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# Agent-Based Modeling of Flood Insurance Futures

## Abstract and Keywords

Agent-based models have facilitated greater understanding of flood insurance futures, and will continue to advance this field as modeling technology develops further. As the pressures of climate-change increase and global populations grow, the insurance industry will be required to adapt to a less predictable operating environment. Complicating the future of flood insurance is the role flood insurance plays within a state, as well as how insurers impact the interests of other stakeholders, such as mortgage providers, property developers, and householders. As such, flood insurance is inextricably linked with the politics, economy, and social welfare of a state, and can be considered as part of a complex system of changing environments and diverse stakeholders. Agent-based models are capable of modeling complex systems, and, as such, have utility for flood insurance systems. These models can be considered as a platform in which the actions of autonomous agents, both individuals and collectives, are simulated. Cellular automata are the lowest level of an agent-based model and are discrete and abstract computational systems. These automata, which operate within a local and/or universal environment, can be programmed with characteristics of stakeholders and can act independently or interact collectively. Due to this, agent-based models can capture the complexities of a multi-stakeholder environment displaying diversity of behavior and, concurrently, can cater for the changing flood environment. Agent-based models of flood insurance futures have primarily been developed for predictive purposes, such as understanding the impact of introductions of policy instruments. However, the ways in which these situations have been approached by researchers have varied; some have focused on recreating consumer behavior and psychology, while others have sought to recreate agent interactions within a flood environment. The opportunities for agent-based models are likely to become more pronounced as online data becomes more readily available and artificial intelligence technology supports model development.

Agent-based models facilitate the enactment of processes and functions of autonomous agents within a system. These populations of agents operate within a society and an environment autonomously, which has the benefit of enhancing understanding of how trends emerge from a diverse population, and how top-down decisions may differentially play out over a diverse population. In the case of agent-based modeling of flood insurance futures, the agent can be any stakeholder that has influence over some aspect of the flood insurance system under investigation. The flood insurance system under investigation can consist of any elements or processes which are impacted, directly or indirectly, by flooding and insurance. Due to this broad definition, how a programmer chooses to develop agent-based models in their research is dependent upon the specific research question. To date, there has been limited, but expanding application of agent-based models specific to the flood. This is due in part to the fact that awareness of agent-based models has grown only recently, though the concept has been in existence since the 1950s. Meanwhile, the opportunities for agent-based models are becoming more pronounced: insurance has become much more heavily influenced by consumer demand and interest, and with growing application of artificial intelligence technology. This means that public behavior, which demonstrates a broad range of rationality within a population, will impact the way in which insurers make decisions and present risk choices to potential policy holders. Concurrently, the conditions which result in flooding, for example changes in climate, are altering, which leads to reduced predictability of flood risk. Both the interactions between policy holders and insurers and the predictability of events mean that agent-based models, capable of modeling from the bottom up due to autonomous agents, have a role to play in supporting the modeling of flood insurance futures.

Agent-based modeling of flood insurance futures requires interdisciplinary research. The knowledge required to build an effective model may span subjects including psychology, computer science, (behavioral) economics, political science, and geography. This broad overview is required as a result of the computer-based nature of agent-based modeling, the physical nature of floods, and the socioeconomic aspects of stakeholders involved in the flood insurance system. Due to this broad spectrum of natural hazard science research, agent-based models of future flood insurance may be suitable for publication in journals whose themes range from artificial intelligence to political governance. The reality of agent-based modeling is that the topic under investigation, including “flood insurance,” will not fit neatly into one system. Interdisciplinarity is necessary, though it poses challenges in that it requires both ensuring modeling is at an efficient level of accuracy and simplicity and that the researcher take a targeted approach to their research questions and methods to achieve the desired outputs.

# Agent-Based Models

An agent-based model is a platform in which the actions of autonomous agents, both individuals and collectives, are simulated. As a tool for simulation, the aim of developing an agent-based model is highly dependent upon the research objective. It is the challenge of the researcher to program a system which will allow a researcher to monitor changing system variables and behavior of entities which may impact outputs, while ensuring outputs can still be understood. This often requires simplification of reality and the extensive application of assumptions, which must also be justifiable.

## The History of Agent-Based Models

To understand why agent-based models are appropriate for modeling flood insurance systems, it is important to understand the history of agent-based models, how their development has impacted the field of hazard research, and the direction agent-based modeling may take in the future.

In the 1950s a need to develop devices which were self-replicating was identified, but was limited by the available computer power (Voinov, 2008). That the output of a system is based on the culmination of its past dynamics was observed and codified (Bellman, 1957), but without computational simulations where simulated time and space could be adjusted and the behavior of agents examined, it was difficult to demonstrate what states of the past had led to the current state of agents and their environment. It had also been shown that following simple rules could lead to a diverse array of outcomes over time (Stonedahl & Wilensky, 2010). Thus, especially in a complex system with many actors operating simultaneously but with a limited footprint of those actions, it was difficult to understand how a population, even with a narrow choice of options, had traversed from one state to another. A model which would allow a researcher to abstractly recreate these populations within a simulated reality would be able to record the actions at each time step of each member of that population (described as an “agent” within an agent-based model). This would enable that researcher to identify particularly elastic points in an agent’s path trajectory and distributions of effects of an environment or other agents over a population, while altering variables to test the sensitivity of systems to stimuli. Such a model would be particularly beneficial in hazard research, where it is not possible, or ethical, to test the outcomes of hazards upon a population. The result of these requirements was that agent-based models were developed; models which simulate behavior and interactions of agents and societies within an environment to examine trajectories to emergent outcomes.

Developed alongside agent-based models were Discrete Event Simulations. Discrete Event Simulation is a “method of simulating the behavior and performance of a real-life process, facility or system . . . [Discrete Event Simulation] models the system as a series of events that occur over time. Discrete Event Simulation assumes no change in the system between events” (Allen, Spencer, & Gibson, 2015). Discrete Event Simulations can jump from one point in time to the next, anchored by events. However, though undoubtedly useful for modeling systems with randomness and uncertainty, the application is best suited to manufacturing systems, distribution systems, and storage systems (Moshref-Javadi & Lehto, 2016). In regard to investigations of individuals or collective entities which interact, especially where that interaction might lead to emergent trends or events, this approach is not necessarily suitable, and agent-based modeling provided an advantage.

