In order to analyse and understand any type of system, however it may be conceived, it is necessary to understand models, what they are; what you can do with them; and why they are so useful. This article briefly lays out the basic principles of system modelling in an abstract way. Although, an abstract discussion on models might seem like a strange place to start if you’re interested in human society and politics, it is worthwhile because they are the foundation stone upon which all analysis is built. If you are unclear about the basics of modelling, you will be unclear about everything.
It is rare, in my experience, to see these principles put forward explicitly. Most people who work with consciously constructed models absorb these ideas through practice. It is common for people to greatly misunderstand these principles, with catastrophic consequences for their ability to analyse and understand everything.
Entities, labels and state
As a first step, let’s imagine that there are some entities and these entities have some sort of state or other. It is common to refer to entities with state as variables, but as soon as you mention the word variable, many people assume that we are talking about something that can be represented by a particular number. I do not wish for people to make such an assumption, as I am talking about a more abstract concept here – so I shall say that we have a set of entities with state.
We give each of our entities a name. These names are arbitrary, mere conveniences so that we can talk about them more easily. In some domains people typically use single characters (x, y, etc) as names for their entities. In other domains, people use words from natural language (e.g. earth, john). In the first case, it is common for observers to assume that the entities are numbers. In the second case it is common to assume that the entities are real things. I wish to avoid both assumptions, as I am working on a very abstract level, so I shall merely refer to them as labels. The important thing is that these labels are arbitrary and are used merely to make it convenient to talk about the entities, they don’t mean anything at all.
We can now say things like: imagine there was an entity called “Pog”. So far so good and, hopefully, hard to misunderstand.
The next thing that we need to do is to define what we mean by the state of the entities. Here things get a little more complicated. By saying that entities have state, we imply that this may vary. So, our entity may have different states and still be the same entity. To nail this down a little bit further, we will say that there are a set of states that an entity can have while still being the same entity. We are not saying anything about the nature of these states – the set of states may be infinite. We are also not saying anything about the dimensionality of the set of states – we may wish to consider them as encompassing multiple dimensions, even infinite dimensions.
Now, we will need to come up with some way of representing the state of the entity – we could use numbers; we could use labels; we could use multiple dimensions that mix labels and numbers. In the general case it doesn’t matter. These are merely ways of representing the different states that the entity can take on.
We can now say things like: imagine there was an entity called Pog, which could be in one of a number of states: “x, happy, 4, $ or French”
I mix together different types of symbols to represent the different states to illustrate the point that we can define the set of states and their labels in any way we like.
Change, state transitions and time
Thus far we have been dealing with a purely imaginary universe that has no correspondence to reality whatsoever. We now wish to introduce restrictions on the ways that states of an entity can change. To do this, we normally need to introduce some concept of time. This creates a connection between the imaginary world we are constructing and the reality of our universe. However, this connection is mostly illusory – in our imaginary world, time is really just an arbitrary dimension by which we impose restrictions on the state of our entities and how they transition from one state to another. We typically refer to it as time because time serves that purpose in our mental model of reality. It is quite possible to use a dimension that does not correspond conceptually to time, but for the sake of simplicity, we will assume that it does. Thus, we will assume that our imaginary world has a time dimension and this time dimension places restrictions on the states of entities and their transitions from one state to another. This time dimension may be a very abstract thing without any units. We could, for example, just say that it is a variable that advances one tick at a time: t, t+1, t+2, t+3…
We can use this dimension to place restrictions on the state of entities in ways such as the following:
- Our entity may be in single state at any given time t
- If our entity is in state French at any particular time, then its next state must be either $ or happy
- Our entity cannot be in state x without being in state happy immediately before.
- If there is an entity in our imaginary world with state 4 there must also be an entity with state French in that world
It should be noted that these are simply examples of restrictions that we impose using the time dimension. These constraints are all arbitrary – we can invent whatever constraints we want for the states of our imaginary entities. We could, for example, allow entities to have any number of simultaneous states and to transition from any given set of states to any other set of states.
Next, in addition to constraints on entity state transitions, we introduce some actual rules to define how the state of entities will change. So, for example, we may impose rules such as the following:
- If our entity is in state $ at time=t, it will be in state French at time=t+1
- If our entity is in state happy at some time, it will be in state x or state 4 at some time in the future
- If our entity is in state 4 it will be in a different state at some time in the future, but will never be in state French
Once again, I am using silly labels to refer to the state transition rules to emphasise their arbitrary, made-up nature. It is also important to emphasise that we do not have to be precise about these rules – they can be very vague in their details. We can say at some time in the future and we can under-specify the rules, so that we only insist upon an entity having one of a set of states, or not having some state. On the other hand, these rules have to be non-ambiguous in order for them to be rules as such – it has no meaning in our imaginary world to say that an entity will be in state sort of happy.
