Peter Dittrich, Thomas Kron and Wolfgang Banzhaf (2003)
On the Scalability of Social Order: Modeling the Problem of Double and Multi Contingency Following Luhmann
Journal of Artificial Societies and Social
Simulation
vol. 6, no. 1
<https://www.jasss.org/6/1/3.html>
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Received: 14-Jun-2002 Accepted: 9-Jan-2003 Published: 31-Jan-2003
Figure 1. Two agents interact by showing signs symbolizing messages. |
1. For each possible activity compute:
(a) expectation-expectation2. activity probabilities w_{AP}=g(w_{AV})
Here, an agent estimates the probability that Agent B expects activity i to be performed next by Agent A. The value is determined by accessing A's memory M_{ego} which has stored responses of Agent A to activities of Agent B. m_{received} is the last activity of Agent B to which Agent A has to respond now. In other words, m_{received} is the number that Agent B displays on its sign. Roughly speaking, the function lookup(M_{ego}, m_{received}, i) returns how often Agent A has reacted with activity i to activity m_{received} in the past. In order to model forgetting, past events long passed are counted less than recent events (see Section 2.12, below).(b) expectation-certainty
In order to calculate the certainty of the future, Agent A possesses a second memory M_{alter}_{}_{}, which stores how the other agent has reacted to A's activities. So, using this memory, A can predict what B might do as a response to A's potential activity i. Consulting A's alter-memory by calling the function lookup(M_{alter}, i) results in a vector (p_{1}, ...,p_{N}_{}) containing N values. A value p_{j}_{} ∈ [0; 1] of this vector is an estimate of the probability that Agent B responds with activity j to activity i of Agent A. The expectation-certainty is measured by the function f_{certainty}. The input of that function is the vector (p_{1},...,p_{N}_{}). The function f_{certainty}_{}_{} returns a certainty of 0.0, if all values of the vector are the same, e.g., (1/N, 1/N, ..., 1/N), since in that case the agent has no information (there is no distinction possible). The highest certainty (value 1.0) is returned for a vector that consists of zeros but a single value 1, e.g., (1, 0, 0, ..., 0). For this work we measure the certainty by the Shannon entropy (Shannon and Weaver 1949):
(1) (See Section 9.5 in the Appendix for alternative certainty measures.)
(c) activity value
The expectation-expectation w^{i}_{EE} and the expectation-certainty w^{i}_{EC} are combined by function f and the result is normalized in order to calculate the activity value w^{i}_{AV}. The parameter α specifies the fraction of EC contained in the activity value. f is a linear sum plus a small additive constant:
(2) with c_{f} being a constant parameter. The addition of c_{f }/ N prevents an activity value from approaching zero in order to avoid artefacts. Division by N assures that the influence of the constant summand does not increase with increasing N. For the experiments presented here we have chosen c_{f} = 0.01 so that there is always at least a small chance for each activity to be selected. In a situation where an agent is rather sure what to do and proportional selection (explained below) is used, there is a chance of about 1% that an activity is selected different from the most probable one. In case of quadratic selection (γ= 2) the "error" probability caused by c_{f} is about 0.2%. (See Section 9.1 in the Appendix for a detailed discussion of c_{f}.)
The activity values are scaled by the selection function g: R^{N} → R^{N}:3. Randomly select activity i such that the probability of activity i is w^{i}_{AP}
(3) The parameter γ allows to control the influence of randomness (see below). Note that y is an exponent in Eq. (3).
(4) |
(5) |
(6) |
In order to simplify the following formalism we define a function which returns a vector of all estimated probabilities for every possible response to message a:
(7) |
Representation: A memory M is represented by a dimensional matrix m_{a}_{,b} called memory matrix. This matrix (m_{a}_{,b}) is manipulated by the following initialization, memorization and lookup procedures:Initialization: The matrix is initialized with (m_{a}_{,b})_{} = 1/N.
Memorize(M, a, b): First, we increase the entry in the memory matrix given by the index (a, b) by the learning rate r_{learn} ^{[7]}:
(8) Then we increase all entries by the forgetting rate r_{forget} divided by the number of activities N:
(9) Finally we normalize every line of the memory matrix:
(10) Lookup(M, a, b): Return the entry of the memory matrix given by index (a, b):
(11)
Second, the expectation-certainty is calculated by using the alter-memory:
and the result is normalized to arrive at the activity values:
(12) |
In the dyadic situation (two agents) order appears for a wide range of parameter settings.
