MOZAIC: New Mathematical Framework Turns Ambiguous Behavioral Theories Into Precise Agent-Based Model Building Blocks

Human behavior sits at the heart of nearly every societal challenge of the Anthropocene, from the cars people buy to the transport modes they abandon. Yet when social scientists try to encode theories of human decision making into agent-based models (ABMs), they run into a stubborn problem: theories written in natural language are inevitably ambiguous, and there are no community standards for translating them into computer code. A new study published in Frontiers in Social Psychology introduces MOZAIC, short for “Mathematical fOrmaliZAtIon for deCision models,” a four-step framework designed to turn behavioral theories into precise, reusable mathematical definitions. The work, led by Stefanie Schutera of Technische Universität Berlin and Freie Universität Berlin, together with Geeske Scholz of the University of Bremen and Sarah Wolf of the Global Climate Forum, aims to give the social simulation community a common language for discussing, comparing, and sharing theories of human behavior.

Agent-based modeling has experienced broad popularity because it allows researchers to implement autonomous agents, such as humans, households, or companies, as individual entities with varying decision-making processes. Each agent makes decisions based on its own circumstances, enabling the exploration of how macro-level phenomena emerge from micro-level interactions. This makes ABM particularly promising for understanding human-environment dynamics under different policy scenarios. However, the methodology comes with unsolved challenges, including calibration and validation procedures, computational efficiency, transparency, and reusability, and, most importantly, adequate representation of human behavior. Many ABMs are implemented using oversimplified approaches to modeling human decisions, and modelers are often unaware of the abundance of behavioral theories accumulated in the social sciences over centuries, or lack the competence to put them into code adequately.

The MOZAIC framework addresses the formalization challenge directly. Social science theories are formulated primarily in natural language, which necessarily leaves room for interpretation when they are translated into computational models. Even the term “formalization” can be interpreted in several ways, referring to graphical representations such as decision trees or flow charts, ontological systems, specific cognitive architectures, mathematical formulae, or pseudocode and code. 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. The authors argue for mathematical formalization as one concrete realization of a common standard beyond source code and natural-language descriptions, reaping benefits such as explicit and self-contained assumptions, generality, and standardization.

Mathematics, the authors note, can itself be considered a language. They cite mathematician George Pólya, who characterized mathematical notation as a “langue bien faite,” a language well adapted to its purpose, concise and precise, with rules that, unlike the rules of ordinary grammar, suffer no exception. Where their use is feasible and suitable, mathematical definitions are precise on one hand and flexible in their level of contextualization on the other. 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. Crucially, the framework requires no mathematical knowledge beyond high school, as it employs only basic concepts such as functions to describe important parts of a model accurately.

MOZAIC guides the formalization process in four steps. First, the behavioral context and theory selection: the model’s context must be delineated, requiring 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. Second, the definition of elementary model components: agents, environment, and interaction structures are specified in mathematical language, including state variables relevant to decision making. Third, the definition of decision model components: possible decision options, a set of decision criteria, and how to evaluate them are specified. Fourth, the mathematical formalization of the theory itself: the social science theory is formalized by defining how the model components from the previous steps are assembled to form the decision model, delivering a decision-making process for agents. In practice, the authors emphasize, there will be a back-and-forth between steps rather than a strict linear progression.

To demonstrate the framework’s usefulness, the team applied MOZAIC to the case of mobility tool ownership decision making within the Mobility Transition Model (MoTMo), an agent-based model developed to simulate mobility behavior in Germany over multiple decades. The purpose of MoTMo is to provide information on long-term dynamics to support informed decision making by policymakers, stakeholders, and citizens. The model should reproduce the past evolution of the number of privately owned vehicles in the study regions, as well as insights about the population’s mobility choices obtained from the “Mobilität in Deutschland” survey studies. The focus on car ownership is motivated by emissions data: motorized individual transport accounts for 75.3 percent of passenger transport emissions in Germany, and the most significant factor in using this mode is availability, defined by car ownership status.

