How Risk Preferences Shape City-State Success: An Agent-Based Model of Resource Management⋆ Andrea Piras1,† , Francesco Bertolotti1,∗,† 1 School of Industrial Engineering, LIUC – Carlo Cattaneo University, Castellanza Varese, Italy Abstract This paper presents an agent-based model to study the dynamics of city-state systems, focusing on the interaction between military and economic actions in a closed environment, with the aim of drawing more general conclusions about risky behaviour with limited resources in a competitive environment. The model includes three types of agents: city-states, villages, and battalions, where city-states are the primary decision-makers that can establish villages for food and recruit battalions for defence and aggression. Simulation data was generated using grid sampling, and analysis suggests that a risk-seeking strategy is more effective in high-cost scenarios if the production rate is sufficiently high. Future work could include memory and trading behaviour to improve the relevance of the model and the generalisability of the results. Keywords agent-based modelling, risk preferences, risk aversion, city-states, computational history, 1. Introduction Social science has a long-standing tradition of using computational methods [1, 2], especially agent-based models (ABMs) [3, 4, 5]. This interdisciplinary approach leverages computational tools and large-scale data collected from various sources to uncover insights into individual and collective human behavior [6]. In this context, multi-agent simulation models are considered to have the capacity to lead to a ”generative” approach [4, 7, 8] and to embody an evolutionary perspective [9, 10]. Thus, in this field, they are considered both a means to perform prediction [11, 12, 13] and to enhance understanding [14, 15, 16] of a phenomenon. Recently, there has been a growing interest in using computational methods to under- stand historical phenomena [17, 18, 19, 20]. Archaeologists have employed multi-agent simulation models to validate their hypotheses regarding excavations [21, 22, 23, 24]. Additionally, various fields, such as the emergence of trading networks in specific areas [25] and the effects of climate change on societal outcomes [26], have utilized this methodology, typically with long simulation time-steps. WOA 2024: 25th Workshop ”From Objects to Agents”, July 8-10, 2024, Forte di Bard (AO), Italy ⋆ You can use this document as the template for preparing your publication. We recommend using the latest version of the ceurart style. ∗ Corresponding author. † These authors contributed equally. Envelope-Open an17.piras@stud.liuc.it (A. Piras); fbertolotti@liuc.it (F. Bertolotti) Orcid 0009-0005-4270-1467 (A. Piras); 0000-0003-1274-9628 (F. Bertolotti) © 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR ceur-ws.org Workshop ISSN 1613-0073 Proceedings Given that war systems have long been considered complex systems [27], agent-based modeling has a tradition of being used to study strategies and action consequences of different kinds of combat systems [28, 29], including real-world armies [30]. Due to its flexibility, it has been applied to both small military units [31] and battles involving tens of thousands of units to assess potential alternative outcomes [32]. Although these models include and analyze tactics to defeat the enemy on the field, this type of competition is tactical rather than strategic, as it omits long-term decisions regarding resources. Walbert et al. [33] present an agent-based simulation model based on empirical data to assess how and why states start a war, considering their network of relationships and wealth accumulation. In this paper, we present an ABM of a city-states system, where cities can perform both military and economic actions [34]. Specifically, there are three kinds of agents: cities, villages, and battalions. The primary decision-making agents are the cities, which can generate villages to produce food and battalions for defense and aggression. Cities consume food to maintain their population level and can generate wealth that can be invested in technological developments. These developments can enhance the efficiency of battalions, food production, or wealth generation. In light of the preceding description, it is possible