LogoGatherer, D. (2002). The Spread of Irrational Behaviours by Contagion: An Agent Micro-Simulation.
Journal of Memetics - Evolutionary Models of Information Transmission, 6.

The Spread of Irrational Behaviours by Contagion:
An Agent Micro-Simulation

Derek Gatherer
1 - Introduction
2 - Methods
3 - Results
3.1 - Simulation 1 - Highly contagious irrational behaviour in a highly susceptible population.
3.2 - Simulation 2 - Contagious irrational behaviour with poorer replication rates.
3.3 - Simulation 3 - Contagious irrational behaviour with possibility of loss of irrational behaviour.
3.4 - Simulation 4 - Making contagion dependent on the outcome of an event
3.5 - Simulation 5 - Introducing the `self-fulfilling prophecy'
3.6 - Simulation 6 - Introducing rational behaviour
3.7 - Simulation 7 - Giving agents the benefit of experience
3.8 - Simulation 8 - Removing successful pairs
3.9 - Simulation 9 - Removing successful pairs from a population with memory
3.10 - Simulation 10 - Introducing a communal cultural information pool influencing conversion decisions
3.11 - Simulation 11- Cultural information pool with replacement of successful pairs.
4 - Discussion


A micro-simulation is described, for rational and irrational strategies in human mating behaviour. The spread of irrational behaviour through a population from a single initial individual, the `contagionist paradigm', is shown to be highly unlikely in most realistic circumstances. An exception to this rule is shown to be where the `self-fulfilling prophecy' phenomenon is exhibited, i.e.. the irrational meme affects the outcome of the mating. Additionally, where the irrational strategy, under conditions of self-fulfilling prophecy, is allowed to co-exist with a rational strategy (i.e.. a strategy based on factual information), both can proceed to fixation, resulting in a population of individuals exhibiting both rational and irrational memes simultaneously. However, where successful pairs are removed from the population, there is a tendency for neither behaviour to persist. Maintenance of either behaviour in the population under circumstances of removal of successful pairs requires a cultural information system, i.e.. one where a common pool of information may be accessed without a requirement for contagion. This implies that contagionist explanations of culture may be strictly limited in their application. Some attempt is then made to generalise the conclusions to financial systems.

1 Introduction

Human behaviour, at both the individual and social levels, is often apparently inexplicable. By contrast, the behaviour of software agents is, by definition, determined by their programmer, at the individual level. Nevertheless, at the social level, collections of software agents may behave as surprisingly as their real world counterparts. The appeal of agent-based social simulation is that one may design agents, of varying degrees of simplicity, that, when allowed to interact with each other, reproduce some of the identifiable behaviour of real human social systems. Some of the more puzzling aspects of human social systems may then be illuminated in the light of the computer simulation.

This paper investigates an agent population exhibiting two kinds of behaviour:

  1. rational - defined as behaviour based on factual information, that maximizes the long-term material welfare of the agent, and
  2. irrational - defined as behaviour based on incorrect premises, that is at best neutral to that agent's interests, and in some cases potentially detrimental.
In a simulation, it may be decided with certainty what constitutes `factual information' and the best interests of the agent. In real life, of course, these can be more difficult to determine with any reliability. It should also be stated that rationality in real organisms is a complex function that requires sophisticated cognitive functions. The agents described here have no AI component, but merely a series of rule-based functions. A considerable literature exists on the circumstances in which maladaptive, or apparently maladaptive, traits may become prevalent in human populations (e.g.. Boyd & Richerson 1985, Durham 1991, Blackmore 1999). In this paper, the behaviours designated `rational' and `irrational' do correspond in many ways to `adaptive' and `maladaptive', respectively. However, the terminology `rational' and `irrational' is preferred since the `irrational' behaviour in this simulation may coincidentally act in the agent's interests, and therefore is not always maladaptive.

These two kinds of behaviour are replicated in 2 ways:

  1. contagion - where behaviour can only be copied directly from one agent to another via social contact (simple `echo' contagion of the behavioural variety - Levy & Nail 1993; those authors' requirement that there should be non-intentionality is not relevant to software objects).
  2. via a common pool of cultural information - an agent can select a behaviour from a set of social norms, without coming into direct contact with another individual exhibiting that behaviour.
The essential concern of this paper is related to that of Doran (1998), in that an effort is made to produce mass behaviour arising from a set of simple behavioural rules. Another methodological inspiration is Ophir (1998), in that pairing events within the agent population are followed by the execution of a variety of calculations affecting the subsequent state of the agents. The agents are reflex agents, as defined by Doran (1998), in that they have attributes which are modifiable according to a set of `condition-action rules', and also trigger actions that modify the attributes of other agents. The model presented is, in Edmonds' (1998) terms, an `abstract, forward' model, in that the interest lies in what kinds of unpredicted group phenomena may emerge out of strictly regimented individual behaviours.