From a pool of roughly thirty theories representing a broad spectrum of human behavioral drivers and usage across scientific disciplines, the authors formalized five example theories: rational actor, bounded rationality, habitual decision making, prospect theory, and descriptive norms. The rational actor theory, or homo economicus, assumes an agent is perfectly informed, acts in self-interest, and makes consistent decisions based on well-defined preferences, formalized through utility maximization. Bounded rationality modifies this by acknowledging limitations in cognitive capacity, information availability, or time. The authors provided two formalizations of bounded rationality: one introducing a gratification threshold so that agents strive for satisfaction rather than maximization, and another removing financial information from the utility evaluation, reflecting evidence that people often underestimate the financial burden of car ownership.

Habitual decision making highlights the role of established behavioral patterns, formalized by allowing agents to enter the decision-making process only when their needs are no longer satisfied or when they experience a mobility milestone, such as obtaining a driving license, a traffic accident, a change of residence, or the birth of a child. Prospect theory, developed by Daniel Kahneman and Amos Tversky, adapts rational actor assumptions to reflect systematic cognitive biases, including loss aversion. In the mobility context, selling a car is interpreted as a loss, so agents who currently own a car will only decide to give it up if the utility of the alternative exceeds that of the car by at least a loss aversion barrier. Descriptive norm theory, based on the work of Robert Cialdini and colleagues, formalizes a fundamentally different mechanism: agents decide on the basis of observable behavior in the population, for example following the majority decision of neighbors within a given radius, skipping the utility evaluation step entirely.

The resulting Mathematical Building Blocks (MBBs) offer mathematical descriptions of the elements and structures of the theories, allowing for comparison and aiming to be reusable in other ABMs by filling in context-specific details. The authors found that the rational actor theory is easy to formalize because mathematical descriptions already exist extensively in economics, making its widespread use understandable. Bounded rationality, by contrast, comes with a broad definition that allows and requires significant subjective interpretation, limiting comparability and highlighting the need to accurately state assumptions. Prospect theory seems promising in guiding how to bound rationality based on empirical findings, while habitual theory does not specify how a new decision is taken when habits are interrupted, meaning a combination with a different decision mechanism is needed. The descriptive norm approach structurally differs from the other theories because it skips the evaluation of available options from an agent’s perspective.

The MBBs are complementary to work on reusable building blocks (RBBs) and the AGENTBLOCKS library, which provides atomic blocks of code of specific mechanisms relevant to many ABMs in a given domain. The authors propose mathematical definitions as formal descriptions that are independent of programming languages, positioned as a stepping stone between social science theories, which are formulated in natural language and general, and model code, which is formal and case-specific. Mathematical definitions are more specific and applicable than graphical illustrations and more general and transferable than implemented code. The authors acknowledge limitations, including that the formalizations are based on a simplified model, that utility functions may not be appropriate for all theories, and that reading mathematical formulae requires concentration and experience. Future work should address the implementation and evaluation of the MBBs within ABM code, potentially creating AGENTBLOCKS based on these MBBs, and applying the MOZAIC framework to more complex models. The authors hope that collaborative, interdisciplinary definitions of MBBs with modelers, social scientists, and mathematicians could facilitate community standards for formalizing social science theories in agent-based modeling.

Subject of Research: A mathematical framework for formalizing behavioral theories in agent-based models of mobility

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

Article References: Schutera, S., Scholz, G., & Wolf, S. (2026). MOZAIC: A mathematical approach to formalizing behavioral theories – applied to an agent-based model of mobility. Frontiers in Social Psychology, 4, Article 1772740. https://doi.org/10.3389/frsps.2026.1772740

Image Credits: AI Generated

DOI: 10.3389/frsps.2026.1772740

Keywords: agent-based modeling, MOZAIC, behavioral theories, mathematical formalization, mobility, decision making, social simulation, car ownership, bounded rationality, prospect theory, mathematical, approach