So, to recap, we have created an imaginary world and populated it with the following:
- A set of entities to which we assign labels
- A set of states which our entities can have, to which we also assign labels
- A set of constraints which govern the transitions that entities’ states may take
- A set of rules which govern the transitions that entities’ states must take
- Normally, we assume that there is a time dimension which we use to separate entity states from one another and to provide meaning to the concept of state changes.
That is it. There is nothing else. The above is a generalisation of all models that have ever existed or will ever exist. We can dress them up, add complex mathematical functions to define the transition rules and transition constraints. We can leave some of the elements out – we can have a model with no entities, or no transitions. We can introduce complex multi-dimensional states for our entities. We can formulate them in entirely different ways. We can do whatever we like with them, but in essence they can all be boiled down to the above.
Okay, there is something else. Normally, we want our imaginary worlds to be internally consistent. That is to say, that we do not want any of our rules and constraints to contradict one another. We want this to be the case, because if our imaginary world is internally consistent, then we can apply the standard rules of logic to it and we can reason about it and derive various things about the behaviour of our imaginary world. And the standard rules of logic encompass a vast arsenal of mathematical and algorithmic weapons which allow us to explore the nature of our imaginary world in minute detail.
The above 5 points can be considered to be the ground rules of an imaginary world that we have created. In mathematical lingo, they are known as axioms. The important thing to get about all of this is that all of this is entirely imaginary and thus cannot be wrong. If it is internally consistent, then we can apply logic to it and, as long as we apply the logic correctly, once again, we cannot be wrong. If it it is not internally consistent, we cannot apply logic to it, but we still cannot be wrong. Because this is an imaginary world, wrongness does not come into it, we can imagine whatever we want and we can apply logic to any given internally consistent imaginary world that we create without any risk of being wrong. We are operating in a world where we have decided exactly what things exist and what the rules are and, as long as we follow the standard rules of logic, it is guaranteed that we will never be wrong. This is one of the reasons why such imaginary worlds are such attractive things, they allow us to reason in such a way that all of our conclusions can be provably correct.
Mapping between reality and models
I emphasise this point so excessively because, in my experience, it is frequently misunderstood. The reason that it is so frequently misunderstood, as with most common misunderstandings, comes down to the imprecise semantics of words in natural language. When we speak of models, in any scientific sense, we are talking about imaginary worlds with arbitrarily defined rules. However, the models we work with are generally models of something that lives in the real world. It is very common for people to confuse the model itself with the assertion that it represents something or other in the real world. For a model to be interesting, it needs a mapping from the real world. Without a mapping from the real world, the model is just an arbitrary imaginary world, with arbitrary imaginary rules and most people aren’t that interested in such worlds in themselves (although there are lots of examples of people who are – string theorists, scholars of Klingon, Middle Earth and Westeros, for example).
Distinguishing models and mappings is important because, while a model cannot be wrong, a mapping from the real world to a model cannot be correct. As soon as we leave the domain of imaginary worlds with their axioms and enter into the real universe, we leave the domain of provability, correctness and wrongness behind us. We enter a fuzzy world where the most we can hope for is better or more accurate or stronger evidence. It is in the mapping between model and reality that problems arise.
In order for a model to be a model of something in the real universe we need a mapping from reality to the model. Through this mapping, we assert that the entities in our model represent entities in reality and that their states represent states of the real world entities. We further assert that the rules of the model represent some aspect of the rules that govern the real world. For the model to be a useful model of something in the real world, the mappings between the real world and the entities in our model must be bi-directional. That is to say that we can take some aspects of reality and map them into entities in our model, then apply the rules of our model to the entities to see how the states of the entities change over time, then map the states of the entities back to the real world entities that they represent.
Theory = Model + Mapping
In such a way, a model plus a bi-directional mapping from reality is a theory. By putting forward a model with a mapping, we are implicitly asserting the following: this model represents reality to such an extent that, if we map from reality into our model, then apply the rules of our model to the entities, then map back from our model to reality, the states that the entities take on in the imaginary world of our model over time, will map to the states that the entities have taken on in the real world. Our model and mapping together are capable of predicting some aspect of reality.
I now introduce a couple of examples of concrete models and discuss the problem of mapping them to reality
We have a single entity called Hat.
