=Paper=
{{Paper
|id=Vol-3261/paper8
|storemode=property
|title=The evolution of risk sensitivity in a sustainability game: an agent-based model
|pdfUrl=https://ceur-ws.org/Vol-3261/paper8.pdf
|volume=Vol-3261
|authors=Francesco Bertolotti,Sabin Roman
|dblpUrl=https://dblp.org/rec/conf/woa/BertolottiR22
}}
==The evolution of risk sensitivity in a sustainability game: an agent-based model==
https://ceur-ws.org/Vol-3261/paper8.pdf
The Evolution of Risk Sensitivity in a Sustainability
Game: an Agent-based Model
Francesco Bertolotti1,* , Sabin Roman2
1
Università Carlo Cattaneo – LIUC, Corso G. Matteotti, 22, Castellanza (VA), 21053, Italy
2
Centre for the Study of Existential Risk, University of Cambridge
Abstract
The research of a balance between the growing pressure of addressing long-term sustainability issues and
the existence of short-term economic and political challenges is one of the main issues of this century.
To contribute to this case, we design a four-players game to elicit this tension. Then, an agent-based
model of this game is developed, to observe the effects and the effectiveness of various behaviours and
strategies, especially regarding risk preferences. An evolutionary meta-model selects the best-performing
agents during multiple generations, and data from the resulting population are collected. The analyses
of the results suggest that environmental factors affect the resulting risk sensitivities.
Keywords
Agent-based modelling, Risk sensitivity, Collapse, Sustainability, Risk Preferences
1. Introduction
The concept of sustainability was present in human societies much before the term became
popular. In the middle age, policymakers of the Republic of Venice were already concerned
about the scarcity of wood related to the consumption of the Venetian forests, which could
preclude the production of new ships crucial for the city’s economy. Hence, they impose some
limitations on employing this resource for private purposes in selected areas to enable forest
renovation and not consume all the life-stock [1]. Since the Brundtland Report was published
[2], there has been an increasing awareness of sustainability issues and the goals of sustainable
development [3]. That essay elicited the trade-off between humanity’s aspirations toward a
better existence on the one hand and the boundaries imposed by nature on the other. In time,
the concept has been re-interpreted and encoded into three dimensions: social, economic, and
environmental [4].
There is ample literature documenting the urgent sustainability problems facing the modern
world [5]. One response to these pressing issues has been to develop more engaging educational
and instructional tools for stakeholders, policymakers and the general public. Serious games
(SSGs) are one such tool. They formalize key aspects of ecological and instructional dynamics
as well as their constraints. Likewise, they provide valuable teaching on how to manage and
WOA 2022: 23rd Workshop From Objects to Agents, September 1–2, Genova, Italy
Corresponding author.
*
$ fbertolotti@liuc.it (F. Bertolotti); sr911@cam.ac.uk (S. Roman)
0000-0003-1274-9628 (F. Bertolotti); 0000-0002-7513-3479 (S. Roman)
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
Type Examples Aims and themes
Learning SD (LSD) Env. Conservation and urban
Samba Role Play development
Industrial Chlorine Transport Natural resource management
Board/card game
Metagame Social technology as a SD public policy
Keep Cool Sustainable process negotiation
River Basin Game Water cleaning
Atollgame Groundwater cleaning
MHP Strategies for SD technologies
RPG/Video game About That Forest Sustainable forest management
Evacuation Challenge Game Natural disaster and emergency planning
The Arcade Wire: Oil God Food sustainability and geopolitics
3rd World Farmer Sustainability of agricultural land use
Climate Challenge Renewable energy sources and politics
Online sandbox game Stop Disasters! Natural disasters prevention
Energyville Sustainable energy supply
Encon City Energy conservation
New Shores: A Game for Democracy Green project management
World Climate Global warming decision-making
Computer simulation game Green City Green urban development
Tragedy of the Tuna Water resources management
The UVA Bay Game Sustainable products and services
Table 1
List of serious games for sustainable development (SD) goals
cope with ever-increasing sustainability challenges [6]. The last decades have seen substantial
growth in the literature on SSGs, with games varying from board games, to computer and
web-assisted gaming, to role-playing or a mix of these elements [7].
Table 1 highlights some examples of serious games developed to facilitate sustainable goals
[8].
In this paper, we propose a sustainability game that involves investments in either renewable,
non-renewable or military sectors, we simulated it with intelligent agents, and we observe how
evolutionary adaptation in a sustainability setting affects risk preferences.