In comparison to Discrete Event Simulations, through modeling autonomous agents, agent-based models are able to capture complex structures and dynamics within systems due to the low-level focus on agents which can act autonomously over time and space. This is because an agent-based model simulates the activities of the lowest element of a system, then culminates all elements to produce an output. With this comes the additional benefit that discrete elements of a system can be isolated, examined, and even altered. To contrast, normally with complex systems it is the outcomes that are most easy to identify; for example, we know the weather we experience, but we do not necessarily know the variables which were spread over a period of time and space that ultimately led to that weather. Taking this example further, agent-based models allow each weather element to act out, and wait to see what mix of wind patterns, heat, and so on, might lead to the particular type of weather experienced. Therefore, by isolating the components which produce outcomes and understanding how those components act and interact, it is possible to assess the outputs of a system in a way which is not always possible when observing reality.

### Cellular Automata

The lowest level of an agent-based model is that of cellular automata. Technically, cellular automata are “discrete, abstract computational systems that have proved useful both as general models of complexity and as more specific representations of non-linear dynamics in a variety of specific fields” (Berto & Tagliabue, 2017). Practically, cellular automata are an undefined element, which a programmer gives shape to by defining its possible characteristics and possible actions. For example, a programmer may define an element as having states such as being risk seeking, risk averse, or risk neutral, and those states may impact the likelihood of an agent adopting (or not) a flood risk management strategy, such as purchasing flood insurance. Whether an agent may adopt that state may be influenced by its experience within an environment, for example, whether the agent been exposed to flood damage, or whether the agent’s neighbor has adopted flood risk mitigation options. Even with very simple characteristics (e.g. being able to perceive risk from multiple perspectives, making choices on risk mitigation, memorizing flood damage, and observing neighbor activity) a single cellular automata has gone from being an undefined element to an agent representative of a simplified abstraction of a consumer with a broad array of outcomes of simple activity. A cellular automaton could also be an entire company, for example an insurer, seeking to maximize profit or minimize risk while operating in a risk environment alongside other insurers and supplying a population of consumer agents. Thus, cellular automata are, in a sense, whatever a programmer defines them to be.

The concept of cellular automaton was first outlined by John Von Neumann in the 1950s. In Von Neumann and Burks (1966), a theoretical device capable of self-replication was conceptualized. These cells operated over time measured in units, were abstract (meaning they were purely mathematical descriptions), and they were able to compute. By 1970 Burks had developed a computer implementation of these cellular automata, and concepts began to be put into practice as computing power increased. Most well known of these is Conway’s “Game of Life” (1970), which consisted of 2D cellular automata. The Game of Life illustrated the diverse array of results that could be output from following simple rules within a game. Thus, cellular automata were demonstrated as useful in facilitating the identification of emergent properties from interacting systems, both complex and simple. The benefits of cellular automata were further displayed with the application of the prisoner’s dilemma model (Axelrod, 1980) within a computer tournament. In this scenario, decision rules for how the prisoners should behave were submitted by entrants specializing in psychology, political science, economics, sociology, and mathematics, and then tested on a computer program. By comparing all of the outcomes of decision rules Axelrod was able to determine that cooperation when the other is cooperating, moderate forgiveness, and optimism about others’ responsiveness create the optimum outcome. Thus, not only was it possible to model flocking and other emergent trends of interacting entities, but also conceptual mimicry of behavior observed in human interactions.

The application of cellular automata to the field of insurance has been natural, suiting the complexities of insurance well; most specifically the interactions between stakeholders, rational and irrational, within the flood insurance system. The concept of flood insurance is simple, however the inputs are numerous and complex; the probability of the hazard occurring, the nature and accurate reporting of the goods insured, the motivations of the policy holder, the information available to all parties to measure the probabilities of the risks, the relationships between insurers and other stakeholders, the benefits to the state of the property-level security net, as well as the benefits of profits from a buoyant insurance industry. Insurance, though simple in concept is complex in terms of its inputs, processes, and outputs due to uncertainty and ambiguity. Due to this, there have been a wide range of applications of agent-based models to insurance and flood insurance. The focus of these has varied from impacts of reputational risk upon insurers (Lin & Jianshuo, 2015), and the impact of penalties for noncompliance with state flood insurance upon house prices (Filatova, Ollikainen, & Tahvonen, 2017), to the interactions between different stakeholders in response to surface water flooding (Dubbelboer, Nikolic, Jenkins, & Hall, 2017) and the integration of consumer behavior in the uptake of insurance (Geaves, Penning-Rowsell, & Hall, forthcoming).

In the 1980s, once games had been developed and the benefits of cellular automata had been demonstrated, user-friendly platforms to run agent-based models were developed. These included, but are not limited to, Repast, NetLogo, and Swarm (North & Macal, 2014). These agent-based modeling platforms had the benefit of allowing those with less programming experience to develop models on user-friendly interfaces. The result is that even those researchers with limited programming experience are able to run a functioning agent-based model. However, beyond these well-known agent-based modeling platforms, there is a broad array of available software (Abar & O’Hare, 2017). Abar and O’Hare (2017) give a review of 85 agent-based toolkits which support those wishing to either develop or use agent-based models, thus providing the reader with a good idea of the benefits of the platforms available. Due to the difficulty in identifying the elements of a flood insurance system which should be modeled, it is then vital that a researcher chooses or develops an agent-based model which allows the components of the system to be expressed in a way which is in line with research requirements, and which is justifiable when submitted to scrutiny.

### Systems Thinking

Due to the many ways that agents can be organized and applied to flood insurance, a prerequisite to agent-based modeling is systems thinking. According to Hall and Day, “any phenomenon, either structural or functional, having at least two separable components and some interaction between these components may be considered a system” (Hall & Day, 1977). Systems thinking is useful for creating efficient agent-based models as it facilitates sharing components of emergence, self-organization, and hierarchies of interacting systems (Abbott & Hadžikadić, 2016). Systems thinking describes what elements of a system are important, while agent-based models allow researchers to examine entities and collectives across simulated space and time to understand likely trajectories of how outcomes of those systems will manifest and play out within and between populations.