(13) |
(14) |
In the following we will use O_{AV} only, since it is very similar to O_{AP} and because O_{AP} taking into account has not lead to different conclusions. (O_{AV} and O_{AP} data can be found in log-file: runName.log)
(15) |
where p_{i}_{,j} is the probability that a randomly drawn agent reacts with activity j to the displayed message i of Alter. M_{i} is the number of agents displaying message i. The matrix (p_{i}_{,j}) can be interpreted as the average behavior matrix of the whole population. In order to get an intuitive understanding of the systems level order imagine the following game: An external player has to predict the activity of the agents. At the outset the player can take a look at the internal state of every agent. Then, in each turn, an agent is chosen randomly, the activity number i on his sign is shown to the player. (Note that the agents are anonymous, so that the player does not know which agent possesses what kind of internal state). Then, again an agent is chosen randomly from the population and the player has to predict the reaction of that agent to the activity number i. For each correct prediction the player receives a point. The state of the agents is not altered during the game, so they are not allowed to learn during the game. The larger the systems level order O_{P} is, the higher is the maximum average score that the player can achieve. For O_{P} = 0 the player cannot perform better than just guessing randomly. For O_{P} = 1 the player can predict the reaction correctly in each turn. Note that this measure makes sense only if the number of agents is large compared to the number of messages actively used. Sociologically we can interpret the value O_{P} as a measure of integration. Integration with regard to the whole society, "social integration", is a very important term in sociological theory even if it is not definitely clear what it means, because the integration of society could be observed from different analytical perspectives (Münch 1997). The core of social integration consists of a situation of society where all of its particles are stably affiliated with each other and build up a unit which is marked-off outwards. So, in modern societies on the one hand we can differentiate between economical integration, the accentuation of exchange, free contracts, and the capitalistic progression of wealth; political integration, the accentuation of the importance of the exertion of political enforcement by means of national governance; cultural integration, the accentuation of compromise by discourse on the basis of mutually shared reason; and solidarity integration, the accentuation of the necessity of modern societies to generate free citizenships in terms of networks of solidarity. On the other hand we can observe the systemic integration (Schimank 1999). The accentuation here lies on the operational closure of the system.^{[9]} If a (activity) system is operationally closed it is in the position of being able to accept highest complexity of its environment, to pick up and to process this complexity in itself without to imperil its existence. We interpret the value O_{P} as a measure of systemic integration in this sense. In all our simulation experiments a high value of O_{P} has indicated a closure of the emerging activity system. This means that the better the agents are able to predict the activities of other agents as reactions to their own activity selections, the more the communication system appears to be operationally closed, that is certain activities follow certain activities.
Figure 2. A complex example of a single run of the dyadic situation. There are N = 64 potential activities. The agents are using expectation-expectation only (a = 0.0). The selection method is something in between proportional and quadratic selection (γ = 1.5). Learning rate r_{learn} = 0.2. Forgetting rate r_{forget} = 0.001. c_{f }= 0.010000. |
Figure 3. Average number of different activities in an interval of 50 time steps for different N. Measurement started at time 500 after the transient phase at the beginning. Simulation time: 1000 time steps for each run. Parameter setting: normal learning rate r_{learn} = 0.2, low forgetting rate r_{forget} = 0.001, proportional selection y = 1. |
Figure 4. Same as in Figure 3, but with a lower influence of randomness (y = 1.5). |
Figure 5. Average number of different activities in an interval of 50 time steps for different a. Measurement started at time step 500, such that the transient phase at the beginning is not considered. Simulation time 1000 time steps for each run. Parameter setting: normal learning rate r_{learn} = 0.2, low forgetting rate r_{forget} = 0.001, number of activities N = 20. |
Figure 6. Average number of different messages (activities) in an interval of 50 time steps for different y. Measurement started at time step 500 after the transient phase at the beginning of runs. Simulation time 1000 time steps for each run. Parameter setting: normal learning rate r_{learn} = 0.2, low forgetting rate r_{forget} = 0.001, number of activities N = 20. |
Figure 7. Average behavior of multi-agent simulations where the Ego-memory is used for calculation of the expectation-expectation, as in the basic dyadic scenario. Parameters: N = 10 activities, y = 2 (quadratic selection), r_{learn} = 0.5, r_{forget} = 0.001, cf = 0.01. Multi agent scenario as described in Section 6.1 |
Figure 8. Same as Figure 7, but Alter-memory is used to calculate expectation-expectation (EE). Multi-agent scenario as described in Section 6.1. |
In each simulation step, do:
- Randomly choose an agent and call it Ego.
- Randomly choose another agent and call it Alter.
- Let Ego observe Alter's displayed message a (equal to Alter's last activity).
Let Ego reacts to Alter's message. (Note that only Ego acts, but not Alter.)- Ego stores its reaction b in its Ego-memory.
Formally, Ego does: M_{ego} := memorize(M_{ego}, a, b).- Alter stores Ego's reaction b in Alter's Alter-memory.
Formally, Alter does: M_{alter} := memorize(M_{alter}, a, b).- Choose n agents randomly and call them observers.
- Each observer stores Ego's reaction in its Alter-memory.
Formally, each observer performs M_{alter }:= memorize(M_{alter}, a, b) where a is the message displayed by Alter and b is Ego's reaction.