This sustainability game can be mapped to a discrete dynamical system that allows us to
explore its outcomes in a systematic way. The methodology utilised is dynamical system
simulation, a discipline with a long tradition of being applied in sustainability settings at least
since the pioneering model called World3 by Jay Forrester generates the data for the Limit to
Growth report in 1972 [9]. More recently, both agent-based and system dynamics models have
been built to face sustainable issues [10, 11].
The research addresses the intertwined relationship between sustainability, renewable re-
source stocks and risk preferences in production decisions. A risk preference is a property of
every decision-making entity, and it defines how it addresses the uncertainty in the outcome of
the decision. For example, one can assume to have two alternatives 𝐴 and 𝐵 with the same
expected results, quantifiable in terms of utility: a risk-averse entity favour the option B (with
lower variability) and a risk-prone entity prefer option A (with higher variability) [12]. An
entity without a preference between 𝐴 and 𝐵 is called risk-neutral. Here, we refer to a non-risk
neutral entity as "risk sensible" and the property of having a risk preference as "risk sensitivity".
This work aims at understanding how risk sensitivity adapts by genetic evolution in a setting
with renewable resources and how this adaptation is affected by environmental factors, with
attention to the dangerousness of the scenario. The results show that non-trivial relationships
exist between these parameters and the resulting risk sensitivities. The generality of the setting
suggests that the conclusions of this research could have some real-world implications regarding
how different entities address risk in complex environments with competition and sustainability
issues.
The paper divides as follows. Section 2 presents the methodology employed for the research,
dividing it into the serious game, its agent-based model, and the evolutionary meta-model used
to generate the results. Section 3 shows and discusses the results, while Section 4 draws some
conclusions.
2. Methods
This section displays the methodologies employed in the research work and contains four
parts: the sustainability game; its agent-based model; the evolutionary meta-model; and the
experimental setting.
2.1. Sustainability game
The game introduced in this paper aims to put players in a competitive environment where
long-term sustainability goals contrast short-term production and defence objectives. In what
follows, the game is described twice: firstly qualitatively, then employing difference equations
(Equations 1-18).
2.1.1. Qualitative description
The primary components of the game are four kinds of blocks, which stands for a type of
resource available. The blocks divide by colours for greater clarity: black, green, red and brown.
Respectively, each type of block represents the following real-world resource:
• the black blocks stand for non-sustainable industrial capacity;
• the green blocks stand for sustainable industrial capacity;
• the red blocks stand for the military capacity;
• the brown block stand for the biosphere.
The game is comprised of 800 turns, so each player plays 200 times. Each turn, a player can
make decisions regarding two aspects: produce new blocks and use the red blocks to attack
another participant. Each player begins the game with a single black block and no other blocks.
Players have a certain integer amount of black, green and red blocks. The brown blocks, which
stand for the stock of natural resources, are shared between all players. A player leaves the
game when one of these four conditions is satisfied. First, it is the sole player remaining in
the game. In that case, it wins. Second, the shared level of the brown blocks is zero. This
Figure 1: Scheme of resource creation in the sustainability game
condition embodies a circumstance in which human activities destroyed the biosphere. In this
situation, all the players lose. Third, a player has no green or black blocks left. In that situation,
it individually loses while the game proceeds. Fourth, at the end of turn 800, the player is still
in the game. In that case, each participant still in the game wins. Intuitively, the game creates
tension between individual short-term and global long-term goals.
The production blocks, which are the black and the green, can be used in the following way,
to produce new blocks every turn.
• 1 black block can be use each turn to produce 1 black block;
• 2 black blocks can be use each turn to produce 1 green block, destroying the black blocks;
• 2 green block can be use each turn to produce 1 green block;
• 2 green block can be use each turn to produce 1 black block, destroying the green blocks;
• 1 black block can be use each turn to produce 1 red block;
• 1 green block can be use each turn to produce 1 red block;
• 2 green blocks can be use each turn to produce 1 brown block;
Each black and red block consume one brown block at each time step. In this way, the more
non-sustainable industrial and military capability a player possesses, the more they destroys
the natural stock. This rule has a single exception, which is that every green block can sustain a
red block, not making it eroding the common stock of brown blocks. So, for instance, if a player
has eight green blocks and ten red blocks, it consumes only two brown blocks per turn from
red blocks maintenance.
The last rule regards the usage of the red blocks, which stands for the military capacity. Red
blocks can be operated to fight another player, and each player can attack only once per turn.
When the two players engage, each one loses a number of red blocks equal to the minimum
number between their red blocks and the opponent’s red blocks. Then, if the attacking player
holds more red blocks than the defending player, it takes other blocks from it. Firstly, it takes
one black block per red block outlasted from the combat. Then, if the number of red blocks
remaining is greater than the black blocks of the defending agents, it takes also a number of
green blocks equal to the minimum amount between the remaining red blocks and the green
blocks owned by the defending player.