The state of this entity has a dimension that has a scale. We will refer to this dimension as pink. We define a transition rule which dictates that Hat’s pink grows over time, according to some function. We can describe this model mathematically as follows:
Hatpink(t+1) > Hatpink(t)
Hat(t+1) = f(Hat(t))
We have 3 entities, labelled:
Joe, Sally, Bob. Each entity has a state dimension that is labelled
fish. Joe also has a state dimension labelled
chips. We define the following constraints:
- There must be no more than
100 fishat any one time in the world.
- Bob can either have
Joemust have less than
The transition rules are slightly more complicated because we introduce a condition:
- If the product of the number of
fishand the number of
Joehas is greater than a number c, then
100 fishand everybody else gets
chips. unless he has
0 fish, in which case, he stays the same.
Both of these models are internally consistent in that none of the rules contradict any of the others. They describe extremely simple imaginary worlds. On the face of it, they are not interesting at all. All the first model does is to describe an entity with a state dimension that increases over time that can be expressed by some function which depends on the entity’s current state.
The second model is a little more complicated. We can do some basic reasoning over it, to figure out how it will behave depending on what the values of the entities’ states might be at any given time t. We discover that we will arrive in a stable state where Bob has all the fish, the only question is how quickly. This depends on the value of c and the rate of increase of Joe’s fish and chips dimensions. The model can be reduced to the simple state-transition / flow chart shown here.
Still, we have no reason to believe that there is anything useful or representative about the model – it’s just a bunch of entities, with silly labels, and a bunch of rules with no apparent significance to anything. If we want to make these models interesting, we need to create a mapping from something in the real world to the entities in the models and to demonstrate that the behaviours of the entities in our models mimic the behaviours of the real world entities that they are mapped to.
But, before talking about mappings, we will change the labels on the models as follows:
Hat => Universe
Pink => Space
Joe => Proletariat
Sally => Capitalists
Bob => Classless Society
Fish => Proportion of the population belonging to
Chips => Antagonism.
The new labels imply a mapping between the model and reality – and the models begin to embody theories. The first model can now be seen to express the theory that the universe will keep on expanding. The second model expresses Marx’s dialectical theory of revolution: increasing antagonism between capitalists and a growing proletariat will lead to a revolution that will usher in a classless society. We can now redraw the flowchart to reflect the new labels.
Our models have not changed. They are still imaginary worlds with arbitrary rules that we have defined. The labelled model is itself no more than the outline of a hypothesis because it is still tied only vaguely to reality. Words have loose semantics and have limited capacity to bridge the gap between the abstract space of the axiomatic model and fuzzy reality. The more precise, less subjective and readily quantifiable the mapping between the model and reality, the more the model becomes a fully fleshed out scientific hypothesis. Because, with precision in mapping comes testability – the ability to accumulate evidence to support the proposition that the model’s dynamics mirror those of reality. We can map from reality to the model, run the model for a period, then map back to reality to see how well the model predicted the real world dynamics.
Revolution in the model
It is worthwhile to briefly analyse the second model above with these new labels to identify how well it maps to reality. It should be noted that the model is, obviously, a huge simplification of Marx – he wrote hundreds of thousands of words on these subjects. However, that is the point of a model, and when it comes to his theory of revolution, the model is more or less an accurate and complete representation of the dynamics he postulated.
What he claimed can be boiled down to the proposition that the dynamics of capitalist economic development were such that the proletariat would continue to grow in relative size and that the class antagonism between them and the capitalists would continue to sharpen until, at some stage, a threshold of revolutionary potential would arise, whereupon a classless society would emerge as the dialectical synthesis of the existing classes.
The problem with the model, is not its fidelity to Marx, nor its internal rules, it is the lack of precision in the mapping between the model and reality. We want to know, first of all, whether the proletariat had indeed shown a continued rise, as a proportion of the population under capitalist economies. On the face of it, this should be easy enough. There is lots of historical census data around and historical studies that have aggregated demographic data. However, what exactly are we looking for? What constitutes the proletariat? How do we measure what proportion of the population belong to it? Similarly, how might we measure class antagonism? Without a reliable way of answering these questions, we have no reliable way of deciding whether our model’s dynamics mirror those of reality. Unfortunately, it is very difficult to extract a definition from Marx’s writings that can happily be mapped to modern society – are service workers proletarians? How about managers and administrators? Are we to include the lumpen-proletariat? These are the sorts of questions that have animated the last 150 years of Marxist theoretical debate without much clarity emerging.
