A complex system is a dynamic system, meaning it evolves over time, comprised of a large number of components that interact with each other, and whose global behavior emerges from the local interactions among its individual parts. Therefore, the behavior and properties of a complex system as a whole cannot be fully understood or predicted solely by analyzing its individual components, thus recalling the concept expressed by Aristotle in his famous phrase: "The whole is greater than the sum of its parts."

Some of the most fascinating aspects of complex systems are:

• Networks: A complex system is comprised of many components and their interactions. For this reason, it can be represented by a network or graph where the nodes are the individual entities of the system and the links between the nodes represent their interactions. For example, a community or city is effectively a complex system whose components are the individual people; therefore, it can be well represented by a graph where the nodes are the people and the links between nodes represent their interactions. 

• Emergent Behaviors: One of the most fascinating aspects of complex systems is their ability to generate emergent behaviors, which are global outcomes that cannot be predicted solely from the analysis of individual components. For example, fractals are examples of emergent behavior, generated by simple iteration rules that produce self-similar geometric structures on different scales. 

• Adaptive Behaviors: Elements of complex systems often exhibit adaptive behaviors, meaning they modify their actions in response to environmental conditions or interactions with other components of the system. This capacity for adaptation can significantly influence the overall evolution of the system and enables complex systems to achieve resilience to potentially harmful external factors. For example, in nature, many animals have developed extraordinary adaptive abilities to survive and thrive in their habitats. One of the most fascinating is the ability to camouflage through color and pattern of their coat. This adaptive behavior is evident in creatures like snow leopards. These felines inhabit the mountainous regions of Central Asia and have a spotted coat that blends perfectly with the rocky and snowy environment of their lands. This color pattern helps the snow leopard to camouflage while hunting or hiding from predators.  

• Self-organization: This concept refers to the tendency of complex systems to spontaneously organize their own structure and behavior without the need for external control. The global structure emerges from interactions among the elements of the system. An example is the formation of patterns in the behavior of a flock of birds or in the movement of people in a crowd. 

• Nonlinear Dynamics: This concept refers to the fact that small changes in initial conditions can lead to significantly different outcomes over time. This makes predicting the behavior of the system difficult and can lead to counterintuitive phenomena. 

Agent-based models (ABMs) are a class of computational models aimed at simulating the actions and interactions of autonomous agents (both individual and collective, such as organizations and groups) in order to assess their effects on the overall system. These models are ideal tools for representing complex systems precisely because they are able to capture all their key aspects.

Imagine being in a busy city. Every person, every car, every traffic light is an agent. Each agent has its own behavior and objectives: cars want to reach a destination, people want to cross the street, traffic lights regulate traffic. Together, all these agents interact to form the overall system of urban traffic.

Agent-based models replicate this concept. Using specific algorithms and rules, scientists can create simulations in which each agent follows predefined rules and reacts to the surrounding environment. This helps understand how the system as a whole behaves and how influencing the behavior of agents can change the entire system.

Thanks to their ability to model the individual entities of a system, their interactions, and simulate the dynamic evolution of the system over time, ABMs can capture the complexity and emergence of adaptive and self-organizing behaviors. Essentially, ABMs allow us to explore and understand the complex world around us, providing a detailed view of the underlying dynamics.

Game theory is a mathematical discipline dedicated to study social interactions among rational individuals whose decisions are interdependent, meaning the outcome of their choices depends not only on their own decisions but also on those of all others. Individuals, in this context, are called players, and assuming they are rational means they can evaluate different available options and choose the one that maximizes their benefit based on their objectives.

Informally, a game represents a situation of social interaction among two or more players and usually includes:

• A set of players interacting with each other.

• A set of actions or choices available to each player, called pure strategies.

• The information each player has when making their decision (i.e. whether other players' actions are known or not).

• A payoff function, which assigns a numerical outcome or benefit to each combination of actions representing the outcome or benefit obtained by each player.

Therefore, in game theory, a "game" is not meant in the traditional sense of entertainment but as an interaction among agents, where each of them must make decisions considering others' actions and the consequences of their actions, and where each agent seeks to maximize their own gain or benefit, taking into account others' behavior.

Game theory gained significant prominence through the work of mathematician John Nash, who won the Nobel Prize in Economics in 1994. Nash significantly contributed to the understanding of many key concepts of game theory and introduced the concept of "Nash equilibrium."