2.1.2. Formal description
Formally, given 𝑏(𝑡), 𝑔(𝑡), 𝑟(𝑡) respectively the number of black, green and red blocks owned
by player 𝑖 at the beginning of the turn 𝑡, and 𝑒(𝑡) the number of brown blocks remaining at
the beginning of turn 𝑡. 𝑏𝑏(𝑡), 𝑔𝑏(𝑡), 𝑟𝑏(𝑡), 𝑔𝑔(𝑡), 𝑏𝑔(𝑡), 𝑟𝑔(𝑡), 𝑒𝑔(𝑡) are the specific decisions of
each player, in term of deciding which resource to produce another resource. 𝑗 is the attacked
player, 𝑎(𝑡) a Boolean variable that define if the player attacks or not during a turn, and 𝑏𝑗 (𝑡),
𝑔𝑗 (𝑡), 𝑟𝑗 (𝑡) respectively the number of black, green and red blocks owned by player 𝑗 at the
beginning of the turn 𝑡. Finally, 𝑏𝑖𝑛 (𝑡) and 𝑔𝑖𝑛 (𝑡) are the variation of each kind of block in player
𝑖 stock related to the blocks stolen from player 𝑗 in case of victory, and 𝑟𝑜𝑢𝑡 (𝑡) is the amount
of red blocks lost by player 𝑖 during a fight, and 𝑟𝑠 (𝑡) the number of red blocks of the player
𝑖 surviving the fight. The stock variations for each kind of stock is outlined by the following
equations:
𝑏(𝑡 + 1) = 𝑏(𝑡) + 𝑏𝑏(𝑡) + 0.5 × 𝑏𝑔(𝑡) − 𝑔𝑏(𝑡) + 𝑏𝑖𝑛 (1)
𝑔(𝑡 + 1) = 𝑔(𝑡) + 𝑔𝑔(𝑡) + 𝑔𝑏(𝑡) − 𝑏𝑔(𝑡) + 𝑔𝑖𝑛 (2)
𝑟(𝑡 + 1) = 𝑟(𝑡) + 𝑟𝑏(𝑡) + 𝑟𝑔(𝑡) − 𝑟𝑜𝑢𝑡 (3)
𝑒(𝑡 + 1) = 𝑒(𝑡) + 𝑒𝑔(𝑡) − 𝑏(𝑡) − 𝑚𝑎𝑥 (𝑟(𝑡) − 𝑔(𝑡), 0) (4)
There are 7 production boundaries, one for each resource-production decision.
𝑏𝑏(𝑡)𝑏(𝑡) − 𝑔𝑏(𝑡) − 𝑟𝑏(𝑡) (5)
𝑔𝑏(𝑡)𝑏(𝑡) − 𝑏𝑏(𝑡) − 𝑟𝑏(𝑡) (6)
𝑟𝑏(𝑡)𝑏(𝑡) − 𝑏𝑏(𝑡) − 𝑔𝑏(𝑡) (7)
𝑔𝑔(𝑡)𝑔(𝑡) − 𝑏𝑔(𝑡) − 𝑟𝑔(𝑡) − 𝑒𝑔(𝑡) (8)
𝑏𝑔(𝑡)𝑔(𝑡) − 𝑔𝑔(𝑡) − 𝑟𝑔(𝑡) − 𝑒𝑔(𝑡) (9)
𝑟𝑔(𝑡)𝑔(𝑡) − 𝑔𝑔(𝑡) − 𝑏𝑔(𝑡) − 𝑒𝑔(𝑡) (10)
𝑒𝑔(𝑡)𝑔(𝑡) − 𝑔𝑔(𝑡) − 𝑏𝑔(𝑡) − 𝑟𝑔(𝑡) (11)
Lastly, the results of the fights are computed according to these boundaries and equations.