The system simulated is that of human mate pairing. It is clear from biological studies that many species are very selective in mate choice, and that there may also be vigorous competition between individuals for mates. As Darwin (1872) recognised, these two factors have resulted in the phenomenon of sexual selection for a wide variety of traits in species ranging from insects to mammals. In some species, mate choice behaviour also has a cultural dimension which may amplify, or conflict with, any underlying innate biological tendencies (Wade & Pruett-Jones 1990; Laland 1994; Freeberg et al 1999). The simulation presented here attempts to dissect how a `rational' mating behaviour, arbitrarily taken to be selection of a mate of same status as oneself, conflicts with an `irrational' mating behaviour, for instance the use of astrology to select a mate. Note that such `astrological' behaviour need not necessarily always be maladaptive, since a good mate may coincidentally be chosen. It is, however, always `irrational' regardless of its adaptiveness. Although astrology is apparently believed by 25% of US adults (Blackmore 1999, p184), it may not necessarily figure very prominently in real human partner choice (but perhaps in other aspects of reproduction, see Goodkind 1993). However, the example is given merely to contrast irrational and rational memes in a simple contagion-based agent-pair interaction system. Generally similar dynamics may occur wherever agents are exchanging information repeatedly in a closed system. Additionally, it should be pointed out that the mated pairs are not engaged in reproduction, or even necessarily in any physical copulation, but rather in partner choice for potential reproduction. In other words, this model takes place within a single generation. Moreover, no genetic components are envisaged, and the behaviours are taken to be totally cultural. Finally, the population is not spatial segmented, and any individual can potentially mate with any other of the opposite sex. Ways in which the model could be expanded to incorporate a combination of genetics, multiple generations, and spatial heterogeneity are considered in the discussion.

2 Methods

The agents have the following attributes:
  1. Gender - male or female
  2. Status - high or low
  3. `Star sign' - A or B
  4. Seek status information, i.e.. act `rationally' - yes or no
  5. Seek astrology information, i.e. act `irrationally' - yes or no
The status and astrology attributes (4 and 5) are thus different memes and not mutually exclusive `allomemes' (Durham 1991). With 5 binary attributes, there are 32 different potential types of agent in the population, that is 16 different types for each gender. Since all pairings in all simulations are between members of opposite genders, there are 120 possible kinds of pairings, i.e. n(n-1)/2, for which rules need to be constructed. Some of the preliminary simulations only use a subset of the attributes and have very simple rules. The patience of the reader is requested for the apparent triviality of the first few simulations, as they are designed to clarify the basic dynamics of the system. From simulation 3 onwards, the system becomes increasingly complicated, as more attributes and complex rules of interaction are progressively applied.

Gender, status and `star sign' (attributes 1 -3) are considered fixed for any individual and cannot be changed. The behavioural states (attributes 4 and 5) relating to status and astrology may be changed under certain circumstances, representing a behavioural change in that agent. A `yes' state for astrology means that the individual will seek to ascertain the star sign of the other individual. Likewise a `yes' state for status results in behaviour to determine the status of the potential partner. The status-`yes' state is `rational' in that status objectively influences the likelihood of success of the pairing (in the simulation, whether or not is does so in real life is quite another matter - see Brown et al 1988; Pizzari 2001). By contrast, the astrology-`yes' state is `irrational' in that star sign has no objective effect on success (see Goodkind 1993, for an empirical example of how astrology may influence human reproductive behaviour).

Agents seek a mate at random from among the opposite sex in the population (as in Ophir 1998). The outcome of the mating (successful or unsuccessful) is determined by a combination of random chance, a background a priori likelihood of success, and various rules dependent on the types of agent involved. Having mated every individual in the population, the cycle repeats. Each iteration, the proportions of the population exhibiting the enquiry behaviours, i.e.. the `yes' states, for status (rational) and star sign (irrational) are recorded. In some later simulations, successfully mated pairs are withdrawn from the mating pool and replaced with naïve individuals, as would be the case if mated pairs were to concentrate on reproduction and cease to look for further partners

Each simulation is produced using Perl scripts which, along with the full list of rules used in the more advanced simulations, are available from http://www.geocities.com/derek_gatherer/supp.htm.