to categorise the strategic attitudes of cities into two broad categories: expansive and conservative. These two scenarios are linked to different risk predispositions that each city decides to adopt. It is reasonable to posit that cities with greater resource availability will adopt more expansive strategies, simply because they are the only ones that can afford to do so. Conversely, we anticipate that cities with limited resources will adopt a more cautious approach, attempting to minimise their exposure to potential external threats. The results of the paper are counter-intuitive and of significant interest for decision- making. City-states can be seen as black boxes that generate resources to buy goods, where resources are food and gold, and the goods are military units and technological investments that increase production rates. Given a fixed resource generation rate, we would expect that if the cost of production is low, the best strategy would be to produce as much as possible, and vice versa for high production costs. However, the results indicate the opposite. We explain this observation by considering the higher value of individual units. When producing and investing are more expensive, each unit has a greater marginal advantage. Therefore, producing more is rewarded with a higher chance of survival. However, if the production rate is too low, this advantage no longer applies because there are insufficient means to achieve adequate production. In behavioral terms, this suggests that a risk-seeking strategy is prefer- able when the cost of investment is high, but this does not apply if the production rate is too low. The paper is structured as follows. First, the methodology is explained, including a detailed description of the agent-based model and the experimental design. Next, the results are presented and discussed. Finally, conclusions are drawn. Figure 1: Model scheduling for a single time-step for a city-state 𝐶𝑖 2. Methodology 2.1. Agent-based model This research paper presents an agent-based model (ABM) that examines the interactions between city-states located within a landscape. ABM is a computational methodology that simulates the behavior of systems by modeling the behavior and interactions of their composing entities [35, 36]. The model depicts a bi-dimensional and topological explicit, i.e. the fact that agents have a certain position in a space, in this case bi-dimensional, system where a limited amount of city-states are competing for space, and where no new city-states can be founded. City-states can produce food by means of villages, and this overall affect the growth level of the population, which has a positive effect on every economical aspects. Also, city-states and can attack each other with military units. No other kind of interaction has been inserted in the model. The purpose of the model is then to observe which kind of cities survives in different environmental setting, and try to draw a more general understanding regarding competition in a close environment with scarce resources. So, the model assesses the different paths that each city-state can take in terms of economic development, military strategy, and resource management to achieve survival and prosperity over a specific period. In the model, three types of agents are present: city-states 𝐶𝑖 , villages 𝑉𝑖 , and battalions 𝐵𝑖 . City-states are the primary decision-makers that undertake various actions. Figure 1 depict their scheduling for a single time-step. Villages are the food producers and provide the necessary supplies to sustain the population of the city-states, which is the driver for the whole economics of the city-state, as depicted in Figure 3. Battalions are recruited by the city-states to defend against external threats or to launch military campaigns against other city-states. Each agent type plays a distinct role in the simulation, contributing to the overall dynamics of resource management, economic growth, and military strategy. Each city-state (𝐶𝑖 ) possesses the state variables depicted in 1, which can undergo both endoge- nous changes, such as the population stock 𝑝𝑖 (𝑡), which represents the number of tax-generating citizens in the city who are also available for enrolment, that increases when a certain amount of food 𝑓𝑖 (𝑡) is