Assessing the Safety of (Numerical) Representation in Social Simulation
Centre for Policy Modelling
Manchester Metropolitan University
The nature and function of representation in social simulation is analysed into the: representational; necessary; and significant. Conditions for a reliable use of a simulation are then formulated. The special case of numerical representation is then considered. Three simulations that use numerical representation at their heart are discussed, one in detail. It is found that small changes in the representation (ones indistguishable using observation) can cause significant changes in the outcomes, and so are unsafe in terms of informing us about the target phenomena.
Introduction – Representation in social simulation
The question of how one represents aspects of social phenomena is a fundamental issue for the field of social simulation. The best way to represent any particular aspect of social phenomena depends on a great many things, including: the goals of the modeller; the data that is available; the techniques and facilities accessible to the modeller; and what is already known (or thought to be known) about the phenomena. Not much is known concerning the cognition of higher animals (including humans). To be precise, there are no general theories of cognition that are useful at the macroscopic level – nothing that would help the modelling of cognition. Thus a modeller who seeks to simulate such actors interacting often has a wide choice in how to represent the relevant aspects of their cognition in their simulations.
Now a basic assumption behind much social simulation work is that it is useful (in some ill-defined sense) to construct simulations, even in the case where it is not known how to represent key elements of the phenomena (e.g. the cognition). In other words, that we can learn something useful from simulations which include arbitrary or guessed elements.
To put it in the bluntest and simplest way, the astounding assumption behind much social simulation is that
our simulations tell us something true even though key elements of our simulations are wrong 
Sometimes the principle of “simplicity” is invoked to justify the modelling choices in these cases, a justification that I and Scott Moss criticise (Edmonds and Moss 2004) as either: misleading or mistaken.
At other times it may be believed (or, more often, merely assumed) that the exact way cognition is modelled is not critical to the significant aspects of simulation results (what has sometimes been called the “statistical signature”). Clearly this does not always hold (e.g. as shown in Edmonds and Moss 2001), and critically depends upon the interpretation given to the simulation (what is important and what is not). On the other hand, our general experience indicates that one can sometimes model/understand phenomena without knowing the full detail of the working of all the relevant parts – for example, we did understand enough about breeding animals to be useful before we knew about genetics. However, these cases tend to involve knowledge of a vague, practical and context-specific nature developed over long time scales, this assumption has been spectacularly less successful with precise, theoretical and general theories developed within single lifetimes (e.g. Neo-Classical Economic models).
Let us consider cases where we think that the exact representation of cognition will not effect the important aspects of the results – where we “plug in” a somewhat arbitrary cognitive mechanism (in the widest sense) to get the simulation to work. Thus we have a simulation whose parts are differently endorsed by the modeller: some aspects will be judged as representative of the phenomena and some not; some aspects will be judged as necessary to cause (within the simulation, and hence presumably in the corresponding parts of the phenomena) the significant aspects of the resulting simulation outcomes.
It is important to understanding the importance and meaning of a social simulation to be clear about the following:
· which aspects of a simulation set-up are representative (and how);
· which aspects of a simulation set-up are necessary to get the significant results;
· which aspects of the outcomes are representative (and how);
· which aspects of the outcomes are those that are caused by the necessary elements of the set-up.
The way aspects of a simulation (either set-up or outcomes) represent social phenomena may either be direct or indirect (via a conceptual model) (Edmonds 2001).
Unfortunately, it must be said that in many papers describing simulations these distinctions are not make clear. There is the suspicion that, in some cases, if these were made clear the simulation would be judged as having much less significance than is presently the case.
Clearly, if a simulation is to reliably inform us social phenomena the following should hold:
1. the aspects of the set-up that are judged as necessary are included within those judged as representative;
2. the aspects of the set-up judged necessary are, in fact, necessary to cause the aspects of the outcomes judged significant;
3. the aspects of the outcomes judged to be significant include those that are judged to be representative.
In particular, non-representative (e.g. arbitrary) aspects of the set-up should not be critical for obtaining the aspects of the outcomes that are deemed representative. In other words, it is critical to any conclusions drawn that any results would not significantly change as a result of changes in the representation of underdetermined elements. The chain is illustrated in Figure 1.