However, even if we take the most restrictive possible definition, it is probable that the prediction will hold in reality – simply because there has been a consistent and protracted decrease in the size of the peasantry over the last couple of centuries and this, by itself, is probably sufficient to prove Marx right when it came to the relative growth of the proletariat whichever way we look at it. When it comes to the other aspect of the model, the results are less favourable. Firstly, although there ware many possible ways in which we might plausibly measure class antagonism – frequency of labour disputes, strikes, votes for socialist parties, etc, it is difficult to know whether they are suitable proxies for what he intended to express. Worse still, whatever proxies we choose, it is difficult to come to the conclusion that class antagonism has indeed sharpened over the last 100 years or so. More problematic still is the prediction of a revolutionary class-synthesis – this was based on nothing more than Hegelian dialectics – and there is nothing in the way of evidence to suggest that it is a pattern that reality obeys.
The creation of models, definition of mappings between models and reality, and investigation of the extent to which models mirror reality is science. The internal rules, entities and states of the model can be arbitrary and invented; they do not matter in terms of its scientific value. What matters is how well the model can dynamically map to and from reality. Extremely strange models, which internally violate some of our most basic assumptions about reality, are useful if they mirror some aspect of reality better than any alternative. The standard model of quantum mechanics, for example, is so counter-intuitive that it is virtually impossible to conceptualise the model’s behaviour. Yet its rules mirror the observed and measured interactions of particles at the lowest level very accurately.
Mapping between logical models and fuzzy reality is never straightforward – reality can never really be measured directly. However, this is less important than one might think. As soon as we have a means of consistently mapping between our model and some aspect of reality, we have a feedback loop which allows us to refine our model, improve our mappings, finesse our measurements to improve the extent to which our model mirrors the dynamics of reality. The goal is not to access truth, it is to produce models that mirror reality more reliably than any alternative.
All analysis of reality is mediated by models. The complex and fuzzy nature of reality must be mapped into some form of a simplifying model before it can be analysed in any way. Human brains have evolved to overlay a mental model upon the world, a model that is mapped to reality via their senses. It is, thus, not surprising that the way that the universe works at the very small and very large level, beyond the perception of our senses, is so counter-intuitive and breaks many of the fundamental assumptions of our mental models. When we consider a group of people as a collection of persistent, discrete individuals, or a brick wall as a solid obstacle, we are as much imposing a model onto a fuzzy reality as we would be if we were mapping them to a mathematical model. A solid obstacle may be a good approximation of a wall if we are thinking about a human walking through it, it is not a good approximation if we are firing a stream of neutrinos at it – a vacuum would be a better approximation in that case. Similarly, human beings are made up of cells which are constantly recycled, they have fuzzy boundaries – is the gut-flora part of the individual? The skin-bacteria? How about the symbiotic viruses in our DNA? A persistent, discrete individual person is a simplifying approximation for certain patterns of particles. Our brains incorporate such a model because it has proved to be a useful approximation which simplifies our analysis and understanding of the world around us.
One can find numerous critiques of the general approach of using simplified models to analyse human society:
- individuals cannot be treated like simple variables
- human society is too complex to be adequately captured by mathematical models
- things that are generally of social interest are too difficult to measure and too subjective to be reliably mapped to entities in a model
- simplifying models look at aspects of reality in isolation, the real universe is so interconnected that the models do not apply.
These objections might be significant, if truth was the goal of our model, but we do not need to aim for truth when we have a feedback loop which allows us to identify when the model better mirrors reality and can aim merely for better mirroring, not truth.
If a model more faithfully parallels the dynamics of reality than any alternative, it does not matter how it chooses to represent the entities, whether it maps individuals to simple variables, complex agents or merges them into aggregations, it is still useful in analysing how the real world will behave. Although reality may be extremely complicated and interconnected, it exhibits definite patterns that allow us to define useful generalisations. Furthermore, modern statistical analysis provides us with sophisticated ways of evaluating predictions and isolating factors even in the presence of complex interference. Applying models to complex, interconnected worlds is difficult, but not impossible and we are getting better at it all the time.
However, criticisms of the use of models in analysing complex, interconnected systems like human society cannot be simply explained away as the consequence of a basic misunderstanding of the relationship between theories, reality, models and truths. There are good reasons why many people are sceptical of the predictions produced by models of human societies. It is very easy to make a spurious but plausible conclusion about some aspect of reality by incorrectly extrapolating the predictions of a model and there are many people who are demonstrably willing to do so, in pursuit of agendas that are not connected to a desire to understand how the world works. Scepticism is always the most reasonable, evidence-based position in response to any claim that a model tells us anything about reality.