For a researcher modeling some aspect of flood insurance, determining which components and interactions to include within a model, and which of those should be variables versus fixed assumptions, constitutes a challenge. A model is a simplification of reality and, therefore, not all aspects of the flood insurance system may be require inclusion in the final model. For example, Geaves et al. (forthcoming) focused on consumer behavior applied to a simplified flood insurance market. Dubbelboer et al. (2017) examined the impact of stakeholder interactions (householders, insurers, banks, and local governments) upon surface water flooding. Both of these examples are concerned with the flood insurance system, but due to the types of research questions posed, the focus of those systems varies significantly. The fact that in the UK flood insurance is bundled into other risks, or that in London there are multiple sources of flooding, does not decrease the utility of these models. However, elements of a flood insurance system should not be pruned without justification or an assessment of the impact on the output of the agent-based model being made through sensitivity analysis. Thus, a model should seek to streamline the system under investigation while highlighting and justifying the assumptions made about that system.

### Artificial Intelligence and Machine Learning

The 21st century brought with it additional computing power, artificial intelligence, neural networks, and machine learning. Agent-based models are often confused with artificial intelligence or machine learning tools. However agent-based models do not necessarily have any learning function and the output data do not have to be analyzed using machine learning algorithms. Artificial intelligence “refers to machines that can learn, reason, and act for themselves. They can make their own decisions when faced with new situations, in the same way as humans and animals can” (Hao, 2018). Agent-based models generally endow agents with simple reactive behavior, not the hyperrational, deliberative agency expected within artificial intelligence models. However, this does not mean that agent-based models do not have the capacity to develop agents which apply artificial intelligence reasoning methods. Artificial intelligence can be used to allow agents within an agent-based model to learn from past events and respond in ways which are not predefined.

Machine learning refers to algorithms which use statistics in the form of supervised, unsupervised, or reinforcement learning to find patterns in data sets. Supervised learning exposes the machine to the characteristics the researcher is looking for so that the machine can then identify these within large data sets. In unsupervised learning, the algorithm identifies likely clusters in data sets, though it does not attribute these to an entity with predefined characteristics. Reinforcement learning is based on amplifying successful model behaviors and minimizing those which do not achieve the desired outcome. Each of these approaches to machine learning has a different strength depending upon the nature of research undertaken and the desired outcomes. Machine learning can be integrated into agent-based models in a number of ways including using machine learning techniques to analyze data output from agent-based models.

An example of integrating machine learning into agent-based models includes Frances, Rubio-Campillo, Lancelott, and Madella (2015), in which a Markov decision process model is integrated into an agent-based model, then contrasted against outcomes of a model with reactive agents with differing motivations. The results demonstrated significantly different outcomes between reactive and Markov decision process models. The Markov decision process falls into the category of reinforcement learning (Littman, 1994). This means that the machine learning paradigm is not supervised but does have a reward signal for successful outcome; feedback may be delayed; and an agent’s action will impact the future information to which it will be exposed. When considering the application of machine learning approaches, a researcher must consider whether the learning style is suitable to the learning approach of the agent. For example, when applying Markov decision processes, a researcher would need to consider whether the agent represented would actually seek to maximize total future reward and whether the reward the agent seeks is reduced flood risk or lowered insurance premiums, and so on.

That reward maximization of a specific reward is what an agent is seeking is an assumption. Adopting this assumption would mean that during a simulation an agent would naturally lean toward “good” behavior, amplifying actions on a trajectory which may not reliably represent agents’ various motivations. This issue is particularly important when taking into account research which demonstrates that when making risky decisions, consumers will become risk seeking in dealing with gains and risk averse in dealing with losses (Kahneman & Tversky, 1979). The reward that an agent seeks may be varied, from guaranteeing security from flooding to having the most prestigious house possible with the best riverside views regardless of flood risk or insurance premiums. What is more, the nature of what an agent perceives as a reward may change over time or in response to an event. This can occur in flood risk households and in government institutions (Johnson, Tunstall, & Penning-Rowsell, 2005). This does not mean that reinforcement learning is not appropriate to agent-based models, but it does mean that a researcher should not equate artificial intelligence algorithms as learning paradigms which will automatically represent the way in which different agents learn.

## The Structure of Agent-Based Models

Typically, there are at least two layers of agent-based model—the universal environment within which a society of agents functions, and the agent. Here, three layers are described, to include the society of agents of which the agent is part. Not all models will use societies, due to the requirements of the model and problem set. However, for more complex models it is useful to use the concept of society to distinguish between different agent populations. Those separate populations may follow different rules and interpret their environment with different motivations to other agents.

### Types of Agent-Based Model

From a sociological framing, Nigel Gilbert (2019) describes three types of agent-based models; abstract, middle-range, and facsimile models. Abstract models aim to demonstrate social processes which lead to a particular outcome, most likely in response to a developed hypothesis. These models are unlikely to be related to a specific case, and, therefore, it may not be possible to observe trends in complex real-world systems. Abstract models require comparison to expected and observable macro-level patterns in order to validate their output, as well as the systematic variation of parameters and associated interpretation of results. In this situation, it is likely that further modeling will occur after trends have been identified, and that this model type will be support the development of “middle-range” theories (Merton, 1968). Merton (1968) dictates that the start point of middle-range theory is an empirical phenomenon and from this abstracts general statements which are then verified by data. Simulations will generally describe the characteristics of a social phenomenon, with at least one aspect of the model requiring calibration against qualitative similarities to complex real-world situations. Finally, Gilbert (2019) describes facsimile models, which aim to mimic reality, often to use it to predict future outcomes under varying scenarios. Facsimile models are intended to provide a reproduction of some specific target phenomenon as exactly as possible. These models are often developed with the intention of making a prediction of the future state of an agent or environment, or to predict what will happen if some policy or regulation is changed. Recreations of complex systems are exceptionally difficult to develop due to the randomness and chaos which exist in most systems. Thus, the most frequent outcome of a model is likely, though not certain, to align with reality. In the case studies discussed in the section “Case Studies,” facsimile models are primarily used in the modeling of flood insurance futures.

### The Agent

As an individual agent, the agent is effectively a shell which becomes an entity through mathematical rules which inform its characteristics and behavior. For example, the agent is able to judge [flood risk], or the agent is able to purchase [insurance]. The ability to judge or purchase is an agent action allowing the agent to become an entity; that is, the agent is an analyst (regardless of proficiency) of flood risk or the agent is a potential consumer of insurance. The nature of the risk the agent is analyzing is likely to be determined by the universal environment or agent society —for example, flood risk—while the mechanisms which inform the agent of the risks so that the agent can weigh the decision might include government agencies, websites, past experience, and so on. The response to that risk is likely to be established within an agent society, with variations in the application of responses possibly varied between agents.