Figure 1. An illustration of sound inference using a simulation
Of course, the soundness (or otherwise) of the steps used to conclude something about the target social phenomena depends upon what is being represented in set-up and outcomes (and how) as well as what is claimed to be the significant causation in the working of the simulation. Also the chain itself can be used in different ways, for example by attempting to reverse engineer information about what the simulation set-up must be like in order to get certain results.
The particular case of numerical representation
So given the analysis above the key issue I wish to address in this paper can be formulated:
why are the difficulties of the simulation task (taken as a whole) exacerbated by the use of numerical representation?
My answer is twofold:
1. numerical mechanisms are often used to represent underdetermined aspects of cognition, i.e. they are not known to be representative;
2. and they are brittle, in that they often cause unintended/misunderstood aspects of the significant outcomes (“artefacts”).
Thus mechanisms such as plausible (but largely arbitrary) functions are used to implement decision making. The classic case of this is the optimisation over a given utility function as in many “economic-style” models (e.g. Thoyer 2000, as criticised by me in Edmonds 2000; as well as the examples below). I have speculated elsewhere about why this is done (Edmonds 2004), but it could be roughly characterised as either tradition or laziness.
The brittleness of many simulations with numerical representation has now been demonstrated in many cases included within the Model-to-model workshops. This contrasts somewhat to the case where inference is done symbolically to obtain closed solutions in an analytic way. There the assumptions are explicitly outlined and one has some confidence that the results are sufficiently reliable. However in this case one has the difficulty of relevance: social phenomena are almost always too complex for analytical techniques to be feasible without introducing assumptions that are so drastic that the relationship with the target phenomena is lost.
Clearly there are several possible defences to the criticism above, including:
· that there is some evidence that the mechanism is representative;
· that the mechanism is not critical for producing the significant outcomes;
· or that there is no other feasible way of implementing the mechanism.
Elsewhere I have argued that the use of numbers is a difficult and dangerous choice (Edmonds 2004) and that this is not always a necessary choice (Edmonds 2003). Here I reinforce these general arguments by reference to some particular simulations.
The Case Studies
In this paper I report on investigations into the role of representation in some well known social simulations: the “smooth bounded confidence” social influence model of (Deffuant, Amblard and Weisbuch 2002); the model of tag-based cooperation of (Riolo et al. 2001); and Hemelrijc’s model of ape dominance interactions (e.g. Hemelrijk 2000). The first of these will be investigated in depth, whilst the other two are only briefly discussed. All of these simulations use floating point numbers to represent key elements of the phenomena they are concerned with. This involves reimplementing versions of these simulations and comparing the significant results that are obtained when using different set-ups – set-ups that can not be ruled out by evidence from the target phenomena (i.e. they are not representative in the above sense).
Case Study 1: Opinion dynamics - the smooth bounded confidence model
In the smooth bounded confidence model the opinion of an individual on an issue is modelled as a real in a bounded interval [-1,1]. This is supplemented by another real from [0, 1] for the uncertainty of the individual. Further elaborations to this in a similar vein have proposed (e.g. empathy). (Deffault et al. 2004) argue that this is an improvement upon the simple binary Ising model due to that model’s lack of realism. However, it is not clear that the opinions of individuals are sensibly representable as real numbers. Rather this method of representation is, in effect, a very strong assumption. Even stronger is the way in which opinions influence each other via particular specified functions.
It is not argued that opinions may not be numerically measured (or, at least, something associated with opinions) via such procedures as surveys or voting. However this a posteriori measurement should not be confused with a casual mechanism in a simulation. In elections the proportion of people voting for left and right-wing parties can be measured, but the opinions of the voters are not simply points on a single dimension but much more complex and effected by a myriad of issues. This is evident due to the fact than substantial proportions of lower income groups vote for conservative parties that reduce the tax of richer groups and provide the lower income groups with fewer benefits.
