Rainer Hegselmann and Andreas Flache (1998)
Journal of Artificial Societies and Social Simulation vol. 1, no. 3, <http://jasss.soc.surrey.ac.uk/1/3/1.html>
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Received: 28-May-98 Accepted: 11-Jun-98 Published: 30-Jun-98
Figure 1: Two-dimensional CA. Left: without edges pasted. Right: with edges pasted. |
Figure 2: Different neighbourhood templates. |
. | [1] |
if x < 1/2, then y = 0 if x >= 1/2, then y = 1 |
if x < 1/3, then y = 0 if x >= 1/3 and x < 2/3, then y = 0.5 if x >= 2/3, then y = 1 |
Figure 3. Top: two possible opinions (0 and 1). Bottom: three possible opinions (0, 0.5, and 1). |
. | [2] |
[2] is the universal transition rule. It is applied sequentially to randomly selected cells, which is to say that the updating procedure used is sequential updating. (Simultaneous updating involves simultaneously applying [2] to all cells.) Figure 4 shows the typical results of simulation runs with 2, 5, 10, 15, and 30 different opinions. At the beginning all possible opinions have the same frequency in the population. Obviously in all cases, opinion patterns and clusters emerge. The structures shown are stable for ever. In the continuous case shown in Figure 4(f), no step function is applied and the dynamics are driven by [1] instead of [2]. It can be proved analytically that these dynamics always lead to ubiquitous agreement when t tends to infinity (Hegselmann et al. 1998).
(a) 2 possible opinions | (b) 5 possible opinions |
(c) 10 possible opinions | (d) 15 possible opinions |
(e) 30 possible opinions | (f) continuous opinions |
color coding of opinions | |
Figure 4: Results of simulation runs with 2, 5, 10, 15 or 30 discrete opinions and with continuous opinions. |
Figure 5: Discrete opinion dynamics with up to 50 possible opinions: Mean frequencies of opinions after convergence. |
. | [4] |
is maximised. d is the Euclidean distance between i and j. w determines how strongly valences are discounted by distance. The greater w, the less the valences are discounted by distance. Sakoda's world is an 8 × 8 checkerboard occupied by two groups, each with 6 members. The members of one group are represented as squares and members of the other as crosses. Sakoda analyses different combinations of attitudes. One of these is called segregation, another is called suspicion. These attitudes are shown in Figure 6.
Figure 6: The attitude combinations segregation and suspicion. |
Figure 7: The dynamics of segregation (Sakoda 1971, p. 127). |
Figure 8: The dynamics of suspicion (Sakoda 1971, p. 126). |
Figure 9. Left: segregation
Right: suspicion. Both dynamics were started with the same random distribution of cells. |
Figure 10. Left: random distribution (Schelling 1971, p. 155). Right: segregation (Schelling 1971, p. 157). |
Preference: <= 90% | Preference: <= 50% |
Preference: <= 30% | Preference: <= 20% |
Figure 11: Experiments with different neighborhood preferences. |
Nevertheless, it is only within the last decade that CA based models have been used more frequently in the behavioural and social sciences.^{[9]} In social psychology, Nowak et al. (1990) have developed a two-dimensional CA model of evolution in attitudes. In economics, Keenan and O'Brien (1993) use a one-dimensional CA to model and analyse pricing in a spatial setting. Axelrod (1984, pp. 158ff.) made the first steps in analysing the dynamics of cooperation within a CA framework, although he did not refer to his model as a CA. Nowak and May (1992, 1993) developed the idea and studied the dynamics of cooperation using a two-dimensional CA with two-person games as building blocks. Bruch (1993) and Kirchkamp (1994) follow the same line, but use different and more sophisticated learning rules. The same framework is used in Messick and Liebrand (1995), who analyse the dynamics of three different decision principles.^{[10]} A substantial number of the artificial societies described in Epstein and Axtell (1996) are based on CA modes. Gaylord and D'Andra (1998) describe a toolkit for CA based modelling of social dynamics using MATHEMATICA.
S : | Saved | D: Drowned | S > D |
M: | Move on | H: Help | M > H |
p_{1} : | probability that player 1 will become needy | ||
p_{2}: | probability that player 2 will become needy | ||
0 < p_{1}, p_{2} < 1 | |||
Figure 12: Support game^{[12]} |
. | [5] |
Provided that the condition in [5] is fulfilled the expected payoff for mutual support is greater than the expected payoff for not giving support to each other. Therefore [5] describes a condition under which the solution of the support game becomes inefficient. If characterised as a matrix game one can easily verify that [5] describes a condition under which the support game turns into a 2-person Prisoner's Dilemma. We will refer to [5] as the PD-condition.