3 Results

3.1 Simulation 1 - Highly contagious irrational behaviour in a highly susceptible population.

The population is initialised with an equal number of individuals of each gender, each of which has randomly assigned status and star signs. Initially, all individuals are in the `no' state for both behaviours. In this first simulation, a single individual is released into the population, exhibiting the `yes' state for the irrational behaviour. A single simple rule is made that individuals only convert from `no' to `yes', and never in the reverse direction (in other words, this is the `biased cultural transmission' of Takahasi 1998, 1999; or the `cultural selection' of Cavalli-Sforza & Feldman 1981). As might be expected, under such a rule, the `irrational' behaviour spreads rapidly throughout the population, essentially doubling in frequency for the first six cycles of pairing. After that point, its rate of increase slows a little as `yes' individuals become more likely to encounter prospective partners who are already `yes' in phenotype. In 3 simulations conducted in a population of 500 individuals, the entire population exhibits the irrational behaviour by cycle 12 in all 3 cases (data not shown). The dynamics are precisely the same as those of a highly infectious, incurable, sexually transmitted disease in a promiscuous population with no immunity. The total takeover of a population by a behaviour is called `fixation', borrowed from population genetics - the opposite state being more obviously termed `extinction'. The sigmoid curve of uncontrollable epidemic spread from a single individual to encompass the entire population is referred to here as the `contagionist paradigm', and has been a regular feature of models of social processes at least as far back as Rashevsky (1949).

3.2 Simulation 2 - Contagious irrational behaviour with poorer replication rates.

Let the irrational behaviour now depend on an element of chance in its replication. Once again the population is seeded with a single individual exhibiting the irrational behaviour. With lower chances of replication at any particular pairing event, the progress to fixation is delayed. However, since an individual, once exhibiting the irrational behaviour, is permanently in that state, eventual fixation is inevitable, even when transmission rates are very low.

Figure 1: Spread of a contagious trait introduced by a single agent into a population of 500 agents, at differing values of probability of transmission per contact, p. Where p = 1, i.e.. the trait is always transmitted, it approximately doubles every iteration, and a population of 500 is almost always fixed by generation 12. These S-shaped curves represent the most trivial case of contagion, and are presented as a reference point for the curves in the subsequent simulations.

Although these curves are sometimes fairly flat, their shape is still sigmoid, and the `contagionist paradigm' still applies. The only variable is whether the contagion is slow or fast. For instance in Fig.1, for p = 0.1, i.e. 10% chance of transmission per contact, it still takes 45 iterations for 20% of the population to have been infected. Thus, for cases of `biased transmission'/ `cultural selection', the contagiousness of a trait is not important to the eventual outcome, ceteris paribus. It is however, precisely this ceteris paribus assumption that is violated in all but the most trivial cases.

3.3 Simulation 3 - Contagious irrational behaviour with possibility of loss of irrational behaviour.

Let the `no' state for the irrational behaviour now also be transmissible. This means that in any pairing event, it is possible that individuals exhibiting the irrational behaviour may be converted to the rational behaviour. Here, the analogy with an infectious disease begins to break down, as contact between two individuals is as likely to `cure' the one as `infect' the other, at p = 0.5. A better analogy is to the random drift of a rare neutral allele frequency in a genetic system (Kimura & Ohta 1978, Kimura 1979). It is clear that under such circumstances, with two mutually exclusive behaviours vying for predominance in the population, the eventual victor will be the one with the greater tendency to convert the other. This is merely a confirmation of intuition, but it is perhaps less intuitive that a corollary of this is that, at p = 0.5, about half of novel contagious behaviours will face immediate extinction in the first generation. In order to illustrate the `random walk' tendencies of the population under such circumstances, the simulation is started with equal numbers of `yes' and `no' individuals. The conversion likelihood is equal in both directions, and 1000 iterations are used. Of 5 populations of 500 individuals beginning with 50% in each behavioural category, after 1000 iterations, 3 of the 5 populations have become fixed for the irrational behaviour and 2 have lost the irrational behaviour entirely.

Figure 2: 5 runs of a simulation in which the likelihood of conversion to a trait and conversion away from a trait are equal in each contact. In a population of 500, each starting with 250 individuals exhibiting the trait, 3 populations become fixed for the trait (blue, brown and turquoise) and the trait becomes extinct in 2 (green and pink). These are the most trivial examples of random walks, and like the S-shaped curves in Figure 1 are given for the purposes of reference.

Random walks are totally random, by definition, but may appear directional. Take for example the pink population in Fig. 2. After a period of quasi-periodic movement upwards, it suddenly turns and declines to zero. As this is an agent simulation, it is known that this is simply a random walk. However, an empirical worker approaching evidence in the other direction, might easily conclude, wrongly, that there must be non-random influences on the trend. In a random walk, at p = 0.5 as given here, any trajectory is equally likely and the odds of any trajectory occurring are 0.5n (i.e.. pn) for any particular trajectory of n steps. Since n is 1000 in simulation 3, there are approximately 10301 different possible trajectories, all equally likely, and each of which has an approximate probability of 10-301. Over an infinite number of n, all trajectories will eventually terminate in fixation or extinction, with equal probability (Kimura & Ohta 1978, Kimura 1979).