𝑟𝑜𝑢𝑡 (𝑡) = 𝑎(𝑡) × 𝑚𝑖𝑛 (𝑟(𝑡), 𝑟𝑗 (𝑡)) (12)
𝑟𝑠 (𝑡) = 𝑟(𝑡) − 𝑟𝑜𝑢𝑡 (𝑡) (13)
𝑏𝑖𝑛 (𝑡) = 𝑎(𝑡) × 𝑚𝑖𝑛 (𝑟𝑠 (𝑡), 𝑏𝑗 (𝑡)) (14)
𝑔𝑖𝑛 (𝑡) = 𝑎(𝑡) × 𝑚𝑖𝑛 (𝑚𝑖𝑛 (𝑟𝑠 (𝑡), 𝑏𝑗 (𝑡)) , 𝑔𝑗 (𝑡)) (15)
𝑏𝑗 (𝑡 + 1) = 𝑏𝑗 (𝑡) − 𝑏𝑖𝑛 (𝑡) (16)
𝑔𝑗 (𝑡 + 1) = 𝑔𝑗 (𝑡) − 𝑔𝑖𝑛 (𝑡) (17)
𝑟𝑗 (𝑡 + 1) = 𝑚𝑎𝑥 (𝑟𝑗 (𝑡) − 𝑎(𝑡) × 𝑟(𝑡), 0) (18)
2.2. Agent-based model
This section presents a simulation model of the game shown in the previous section. Equations 1-
18 formalize the game dynamic and its boundaries for every turn. However, the most interesting
element is the players’ decision-making related to the allocation of the resources (e.g., variables
𝑏𝑏(𝑡), 𝑔𝑏(𝑡), 𝑟𝑏(𝑡), 𝑔𝑔(𝑡), 𝑏𝑔(𝑡), 𝑟𝑔(𝑡), 𝑒𝑔(𝑡)) and the aggression to other players (e.g., variables
𝑎𝑖 (𝑡) and 𝑗). For this reason, the model is not simulated by means of a classic stock-flow model
but using an agent-based technique, which suits the representation of individual behaviours
and appraises their effect on the overall system [13]. The agent-based model includes a single
kind of entity, which is the player. In a single game, there are 4 agents. Each simulation runs
for 200 time steps, which means that each agent gets to decide 200 times. Figure 2 depicts the
scheduling for each time step of the simulation model, which has three main phases: production
phase, aggression phase and computing black blocks. The last stage is essential, consisting
solely of two activities: computing the current number of brown blocks (see Equation 4) and
terminating the game when the number of brown blocks is equal to or lower than zero. The
other two phases are less trivial and embody agents’ decision-making.
Agents decide according to a tuple of 12 parameters, shown in Table 2. The parameters divide
into three categories. 𝑝𝑏1 , 𝑝𝑏2 , 𝑝𝑏3 , 𝑝𝑏4 , 𝑝𝑏5 , 𝑝𝑏6 , 𝑝𝑏7 influence the allocation of resources (e.g.,
the values of the variable bounded by Equations 5, 6, 7, 8, 9, 10 and 11). Hence, the higher one
of these parameters, the higher the possibility that one of the related input blocks (e.g., a black
block for 𝑝𝑏2 ) is employed to produce the related output block (e.g., a green block for 𝑝𝑏2 ). 𝑝𝑎1 ,
𝑝𝑎2 , 𝑝𝑎3 affect the decision-making regarding which other agent to attack. This decision is
Figure 2: A single time-step of the agent-level simulation process
Table 2
Behavioural parameters
Name Description Allowed values
𝑝𝑏1 Propensity in producing black blocks from black blocks 𝑝𝑏1 ∈ [0, 1]
𝑝𝑏2 Propensity in producing black blocks from green blocks 𝑝𝑏2 ∈ [0, 1]
𝑝𝑏3 Propensity in producing green blocks from black blocks 𝑝𝑏3 ∈ [0, 1]
𝑝𝑏4 Propensity in producing green blocks from green blocks 𝑝𝑏4 ∈ [0, 1]
𝑝𝑏5 Propensity in producing red blocks from black blocks 𝑝𝑏5 ∈ [0, 1]
𝑝𝑏6 Propensity in producing red blocks from green blocks 𝑝𝑏6 ∈ [0, 1]
𝑝𝑏7 Propensity in producing brown blocks from green blocks 𝑝𝑏7 ∈ [0, 1]
𝑝𝑎1 Propensity of attacking a player with black blocks 𝑝𝑎1 ∈ [0, 1]
𝑝𝑎2 Propensity of attacking a player with green blocks 𝑝𝑎2 ∈ [0, 1]
𝑝𝑎3 Propensity of attacking a player with red blocks 𝑝𝑎 3 ∈ [0, 1]
𝑟𝑠𝑟 Risk sensitivity regarding resource production 𝑟𝑠𝑟 ∈ [0, 1]
𝑟𝑠𝑎 Risk sensitivity regarding attack 𝑟𝑠𝑎 ∈ [0, 1]
affected by the number of blocks of a colour the possible targets own. For instance, an agent
with a high value of 𝑝𝑎3 is likely to attack an agent 𝑗 with a high number of red blocks (see
Table 2). Lastly, 𝑟𝑠𝑟 affects the allocation between sustainable and non-sustainable resources
and 𝑟𝑠𝑎 influences the rate of aggression. An agent with 𝑟𝑠𝑘 > 0 is considered risk-averse
regarding the feature 𝑘. Oppositely, if 𝑟𝑠𝑘 < 0, the agent is risk-seeking. Finally, for 𝑟𝑠𝑘 ≈ 0,
agents are risk-neutral. 𝑟𝑠𝑟 and 𝑟𝑠𝑎 are parameters of an agent, not states: consequently, they
do not change during the simulation. Hence, agents in this model cannot learn. Nevertheless,
agents can adapt their behaviour to the specific environmental features using these two risk
sensitivities parameters, which do not vary during the simulation but allow agents to modulate
their actions according to specific conditions. This implementation lets the agent potentially
hold non-trivial strategies (e.g., a combination of different behavioural parameters) without
making any preliminary assumptions regarding their decision-making style.