Micro-economic theory and its approach to modelling is responsible for the greatest damage to the idea of studying human society through axiomatic models. Much of the problem can be traced back to the start of the discipline in the 19th century. The theoretical goals of the field were to demonstrate that an optimal equilibrium would emerge from the uncoordinated interactions of individuals through a market mechanism rather than to study the dynamics of actual economies. In order to demonstrate such outcomes, it is necessary to introduce a variety of rules into the model which break any implied mappings between the behaviour of the agents in the model and people in all possible configurations of the real world. Without a solid and reliable mapping to the real world, the model is merely an abstract intellectual toy and should not be considered to say anything about anything in the real world. It is easy to become lost in such a model, to focus on elucidating its behaviour to such an extent that the mapping between the model and reality becomes forgotten. This would be fine in itself – there’s nothing wrong with exploring the contours of abstract models – were it not for the propensity of economists to propose real world policies on the basis of such models. One simply cannot say that “this rule produced the following outcome in a world of agents with perfect information and complete, pre-determined indifference curves, therefore changing that law in the real world will be likely to have a particular outcome”. The unfortunate outcome is a generalised suspicion of the use of formalised models to represent societies, even when they are tightly bound to real world phenomena and produce solid predictions.
The inescapable nature of models
The fundamental problem with all critiques of the use of models to analyse human societies is that there is no alternative. All analysis and understanding is mediated through models. Our brains have evolved to map the chaotic jumble of sensory input they receive into simplified models of the world to allow us to better navigate our world, survive and reproduce. Our brains create models populated with integral, individual humans, not the cells and molecules that constitute them. We perceive brick walls as solid obstacles, not swirling masses of indeterminate probability distributions. Above and beyond the models that evolution has wired into our brains, we apply a multitude of models to provide further order to our complex social worlds. We divide people into friends, enemies and rivals, natives, foreigners, and immigrants, men and women, workers and employers and so on. Not only do we use such models to understand the world, we also interpret the world through them. If we see two individuals fighting, depending on what models we employ, we may see it differently if they are, for example, a man and a woman, a foreigner and a native, a worker and a policeman. We see a man beating a woman, a cop oppressing an immigrant or similar and the perception itself is imbued with attributes of our model, which are not derived from the scene itself .
People typically employ mental models that they find most useful for interpreting and simplifying the world as they experience it. They may, for example, deploy a model of gendered behaviour which defines how men and women should behave in any given situation. Frequently such models are built upon at least a vague correspondence between the model and reality. However, the implicit models that people apply to the world are at least partially subconscious, rarely directly tested or evaluated and are often capable of resisting large quantities of negative evidence. Furthermore, large-scale societies with complex integrated social systems are a relatively recent phenomenon in human evolution – stretching back a mere 12,000 years. Until much more recently, most of the population would have had relatively few interactions beyond their immediate, local community. Therefore, the evolved mental models that people instinctively apply to the world around them often do not map well to the modern, large-scale real world society they are trying to understand.
For example, it is common to hear commentators analysing relations between nations in terms of inter-personal relationships (France are our friends), or analysing broad populations through the prism of personality or psychology (as a nation we can’t make up our mind), or through the model of some idealised family. Models from familiar worlds are happily applied to domains where they have neither a useful mapping to the model nor any evidence of the model’s ability to mirror reality.
Thus, it is impossible to escape the use of models in analysis. By being explicit about our models and their mappings to reality and requiring that their validity be proven by demonstrating that they can mirror reality better than any alternative models, we move our analytic framework into the foreground, where it can be examined, tested and proven. Without explicit models, we are limited to the instinctive models that live in our heads and they are frequently wrong.
However, models are imaginary worlds with arbitrarily defined rules. For the model to become a hypothesis, it needs to be grounded through a reliable mapping to reality. The more precise and less subjective this mapping is, the more useful the model. Words and labels are too semantically loose to serve this purpose on their own – if different people can interpret the mapping in different ways and yield different results, then the model is not reliable. The closer that we can get to an objective measuring process, the more tightly bound to reality our model will become.
Finally, once a model has a reliable binding to reality via a non-ambiguous mapping we must demonstrate that its dynamics parallel those of reality – at least better than any alternatives. Once we can do this we have a proper theory. Nothing else really matters – it doesn’t matter if what the model’s internals look like, it doesn’t matter what entities are mapped into and out of the model, as long as it exhibits dynamic parallelism and can be reliably mapped to reality, we can use it to understand the world.