Theoretically there is no limit to the nature of an agent or the decisions an agent can make. However, usually an agent is a reactive entity, that is, the agent is programmed to respond to the environment following preset rules. Despite this, as discussed in the section “Artificial Intelligence and Machine Learning,” an agent could be programmed to act deductively, writing its own rules based on experiences and opportunities. Most models to date, and all models within the flood insurance agent-based model field have been reactive, that is, if [occurrence] then [action], though there are levels of complexity of response. For example, Geaves et al. (forthcoming) allowed an agent to adopt one of three consumer characters informed by field observations. These characters included agents which made judgments based on past flood exposure, past insurance cost incurred, and availability of savings:

• if (simulationTime >0) then
•   if (self.NumberOfFloods >= simulationTime) then
•     self.rp = self.ReferencePoint_AgentExperience
•   else
•     if (self.insurance_count >= simulationTime) then
•      self.rp = ReferencePoint_InsuranceCost
•     else
•      self.rp = ReferencePoint_AgentSavings
•     end
•   end
• else
•   if (self.NumberOfInsurancePurchases >= simulationTime) then
•      self.rp = ReferencePoint_InsuranceCost
•   else
•      self.rp = ReferencePoint_AgentSavings
•   end
• end

Depending upon their state, agents were then subject to differing weightings of how they perceived flood risk and whether the agent perceived potential outcomes as gains or losses. Though this approach adds an additional complexity to consumer agents, the agent is still following simple rules. However, there are a plethora of opportunities to use artificial intelligence within agents, and also to apply this to flood insurance systems. For example, the Markov decision process, described in the section “Artificial Intelligence and Machine Learning,” would be well suited to the problem set invoked by an agent observed to demonstrate different behaviors depending upon certain thresholds, and those behaviors then showing decreasing elasticity over time.

An agent is also able to have a memory, and may not be required to reassess its environment for each round of the agent-based model. This allows for an agent to make decisions based on past decisions and the current environment as opposed to the environment at the beginning of the model run. For example, in Brouwers and Verhagen (2003) agents were able to memorize consumer choices that had been successful. However, the way in which memory is approached by each stakeholder varies and different stakeholders have different mechanisms through which memory is computed. For example, an insurer will likely record all events which impact its operations, for example flood events, investments, claims, and so on. These will be recorded methodically, and likely in a way which does not have preferential retrieval. In comparison, human memory is not as precise or effective, and this is especially the case for traumatic events such as flooding. In comparison, a householder may not remember what they had for dinner yesterday, though they may be able to remember clearly a flood event which occurred 30 years previously. As will be discussed in the section “Agents with Bounded Rationality,” this has a significant impact upon behaviors in flood environments, especially relating to flood insurance, and will vary across a population depending upon its previous exposure to flood events.

### Societies of Agents

The agent is part of a society, and each agent within this society may end up taking different path trajectories in their model cycle. The society lays out the “social norms” as it were, which are called universal variables: for example, all agents must have a bank account or council tax band, though the amount determined or specific council tax band is set within the individual agent. A model may have more than one “type” of agent, with these explored in the section “Modeling Insurance Stakeholders.” In a flood, system stakeholders might include the insurer, the consumer, the local government, or even banks. This means that instead of a society being representative of a group of policyholders, the society may consist of a group of insurers, banks, or a local government authority. Each of these societies will have different likely characteristics, motivations, and methods of operation. As referred to in the section “The Agent,” there may also be different mechanisms of gaining, storing, and memorizing information, which will impact the agent’s level of rationality. On a practical level, splitting groups of agents into societies is useful as it facilitates clarity over programming, which supports the identification of possible errors and the development of a systematic sensitivity analysis.

### The Universal Environment

“An agent must be somewhere, and that somewhere gives the agents input through sensors and receives the output or effects of the agent action. This somewhere, in which the agents ‘live,’ is the environment, and it contains all the information external to the agent used in the decision making process and provides a structure of space for agent interaction” (van Dam, Nikolic, & Lukszo, 2013). The agent and societies function in a 1D (e.g. structural), 2D (e.g. a map), or 3D environment. The number of dimensions of the environment may influence factors such as the characteristics of the flooding the agent may be exposed to, the mechanisms through which damage to property is incurred, and the probability of being exposed to a feature of that environment, for example, a flood. The number of dimensions of space which are applied within an agent-based model is dependent upon the spatially dependent nature of the problem and the requirement to visualize the outputs.

For example, Geaves (2016) developed an agent-based model within a 1D environment. The justification for this was that all of the agents were those already considered at high risk, and therefore their chance of flooding must be greater than a specific threshold, which, in this case, was predefined by the UK government. Therefore, the research question was not concerned with whether the households would flood, as because of the nature of the agents described, property damage incurred by flooding would occur at some point in time according to a predefined probability. The activities of the agents, in terms of purchasing different flood risk mitigation options, also did not have to occur in 2D. Therefore, in this case of a flood insurance agent-based model, 1D was appropriate.

Within the universal environment of a 1D agent-based model, the universal features might be as simple, as follows:

• ROUNDS= 30—number of rounds simulation is run
• FLOOD_PROBABILITY= 0.01—probability of flooding
• POLICY_LENGTH= 25—length of government initiative
• OPTION_COST= -4700—cost of mitigation options

That these features are in a universal environment does not mean that the agent or society cannot alter their experience of those features, or that these feature could not be placed within the agent environment. For example, if a householder installs a property-level protection measure, for example, a floodgate, to reduce the likelihood of water entering a house during a flood, it does not mean that all agents in the society would then experience reduced flood risk. The flood risk is only reduced for that agent. Therefore probability of flooding needs to be captured in both the universal environment and the agent environment. Likewise, there are some features that could be variable within an agent, such as cost of flood mitigation options, or even the length of policy which could be altered by the government agent. It is up to the researcher to create a model which is a balance between possibility and likelihood.

