utility function in the formalization of four out of five theories. In some cases, a utility function might not be appropriate or even impossible to define. Suggesting solutions for such cases remains an open research question. Fourth, this paper did not touch on the topic of evaluation or validation of MBBs. Fifth, and this may be the most fundamental point, reading and grasping mathematical formulae requires a high level of concentration, sometimes patience, and especially experience or habituation. Graphical representations are generally considered much more intuitive, and pseudocode seems to be the most chosen formal option in the ABM community. However, comparing a formula such as Equation 10 with a piece of pseudocode expressing the same equation, which one of the two representations is judged to be more accessible will depend on personal preference and training and hence, MBBs can offer an additional tool to communicate, refine, and share ABM to its diverse audience.

Future work should address these limitations. A logical first step is the implementation and evaluation of the MBBs presented within the ABM code. Creating AGENTBLOCKS based on these MBBs would be an interesting exercise. Implementing MBBs could indicate whether and when the stepping stones would be useful for translating a theory to a concrete ABM. Moreover, the resulting ABM could be employed for validating the formalizations to some extent⁸. Within a given model, one might, for example, compare several MBBs to see which one produces the best fit of the model with real-world data. Moreover, it would be interesting to test the inverse modeling approaches described by Epstein (2023), and Schubotz et al. (2025). These approaches create and explore different structures and parameters of rule-based models that match certain output data. Connecting MBBs with inverse modeling could be fruitful in two ways: either to use inverse modeling to scrutinize one MBB (allowing the algorithm to produce different formalizations, e.g., formulae) with respect to given data; or, to do the same with the order of several MBBs (ordering MBBs or adding/deleting MBBs). Another way of using MBB in the future could be their integration into larger frameworks that compare and relate social science theories to each other (such as, e.g, West et al., 2019, did for health-psychology theories) or integrating them into larger theoretical or conceptual frameworks in a systematic manner.

Next, applying the MOZAIC framework to more complex models could produce MBBs that may be more useful for current modeling efforts in the context of societal challenges. At the same time, this would provide a testbed for the MOZAIC framework itself and could identify potential improvements needed to address some of the aspects of ABM disregarded here. Further work on the MOZAIC framework itself would probably be most successful if it were interdisciplinary and iterative. Collaborative definitions of MBB with modelers, social scientists and mathematicians could ensure that all necessary fields of expertise are included in producing mathematical formalizations of social theories for models. A community-based process for reviewing MBBs could then facilitate interdisciplinary learning to create formalizations that are accessible to researchers of various backgrounds. Clearly, well-defined formalizations can improve the modeling of human behavior by establishing a common language to facilitate meaningful discussions. Hopefully, mathematics can provide a contribution to such a language.

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Footnotes

Edited by: Davide Secchi, Paris School of Business, France

Reviewed by: Angelica Sousa Da Mata, Universidade Federal de Lavras, Brazil

Robin Purshouse, The University of Sheffield, United Kingdom

Footnotes

Note that, in much of software engineering, algorithms that are coded and hence become libraries, reusable as modules in various programs, have been available in mathematical formulation beforehand, sometimes for long periods of time.

For an overview of available frameworks and guiding lists of questions to select a theory, see: Schwarz et al. (2020); Wijermans et al. (2023a).

Moreover, a set of real numbers can be represented on a computer only up to machine precision, resulting in finite sets of options in the computational model, anyhow. In real-world examples, some smallest unit will also prevail in this case, e.g., a cent in monetary terms or a gram of flour.

This theory has been thoroughly discussed in the context of ABM in previous works (see, e.g., Muelder and Filatova, 2018; Ajzen and Klobas, 2013; Klabunde et al., 2015). Its high degree of elaboration might lead to confusing rather than illuminating results when combined with the (overly) simplistic model used here.

Reflexivity means that each option is at least as preferred as itself, Completeness means that any two options can be compared by the preference ordering, Transitivity means that if x_*i*≼x_*j* and x_*j*≼x_*k*, then also x_*i*≼x_*k*. With these conditions, a preference ordering is called rational.

Note that for some criteria, such as time needed and costs, the standard assumption would be less is better contrary to what was stated above, but this can be taken care of.

In fact, in MoTMo, these milestones are given by a demographic evolution that is also part of the model, but for simplicity it was not included in the present setting.

The term “validation” here needs to be taken with a grain of salt: as pointed out by Antosz et al. (2023), “any conclusions about theories put into an ABM is relative to a host of other added assumptions” that are part of the model itself.