2.3. Evolutionary meta-model
Intuitively, the decision-making process depicted in the previous section is not optimal and
is not supposed to be. There could be many other ways to design and implement intelligent
agents that play this game. Nevertheless, the scope of this study is not to identify the best
possible strategy but to observe how the risk sensitivity evolves in repeated matches. This
section depicts how we design and implement the evolutionary tournament.
Table 3
Parameters of the evolutionary meta-model
Name Description Allowed values
𝑛 Number of agents 𝑛 ∈ [4, ∞)
𝑤 Weight of surviving in computing the score for each game 𝑤 ∈ [0, 1]
𝑔 Number of games per generation 𝑔 ∈ [4, ∞)
𝑠 Survival rate per generation 𝑠 ∈ [0, 1]
𝑚 Mutation rate 𝑚 ∈ [0, 1]
Figure 3 presents the evolutionary meta-model, consisting of the following components.
At the beginning of the meta-model, a population of 𝑛 agents (where 𝑛 is a parameter of the
meta-model) is initialized. For each agent, the behavioural parameters presented in Table 3
are sampled from a uniform distribution. A uniform distribution is selected because all the
allowed values vary between 0 and 1, and the sampled values do not cover different orders of
magnitude. Then, the model process enters the first loop, consisting of three activities: creating
random teams of agents, simulating the game and computing the score. For every match 𝑚𝑔,
agents gather into randomly-selected groups of 4 members, and each group enters into the
agent-based simulation game described in the previous section. According to the round result,
each agent collects a score in two ways: surviving until the end of the simulation (a shared
victory) or being the sole agent to remain alive in a given intermediate time step (a lonely win).
Each of these conditions provides a Boolean outcome 𝑏𝑠 and 𝑏𝑙 . The outcomes are weighted by
a parameter 𝑤, the importance of surviving until the end of the simulation, so that the score
of a single match is 𝑆𝑚 𝑔 = 𝑏𝑙 × 𝑤 + 𝑏𝑠 × (1 − 𝑤). if 𝑤 = 0, only the part related to victory
(e.g., remains the only agent alive) matters, and if 𝑤 = 1, only staying alive until the end of
the simulation is relevant to computing the final score. Later, the new score cumulates to the
total score of the generation: 𝑆(𝑚𝑔) = 𝑆(𝑚𝑔 − 1) + 𝑆𝑚 𝑔, with 𝑆(0) = 0 The number of
games per generation 𝑔 is a parameter of the evolutionary meta-model. When the number of
matches played in a generation 𝑚𝑔 = 𝑔, the generation ends, and the evolutionary selection
starts. The evolutionary process creates a set of agents (the so-called evolutionary wheel), where
each agent is replicated a number of times proportionally to its score. Every generation has a
survival rate of 𝑠 (a parameter of the meta-model). It means that [𝑛(1 − 𝑠)] agents dies during
each selection process. Consequently, they need to be replaced with new [𝑛(1 − 𝑠)] agents. So,
then a random number of [𝑛(1 − 𝑠)] agents is taken from the population and removed from
the evolutionary meta-model. At the same time, an equal number of new agents is generated
and added to the meta-model. Each new agent has two parents, randomly selected from the
evolutionary wheel. So, the higher the score from the generation, the higher the probability of
having offspring. The behaviour parameters (see Table 3) of each new agent are the random
combination of the behavioural parameters of the parents. Mutations occur with a rate 𝑚, a
parameter of the meta-model. After the evolutionary selection, there are two possible cases. If
𝑔𝑚 < 𝑙, where 𝑔𝑚 is the number of the current generation, the previous stages repeat for a
new generation. Contrarily, the meta-model is terminated.