As flooding is a spatially dependent phenomena, there are many cases where an agent-based model will be required to be 2D or even 3D: see for example Dubbelboer et al. (2017). In this case the environment is likely to be a matrix of cells (usually squares or cubes, though depending upon the level of sophistication required and the nature of the hazard, it could consist of other shapes), with each cell holding its own properties. A basic example is outlined below, where flood risk is categorized according to colors on a map overlay:

• —///////////////////////////////////////
• —Making cellular space
• —///////////////////////////////////////
• local cs = CellularSpace{
•      database= DB_PATH..”FloodMap.mdb”,
•      theme= “cells5x5”,
•      where= “(landUse = 001 OR landUse = 002 OR landUse = 003)”..
• }
• —///////////////////////////////////////////////
• —Assigning properties to types of land cover
• —///////////////////////////////////////////////
• forEachCell(cs, function(cell)
•      cell.cover = “noData”
•      if(cell.landUse == 001) then cell.cover = “river”end
•      if(cell.landUse == 002) then cell.cover = “floodplain_1”end
•      if(cell.landUse == 003) then cell.cover = “floodplain_2”end
• end)
• forEachCell(cs, function(cell)
•      cell.local_prob = “noData”
•      if(cell.landUse == 001) then cell.flood_probability = 1 end
•      if(cell.landUse == 002) then cell.flood_probability = 0.02 end
•      if(cell.landUse == 003) then cell.flood_probability = 0.01 end
• end)

In the case above, an agent will exist within a cell. Each cell is associated with a kind of cell cover (river, floodplain_1, floodplain_2) according to, in this case, a color on a map (e.g. 001, 002, 003). That categorization of the cell then impacts the universal probability of the cell being susceptible to a flood. This does not mean that an agent cannot either modify that cell or move cells. However, the changes or decisions made by the agent will not necessarily impact the universal environment (unless they are programmed to); only the agent itself. In the case that a researcher wanted to see the spatial distribution of results, or cumulative impacts of agent activities upon a spatially informed hazard, for example a flood, then a 2D environment may be of use.

A universal environment with a spatial dimension has the benefit (if required for the research question) of placing agents in such a way that their interactions can be spatially dependent. This may be useful in scenarios where there is a trust function and a consumer is more likely to adopt a flood risk management strategy if their neighbor does. Van Dam et al.(2013) describe three ways in which this structure can be defined within the universal environment; soup, space, small-world networks, and scale-free networks. These different structures can be applied depending upon the type of flood insurance problem.

### Rules and Decision Making

Agent-based models are concerned with behavior and interactions which are determined by rules. In the section “Artificial Intelligence and Machine Learning,” it has already been discussed how the nature of rules can actually define the character of the agent itself. Therefore when designing an agent-based model it is useful to consider agents as rational, or as having bounded rationality or irrationality. This is particularly pertinent when considering flood systems which have agents with different mechanisms and powers to understand and process information, act, memorize, and control universal outcomes. These will be explored for each primary flood insurance stakeholder in the section “Modeling Insurance Stakeholders.” Depending upon levels of rationality, if a particular agent has a limited elasticity in the range of behaviors, then following appropriate sensitivity analysis, it may be more logical to program that agent as model parameters as opposed to an agent. For example, if the government policy is unlikely to change for the interested time frame across which the model is to play out, programming the government policies as parameters within the universal environment instead of creating government agents may have benefits: for example, simplification of the model, and less chance of blackboxing a model and then having difficulty locating errors or understanding outputs. Thus, the nature of rules are not only important in developing the level of rationality of the agent, but also in determining whether that stakeholder should be represented as an agent or as a rule.

Rationality and rules within the agent-based modeling context do not mean that decisions are based on reason, but that behavior is consistent, with agents weighing different potential outcomes that maximally reach their goals. Models of rationality do not necessarily align with how decisions will be made by individuals or collectives in reality, and, importantly in modeling flood insurance systems, how decisions are made varies between stakeholders. For example, Verschure and Althaus (2010) described how traditional methods of artificial intelligence were based on the notion of knowledge level, whereas new approaches to artificial intelligence are based on embodied systems which interact with the real world, for example Distributed Adaptive Control. This means that previous levels of rationality of machines were based on the level of knowledge of those machines; however, new models assess also the type of cognition which goes into making that knowledge. It is clear that how a human memorizes, recalls, and changes behavior due to a severe flood event will be very different to how an insurance company memorizes events, and that, even between consumer agents, the mechanisms by which an event is recalled may vary. Therefore, in an agent-based model, it may be more accurate to create a rule about the agent’s cognition processes instead of rules about desired outcomes.

To demonstrate such an approach, Verschure, Maffei, Santos-Pata, Marcos, and Sanchez-Fibla (2015) developed a model to increase the efficiency of foraging. In this model, agents were given rules such as taking decisions on internal states, (e.g. hunger or thirst), relying on associative learning and landmarks with locations of resources and spatial memory. Upon increasing efficiency the agent was provided with a reward. To be able to gain that reward the agent had to know what kind of resource to use to make decisions relating to knowing an internal state and collecting the appropriate resource, navigate while avoiding obstacles, and gain and utilize specific knowledge (e.g. cue and associations) and contextual knowledge (e.g. space and trajectory). In this case the agent has three kinds of rules; reactive, adaptive, and contextual. Thus, when rationality is expressed through processes as opposed to knowledge levels it is possible to encompass a wide variety of agent needs. This undoubtedly has benefits in flood insurance models, where goals of agents may be varied—for example minimizing flood loss, maximizing profit, reducing flood insurance premiums—and elastic depending upon factors such as exposure to flooding.

The methods of approaching rules will depend on the agents modeled and the way in which those agents make decisions. For complex decision making, few consumers adopt explicit calculus to help them with their decision making. Instead they may apply some kind of heuristic decision making or compensatory decision making. These forms of decision making are likely to lead to a good, if not optimum outcomes, which over the course of an agent-based simulation could lead to significantly different outcomes when compared to rational decision making over time. Herbert Simon (1955) developed the concept of bounded rationality in “A Behavioral Model of Rational Choice.” Within this work he describes how limiting searches and satisficing (a compromise of satisfying and sufficing) were the main ways in which consumers made sense of difficult decisions. For example, when buying a property there are usually many options available. A potential buyer then is required to search and narrow these options down to a manageable set. This might be done by assessing factors such as proximity to workplace, schools, or family, and then mediated by issues such as closeness to a motorway, risk of flooding, levels of crime in the area, and so on. To assess the benefits of purchase a house consumer’s behavior might range from creating detailed cost–benefit assessments to prioritizing one feature, or even just making a snap decision on a whim. Especially when conceptualizing a population with diverse methods of decision making, creating agents with bounded rationality can support the development of accurate models.