available in the city to cover the food needs of the citizens and soldiers stationed to defend the city, and exogenous changes, related to the interaction processes between city-states. The gold 𝑔𝑖 (𝑡) of the city-state (𝐶𝑖 ) is linked to the population by a positive dependence on the fact that this increases with the collection of taxes in direct proportion to the number of citizens in the city. The variables 𝑤𝑖 (𝑡), 𝑐𝑡𝑖 (𝑡), 𝑚𝑡𝑖 (𝑡), and 𝑐𝑑𝑖 (𝑡), which respectively represent the general wealth level of the population, the technological level in the civil field, the technological level in Figure 2: Graphical interface of the model Name Description 𝑔𝑖 (𝑡) Gold in the city 𝑓𝑖 (𝑡) Food in the city 𝑝𝑖 (𝑡) Population of the city 𝑤𝑖 (𝑡) Wealth of the the city 𝑐𝑡𝑖 (𝑡) Civil technology of the city 𝑚𝑡𝑖 (𝑡) Military technology of the city 𝑐𝑑𝑖 (𝑡) Defence of the city Table 1 List and description of state variables for a city-state 𝐶𝑖 the military field and the city’s defences, only undergo positive increments whenever the city decides to embark on a development phase compatible with the available resources. Figure 3 exemplifies the economic dynamics of a city-state agent, highlighting the dependencies that the different state variables have on each other. City-states are decision-making entities. In this sense, they are undertaking a decision at each time-step, regarding in which kind of activity to invest the resource, or if to create villages or battalions, or how to use the battalions. The economic phase of a city-state 𝐶𝑖 decision-making Figure 3: Graph of functional dependencies depicting the economical dynamics of a city-state 𝐶𝑖 Name Description Allowed Values 𝑝𝑣𝑖 Preference to found a village 𝑝𝑣𝑖 ∈ [0,1] 𝑝𝑐𝑡𝑖 Preference to invest in civil technology 𝑝𝑐𝑡𝑖 ∈ [0,1] 𝑝𝑚𝑡𝑖 Preference to invest in military technology 𝑝𝑚𝑡𝑖 ∈ [0,1] 𝑝𝑤𝑖 Preference to invest in wealth 𝑝𝑤𝑖 ∈ [0,1] 𝑝𝑑𝑖 Preference to invest in defences 𝑝𝑑𝑖 ∈ [0,1] 𝑝𝑏𝑖 Preference to recruit a battalion 𝑝𝑏𝑖 ∈ [0,1] 𝑝𝑝𝑖 Preference to send protecting troops 𝑝𝑝𝑖 ∈ [0,1] 𝑝𝑚𝑖 Preference to organize a mission 𝑝𝑚𝑖 ∈ [-0,1] 𝑝𝑣𝑎𝑖 Preference to attack a village 𝑝𝑣𝑎𝑖 ∈ [0,1] 𝑝𝑐𝑎𝑖 Preference to attack a city 𝑝𝑐𝑎𝑖 ∈ [0,1] Table 2 List and description of strategic parameters for a city-state 𝐶𝑖 divides into two phases. First, a city-state collect gold and food based on their gold-rate and population values and the village production. Then, a city-state decides if to improve wealth, technology, or defense, to build a battalion, or to found new villages. The military phase instead consists in the decision of what to do with the battalion: the city-state can organize missions to directly attack enemy cities or their villages, or alternatively, it can send battalions to defend a village and protect it from possible enemy attacks. Each decision can be trigger by two elements: a specific internal or external condition, and a set of behavioural parameters. Behavioural parameters can hence be divided into two categories: strategic parameters (2) and the tactical parameters (3). The strategic parameters can assume a value 𝑥𝑖 ∈ 𝑅 ∶ 𝑥𝑖 ∈ (0, 1) ∧ ∑ 𝑥𝑖 = 1, with the exception of 𝑝𝑣𝑎𝑖 and 𝑝𝑐𝑎𝑖 , which can value 𝑦𝑖 ∈ 𝑅 ∶ 𝑦𝑖 ∈ (0, 1) ∧ ∑ 𝑦𝑖 = 1. This difference is due to the Name Description Allowed Values 𝛼1 Coefficient of target decision regarding enemy’s defence 𝛼1 ∈ [-1,1] 𝛼2 Coefficient of target decision regarding enemy’s number of battalions 𝛼2 ∈ [-1,1] 𝛼3 Coefficient of target decision regarding enemy’s distance 𝛼3 ∈ [-1,1] 𝛼4 Coefficient of target decision regarding enemy’s military technology level 𝛼4 ∈ [-1,1] 𝛼5 Coefficient of target decision regarding enemy’s gold 𝛼5 ∈ [-1,1] 𝛼6 Coefficient of target decision regarding enemy’s food 𝛼6 ∈ [-1,1] 𝛼7 Coefficient of target decision regarding enemy’s population 𝛼7 ∈ [-1,1] Table 3 List and description of tactical preference parameters for a city-state 𝐶𝑖 Name Description Allowed Values 𝑁 Number of starting city-states N ∈ [5, 20) 𝑏𝑠𝑐 Base battalion recruitment cost 𝑏𝑠𝑐 ∈ [20000, 2000000] 𝑝𝑔𝑝 Person gold production 𝑝𝑔𝑝 ∈ [1, 1000] 𝑏𝑠𝑝 Base village food production 𝑏𝑣𝑝 ∈ [1 , 1000] Table 4 List and description of environmental parameters fact that 𝑝𝑣𝑎𝑖 and 𝑝𝑐𝑎𝑖 pertain to the city’s preference to directly attack enemy cities or their villages. These values are subordinate to the