The criticality of this method of representing opinions is evaluated by comparing the results of (Deffault et al 2002) and similar models with very slight modifications. We will call the Smooth Bounded Confidence model of (Deffault et al 2002), the SBC model.
In the SBC a normal function, , is used as the definition of “influence” (between two individuals who interact whose opinions are at a distance, dist, from each other and where the affected individual has uncertainty, u). This is a fairly arbitrary function, if one replaces it with a similar function one gets very different results. The two functions are illustrated below in Figure 2. The factor of 1.361 in fu was chosen so that the functions had the same area and hence, roughly, the same overall “pulling power” (but with a slightly different shape). I will call the variant of the SBC model that replaces the gu function by the fu, the “inv cube” model.
Figure 2. The gu (black) and fu (grey) functions compared for dist in [0,1] (where u=2). fu is slightly more leptokurtic than gu. Importantly this means that at greater distances fu retains some (small) significant value.
These functions are pretty close to each other, certainly close enough that we will probably not be able to tell which is a better model of the extent of influence between two individuals from direct observations of those individuals. However the difference in the outcomes from the interactions in the model is significant – one gets qualitatively different results. Figure 3 shows the output from (Deffuant et al. 2002) model using the gu function to moderate the influence of individuals. This model starts off with two groups of 5 extremists with opinions –1 and 1 and low uncertainty (u=0.05) and 100 moderates of a higher uncertainty (in this case u=0.6) with a random distribution of initial opinions. All individuals interact once with a random other each time period. The result is that the extremists “pull” the other opinions towards an extreme (see Figure 3 – the results are then constant from then on).
Figure 3. A typical outcome of opinion dynamics model using the gu function (i.e. the original SBC model), where the moderates have an uncertainty of 0.6.
Compare this with the results from the same model, where the only change is due to using gu rather than fu. A typical run is shown in Figure 4. The result is that this time, in the end, the moderates “pull” the extremists into the centre.
Figure 4. The corresponding typical outcomes from the opinion dynamics model using the fu function (otherwise the same as that illustrated by Figure 3.
Similarly if one takes the model using gu and adds a very small amount of noise to the process, one gets different results again. Changing this model so that there each time period there is a 0.1% chance of each individual’s opinion having some random Gaussian noise (SD 0.3) added. Thus we are changing the model so that (on average) out of about 100 people someone only changes their mind due to an otherwise unmodelled reason only once every 10 periods. This is a very conservative amount of noise – it is likely that, in reality, the corresponding rate is much higher! The outcomes now change again, a typical run is shown in Figure 5. Here one has the initial polarizing affect seen in the original model, but then some interesting dynamics later. Over time enough individuals with “mutated” opinions collect to form intermediate groups – that is new moderate groups are continually spawned.
Figure 5. A typical set of outcomes (using the SBC gu function) where a low level of noise is added (a 0.1% probability of mutation for each individual each cycle having Gaussian noise of SD 0.3 added)
To show that these differences are real and I have not merely been lucky in the above examples, I ran the SBC model 25 times for every combination of: one of five functions (SBC, inv cube, linear, 1-SBC and constant) in both bounded and unbounded versions; moderate uncertainties of 0.1, 0.2, …, 0.7, 0.8; and probabilities of mutation of 0, 0.001%, 0.01%, 0.1%, and 1%. That is a total of 10,000 runs of the model with 110 individuals to 10000 cycles. The functions are illustrated in Figure 6 – the bounded versions of these functions are the same but are truncated to zero for all distances greater than 3 times the uncertainty.
Figure 6. The different influence functions tested for scaled distances in [0,1]. Bounded versions of the functions are the same but truncated to zero for distances above 3 times the uncertainty.
A summary of the results are shown in Figure 7. This shows the average index of extremism over the last 1000 of the 10000 cycles. This index is the size of the greatest group times its distance from the centre (i.e. its absolute value). Thus if the run ends with many different groups or they are all in the centre the value is small. Several paragraphs of „Current text“ follow.