Figure 9. Same as Figure 8, but with n = 3 observers. For a = 0 (EE only), systems level order is scalable. As in Figure 8, the Alter-memory used to calculate the expectation-expectation. Multi-agent scenario as described in Section 6.13. |
An activity system consists of a set of activity symbols^{[14]} (subset of {1, 2, ..., N}) where activity symbols within that set mainly produce activity symbols within that set, and every activity symbol in that set is produced by activity symbols of that set. This can be expressed more formally^{[15]}, e.g.: The set O,O ⊆ {1,2,...N}, is called activity system, if and only if (1) for all v_{1}∈O and v_{2} ∉O,w(v_{1}, v_{2}) ≤ r, (property of closure), and (2) for all v_{2}∈O there exists v_{1}∈O such that w(v_{1}, v_{2}) > r (property of self-maintenance).
Figure 10. Typical activity system that has emerged in a simulation experiment with 20 agents, 64 possible activities, and no observers. A node represents an activity. All nodes without any incoming edges are removed, except for one (upper right corner of the diagram), which should illustrate how the removed nodes are connected to the "inner" active network. Parameters: y = 2, N = 64, a = 1:0; r_{learn} = 0.2, r_{forget} = 0.001, M = 20. No observers. Ego-memory used for EE calculation. Threshold = 0.01. The corresponding single run is shown in the Appendix, Figure 13. |
Figure 11. Example of an activity system that has emerged in a simulation experiment with 10 agents, 10 possible activities, Alter-memory used for EE calculation, and no observers. A node represents an activity. The sub-set of nodes {1, 5, 6} can be interpreted as an autopoietic (sub-)system. Parameters: y = 2, N = 10, a= 0.0, r_{learn} = 0.2, r_{forget} = 0.001, M = 10. No observers. Alter- memory used for EE calculation. Edges with weights smaller than r = 10% have been omitted for clarity. The corresponding single run is shown in the Appendix |
^{2} The term "entity" denotes what Luhmann (1988) called "Ego" and "Alter" in Chapter 3 (p. 148), and Parsons called "actor".
^{3} One of Luhmann's basic assumptions is that both actors are interested in solving this problem. Luhmann (1984, p. 160): "No social system can get off the ground, if the one who begins with communication, cannot know or would not be interested in whether his partner reacts positively or negatively." But the question remains: Where does the motivation (interest) come from? According to Luhmann, an answer should not consider actor characteristics (like intentions) as starting point for system theory. We think that Luhmann falls back to his earlier anthropological position (see Schimank (1988, p. 629; 1992, p. 182)) and assumes a basic necessity of "expectation-certainty", that is, that Alter and Ego want to know what is going on in this situation. A fundamental uncertainty still remains and takes further effect in the emerged systems as an autocatalytic factor. See also the approach to formulate "double contingency" from the perspective of a communication network as provided by Leydesdorff (1993, pp. 58f.)
^{4} For new simulation experiments about the genesis of symbolic generalized media, see Papendick / Wellner (2002).
^{5} We prefer to use the term "activity" because Luhmann's concept of communication is much more complex than the communication processes among the agents in our model; see also Hornung (2001). We cannot use the term "action" as understood by Parsons, because our agents do not show a meaningfully motivated behavior that is oriented toward other agents according to certain goals, means, and a symbolic reference framework rooted in the situation. Therefore, our agents decide to do something we call "activity" - no more and no less. For Luhmann every communication consists of a selection triple based on the three distinctions (1) information, (2) transmission (German: Mitteilung), and (3) understanding (see Luhmann (1984, Chapter 4)). Our agents do not communicate in that sense, but we could take their interaction as an abstract model of communication dispensing from the distinction between message, information and meaning. Since it just represents the transmission of information, it is not a accurate model for Luhmann's communication concept. In this contribution, however, we are just interested in the process of order formation and have removed many details for the sake of simplicity.
^{6} Storing an event (a, b) in memory M is achieved by calling M':= memorize(M, a, b). M' is the new memory (or the new state of the memory), which is created by inserting (a, b) into M.
^{7} With increasing learning rate, the number of different messages, used by the agents, decreases (see Section 9.2 in the Appendix). Qualitatively, the relative behavior of the model is independent of the choice of the learning rate (above 0.2).
^{8} Alternatively we can measure the number of different activity- reaction pairs (a, b) occurring in a time interval.
^{9} Another important meaning of systemic integration in sociology is the integration of different social systems within society.
^{10} Note that in our simulation there is always a small chance for every activity to be selected.
^{11} "More or less randomly" means that there are some very small memory traces left. Thus the reaction is not fully dictated by chance. Note also that when we add expectation-certainty the situation will be different.
^{12} Recall that high systems level order means that just by observing the interactions among (non-learning) agents, it is possible to predict, how an agent in the population will react, without knowing his internal state nor his identity.
^{13} This non-linearity is not a general phenomenon, because it does not appear if using a different memory type like the linear degenerating memory (type 04), see Appendix Section 9.4.
^{14} Activity symbols are equivalent to an activity.
^{15} Note that we define "activity system" operationally for the purposes of our discussion here. The definition should show what can be interpreted as an activity system in our model. And the formal form should make that as clear as possible. It is, however, not a general definition of "activity system".
^{16} Here, "entity" refers to Ego and Alter in Luhmann's (1984) explanation of the situation of double contingency. Note that an "entity" needs not to be a human person.
^{17} In terms of Hempel and Oppenheim (1948).
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