A Nash equilibrium occurs when each player, acting rationally, chooses the best possible response to the actions of other players. This concept is crucial in the analysis of game strategies and social interaction situations.

There are several types of games studied and recognized. Three of the most important ones are:

• The Prisoner's Dilemma: Two suspects are accused of committing a crime. The police arrest them and place them in separate cells, preventing them from communicating. They are given two choices: confess or remain silent. It is also explained to them that:

  • If only one confesses, accusing the other, the one who cooperates avoids punishment while the other is sentenced to 10 years in prison.

  • If both confess, accusing each other, then each of them is sentenced to 5 years in prison.

  • If both remain silent, then each of them is sentenced to 6 months in prison.

Both players try to minimize their sentence, so they reason as follows: "My partner can confess or remain silent. If he confesses and I remain silent, I get 10 years, whereas if I also confess, I get 5 years. If, on the other hand, he remains silent, then if I confess, I am free, while if I remain silent, I get 6 months. Therefore, whether my partner confesses or not, to minimize my sentence, it is better for me to confess." This implies that both suspects will confess.

• The Stag-Hunt Game: Two hunters face a dilemma; they go hunting and must decide whether to hunt a stag or a hare. Their decision must be made without knowing the other's decision, keeping in mind that successfully hunting a stag requires both to choose it as a target and therefore cooperate together, while for the hare the efforts of one man alone is sufficient. The game also specifies that the hare represents a less satisfying prize than the stag, which instead represents a better meal, although it will be divided between the two hunters. Therefore, each player can choose between two strategies: "Hunt the stag" and "Hunt the hare." In this context, each player will choose to hunt the stag when the other also chooses the same strategy, and similarly for the hare strategy.

• The Hawk-Dove Game: The game is defined by imagining two animals fighting for a limited resource of food and having two possible strategies: "Hawk" and "Dove." The Hawk strategy is more aggressive and violent, while the Dove strategy is more passive and docile. If both players choose the Hawk strategy, they will fight until one is defeated and the other wins the resource, whereas if both choose the Dove strategy, they will avoid the fight and share the resource. Finally, if one player chooses the Hawk strategy and the other chooses the Dove strategy, the former will get the entire food resource for itself, while the latter will flee. In this context, each player will prefer to have an aggressive strategy when the other is more docile and vice versa.

In conclusion, since game theory allows describing strategic behavior among rational individuals in social interaction situations, it emerges as one of the most effective techniques to represent how agents make decisions within an ABM.

The decision-making process of an agent-based model is of fundamental importance because the emergent behaviors at the macroscopic level of the studied complex system often depend on the decisions made at the microscopic level by the agents. For this reason, game theory allows exploring how such decisions influence the collective behavior of the studied complex system.

Agent-based models (ABMs) have been widely employed in the field of sociology, proving to be powerful tools for exploring interactions among agents and understanding how individual actions influence social outcomes. This leads to the use of such models in the context of environmental sustainability, as they allow for analyzing whether agent-agent and agent-environment interactions at the microscopic level can generate sustainable scenarios at the macroscopic level.

In particular, using ABMs, we can build models where agents represent all people belonging to a community or city (residents, students, workers, tourists, visitors, etc.), who make continuous decisions that affect urban sustainability, ranging from choices in water and energy consumption to waste management.

The decision-making process can be described using game theory, as we have seen in the previous paragraph. More precisely, each agent in the model is a player who can choose between two game strategies: cooperation and defection. Cooperation is a concept widely studied in game theory and can be defined as the act of collaborating together to achieve a common goal.

In the context of environmental sustainability, cooperation is fundamental because it implies the active involvement of all entities belonging to a community in adopting behaviors that promote conservation and responsible management of natural resources, pollution reduction, and combating climate change.

An ABM in this context allows for a detailed analysis of decision-making and interaction dynamics within an urban community, understanding how the individual actions of agents influence urban sustainability at the macroscopic level. Using game theory to describe the decision-making process of agents, it is possible to examine how cooperation and defection at the individual level influence the overall behavior of the community.

[1] Link. La scienza delle reti - Albert László Barabási

[2] Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engeneering - Steven H. Strogatz

[3] Simulazione sociale. Modelli ad agenti nell'analisi sociologica - Flaminio Squazzoni

[4] Agent-Based and Individual-Based Modeling: A Practical Introduction - Railsback S.F. & Grimm V. 

[5] Teoria dei Giochi - Pierpaolo Battigalli

[6] An introduction to game theory - Martin J. Osborne