Figure 7. average index of extremism for different levels of moderates' uncertainty and probabilities of opinion mutation. Rows are for different functions (from top to bottom: SBC, inv cube, linear, 1-SBC and constant). Columns are for unbounded (left) and bounded (right) versions. The different colours are for different probabilities of mutation (blue-0 up to red-1%).
There are several interesting points to note in the results illustrated in Figure 7. Firstly, increasing noise (in the form of the probabilistic mutation of opinions as described above) decreases the extremism that results. A probability of 0.1% makes a substantial difference (see Figure 5) allowing the formation of intermediate groups and a probability of 1% almost completely eliminates any coherent tendency (including that to an extreme). Secondly, bounding the SBC and linear functions (1st and 3rd rows, left and right) makes no essential difference to the results – this is unsurprising since these functions essential fall to zero before they can be bounded anyway. Thirdly, for all the other functions (inv cube, 1-SBC and const) it makes a substantial difference to the results (for levels of noise at 0.1% and below). Fourthly, (as noted by Deffuant, Amblard, Weisbuch and colleagues in their papers on this) that an increase in the moderate’s initial uncertainty increases the extremism that results, but only for effectively bounded functions and low (<0.1% probability) levels of noise.
Thus we can see that the SBC model of opinion dynamics is vulnerable to slight changes in the model, e.g. that between using the SBC and inv cube function. This change is so small that one could probably not be distinguished between them by any direct empirical observations of how individuals actually influence each other when they interact. This means that any cross-validation (in the sense of (Moss and Edmonds 2005) is not possible. Thus, if we are to preserve the structure of the SBC model, we are left with only the outcomes that can be meaningfully validated against what is observed. Because of its brittleness we can not use this model to explain the outcomes in terms of what is known (or suspected) about the workings of individuals’ opinions.
In other words we are left with trying to infer something about the working of individual’s opinions when they interact in pairs from our validation of the outcomes. However, if different versions of the model (e.g. using a fundamentally different underlying representation of opinions) can produce the same outcomes (at least with the same distinguishable outcomes in terms of what is qualitatively known/observable of groups of individuals) then we can not even do this. However (Stauffer et al. 2004) essentially do this by producing a discretised version of the SBC model and getting the roughly the same (and qualitatively indistinguishable) results. Thus we are left with the conclusion that we can not infer anything using the SBC model – it is fundamentally unsafe for inference in any direction.
I would argue that the reason it is unsafe, is that the opinion mechanism has no direct reference to anything directly observable in terms of what individual’s do or the nature of their opinions. Thus there is no way of knowing whether, for example, we should add noise or not (and if so how much of what kind). Similar experiments with multidimensional binary or continuous representations of opinion suggest that similar divergences occur in the presence of small amounts of noise and/or when there is very weak interaction between the dimensions, however that research is still at an early stage.
One of the problems here is that it is not clear what in the model the authors intend as essential to the model and what they think is incidental (done simply to get the model working but thought irrelevant to the results). More information about the intermediate conceptual model, which these models are really about (Edmonds 2001) would help clear up these issues. The results shown in Figure 7 would suggest that the results are fairly independent of the exact shape of the influence function except for its sharp boundedness (its “meta-stability”). Thus I would suggest that this unnecessary and unrepresentative (in the above senses) numeric functions should be eliminated in terms of a simple model based on the simple principle that one is influenced only by those with similar opinions as oneself.
Case Study 2: Tag-based models
(Riolo et al. 2001) presents a model of social interaction where individuals each have a “tag” – a socially observable cue, in a “prisoner’s dilemma” type cooperation game, following a suggestions in (Holland 1983). In this, the tag of each individual is represented as a floating point number, t, in [0, 1]. Each individual also has a “tolerance threshold, T, represented in the same way. The idea is that individuals as potential donors, D, are randomly paired with others who are potential recipients, R. If the difference in their tags is strictly less than the first individual’s tolerance (|tD - tR| £ TD) it donates some resources to the other, with no direct recompense. The tolerance value represents how selfish the individual is – a low value means they donate to few and a high value means they donate to a wide range of others.
Surprisingly this model results in a high rate of donation and a small but significant average tolerance value. The statistics for a typical run are shown in Figure 8.
