I wonder whether attachment styles represent local equilibria? By this I mean that attachment styles might capture behavioural strategies that work in local contexts, or specific ecologies. As different relationships will have different utilities for individuals you might expect some flexibility such that styles can change.
A quick look-up on attachment styles and found a paper by Oskis and colleagues in which the following styles were listed:
Avoidant - which relates to mistrust, constraints on closeness, fear of rejection, high self reliance and anger.
Anxious - which relates to high desire for company and a fear of separation.
Caregiver support - which relates to active emotional support and felt attachment.
This is enough to enable a simplifying assumption, which in turn will help explore formal possibilities.
So, let's assume two attachment styles – Avoidant and Anxious. One way to consider these strategies is as points on a continuum of social stickiness. We can assume that social interactions are viscous, such that individuals stick together, and that the viscosity of a given group will vary as a function of individual stickiness. Thus, an Avoidant individual is low on stickiness (slippery) whilst an Anxious individual is high on stickiness. This could be mathematically represented on a scale from 0-1, where 1 is Anxious. However it might be that stickiness is variable within an individual, such that the stickiness quotient will range between 0 and 1 as a function of some other parameter. For now, this other parameter can be regarded as extrinsic to the individual and as pertinent to a single utility that the individual is seeking to maximize. We might denote this property as follows:
Where P is an individual person, s is their stickiness quotient, on a scale of 0-1. This compound is a little informal – I doubt we would need the numerical value in any subsequent algebraic equations – but I hope it is clear what is meant.
At this point we can either think of s values as fixed within an individual or as changeable. If the former then we should be interested in how many of each value are present in the population, and how interactions between each type will play out. If the latter, then we can see s as a conditional trait that responds to external circumstances. The experience of external circumstances will lead to variable ratios of s values in the population, but s values themselves will be external circumstance for others.
From the descriptions above, we might hypothesize that a key utility would be cooperation. In other words as s reduced in value, an individual would be less likely to engage in a cooperative interaction with another individual. The most likely reason for this is because low stickiness (slippery people) would reduce the encounter rate with other individuals. A simple agent-based model might have it that any social contact would lead to a productive social exchange, maximizing both agents’ utilities. I would model this as fitness, perhaps using energy gain as an alternative currency, and then having an energy threshold above which asexual reproduction could occur, producing one copy – the game would be to acquire enough energy for reproduction through social encounters. This would have to be contrasted with a solitary strategy in which agents could still maximize utility through their own actions. For this we would need some form of individual behaviour that acquired energy. One point of interest would then be the differential rates of energy gain as a consequence of the value of individual stickiness quotients.
Patch exploitation gathers energy and the game could be that cooperative action better exploits patches. A ratio could be established between the number of individuals working on a patch and the individual yield (rather like a public goods game with no free riders at this point). The shape of that pay-off curve would also impact on s values.
We have a basic model. But a psychologist might argue that our Anxious end point for s is actually better described as Securely Attached. This is because s=1 leads to total stickiness and cooperative association. Whilst this is a desire of the Anxious typology, there is also a fear of separation that might be costly. This could be modelled in a number of ways. For example, we might build in a parameter for the amount of social interactions. It could benefit an individual to have a number of social associations to maximize fitness. High s will allow social cohesion, but we also want an assortment variable, a, which will vary from 0-1, where 1 is even assortment throughout the group and 0 is association with <1 other individual. Thus if s is high and a low, the individual is eager to attach, but relentlessly attaches to a few people – sticking to only them, and not risking other social interaction. Individuals are now thus characterized:
Again, a could be dependent upon extrinsic variables. For example, if the risk of defection within a cooperative interaction was high, it might pay to reduce a and engage in close work with a few individuals. Those individuals could be chosen for their cooperative behaviours, or they might be cultivated through extended tit-for-tat interactions. The intuition here is that if P s,a is high on both variables, thus sticky and seeking maximum assortment, the population size will play a crucial role. A large population size, N, will lead to an increased possibility of one-shot interactions between individuals, which will stabilize defection, not cooperation. Technically, we might say that a closed population of any size, interacting over many cycles but with an unknown limit to those interactions, will lead to cooperation through tit-for-tat. This is the w parameter in standard game theory – which is a discount function. But a more realistic model would have to include cognitive constraints, such as the ability to remember interactions and individuals. (This might lead to an empirical question around attachment style, social network size and structure and social cognition – a regression analysis of data on these facts should be revealing.) Memory, or rather forgetting, could be modelled probabilistically such that at each interaction an individual takes a bet on the value of w. I suspect this is what we do in fact, and our values are influenced by dispositions such as s and a, but also by social structures. I would further predict that someone who routinely bets on w being very low would be more likely to exhibit standard paranoid ideation on various psychometric tests. The preceding description might be a start for functionally capturing such a thinking style.