The sensitivity of such a situation to variation in p may be illustrated by considering what happens when p deviates from 0.5. Effectively, even small deviations away from 0.5 will pull the random walk rapidly to fixation or extinction.

Figure 2b: 5 runs of a simulation in which the likelihood of conversion to a trait and conversion away from a trait are equal in each contact. Again a population of 500 is used, starting with 250 individuals exhibiting the trait. Varying values of p, the probability of conversion to the trait, are given.

The rapidity of fixation and extinction is almost too fast to see over a timescale of 1000 iterations in Fig. 2b, so the early stages of the 5 simulations are shown again in Fig. 2c. Where p <= 0.4 the trait barely survives 20 generations.

Figure 2c: The early stages of the simulations given in Fig. 2b above.

In general, the probability of a random walk, where p is not equal to 0.5 are pn+(1-p)m, where n is the number of incremental turns and m is the number of decremental turns. Therefore where p is not equal to 0.5, not all walks are equally probable, and the walks are quasi-random. This much is trivial, once again, but illustrates how easily cultural traits may exhibit chaotic dynamics, especially if p is variable from generation to generation (dependent perhaps on other cultural factors). Under such circumstances, a predictive memetics would become virtually impossible, as cultural traits would swing wildly in frequency with apparently no rationale (perhaps a familiar scenario for empirical observers of culture). However, the situation does become more predictable as other factors are added to the simulation.

3.4 Simulation 4 - Making contagion dependent on the outcome of an event

So far, the irrational behaviour considered has been purely (and arbitrarily) contagious. Now a complication is introduced in that propagation of the irrational contagious behaviour depends on the perceived success of the mating by the pair involved. As described above, the population is divided into two arbitrary star signs. These are random labels with no function other than to divide the population into two random groups. Therefore any behaviour based on such divisions is by definition irrational, for example, the discussion of star sign with prospective partner. See Lindeman (1998) and Delfabbro & Winefield (2000), for some real life examples of irrationality in event interpretation.

Let the outcome of pairings in each iteration be recorded. Let also the chance of success in each pairing be s (0 <= s <=1), the a priori success probability. A random number generator is used to generate a number between 0 and 1. If this random number is less than s, the pairing is deemed to have been successful. The following rules then apply:

  1. If an `astrological' individual pairs with a `non-astrological' individual, and they are of divergent star signs, and the pairing is unsuccessful, the `non-astrological' individual will convert to `astrological' (because that is what the irrational hypothesis predicts, and spurious `evidence' has apparently been provided).
  2. If an `astrological' individual pairs with a `non-astrological' individual, and they are of divergent star signs, but the pairing is successful, the converse will ensue, i.e.. the `astrological' individual will convert to `non-astrological' (because the irrational hypothesis has been falsified in this instance).
  3. If an `astrological' individual pairs with a `non-astrological' individual, and they are of compatible star signs, and the pairing is successful, the `non-astrological' individual will convert to `astrological' (again because the irrational hypothesis has been spuriously `verified').
  4. If an `astrological' individual pairs with a `non-astrological' individual, and they are of compatible star signs, but the pairing is unsuccessful, the `astrological' individual will convert to `non-astrological' (because the irrational hypothesis has been falsified in this instance).
In other words, the outcome of the event will either be as predicted by the irrational individual or will not be so. The success or failure of the prediction determines the likelihood that one or the other individual will be converted to the behaviour pattern of the other. Note that pairings between two irrational, or two rational, individuals are not considered, as contact is required with an individual of the opposite behavioural type in order to produce behavioural modification. This is what is meant by a purely `contagionist' mechanism, since contact between individuals is required for `transmission' of behaviour.

Under such circumstances, it is again, as in simulation 3 above, virtually impossible for the irrational behaviour to colonise the population from a single individual. That is because the rules above are balanced, i.e.. the number of events causing a conversion in one direction is equal to the number of events causing conversion in the opposite direction (rules 1 & 3 versus 2 & 4).

As in the previous simulation, 5 populations are allowed to run for 1000 iterations with the initial frequency of the irrational behaviour at 50%. The chance of success for each pairing is also set at 50%.

Figure 3: 5 populations of 500 individuals, with initially 250 individuals exhibiting the irrational behaviour, and the transmission of the behaviour dependent on the outcome of pairings. Although the dynamics of this simulation are strictly rule-based, and those of simulation 3 dependent on binomial chance, the population trajectories produced by the two systems are indistinguishable.