Figure 8. The donation rate and the tolerance value in a typical run of the Riolo et al. model
However it turns out that if one changes the condition of donation from |tD - tR| £ TD to |tD - tR| < TD then occurrences of donation all but disappears. Figure 9 shows the output from a typical run in this case. After an initial peak of donation it quickly dwindles to insignificant levels and the tolerance values converge on a zero tolerance. It turns out that, in this model, the threshold tolerance values are not important, for the “engine” of cooperation if a large set of individuals with exactly the same tag values (“tag clones”). Members of his set are forced to donate to each other in the original model but not in the modified model. This fact was “masked” by the representation of tags as floating point numbers with different degrees of tolerance. It is much less misleading to use a (large) set of discrete labels as tags and a single bit for tolerance. Doing this involves loosing nothing of the results and gains much in terms of clarity. A much fuller investigation of this model is reported in (Edmonds and Hales 2004).
Figure 9. The donation rate and the tolerance value in a typical run of the Riolo et al. model, but with strict tag comparisons.
Note that, as in the opinion-dynamics case, it is not impossible to use floating point numbers to represent aspects of social processes, but that (1) the complexity of numerical representation can act to “mask” assumptions upon which models work; (2) numerical representation can have unintended affects, creating model “artefacts”; and (3) often the full range of properties of such numbers are not needed to represent what they do –more representative models are often possible.
In (Edmonds 2005) I exhibited a tag model which has the same representation of tags and tolerances as that above (but where the totally selfish case is allowed as in the second case above). However this resulted from the exploration of more complex (multi-dimensional and/or discrete) representations of tags that might have a more obvious genetic interpretation – what I discovered (and this tallies with the experience of David Hales in many papers – see www.davidhales.com) is that, in the case of tags, the representation of tags, the distance function etc. do not make a substantive difference to the results. As a result of this I felt justified in using a numeric representation since it was not necessary to the results. This is one benefit from following a KIDS rather than a KISS approach (Edmonds and Moss 2005).
Thus with tag-based methods of group formation, it seems that one can use a reasonably wide range of mechanisms for tags and that what is significant about this mechanism are things like: that it is possible for individuals not to have to be cooperative; that new groups form relatively easily compared to the rate at which old ones are invaded; and that once invaded old groups die. The earlier use of numerical functions in such as (Riolo et al. 2001) served to mask these underlying distinctions leading to misunderstandings, e.g. the “Tides of Tolerance” interpretation of (Sigmund and Nowak 2001). This contrasts strongly to some of the simulations of David Hales (e.g. Hales 2001) where tags are simply represented by one of a set of unique labels.
Case Study 3: Hemelrijc’s model of ape dominance interactions
Finally Hemelrijk’s model of ape dominance interactions is examined. A key aspect of this model is that each individual (standing for a single ape) has a floating point number associated with it to indicate its dominance. During the questions after her presentation of her work (Hemelrijk 2003) at ESSA 2003 she said that whilst some intransitivities are occasionally observed (i.e. simultaneously A >B, B>C, and C>A) it did appear that (on the whole) the apes did act as if each individual had a dominance level which was discernable by others. The exact empirical grounding behind this claim is not entirely clear. However (even though the intransitivity and the claim are not entirely compatible) this claim has been given justification on empirical grounds (i.e. it is descriptively accurate) and thus brings the numerical mechanism within the representational.
In this model apes move around on a 2D surface. If they get sufficiently close one (or both) may consider instigating a “dominance interaction” with the other. Before doing this they mentally try this out to see if they might win it, if they do they engage in such an interaction, otherwise they signal submission to try to avoid this.
I reimplementated the model as reported in (Hemelrijk 2000, 2003). However this replication has been done using a very different language and framework than that of the original and I have not been able to get some of the results that Hemelrijk reports. Despite several requests to her (and others) I have not been able to obtain the code to read and check the necessary details.