Intuitively, the purpose of the evolutionary meta-model is to observe how the behavioural
Figure 3: Process of the evolutionary meta-model employed for the experiments
Figure 4: Black box diagram of the experimental setting
parameters of agents adapt under evolutionary pressure. In this research, we studied the two
risk sensitivities parameters. In the next section, we present the experimental setting used to
explore the evolutionary adaptation of this model.
2.4. Experimental setting
The structure of this meta-model allowed us to perform experiments on two different groups of
parameters: the parameters of the agent-based model and the parameter of the evolutionary
meta-model. The experimental setting was designed to collect data on the effect of parameter
variations on behavioural parameters (see Table 5), but for the purpose of this research, only the
risk-sensitivity parameters are then analyzed. Also, experiments regarding the numerosity of
the agents in the simulation did not provide any interesting results. So, they are not presented
here. Figure 4 depicts the experimental setting with a black box diagram. For greater clarity,
the input and output at the core of this research are bold.
An evolutionary meta-model run for every value of the input parameter of interest. Table 4
Table 4
Default experimental parameters
Name Description Default value
𝑛 Number of agent in the meta-model 400
𝑤 Weight of surviving in computing the score for each game 0.5
𝑔 Number of games per generation 100
𝑠 Survival rate per generation 0.9
𝑚 Mutation rate 0.02
𝑒(𝑡0 ) Initial number of brown blocks 1000
Table 5
Experiments performed
Name Variable varied from default Values investigated
Default None None
Experiment 1 𝑒(𝑡0 ) 100, 200, 400, 800, 1600, 3200, 6400, 12800, 25200, 51200
Experiment 2 𝑠 0.1, 0.2, 0.3.0.4, 0.5, 0.6, 0.7, 0.8, 0.9
Experiment 3 𝑤 0, 0.1, 0.2, 0.3.0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1
presents the four experiments performed with their target parameter variations. Table 5 shows
the default configuration of the parameters when it is not specified differently.
The agent-based model and the evolutionary meta-model were implemented using Python
3.9, without any specific framework for agent-based modelling or genetic algorithm, and every
simulation run on a Windows machine equipped with a 3.30GHz Intel(R) Core(TM) i5-4590
CPU and 4.0 GB RAM.
3. Results and discussion
This section presents and discusses the results of the experimental setting depicted in section
3.4. More precisely, it exhibits the effect of the variation of three different input parameters of
the final risk sensitivities (𝑟𝑠𝑟 for resources and 𝑟𝑠𝑎 for actions) of the population of agents
resulting from the evolutionary process. In the section, the same presentation structure is
followed for each parameter. Four figures are displayed, illustrating the behaviour of the mean
of risk sensitivities (two-line plots, respectively of 𝑟𝑠𝑟 and 𝑟𝑠𝑎 ) and the distribution of the two
risk sensitivities for different values of the input parameter.
Figure 5 and Figure 6 show the presence of non-linear behaviour of the mean risk sensitivities
of agents with the variation of the number of initial brown blocks. Figure 7 and Figure 8 confirm
Figure 5 and Figure 6 by presenting the variation of their distribution with 𝑒(𝑡0 ). This parameter
is a proxy of the dangerousness of the environment. One can consider the game dynamic
illustrated by Equations 1-18: a small number of brown blocks leads to a higher probability of an
early endgame due to life-stock consumption. More precisely, since there were 4 players and a
black block consumed 1 brown block per turn if all the players started the game by doubling the
black blocks available, it took only 6 turns to consume all the stock of brown blocks. Oppositely,
Figure 5: Mean of resource risk sensitivity Figure 6: Mean of attack risk sensitivity
with the number of initial brown blocks with the number of initial brown blocks
Figure 7: Distribution of resource risk sensitivity Figure 8: Distribution of attack risk sensitivity
with the number of initial brown blocks with the number of initial brown blocks
the higher 𝑒(𝑡0 ), the longer the time available for deploying different strategies (for instance,
attacking or converting to green blocks). Consequently, populations tend to be risk-neutral for
𝑒(𝑡0 ) ≈ 0 whereas it was unlikely for an agent to survive. Therefore, the selection process is
less efficient, resulting in "biased" populations. For higher values of 𝑒(𝑡0 ), its effect differs for
𝑟𝑠𝑟 and 𝑟𝑠𝑎 . For the resource risk sensitivity 𝑟𝑠𝑟 , the higher the number of initial brown blocks,
the more risk-seeking the resulting population. If the initial stock of the brown blocks was high
more agents in the game could pursue a non-sustainable strategy without risking destroying
the brown stock. Consequently, risk-seeking manners became more and more effective with
the growth of the initial brown blocks until 𝑒(𝑡0 ) got to a level over which it had no effect
anymore for the reason that the initial stock of brown blocks was enough to consent the agents
to survive without putting attention to the variations in the common stock. This result could
have probably changed by increasing the number of turns of the game on the grounds that the
longer the game proceeds, the higher the initial level of brown blocks needed to generate a
more elevated 𝑟𝑠𝑟 . On the other hand, the attack risk sensitivity 𝑟𝑠𝑎 initially rose with the
number of the initial brown blocks. The variation in the aggression expected return explains
this result. When 𝑒(𝑡0 ) is low, an addition in the value of the parameter made the selected
population more favourable to risk because in that setting, the only way to get points more
points than other players (and consequently, an evolutionary advantage) was to attack often
and force them out of the games before the level of brown blocks got to 0 and everybody loses.