Oskis, A., Clow, A., Hucklebridge, F., Bifulco, A., Jacobs, C. & Loveday, C. 2013 Understanding alexithymia in female adolescents: The role of attachment style. Pers. Individ. Dif. 54, 97–102. (doi:10.1016/j.paid.2012.08.023)
A quick look-up on attachment styles and found a paper by Oskis and colleagues in which the following styles were listed:
Avoidant - which relates to mistrust, constraints on closeness, fear of rejection, high self reliance and anger.
Anxious - which relates to high desire for company and a fear of separation.
Caregiver support - which relates to active emotional support and felt attachment.
This is enough to enable a simplifying assumption, which in turn will help explore formal possibilities.
So, let's assume two attachment styles – Avoidant and Anxious. One way to consider these strategies is as points on a continuum of social stickiness. We can assume that social interactions are viscous, such that individuals stick together, and that the viscosity of a given group will vary as a function of individual stickiness. Thus, an Avoidant individual is low on stickiness (slippery) whilst an Anxious individual is high on stickiness. This could be mathematically represented on a scale from 0-1, where 1 is Anxious. However it might be that stickiness is variable within an individual, such that the stickiness quotient will range between 0 and 1 as a function of some other parameter. For now, this other parameter can be regarded as extrinsic to the individual and as pertinent to a single utility that the individual is seeking to maximize. We might denote this property as follows:
P s 0-1
Where P is an individual person, s is their stickiness quotient, on a scale of 0-1. This compound is a little informal – I doubt we would need the numerical value in any subsequent algebraic equations – but I hope it is clear what is meant.
At this point we can either think of s values as fixed within an individual or as changeable. If the former then we should be interested in how many of each value are present in the population, and how interactions between each type will play out. If the latter, then we can see s as a conditional trait that responds to external circumstances. The experience of external circumstances will lead to variable ratios of s values in the population, but s values themselves will be external circumstance for others.
From the descriptions above, we might hypothesize that a key utility would be cooperation. In other words as s reduced in value, an individual would be less likely to engage in a cooperative interaction with another individual. The most likely reason for this is because low stickiness (slippery people) would reduce the encounter rate with other individuals. A simple agent-based model might have it that any social contact would lead to a productive social exchange, maximizing both agents’ utilities. I would model this as fitness, perhaps using energy gain as an alternative currency, and then having an energy threshold above which asexual reproduction could occur, producing one copy – the game would be to acquire enough energy for reproduction through social encounters. This would have to be contrasted with a solitary strategy in which agents could still maximize utility through their own actions. For this we would need some form of individual behaviour that acquired energy. One point of interest would then be the differential rates of energy gain as a consequence of the value of individual stickiness quotients.
Patch exploitation gathers energy and the game could be that cooperative action better exploits patches. A ratio could be established between the number of individuals working on a patch and the individual yield (rather like a public goods game with no free riders at this point). The shape of that pay-off curve would also impact on s values.
We have a basic model. But a psychologist might argue that our Anxious end point for s is actually better described as Securely Attached. This is because s=1 leads to total stickiness and cooperative association. Whilst this is a desire of the Anxious typology, there is also a fear of separation that might be costly. This could be modelled in a number of ways. For example, we might build in a parameter for the amount of social interactions. It could benefit an individual to have a number of social associations to maximize fitness. High s will allow social cohesion, but we also want an assortment variable, a, which will vary from 0-1, where 1 is even assortment throughout the group and 0 is association with <1 other individual. Thus if s is high and a low, the individual is eager to attach, but relentlessly attaches to a few people – sticking to only them, and not risking other social interaction. Individuals are now thus characterized:
P s,a
Again, a could be dependent upon extrinsic variables. For example, if the risk of defection within a cooperative interaction was high, it might pay to reduce a and engage in close work with a few individuals. Those individuals could be chosen for their cooperative behaviours, or they might be cultivated through extended tit-for-tat interactions. The intuition here is that if P s,a is high on both variables, thus sticky and seeking maximum assortment, the population size will play a crucial role. A large population size, N, will lead to an increased possibility of one-shot interactions between individuals, which will stabilize defection, not cooperation. Technically, we might say that a closed population of any size, interacting over many cycles but with an unknown limit to those interactions, will lead to cooperation through tit-for-tat. This is the w parameter in standard game theory – which is a discount function. But a more realistic model would have to include cognitive constraints, such as the ability to remember interactions and individuals. (This might lead to an empirical question around attachment style, social network size and structure and social cognition – a regression analysis of data on these facts should be revealing.) Memory, or rather forgetting, could be modelled probabilistically such that at each interaction an individual takes a bet on the value of w. I suspect this is what we do in fact, and our values are influenced by dispositions such as s and a, but also by social structures. I would further predict that someone who routinely bets on w being very low would be more likely to exhibit standard paranoid ideation on various psychometric tests. The preceding description might be a start for functionally capturing such a thinking style.
Oskis, A., Clow, A., Hucklebridge, F., Bifulco, A., Jacobs, C. & Loveday, C. 2013 Understanding alexithymia in female adolescents: The role of attachment style. Pers. Individ. Dif. 54, 97–102. (doi:10.1016/j.paid.2012.08.023)
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