# Agent-Based Models for Insurance

## Modeling the Flood Risk Environment

Flood insurance is a tool which expands beyond the recompense of assets for a policyholder. Insurance is a profitable industry in which a wide variety of skilled and unskilled workers are employed. The reserves from premiums received are frequently channeled into providing long-term investment finance, with positive outcomes for the real economy. Furthermore, insurance can provide a state function by supporting the government in providing a stable social and economic platform from which a society can function. In short, insurance has multiple benefits across tactical, operational, and strategic levels, involving multiple stakeholders. The primary objective of insurance is not its only use, with secondary functionality, such as finance investment, being of equal importance to the functioning of a state. Yet flood insurance is a risky decision investment for all stakeholders, with many variables which can augment ambiguity and uncertainty.

### The Flood Hazard

The initial uncertainty to an insurer is the likelihood of flooding in a given area. Knowing the chance that flooding will occur allows the insurer not only to make a proper risk assessment, but also to apply management approaches, such as bundling of risks, reinsurance, or public–private partnerships. Flooding, though also impacted by non–weather related phenomena, for example burst water pipes or water channel disruption, is generally a product of weather. Weather can be defined as the daily variations of environmental conditions which culminate to produce the climate. Though climate has usually been associated with long-term prevailing weather patterns, there are anomalies in climate, some of which are predictable, e.g. El Nino (Kripalani & Kulkarni, 1997), and others which are less so. Within the last 50 years, evidence has arisen that these long-term climates may also be changing, leading to different outcomes depending upon drivers of the climate in the area (Parmesan & Yohe, 2003).

Agent-based models are unlikely to focus on solely the impact of weather or changing climate, as agent-based models are primarily concerned with interactions. Therefore, in a model involving flood risk, the climate or weather are likely to be part of the universal environment upon which the agents act, as opposed to being an agent itself. Depending upon the purpose of the agent-based model, a programmer will approach the application of weather, climate, and water flow within a model differently. An agent-based model assessing long-term shifts in insurance sector bundling of risks in response to climate change would likely require a detailed climatic model capable of producing multiple scenarios of changing climatic hazards between countries (Moss, Pahl-Wostl, & Downing, 2001). Others may focus on a specific drainage basin or area of a country. For example, Brouwers and Verhagen (2003) examined the Upper Tisza region of Hungary and only modeled flood events caused by breached levees, despite the existence of groundwater flooding, because such flooding was not included in the insurance policy. Even more restrictive of the represented system, a model that is dealing with a specific group of high-flood-risk policyholders may only require a probability of damage as opposed to a climate-change or flood risk model, for example Geaves et al. (forthcoming). Both examples require the integration of climate into the model, but the complexity of the simulation is dependent upon the aspect of insurance which is being researched.

### The Vulnerable Property

Following uncertainty of whether water will flood a property, the next area of interest for insurers is the type and contents of that property. This information informs the probability of damage to that property, the recoverability of property and, in turn, the value of property which could be and is likely to be damaged in the event of a flood. Types of property could vary from agricultural through commercial to residential, and within these categories further variations exist. For example, the concerns regarding flooding of rice paddies versus flooding in pastoral or wheat farming diverge significantly. Likewise, a flood-resilient property with little furniture would be treated differently to a timber-framed house fitted out with extensive and unprotected electrics. To support insurers, some countries have publications such as the Multi-Coloured Manual on Flood Risk (Penning-Rowsell et al., 2014), which lays out how variables of property impact the likely damages incurred. If householders report these assets transparently to insurers these estimates will inform the value of assets that may be impacted by flooding. Therefore, with knowledge of the weather (which informs the hazard) and knowledge of the assets to be insured (which in part dictates the vulnerability), the insurer should have a good idea of overall risk.

In an agent-based model which requires a representation of a property, the property within the agent-based model might be structured in a number of ways; either in the universal environment, society of agents, or the agent. If all properties within the society were to be of consistent design and were not going to change within the simulation, then the parameters of the property might be programmed into the universal environment. The property may also be fixed but then chosen by an agent, in which case, it would also be placed within the universal environment. More likely however, property would be placed within the society of the agent, but with the potential for the agent to adapt the property. For example, there may be a society of agents who represent farmers with standard flood-damage-prone crops, and another society of agents representing insurers offering reduced premiums if farmers adopt flood-resistant crops. In this case, the property is placed within a society of farmers, and not in the universal environment because the property can be altered by one society, but only influenced by another. Thus, how the insured assets are approached within an agent-based model depends upon the aspect of the system which is under investigation by a researcher.

## Modeling Insurance Stakeholders

### Rational Agents

This section addresses the nature of rational agents in flood insurance and the benefits of agent-based modeling to stakeholders considered “rational,” and then hypothesizes approaches of integrating rational agents into models. Rational agents have clear preferences, they understand uncertainty and are able to calculate that uncertainty derived from scientific methods, and from the options available to them these agents will chose the optimum outcome. Within the flood insurance system agents which may fall into the category of “rational” include insurers, banks, and governments. However, consumers could be argued to be rational, just as insurers could be argued to act within bounded rationality; it is up to the researcher to justify how an agent is framed.

One rational method to find the value of a risky decision is to calculate the Expected Monetary Value (EMV). EMV is a product of probability p and the outcome of the option x to inform the value of a gamble $EV(G)$:

$Display mathematics$

An agent following clear rules to maximize the probability of achieving an optimum outcome might be considered rational. Yet, when posed with decision making under conditions of risk, it may not be possible for the agent to know the optimum choice in advance due to a variation in likely occurrences. If flood insurance is applied to the EMV from the perspective of the insurer it is clear that neither the probability p of damage nor outcome x of a flood event can be certain until the event occurs. This is due to the uncertain nature of flood events, of the accuracy in reporting of assets likely to be damaged, and of the actual damage following an event.

Insurers may not only be the agents within models, but those seeking to benefit from agent-based models of flood insurance systems. Though insurers are by most accounts rational agents, due to the complexity of the financial markets, climate systems, and human systems in which they can exist, it is possible for them to achieve suboptimal results. This instability is further amplified as insurance markets are subject to greater consumer feedback and interactions due to machine learning software supporting the operations of many insurance companies. Thus, there are increasing sources of complexity and, therefore, instability, and these variations can now occur more rapidly than at any previous point in history. For example, in September 2008 when the US government had to bail out the world’s largest insurer, AIG, for \$85 million due to issues in the housing market (Sjostrom, 2009). Flood insurance, therefore, should not be considered as a system acting in isolation, but within other systems whose activities may force suboptimal outcomes for the insurer. Agent-based models of rational agents can be of benefit in supporting capital allocation, pricing, and other areas of business management through modeling of trading algorithms, hedgers, and speculators in transactions. For these models the combination of rational agents, due to the complexity of the trading environment, can lead to complex aggregate outcomes.