3.5 Simulation 5 - Introducing the `self-fulfilling prophecy'

The above example demonstrates that an irrational behaviour, even when based on `evidence', is likely to follow a random walk within a population. Such a randomly walking trait is very unlikely to be able to colonise a population if there is initially only a single individual manifesting it. However, in simulation 3 above, it is assumed that the success or otherwise of every pairing was due to random chance. On the basis of this random chance, various rules determined the tendency of individuals to convert from one behaviour to the other. Let it now be imagined that the irrational behaviour affects the outcome of the mating (see Downey et al 1998 for a real-life example). The rules remain the same, but likelihood of success of a pairing is not equal for all pairs. If those individuals enquiring about the star sign of a prospective partner are less likely to have a successful pairing with individuals of incompatible star signs, and more likely to have successful pairing with those of compatible star signs, then this constitutes a `self-fulfilling prophecy'.

Bias is defined as the extent to which astrology will interfere with the potential success of a relationship on a scale of 0 to 1, where the pair have incongruent star signs, and one or both of the pair exhibit the astrology behaviour. This is programmed in the following manner. The a priori likelihood of success of a relationship is s, and bias is b, both are between zero and 1, and both are set in advance. A random number generator is again used to generate a number between zero and 1. Without bias, if that random number is less than s, the pairing is successful. With bias, b is subtracted from s. This reduces the likelihood that the random number will be less than s, and thereby `loads the dice' against success. Where the pair have congruent star signs, and one or both of the pair exhibit the astrology behaviour, bias acts in the other direction, to increase the chance of success. This is programmed by adding b to s, rather than subtracting it as above, and therefore increasing the likelihood of success. Where b-s is less than zero, the relationship is doomed to failure, regardless of the number produced by the random number generator. Likewise where b+s is greater than one, the relationship is guaranteed to succeed.

Where b = 0.5, populations run to fixation fairly smoothly for most values of s, provided the novel irrational meme can survive the first few iterations. This `self-fulfilling prophecy' thus changes the irrational behaviour from a randomly walking trait into a colonizing one. The `contagionist paradigm' of steady sigmoid curves, reappears.

Figure 4: An irrational trait originating in a single individual, colonising a population of 500, running to fixation in under 25 generations in 3 separate runs (red, green and blue). 5 runs of this simulation were performed, of which the other 2 resulted in immediate extinction of the trait at the first iteration. This is clearly an S-shaped curve, demonstrating that, once the novel meme has survived the first few iterations, the `self-fulfilling prophecy' mechanism turns a random walk into an irresistible tendency. Parameters were s=0.5, b=0.3.

3.6 Simulation 6 - Introducing rational behaviour

Up to now, only the presence or absence of the irrational astrology-based behaviour has been considered. Now, let individuals also assess potential partners according to the rational criterion of `status'. It is unimportant how this is defined, except that it is necessary that this be a genuine factor contributing to the success of pairings. Whereas the success of pairings of partners of differing or compatible astrological sign was purely a function of chance alone, or chance coupled to the `self-fulfilling prophecy' phenomenon, the success of pairings of different status groups has a fixed contribution to overall pairing success, independent of star sign or any behaviours. Greater pairing success for compatible status groups is therefore objectively true. That is why the status behaviour is regarded as rational.

Status effects can co-exist with irrational bias effects. In the interaction of the 2 strategies, the following general rules apply. The full set of precise rules are available on http://www.geocities.com/derek_gatherer/supp.htm.

  1. The status of paired agents is compared.
  2. The star sign of paired agents is compared.
  3. The rational and irrational behaviours for the paired agents are compared.
  4. If status is compatible, a priori likelihood of success is doubled (irrespective of whether or not either agent is rational)
  5. If at least one agent is irrational, the `self-fulfilling prophecy' phenomenon interferes with the a priori likelihood of success up or down, accordingly.
  6. The success of the pairing is determined, using the random number generator.
  7. Based on success or failure, the agent states for the rational and irrational behaviours are altered.
  8. However, behaviours only change if an agent is paired with an agent exhibiting the appropriate behaviour (contagion still rigidly applied).
These rules allows both rational behaviour (relating to status) and irrational behaviour (relating to star sign) to enter the population. The two are not mutually exclusive, and therefore separate memes rather than `allomemes' (Durham 1991). Which meme propagates faster depends on the relative degrees of bias (the motor of the self-fulfilling prophecy phenomenon for irrational behaviour) and objective status-related increase in the likelihood of successful pairing. Indeed where a priori likelihood of success, s, is low and bias, b, is high, the rational behaviour tends to extinction.

Figure 5: Progress to fixation or decline to extinction of rational behaviour under varying conditions of `self-fulfilling prophecy' bias, b, from 0.6 to 1, with an a priori likelihood of success of pairings of s = 0.5. In all runs, the irrational behaviour ran to fixation in 10 iterations.