Using this imperfect implementation I tested a couple of variants in the way the update function is performed. This is the “suspicious” part of the simulation, since the dominance number is primarily used to implement a total dominance relation between the individuals in the model, and yet arithmetic operations are applied to them to update this after a dominance interaction. However, despite producing minor variation in the overall levels of dominance, the substantive results in other respects were not significantly altered. This indicates that either: (a) the representation is fairly robust w.r.t. to its intended interpretation or (b) that I have not tried enough variants. In any case the model survived this attempt to probe its weaknesses, for I did not come up with any evidence against it in this respect. This, strength may be related to the fact that the representation is claimed to be representative.
Consider the case where there is a simulation where there are elements that are unrepresentative (unconstrained by information from knowledge about the workings of the target domain) yet where minor changes in these elements (i.e. changes that are still consistent with that knowledge) significantly change the representative outcomes. In this case, in what way can the simulation be said to usefully inform us about its target social phenomena?
Consider the various ways in which one could try to use a simulation in such a case:
1. To predict something about the observed outcomes from the simulation outcomes. This would depend upon being able to identify that the set-up of the simulation is the one relevant to the observed so that one can the corresponding outcome can be obtained from the simulation. However this is not possible since we do not know which set-up is the right one since they are not determinable using information from the target phenomena.
2. To explain something about the working of the mechanisms in what is observed from the unfolding of the mechanisms in the simulation. This would depend on there being a strong relation between the unfolding of the process in the simulation and that in the target phenomena. However, in this case we have no reliable reason to suppose that this is the case (even if the significant outcomes are the same). It could be that the flexibility we have in choosing the mechanism is enough to allow us to (inadvertently) ‘fit’ the outcomes.
3. To reverse engineer something about the underlying mechanism (what corresponds to the set-up in the simulation) from what outcomes are observed to occur in the phenomena. This would depend upon being able to tell that the class of mechanisms that produce a known outcome, where all other necessary parts of the simulation set-up being determined from knowledge about the target. This is possible (albeit unlikely in the extreme) if there were no parts of the necessary mechanism that were not representational and something like the envelope of possibilities left were checkable.
4. To establish the behaviour of a class of models (a sort of ersatz maths). This is easy to do in the sense that any time one runs a simulation one establishes something about the relation between set-up and outcomes is established. However to do this in a way which is at all useful to others is quite difficult. In particular, if others are to be able to use the results the necessary core (and corresponding significant aspects of the results) has to be sufficiently reliable, clear and general. Usually authors do not try to justify their simulations solely on this basis (due to its difficulty), but rather fudge the issue by also relying on some weak but vaguely defined relevance to social phenomena.
Using floating point numbers for things in simulation models is easy – the apparently simple, quick way of getting things working. It is also attractive in that there is a wide range of mathematical analytic and statistical techniques that can be brought to bear. However, numbers (and especially floating point numbers) are deceptively tricky things. They have all sorts of counterintuitive properties, which in particular circumstances can reveal themselves in model outcomes. Whilst there are technical solutions that can help avoid these traps (Polhill, Izquierdo and Gotts 2005), the only way to truly avoid such "model artefacts" is to implement (i.e. model) the processes in one’s model in a causally and descriptively adequate manner. Then any unexpected results that are obtained may be checked (at least in a rough qualitative way) with what is observed.
Thanks to all those with whom I have discussed and developed these ideas over the years, including: Scott Moss, David Hales, Juliet Rouchier, Luis Izquierdo, Gary Pohill, Guillaume Deffuant, Frederick Amblard, Rosaria Conte, Nigel Gilbert, Matt Hare, Anne van der Veen, Rodolfo de Sousa, and Olivier Barthelemy.
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 Sometimes there is specific (albeit vague, anecdotal and unreliable) evidence as to the contents of cognition based on the accounts given by subjects themselves and the interpretations of others by stakeholders and experts – evidence that has been traditionally discounted as “unscientific”.
 Clearly softer versions of this are possible, e.g. swapping “true” for “useful”, but this does not fundamentally alter the sheer implausibility of the assumption.
 Although, integers were used for these labels in the simulation, none of their arithmetic or comparative properties were used – an alphabetic system could be substituted without changing the simulation behaviour.