Figure 9: Mean of resource risk sensitivity Figure 10: Mean of attack risk sensitivity
with survival rate with survival rate
Figure 11: Distribution of resource risk Figure 12: Distribution of attack risk
sensitivity with survival rate sensitivity with survival rate
Nevertheless, as soon 𝑒(𝑡0 ) was high enough, risk-seeking behaviour regarding aggression got
less and less fit by cause of the increased chances of surviving until the end of the game. So, it
suited more to have a conservative approach and attack exclusively when the number of red
blocks was high enough, not taking the chance of exposing to counterattacks and being kicked
out of the game. This growth did not last for any values of 𝑒(𝑡0 ). Figure 6 shows that when
𝑒(𝑡0 )12800, 𝑟𝑠𝑎 started to decrease, because the probability of survival of the population did
not vary anymore with 𝑒(𝑡0 ). Since 𝑤 = 0.5 (see Table 4), being a little bit more risk-seeking
regarding the aggression led to an advantage: it permitted agents to win some matches that
otherwise would have ended in a draw, taking away the possibility for other agents to increase
their score, which was evolutionary advantageous.
Figure 9 and Figure 10 show the presence of non-linear behaviours of the risk sensitivities
with the variation of the survival rate. This parameter stands for the harshness of selection in
the evolutionary process. The lower 𝑠, the smaller the amount [𝑛(1 − 𝑠)] of agents that passed
from one generation to another in the meta-model. It had a first implication, observable in
Figure 11 and Figure 12, which is the variety in the dispersion of 𝑟𝑠𝑟 and 𝑟𝑠𝑎 in the populations.
When 𝑠 < 0.8, the variability of the risk sensitivity was smaller. Since we utilised 𝑠 = 0.9 as the
default value, Figure 7, Figure 8, Figure 15, and Figure 17 present wider probability distributions.
Figure 9 suggests that 𝑠 had a positive effect on 𝑟𝑠𝑟 . More precisely, the higher the share of
agents surviving a generation, the higher the probability that an agent generated offsprings
Figure 13: Mean of resource risk sensitivity Figure 14: Mean of attack risk sensitivity
with survival weight with survival weight
Figure 15: Distribution of resource Figure 16: Distribution of attack
risk sensitivity with survival weight risk sensitivity with survival weight
even if 𝑆 ̸= 0 at each generation, since the number of chances grew with 𝑠. To obtain 𝑆 ̸= 0, an
agent had to survive until the endgame, but it was not necessary to be a sole winner. Therefore,
the incentive to invest in non-renewable resources was lower and lower.
Figure 10 rather depicts a non-linear relationship between the 𝑠 and 𝑟𝑠𝑎 . The cause is
similar to the one for 𝑒(𝑡0 ). When 𝑠 was low, to attack or not attack does not substantially
alter the probability that genes would be passed to offspring considering a high number of
agents are selected at each generation. Differently, with the growth of 𝑠 fewer agents were
selected on the evolutionary wheel every time. Then, a greater 𝑟𝑠𝑎 returns to be advantageous
as in the default case. Finally, when 𝑠 ≈ 1, the majority of the agents survive to the next
generation. Consequently, risk aversion regarding aggression became less and less advantageous
as [𝑛(1 − 𝑠)] is smaller, and a high value of 𝑆 is required to reproduce. Therefore, lower risk
sensitivity meant a diminished probability of shared victories and a consequent increase in the
reproducing chances.
Figure 13 and Figure 14 show the relationship between risk sensitivities and 𝑤, which
embodies the relevance in the evolutionary process of a shared or a lone victory: the higher 𝑤,
the more weight has to survive until the end of the simulation with other agents rather than
being the sole agent to win the game.