3.7 Simulation 7 - Giving agents the benefit of experience

The previous simulations 3 to 6, have assumed that conversion events are dependent on the success of immediately preceding pairings. Perhaps more realistically in terms of human society, individuals will withhold judgement until several experiences have accumulated. Let the agents now have a threshold, t. Experiences liable to cause conversion to rationality or irrationality are accumulated, and the appropriate conversion takes place when the threshold is exceeded. The experience counter is then reset to zero. In other words, at a threshold of, say, 3, an agent will convert to rational behaviour only after 3 appropriate contributory experiences.

This decreases the volatility of the system, and promotes the settling of the population settles into a stable state. The equilibrium position is determined by the relative values of s and b, and the speed with which it is obtained is dependent on the threshold for conversion, t.

Figure 6: Progress of rational behaviour in the population under the same conditions as in figure 5 above (s constant at 0.5 and b run at 0.5 to 1), but with a threshold of 3 introduced. This delays fixation of rational behaviour under conditions of low bias, but also prevents its immediate extinction under conditions of high bias.

3.8 Simulation 8 - Removing successful pairs

All the previous simulations recycle the same individuals in a vulgar imitation of Schnitzler (1982). If successful pairs no longer seek further mates, as seems more realistic in human terms, let successful pairs now be replaced in the population by naïve individuals. Note that this is done based on simulation 6 - with possibility of individual expressing both rational and irrational attitudes, and without allowing any accounting of previous experience (naïve individuals have, by definition, no experience).

Where the a priori likelihood of success, s, is high, individuals are removed so rapidly from the population that neither the rational nor irrational behaviour is very persistent. Lowering s to 0.01, and with high self-fulfilling prophecy bias, b, it is possible to retain them for some longer time. However, the influx of new individuals wins out eventually.

Figure 7: Replacement of successful pairs with naïve individuals rapidly removes both behaviours, rational and irrational, from the population. Compare with Figure 4. The same parameters are used, s = 0.5, b = 0.3, which would normally generate a powerful "self-fulfilling prophecy" for the fixation of irrational behaviour. However, here the arrival of waves of naïve agents overwhelms the capacity of the contagion system.

3.9 Simulation 9 - Removing successful pairs from a population with memory

Simulation 7 showed that allowing agents the benefit of experience before changing strategy, caused changes to be subdued, both up and down. Simulation 8 showed that removal of successful pairs extinguishes both behaviours. This simulation combines 7 and 8 to see if memory can maintain the behaviours in such a population. However, the dynamics are the same as in simulation 8 (data not shown). This is only to be expected, as when individuals rely on their own experience, but cannot share it with the rest of the population, that experience is no longer available when the individual leaves the population.

3.10 Simulation 10 - Introducing a communal cultural information pool influencing conversion decisions

All previous simulations 1 to 9 have only allowed individuals to change their behaviour based on the behaviour of individuals with whom they come in contact. These models have thus been strictly contagionist models. Let the individuals now draw on a communal store of previous experience. This communal reference is built by recording the successes of pairings of different types. This is slightly different to simulation 8, in that there individuals were allowed to make decisions on the basis of their own experience only. Now recourse is permitted to knowledge of the experiences of other individuals. Individuals can convert without contact with another individual, if the cultural assessment is that rational or irrational behaviour is correct. This is similar, but not exactly identical to, the `conformist transmission' of Henrich & Boyd (1998).

Thus rules 7 and 8 from simulation 6, above, are now altered to:

7.  Based on success or failure, a contribution to the communal information pool is made. The agent states for the rational and irrational behaviours are still altered in the usual way if an agent is paired with an agent exhibiting the appropriate behaviour.
8.  But now additionally, behaviours can change even if an agent is not paired with an agent exhibiting the appropriate behaviour, subject to consulting the communal information pool for the latest consensus.
Thus the communal information pool does not replace contagion, but supplements it. This simulates an element of `persuasion'. Agents still convert under the influence of other agents, both rationally and irrationally, if conditions are correct. However, where no persuasion is exercised, agents can still alter their behaviour if conditions are correct, and the cultural norm suggests they should. (Perhaps this violates Levy & Nail (1993)'s requirement for an absence of intentionality in a true contagion system, but I am unsure of what intentionality means in terms of software agents.)

This helps to maintain steady random walks very near the starting values, after an initial movement upwards or downwards (dependent on starting parameters) as the contagion effect begins, before the communal information pool helps to correct it.

Figure 8: Non-contagion-dependent cultural system. The self-fulfilling prophecy maintains some degree of irrationality, but the ability of agents to convert away from irrationality without contact with a rational agent prevents it from running to fixation. Again, s=0.5, b=0.3.

Note that in Fig.8 the starting parameters are s=0.5, b=0.3. By reference to Fig. 5, it can be seen that these would normally cause both memes to run to fixation. This is indeed the initial tendency (note the initial rapid upsurge in both blue and red lines), but the communal information then begins to pull the meme frequencies down towards a quasi-random walk around an equilibrium level.