value of 𝑝𝑚𝑖 , which represents the city-state’s preference for organizing offensive missions. Once the mission has been organized, the city must choose which type of target to direct its attack towards. These parameters determine the strategy each city decides to undertake on the resource management. For instance, if 𝑝𝑣𝑖 = 0.2, it means that the probability for a city-state 𝐶𝑖 to build a new village during the economic phase of the decision-making process, and only when the option is available, is 𝑃(𝑣) ∝ 0.2. The tactical parameters can all assumes the value 𝑧𝑖 ∈ 𝑅 ∶ 𝑧𝑖 ∈ (−1, 1), and are used to decide which enemy to attack in the moment where the decision to attack has already been taken. Each of these parameters acts as a multiplier on specific characteristics of the enemy 𝐶𝑖 with which the 𝐶𝑖 interacts. The sum of these values determines a final score, where the 𝐶𝑖 will choose to attack the 𝐶𝑖 with the highest score. Each value is compared with the total amount present on the map. For example, 𝛼2 multiplies, for each 𝐶𝑖 , the number of 𝐵𝑖 it possesses divided by the total number of battalions present on the map. This helps to return a value of the target’s ”danger level.” The choice to vary these values between -1 and 1 was driven by the desire to better explore which characteristics of the target cities were taken most into account. Each 𝐶𝑖 will have its own unique set of preferences, assigning different positive or negative importance to various aspects. These parameters play a pivotal role within the model: given that attacking is the only way of interaction in the model, and that each set of parameters is unique for each city-state 𝐶𝑖 , they are regulating the decision of the target, and so it makes the way in which the economics output of two city-state agents are tested. Finally, there are some environmental parameters of interest (see 4), such as the initial number of city-states 𝑁, the rate of production of the two resources (respectively 𝑝𝑔𝑝 for the gold and 𝑏𝑠𝑝 for the food), and the cost of production of a battalion 𝑏𝑠𝑐. Figure 4: Black box diagram of the experimental setting 2.2. Experimental design To implement the model described in the previous paragraph, we used NetLogo 6.3.0. This software was chosen for its simplicity and because the number of agents in the model was limited, eliminating the need for high performance computing. The experiments were conducted using NetLogo’s BehaviorSpace module, which facilitates grid sampling. Through 1250000 simulations, a wide range of scenarios was analyzed. This number of repetitions was sufficient to ensure statistically robust results and allowed us to explore the effects of various input variables on the interactions between cities, villages, and battalions through simulation data analysis. The grid sampling exploration was performed by sampling four key inputs, with each input variable varied across a specified range to cover both extreme and moderate values. Each variable was collected from a uniform random distribution. The decision to adopt a random grid sampling system was guided by the fact that, not knowing what result to expect a priori, it was considered the best way to examine as many combinations as possible and discover interesting patterns within the model. In future developments of the model, one could explore using, for example, a genetic algorithm to find the best possible strategy within the pool of numerous combinations available. For each simulation run, data was collected on key outcome variables for the surviving cities, enabling the generation of various statistical analyses that could provide insights into how different environmental parameters influenced the overall dynamics of the system. The data was analyzed and processed using Python 3.11.3 in a Jupyter Notebook. These experiments allowed us to observe how different scenarios impact cities’ preferences, resource management, and overall economic and military dynamics. 