Rational agents, such as insurers, can also have different interpretations of maximizing outcomes, and these may alter as an insurance companies grows. For example, some insurers may focus upon increasing or maintaining their market share, while others might seek to specialize. Such models may be of use not only to insurers seeking to identify suitable future investments but also to local governments seeking to apply regulation or encourage certain behavior on the part of insurance companies. For example, in the UK the government subsidizes the flood element of bundled household insurance under a scheme called Flood Re (Department for Environment, Food and Rural Affairs, 2013). This scheme will last for 25 years and will inevitably lead to a response in operations by insurers. How the insurance market might respond to policy instruments applied to risk is unknown; however, agent-based models could support future predictions.

### Agents with Bounded Rationality

As discussed in the section “Rules and Decision Making,” some agents demonstrate behavior outside of what is purely rational more frequently than others. This is particularly the case for householders in areas of flood risk, where it is more likely that householders will display “bounded rationality.” The decision to purchase insurance can be overwhelming for a householder, and the behavior of a consumer agent may not lead to optimum outcomes. Therefore, as programmers it is not necessary to design an agent with perfect methods of gaining, processing, and acting on information—the integration of consumer behavior into an agent-based model can be tailored to the research problem set which may require either an idealized or more accurate simulation of reality. This section address the options available to householders, the causes of bounded rationality, and how these can be approached within an agent-based model.

Within an agent-based model it is important to understand the options available to the agent, in this case a householder, and the decisions they will need to make. Flood risk management decisions also involve uncertainty and ambiguity due to the difficulties of understanding flood probability, the often clustered nature of flood events, and the different forms and causes of flooding. Neither householders nor insurers can be certain of the outcomes of their investments in advance; most years no insurance claims will be made or a flash flood may occur, limiting the preparation time available to install property-level protection. From the flood itself to the installation and visibility of flood risk management measures, both flood events and mitigation options elicit strong emotions; raising fears, questioning the notion of “home,” and potentially leaving a householder with lasting regret if an inappropriate decision is made. Poor decisions can lead to irreversible outcomes, such as the loss of a home, and the loss of financial security. Yet, the need for householders to make good decisions in their flood risk management investments is becoming increasingly pertinent.

Householders want to make the right decision when it comes to investing in their household. The stakes invested in a property are often the largest single expense an individual will incur in a lifetime, and, therefore, it would be assumed that householders are not influenced by behavioral biases which might lead to a suboptimal decision. However, the evidence presented below indicates that even with large investments, such as a property, householders are prone to making suboptimal decisions (Department for Environment, Food and Rural Affairs, 2013; Harries, 2008). Household flood insurance decisions are high-stakes, important decisions, with a great deal of information available (e.g. on Environment Agency websites), and, increasingly a whole raft of volunteer organizations that support householders in making them. Yet, householders continue to make what appear to be objectively bad decisions.

The behavior and choices of a householder investing in flood insurance can play an important role in state-level flood risk management as not all floods can be prevented with larger-scale structures, leaving householders with residual risk to manage. Unmanaged floods can lead to emotional distress, loss of assets, disruption to daily routine, unemployment, and even death (Mason, Andrews, & Upton, 2010). When timing of a flood event coincides with milestones, the impacts can last a lifetime. Flood risk unmanaged by a mass of individuals within a community can lead to social blight as community services are disrupted or destroyed and houses become uninhabitable and cannot be sold (Lamond, Proverbs, & Hammond, 2010). Compounding these trends is the moral hazard associated with the unfairness of the distribution of impacts of a flood, which are weighted against the poor (Johnson, Penning‐Rowsell, & Parker, 2007).

There are a variety of ways that a household can manage flood risk, spanning prevention, protection, and recovery. The primary prevention measure is to avoid living in a flood risk area. However, once residing in a house at risk from flooding, a householder can purchase property-level protection which can reduce the probability of water entering the building. These measures generally include equipment such as flood doors; however more invasive measures such as raising a house have been reported (Harries, 2008). Though these strategies reduce the likelihood of flood damage, if the option fails or its capacity is exceeded then the cost of damage may be the same as if there were no prevention measures installed. Householders can also protect their assets by purchasing property-level resilience measures, such as using flood-resistant materials or raising electrical sockets to a greater height than the likely level of floodwater. These measures do not reduce the probability of water entering a house, but may reduce the costs incurred if it does. Finally, insurance can assist in the recovery by spreading damage costs over time and across an insurance pool containing multiple other risks.

The household management measures available are comprehensive but are not always adopted, may be applied inappropriately, or may not be applicable to the flood context. Therefore, to avoid social blight, authorities seek to encourage the appropriate use of options through incentives and support. For example, a flood door is not always a permanent structure or automated, and residents are required to attach the flood door prior to a flood. In a region of flash flooding this can be difficult to time, and support can come from local authorities texting residents prior to a flash flood. Insurance on the other hand is usually a necessity and must be purchased to access a mortgage. However, if an insurance premium is beyond what a householder can afford, then householders run the risk of being unable to access insurance or a mortgage; making their house both uninhabitable and difficult to sell. With increasing areas at risk from flooding, it is not of benefit to a society to have swathes of uninsurable properties. However, it is also not of benefit to insurers to provide insurance to a house that is likely to flood. As such, state support has had to be given to both householders purchasing insurance and to the insurer. Yet, the realization that hard flood defenses are not always viable has led to an emphasis on the public managing future flood risk alongside larger-scale government-funded schemes (Ball, Werritty, & Geddes, 2013).

There is significant evidence to indicate that though not necessarily irrational, householders at risk of flooding often chose suboptimal options, and that this trend is augmented in low-income households. For example, 91% of homes in the UK are insured with buildings cover across the UK; however, in low-income houses, this drops to just 29% (Department for Environment, Food and Rural Affairs, 2013). From those that are insured, 6.8 million are underinsured, meaning that the cover they have purchased is not in line with the goods they possess. This, however, does not necessarily indicate lack of rationality. That low-income households have not purchased buildings insurance is more likely to be a result of competing priorities as opposed to lack of logic. Likewise, that a household is underinsured may not be suboptimal as they may have the savings to make up for the loss of uninsured assets, and, if they are not paying transaction costs on these savings, that is arguably a better option.