3.11 Simulation 11- Cultural information pool with replacement of successful pairs.

This is a combination of simulation 10 with simulation 8. In simulation 8 both behaviours were lost as naïve individuals took over the system. However, with the common cultural information pool, the behaviours persist.

Figure 9: Same conditions as simulation 10, but allowing replacement of successful pairs. The initial dip is caused by the arrival of the first set of naïve agents in the first cycle or so before the cultural system has adequate data concerning the likely success of various pairs.

As in simulation 10, the cultural pool requires a few iterations to establish its influence, this time after a very short initial spell (almost invisible on Fig. 9) where the downward tendency of simulation 8 is repeated. This precipitate drop recovers almost immediately as the common cultural pool begins to exert its influence.

4 Discussion

The main conclusions of the paper are that:
  1. the `contagionist paradigm', i.e.. the common sigmoid curve representation of a memetic epidemic, is a very special case, requiring a fairly contrived set of system parameters in order to be produced.
  2. the `random walk', i.e.. an apparently stochastic meandering of meme frequency over time, is the more likely situation, even when the underlying parameters are far from random - a frequent variation is a pseudo-random walk around an equilibrium level.
  3. a population with a high turnover of agents (and hence a high proportion of naïve agents at any one given time) cannot maintain either of the described behaviours, rational or irrational, i.e.. those meme frequencies drop to zero, without recourse to the use of a cultural information pool.
Many additional variables could be added to the simulation. For instance, Ophir (1998) gives his agents age attributes, mates them at age 30, and has them expire at a random age between 60 and 100, after producing a single family of progeny. This allows discrete iterations to be replaced with more realistic overlapping generations. However, Ophir's agents are monogamous, and reproduce only once, the emphasis of Ophir's work being the distribution of economic resources rather than mate pairing behaviour. Also, Pearson & Boudarel (2001) have recently devised a matrix based system for agent pair interactions, which could be incorporated into the pairing process described here.

The strictly contagionist person-to-person replication of both the rational and irrational behaviour is shown by simulation 8 to be unstable in the face of removal of individuals from the population. Indeed, even when all individuals are retained in the population, irrational behaviour requires the `self-fulfilling prophecy' phenomenon to spread through the population, as illustrated in the difference between simulations 4 and 5. Self-fulfilling prophecies have been empirically demonstrated in human mate pairing (involving `rejection-sensitivity' rather than astrology, but the basic principle is the same - see Downey et al 1998).

The only way to maintain the irrational behaviour in the population in the face of high levels of replacement is to use some kind of common information pool system. This, however, requires that the agents choose to refer to the information system for guidance. If, for some reason, the population in simulation 11 neglects to maintain the information system, the dynamics of simulation 8 will begin to take over and both memes will slide towards extinction.

It is clear that a `random walk' pattern of meme incidence over the iterations is more common than the `contagionist paradigm' sigmoid curve, which only occurs in rather `pure' situations. Random walk-like effects may be genuinely random because of the stochastic nature of the system (simulation 3), because of balanced rules in a deterministic system (simulation 4), or be anchored pseudo-random walks because of fluctuations around an equilibrium (simulations 10 and 11). Good `epidemiological' sigmoid curves in culture seem to require either arbitrary contagiousness (simulations 1 and 2), or a powerful selective force (the unopposed self-fulfilling prophecy of simulation 5). Some aspects of human life may be arbitrarily contagious, for example crowd hysteria (see reviews by Levy & Nail 1993; Marsden 1998). Several authors have worked within the `contagionist paradigm', producing models of traits that spread in an epidemic manner. For instance, Takahasi (1998, 1999) refers to `biased cultural transmission', Cavalli-Sforza and Feldman (1984) to `cultural selection', and Sperber (1985, 1996) to the `epidemiology of representations'. In these cases, it is taken for granted that the epidemic trait is either arbitrarily contagious or, in the case of Cavalli-Sforza and Feldman's `cultural selection', that some powerful, and usually unspecified, psychological factor is promoting the spread of the trait. The self-fulfilling prophecy phenomenon of simulation 5 would presumably therefore be a case of `cultural selection'. Another example might be `prestige-based transmission' (Heinrich 2001) where certain individuals are far more widely copied than others, or `emotional selection' (Heath et al 2001). Blackmore (1999) gives extensive further examples. Cavalli-Sforza and Feldman (1984) also refer to `natural selection' on culture, where a cultural trait spreads through genuine adaptiveness in the absence of either arbitrary `biased cultural transmission' or psychological `cultural selection'. Powerful natural selective forces of this kind do indeed also operate in culture, resulting in some sigmoid incidence curves for possession of, in modern times, mobile phones and other useful paraphernalia (Rogers 1995), or, in the Bronze Age, for bronze knives (Renfrew 1987). Hewlett and Cavalli-Sforza (1983) describe an epidemic spread of the use of the crossbow among Pygmies, conferring a great hunting advantage over the traditional bow and arrow.