3. Results and discussion Figure 5 illustrates the share of 𝐶𝑖 that survived at the end of the simulation relative to the starting number 𝑁. The graph suggests that often only a low share of 𝐶𝑖 survives, with this share gradually decreasing in frequency as the survival rate increases. However, there is a noticeable increase in survival values close to 1, indicating that specific parameter combinations exist where all the city-states could survive. It is interesting to observe how the behavioral parameters of the city-states change with environmental inputs, which depict an elementary form of fitness to the environment and suggest the best behavior under certain conditions. The following analysis, depicted in Figure 6a, Figure 6b, Figure 6c, and Figure ??, involves Figure 5: Histogram of frequency of percentage number of survived 𝐶𝑖 respect to 𝑁 plotting a behavioral output on the y-axis, while observing the co-effect of two different inputs: one on the x-axis and the other used to divide the data into three clusters by tertiles, which boundaries are respectively called 𝑡1 and 𝑡2 for each variable. These graphs represent on the y-axis the average values of 𝑝𝑣 and 𝑝𝑚, indicated respectively as 𝐸[𝑝𝑣] and 𝐸[𝑝𝑚]. Figure 6a depicts the relationship between 𝑏𝑠𝑐 and 𝑝𝑣, clustered by 𝑁. For all values of 𝑁, 𝑝𝑣 initially increases with 𝑏𝑠𝑐, exhibiting different peaking points followed by a subsequent decrease. This non-monotonic behavior varies with 𝑁: the higher the number of city-states, the greater the preference for founding villages. Larger 𝐶𝑖 seem to sustain a higher preference for a higher cost longer than smaller 𝐶𝑖 . This can be connected to the varying success of different risk-related attitudes. Notably, as 𝑏𝑠𝑐 increases, a more expansive and risk-prone strategy emerges, aiming to seize as much territory as possible by founding villages until the area is saturated. Additionally, since each 𝐶𝑖 can only perform one action per turn, it exposes itself to the risk of enemy offensives targeting its villages. This occurs because the city-state would be less protected due to its lower 𝑝𝑏𝑖 in favor of 𝑝𝑣𝑖 . Figure 6b shows the relationship between 𝑁 and 𝑝𝑣, clustered by 𝑝𝑔𝑝. It is observable that for 𝑝𝑔𝑝 > 𝑡1 , there is an equal increase in 𝑝𝑣 with 𝑁, although with a different intercept. On the other hand, when 𝑝𝑔𝑝 < 𝑡1 , there is almost a null trend. Similarly to what was mentioned for the previous graph, the focus is indirectly on the cost of the external environment. A high number of 𝑁 on the map leads to greater resource scarcity, making these resources more valuable. Consequently, the propensity to expand increases as the number of 𝑁 grows. However, this reasoning does not seem to apply when the 𝐶𝑖 ’s ability to generate resources remains excessively low. The link between 𝑏𝑠𝑐 and 𝑝𝑣, clustered by 𝑝𝑔𝑝, is depicted in Figure 6c. It is possible to observe that when 𝑝𝑔𝑝 > 𝑡1 , there is a non-linear growing relationship between 𝑏𝑠𝑐 and 𝐸[𝑝𝑣], which saturates after a certain level. For 𝑝𝑔𝑝 < 𝑡1 , this saturation occurs much earlier, and the values of 𝐸[𝑝𝑣] start decreasing notably even for low values of 𝑏𝑠𝑐. In this sense, economic (a) Line plot of 𝐸[𝑝𝑣] of 𝐶𝑖 related to 𝑏𝑠𝑐 clus- (b) Line plot of 𝐸[𝑝𝑣] of 𝐶𝑖 related to 𝑁 clustered tered by number of 𝑁 by 𝑝𝑔𝑝 (c) Line plot of 𝐸[𝑝𝑣] of 𝐶𝑖 related to 𝑏𝑠𝑐 clus- (d) Line plot of 𝐸[𝑝𝑚] of 𝐶𝑖 related to 𝑏𝑣𝑝 clus- tered by 𝑝𝑔𝑝 tered by 𝑏𝑠𝑐 Figure 6: Line plot of 𝐸[𝑝𝑣] and 𝐸[𝑝𝑚] related to different clusters and x-axis strength seems to buffer the impact of 𝑏𝑠𝑐. This chart effectively illustrates the relationship between environmental cost and internal production. As 𝑏𝑠𝑐 increases, so does the propensity for expansion by the 𝐶𝑖 . However, as expected, when productivity is excessively low, this propensity drops drastically since the 𝐶𝑖 is not able to sustain such high costs. Figure ?? depicts the relationship between 𝑏𝑣𝑝 and 𝑉𝑖 , clustered by 𝑝𝑚. For all values of 𝑏𝑠𝑐, 𝑝𝑚 initially increases with 𝑏𝑣𝑝, then peaks and either plateaus or decreases. Interestingly, for low values of 𝑏𝑠𝑐, this pattern differs significantly. We notice a particular balance that aligns with the previous statements. As we have learned, the expansive effect increases with 𝑏𝑣𝑝. In this case, we observe the 𝑝𝑚𝑖 values. Filtering by 𝑏𝑠𝑐, we see how this phenomenon is accentuated. However, in the case of low 𝑏𝑠𝑐, despite the increase in 𝑏𝑣𝑝, the 𝑝𝑚𝑖 tends to decrease slightly. When the 𝑝𝑔𝑝 is higher, 𝐶𝑖 tend to have higher 𝑝𝑣𝑖 and can tolerate higher costs, whether for soldier recruitment or otherwise. Larger 𝐶𝑖 tend to sustain higher 𝑝𝑣𝑖 for longer and can support higher costs better