A trend which is not rational is the limited number of households who knowingly live in an area of flood risk and have not been flooded but choose not to invest in protection measures. Plausible reasons for this are living in a prison of experience (i.e. underweighting probabilities until experiencing an event); reimagining their property as an experience as opposed to an asset and therefore categorizing it as a secure “home,” for example ontological security (Dupuis & Thorns, 1998); and perceiving a narrow escape from a flood experience in a flood risk area as an unexpected gain and therefore risk seeking in gains. These trends are described by theories of behavioral economics, particularly prospect theory (Kahneman & Tversky, 1979), with Harries (2008) describing observed behavior in flood risk areas.

Plausible motivations for irrational behavior also covered the concept of image; people wanted their house to appear homely and safe and rejected measures which reminded them that their home was not in fact secure (Dupuis & Thorns, 1998). Behavioral economics reiterates this by defining the interacting nature of experiences and consumer purchases—a house purchase crosses the boundaries of these, being both an investment and an embodiment of the lifestyle a householder wishes to experience. This is why riverside properties can command prices 50% higher than those of similarly constructed properties; it is an experience which is being purchased. Therefore, when looking at modifying their property, householders may make suboptimal decisions by avoiding property-level protection installation due to the value they apply to their home experience.

Traditionally, the concept of rationality, and therefore bounded rationality, was associated with levels of knowledge of the agent (Verschure & Althaus, 2010). Agents were then tasked with following a rule, and were deemed more rational if they succeeded in a task. However, with artificial intelligence being increasingly applied within agent-based models, rationality can be considered the ability to gather, process, and act appropriately upon different types of information, and, instead of goal-orientated rules, rules can be designed to be learning processes. Bounded rationality within agents is likely to change in how it is expressed within agent-based models, and those changes will be more representative of how agents learn, resulting in greater accuracy of models.

# Case Studies

The approaches used by a researcher to integrate behavior into an agent-based model vary and are dependent upon agent research questions. Research questions used in agent-based models are ones which seek to explain commonalities observed at societal and macro level (Gilbert, 2019). Below are selected case studies, with further reading available in the “Further Reading” list.

## Stakeholder-Focused ABM

Stakeholders, which may be described as agents with specific or bounded characteristics, can shape and create an environment, affecting the behavior of other agents. Therefore, a researcher may prioritize analyzing emerging behavior as a result of the interactions of agents, as opposed to recreating the psychological processes to describe an agent. This is demonstrated by Jenkins et al. (2016) who developed an agent-based model “designed to assess the interplay between different adaptation options; how risk reduction could be achieved by homeowners and government; and the role of flood insurance and the recently launched flood insurance pool, Flood Re, in the context of climate change” (p. 1). The agent-based model focused on surface water in London and included six different agents; people, houses, an insurer, a bank, a developer, and local government. The results indicated that wider mechanisms of managing and reducing flood risk at household level were required beyond insurance. Unlike Brouwers and Verhagen (2003) or Geaves et al. (forthcoming), Jenkins et al. (2016) did not apply a model to frame behavior, such as prospect theory or the Consumat model, but sought to model the emergent behavior from the interaction of agents and the outcomes of their decisions. Thus, rather than applying a psychological model to measure emergent changes in consumer behavior, the emergent behavior arose from the model.

## Behavior-Focused ABM

Research questions may be primarily concerned with the actions of one stakeholder within the system, and therefore focus on developing one type of agent, the behavior of which will change throughout the simulation. An example of this is Brouwers and Verhagen (2003) who apply the Consumat model framework to create a richness of behavior across the agent population of flood risk households purchasing insurance. Similarly, Brouwers and Verhagen (2003) apply their agent-based model to the Upper Tisza area in Hungary, a region which is described as “relatively poor.” The Consumat model was developed by Wander Jager and Marco Janssen (Jager & Janssen, 2012) with the aim of developing a meta-theory which encapsulated a number of psychological theories relating to consumer behavior. Within the model agents can conduct four processes (repetition, deliberation, imitation, and social comparison), and which of these is used depends upon an agent’s certainty and/or satisfaction. In Brouwers and Verhagen’s (2003) agent-based model, agents apply one of these four processes depending upon certainty of flooding and past satisfaction with flood insurance. The results of the Consumat model were shown to be more dynamic than the model where behaviors were not taken into account.

Another example is that of Geaves and Penning-Rowsell (2016), who noted that behavior relating to the purchase of flood insurance varied in line with social deprivation. Concurrently, in the UK, increased emphasis was being put on the individual to protect themselves from flooding, and a government initiative was developed to subsidize insurance. However, with differential behavior of insurance consumers depending on social deprivation, research questions were developed to (a) understand whether a subsidy on flood insurance would provide similar benefits to individuals across a varied population, (b) identify which behavioral traits explained differential benefits, and (c) explore initiatives that might mediate these trends (Geaves et al., forthcoming). The results found that insurance subsidies did benefit certain groups of a population, but that this could be mediated through council tax precepts and property-level protection subsidies. In this example, an agent-based model was ideal as different observed behaviors could be applied to a simulated environment where the government initiative could be adjusted. However, research questions can vary significantly, and this will impact the way in which the agent-based model is structured.

# Conclusion

An agent-based model of flood insurance can be considered as a system which integrates some components or interactions related to flooding and insurance, and which runs these through cellular automata over time. The aim is likely to be the analysis of the impact of variations of these components and interactions. Applications of agent-based models to flood insurance are few, but expanding, and also varied as a result of the limited duplicity of the aims and conceptualization of the agent-based models developed to date. It is likely that the number and type of agent-based models of flood insurance systems will grow, and that with the application of machine learning techniques these will become more accurate through recreating the processes which lead to the outputs.

The strength of using agent-based models to examine flood insurance is that the model can look at emergent properties of diverse agents over time. This is especially useful for complex systems, such as flood insurance, and also provides an opportunity to test changes to the system which it may not be possible to produce in reality. Meanwhile, difficulties still arise. These include the necessity of simplifying the system which is under examination. Research is often concerned with specific and narrow fields, yet flood insurance systems are expansive, unpredictable, and varied. This causes difficulties in terms of being able to describe a model in enough detail for it to be an accurate conceptualization of components and interaction.

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