Granted that cultural epidemics can, in theory, occur where there is natural or cultural selection (and certainly do occur for cases of natural selection), the question here is: to what extent could the contagionist paradigm be applied to (non-hysterical) irrationality? If most irrationality is disadvantageous (and therefore unlikely to be `naturally selected', sensu Cavalli-Sforza & Feldman, as an adaptation), and it is not the result of hysteria (and therefore not likely to have greatly biased transmission over a short time period), and the human society concerned has ample recourse to a pool of experience of previous members, then the spread of irrationality in an epidemic manner would seem to be unlikely. The contagionist paradigm therefore seems to be an inadequate model for the long term persistence of irrational behaviour in human populations.

For instance, in a stock market system, agents could predict the future price of shares, and then bid for them irrationally (perhaps again based on astrology) or rationally (based on profit indicators). If the purchase of shares causes their price to be elevated, this constitutes a kind of `self-fulfilling prophecy' phenomenon. This would tend to result in irrational bidding spreading throughout the entire investment community. The fact that this does not happen too often, is perhaps due to there being a well established communal information pool about the success of various previous investments, on which current investment decisions can be based. As in simulation 10, this would prevent even a strong `self-fulfilling prophecy' of irrational share inflation from taking over entirely. Interestingly, Caginalp et al (2001) have identified lack of open factual information as a contributory element in `bubbles' created during lab simulations of stock market trading. Such a situation would promote information exchange on a person to person basis and therefore lead to a greater susceptibility to contagion effects. Study of real markets has also led to the conclusion that the "level and nature of information available to dealers, and social communication networks" are important (reviewed in Marsden 1998, section 2). Empirical data shows that random walking does tend to predominate over long periods of time (Malkiel 1985). It is also interesting that although the common information pool helps to prevent irrationality running to fixation, it does not eliminate it. Just as evolution in financial markets does not lead to maximal market efficiency (Frank 1999), evolution in the mating game, or in any other area of human conduct, does not lead to maximal rationality. Heinrich (2001b) presents an interesting example of the persistence of bottle-feeding in parts of the world where breast-feeding would result in lower infant mortality. One might also cite Rogers and Shoemaker's classic example of the determination of Peruvian villagers not to boil drinking water despite extensive health education and the established use of boiling for other food purposes (Rogers & Shoemaker 1972), or recent discussions of `altruistic punishment' (Fehr & Gachter 2002) and the revival of the `tragedy of the commons' debate (Ostrom et al. 1999, Henrich et al 2001a). Henrich (2001b) finds that the persistence of maladaptive behaviour is most likely when: a) there is high probability that environmental information is inconclusive, and b) there is biased transmission of the maladaptive trait. This is entirely consonant with the simulations presented here. Interestingly and rather controversially, Henrich goes a step further and suggests that even adaptive traits require biased transmission in order to produce S-shaped curves. This is a radical departure from the fundamental theoretical observation that natural selection of an advantageous trait is analogous in form to an epidemic, and produces an S-shaped curve (Fisher 1930). It is not appropriate to discuss this any further here other than to note that the present observations concerning the cultural information pool and the relative rarity of situations in which the contagionist paradigm applies, are consonant with the first part of Henrich's (2001b) thesis.

One point not covered in the simulations is the issue of spatial heterogeneity within the population. The simulations could be modified to give each individual a geographical location. Partner choice, and the extent of the cultural information pool, could thus be limited by geographical isolation. One might also permit the high status individuals greater geographical mobility than those of the low status. Multi-generational effects might allow some of the successfully mated individuals to produce offspring for reintroduction into the population at a later date. Incorporation of a genetic component might also be permitted, with some individuals being inherently more or less sceptical than others.

In summary, the contagionist paradigm is a special case scenario, only applicable to situations of:

  1. hysteria (Levy and Nail 1993),
  2. strong psychological distortion of transmission mechanisms (Cavalli-Sforza & Feldman 1981; Takahasi 1998, 1999), or
  3. powerful natural selection of an advantageous cultural trait (Cavalli-Sforza & Feldman 1981).
According to Henrich (2001b), the last of these three may also be suspect unless 2) also applies. Since 1) is rare, and 2) is often difficult to identify empirically (and in the worst cases is often no more than a post hoc explanation when no other can be found), it is submitted that contagionism has limited range as an explanatory tool of human culture.


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The author thanks the staff of the Sid Martin BDI, Alachua, and the Univ. of Florida Library for their kind hospitality and assistance during his stay.
© JoM-EMIT 2002

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