Figure 13 and Figure 15 present a positive relationship between 𝑤 and 𝑟𝑠𝑟 , while Figure 14
and Figure 16 depict an analogue behaviour for 𝑤 and 𝑟𝑠𝑎 . In both cases, this result explains
the effect of risk sensitivity parameters on the probabilities of shared or lone victory. More
precisely, the linear relationship suggests that the more risk-seeking agents, the less likely were
to get a shared victory. This conclusion derives from the growth of 𝑤 and the meta-model
design. If 𝑤 affected 𝑆 and 𝑆 influenced the chance of reproducing, for 𝑤 ≈ 0 the values of 𝑟𝑠𝑟
and 𝑟𝑠𝑎 selected from the evolutionary process were the ones that maximize the probability
of a sole victory. Differently, with 𝑤 ≈ 0 the resulting 𝑟𝑠𝑟 and 𝑟𝑠𝑎 maximize the chances of a
shared victory.
Interesting, for 𝑟𝑠𝑎 the variation of 𝑤 generates a change of sign, which is a variation in
the overall behaviour. Figure 14 shows that the overall population of agents is risk-prone for
𝑤 < 0.4 and risk-averse for 𝑤0.4.
4. Conclusions
This paper presents an agent-based model of a simple sustainability game in which agents decide
according to risk-preference parameters and an evolutionary meta-model. The research aims at
showing how risk sensitivities evolve and how this evolution is affected by some environmental
parameters such as the level of natural stock at the beginning of the simulated game (e.g., the
brown blocks), the surviving rate per generation and the weight related to the kind of victory
(lone victory or shared victory). The results show that multiple non-linear relationships exist
between these parameters and the resulting risk sensitivities. While it is an abstract model, the
conclusions can have real-world implications on how different entities address risk in complex
environments.
The analysis focuses on the evolutionary adaptation of risk preferences. Nevertheless, the
same simulation model could be employed to study different features. So, future research
includes the analysis of the adaptation of other behavioural parameters, as well as the analysis
of the co-effect of meta-model parameters on the output variables, for instance, through model
exploration strategies such as grid sampling. Learning mechanisms could also be included in
the meta-model so that agents do not only adapt by evolution but also by getting information
from the environment and the actions of other agents. Finally, the model could be analysis also
per se, and the effect of different strategies could be investigated, stand-alone or in relationship
with other specific strategies.
References
[1] K. Appuhn, Inventing Nature: Forests, Forestry, and State Power in Renaissance Venice,
The Journal of Modern History (2000). doi:10.1086/318548.
[2] G. H. Brundtland, Our common future—call for action, Environmental Conservation 14
(1987) 291–294.
[3] K. F. Wiersum, 200 years of sustainability in forestry: Lessons from history, Environmental
Management (1995). doi:10.1007/BF02471975.
[4] T. Kuhlman, J. Farrington, What is sustainability?, 2010. doi:10.3390/su2113436.
[5] J. D. Sachs, The age of sustainable development, Columbia University Press, 2015.
[6] H. Dieleman, D. Huisingh, Games by which to learn and teach about sustainable devel-
opment: exploring the relevance of games and experiential learning for sustainability,
Journal of cleaner production 14 (2006) 837–847.
[7] P. Hallinger, R. Wang, C. Chatpinyakoop, V.-T. Nguyen, U.-P. Nguyen, A bibliometric
review of research on simulations and serious games used in educating for sustainability,
1997–2019, Journal of Cleaner Production 256 (2020) 120358.
[8] M. Stanitsas, K. Kirytopoulos, E. Vareilles, Facilitating sustainability transition through
serious games: A systematic literature review, Journal of cleaner production 208 (2019)
924–936.
[9] W. W. Meadows, Donella H; Meadows, Dennis L; Randers, Jørgen; Behrens III, The Limits to
growth; A report for the Club of Rome’s project on the predicament of mankind, Universe
Book, New York, 1972. URL: https://search.library.wisc.edu/catalog/999473210902121.
[10] S. Pedell, L. Sterling, Agent-based modelling for understanding sustainability, in: Lecture
Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence
and Lecture Notes in Bioinformatics), 2011. doi:10.1007/978-3-642-25044-6_32.
[11] N. C. Onat, M. Kucukvar, A. Halog, S. Cloutier, Systems thinking for life cycle sustainability
assessment: A review of recent developments, applications, and future perspectives, 2017.
doi:10.3390/su9050706.
[12] S. Mishra, Decision-Making Under Risk: Integrating Perspectives From Biology, Eco-
nomics, and Psychology, Personality and Social Psychology Review (2014). doi:10.1177/
1088868314530517.
[13] E. Bonabeau, Agent-based modeling: Methods and techniques for simulating human
systems, Proceedings of the National Academy of Sciences of the United States of America
(2002). doi:10.1073/pnas.082080899.