than smaller 𝐶𝑖 . This indicates economies of scale and possibly better resource distribution and management in larger 𝐶𝑖 . There is a noticeable cost tolerance threshold in both 𝑝𝑣𝑖 and 𝑝𝑚𝑖 . Beyond certain 𝑏𝑠𝑐, preferences decline, indicating a balance point in economic and operational planning. Higher production, both in villages and gold, positively correlates with higher preferences up to a point. However, after certain production levels, the incremental benefits reduce, suggesting optimal production ranges for maximizing preferences. (a) Radar plot of the mean values of tactical pa- (b) Radar plot of the mean values of tactical pa- rameters for all the 𝐶𝑖 lasting in a simulation rameters without 𝐸[𝑎3] for all the 𝐶𝑖 lasting (in red), compared with the related to ex- in a simulation (in red), compared with the pected values (in blue) related to expected values (in blue) Figure 7: Radar plots of tactical parameters In these proposed charts, it was decided to compare the expected values of some parameters with the actual average values observed in the 𝐶𝑖 that survived at the end of the simulations. The expected values of the parameters were calculated based on the range of admissible values by definition. The tactical parameters can take any value in the interval [−1, 1] with equal probability distri- bution. Therefore, their expected value is 0. Figure 8: Radar plot of the mean values of strategical parameters for all the 𝐶𝑖 lasting in a simulation (in red), compared with the related to expected values (in blue) A similar reasoning was applied to the strategic parameters. The strategic parameters can take any value in the interval [0, 1] with equal probability, but the sum of these parameters must equal 1. For this reason, since there are 8 primary strategic parameters, the expected value for each of them is 1/8, or 0.125. Figure 7a illustrate the differences between the expected values of tactical parameters if the environment had no effect on the simulation, and the actual average values obtained from simulations, for strategic and tactical parameters. It is shown that the 𝐶𝑖 have maintained values close to those expected for all 𝛼𝑖 except for 𝛼3 . This markedly negative value indicates an aversion on the part of the 𝐶𝑖 to selecting targets located farther away. Figure 7b was created by removing 𝐸[𝛼4 ] from the visualization to better appreciate the dif- ferences of the other tactical parameters relative to their expected values. In the graph, all values are slightly negative and thus below the expected value of 0. However, the 𝛼4 parameter, although only slightly, stands out as the most negative, highlighting it as the second major contributing factor in determining the target for missions by the cities. Figure 8 shows the average preferences of actions that each 𝐶𝑖 can take, comparing them with their expected values. It is possible to appreciate how 𝐶𝑖 have significantly prioritized the 𝐸[𝑝𝑣] at the expense of almost all other preferences. Only the 𝐸[𝑝𝑚] is slightly above the expected value. This graph offers one final insight. Previously, we discussed more conservative or expansive strategies. We notice how the two main expansive preferences, 𝑝𝑣𝑖 and 𝑝𝑚𝑖 , stand out compared to the others in terms of the 𝐶𝑖 preferences. This tends to indicate a greater preference among 𝐶𝑖 for an expansive strategy, which evidently tends to perform better in different scenarios. 4. Conclusions The ABM of a city-states system presented in this paper aims to analyze the different mixes of preferences and, consequently, the possible strategies that the primary agents of the model, the city-states, might choose to exploit. The objective of each city-state is survival, which can be achieved through absolute conquest or partial coexistence with other city-states. The study of these dynamics has revealed a particular pattern. As previously stated, the expansive attitude and resulting risk propensity of city-states emerge counter-intuitively in response to external environmental and internal resource characteristics. An increase in productive capacity leads city-states to adopt a more aggressive stance toward their neighbors. One might expect similar behavior when the cost of external goods is particu- larly low. 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