An iterated support game has a cooperative solution if and only if, for both players, it holds that:
. | [6] |
[6] is based on the assumption that it is only in cases of unilateral emergency that one player can infer whether the other is cooperating or defecting. I will refer to [6] as the COOP-condition.^{[14]}
Support Networks
Basicsinteraction window: | von Neumann neighbourhood. |
migration window: | 11 × 11 with the given position in the centre. |
world: | 21 × 21 (torus). |
Class structure
individuals per risk class: 35, giving a total of 315 individuals and 136 empty sites.
Payoffs and minimum level
Saved = 5, Drowned = 1, Move on = 7 and Help = 6 so (S - D) / (M -H) = 4 and the minimum level is 50%.
Probabilities for getting migration options
0.05 | => | a = 0.903 (first experiment) |
0.10 | => | a = 0.810 (second experiment) |
0.15 | => | a = 0.723 (third experiment) |
Use of migration options
Figure 13: Initial conditions
Figure 13: primordial soup | Figure 14. First experiment: chance for migration 5% | |
Figure 15. Second experiment: chance for migration 10% | Figure 16. Third experiment: chance for migration 15% |
Figure 17: PD and COOP-condition for (S-D)/(M-H) = 4. |
The last point - a direct consequence of [6] - may seem surprising at first glance. The explanation is that two individuals of extreme risk classes are facing a problem which members of intermediate classes only experience to a much lesser degree. It is very likely that two members of the least needy risk class will not need help. Two members of the most needy risk class will probably be in need of help simultaneously and thus neither of them will be in a position to help the other. In both cases, unilateral emergencies become rare events. For this reason, the expected utility of mutual support decreases at the extremes. At the same time, it becomes more difficult to find out whether other individuals are cooperating or defecting. Therefore, probabilities of stability which makes support relations effective for intermediate risk classes may not be sufficient for the extreme risk classes.
Figure 18: The colours indicate the probability (a^{+}) that has to be exceeded to meet the COOP-condition [6] for a given pairing of risk classes. Black indicates that the PD-condition [5] is not met for that pair of risk classes. As before (S-D)/(M-H) = 4. |
RISK CLASS | quite bad | middle risk | quite good |
quite bad | COOP-condition tends not to be fulfilled. | COOP and PD-condition tend not to be fulfilled. | |
middle risk | COOP and PD-condition can comparatively easy be fulfilled. | ||
quite good | COOP and PD-condition tends not to be fulfilled. | COOP-condition tends not to be fulfilled. | |
Figure 19: Problems of pairing. |
. | [7] |
Figure 20: Percentages of satisfied agents in different classes. Left: first experiment. Right: second experiment. |
Figure 21. Average network dividend in all classes over 2500 periods. Top: first experiment. Middle: second experiment. Bottom: third experiment. Colours indicate classes. |
. | [8] |
Obviously, the lower p_{i}, the higher . Let be the total payoff for a member of class i in period t. Then:
[9] |
is what a member of class i gets in period t due to successfully established support relations. Let's call the network dividend. Figure 21 shows the development of the average network dividend for all 9 classes over 2500 periods for the three experiments.
^{2} The model is basically a special case of French (1956), Harary (1959) and Abelson (1964). See also Nowak et al. (1990) and Hegselmann et al. (1998).
^{3}As states we use a finite number (n) of real valued opinions. It would be just as easy to represent opinions by integer values but this would not affect any of the arguments presented in the paper.
^{4}Obvously, the step functions in Figure 3 are not the only method for obtaining a finite number of possible opinions from an infinite set of real valued opinions. For example, it might be more realistic to assume that a particular opinion is adopted with a probability that declines with the distance between that opinion and u_{i}^{*}(t+1). However, we do not investigate this possibility, because this paper focuses on illustrating the CA framework rather than demonstrating the empirical adequacy of a particular model.
^{5}That includes the possibility of modelling and understanding micro/macro feedback loops within the CA framework.
^{6}For a discussion of artefacts related to updating procedures see Huberman and Glance (1993) and Hegselmann (1996).
^{7}This model plays an important role in Lehman (1977), an early introduction to computer simulations for social scientists.
^{8} Another early application of checker board modelling is Hägerstrand (1965) where it is used to analyse the diffusion of new agricultural techniques.
^{9}This is due to the fact that those models ask for a great deal of iterated computation even if very simple assumptions are used. It has become progressively easier since user-friendly programming languages and powerful debugging tools have been created. Nowadays corresponding hardware and software is also available to social scientists.
^{10} For recent overviews of research on social dilemmas see Schulz et al. (1994) and Liebrand and Messick (1996).
^{11} The model was originally developed in Hegselmann (1994a, 1994b).
^{12} In what follows, it is always assumed that D < M. When D > M we get support games in which those in need of help are those with favourable opportunities which they cannot take advantage of without support from another actor.
^{13} But notice that there are also many other equilibria.
^{14} The proof of [6] only exists as a manuscript available from the author.
^{15} This implies that we can assume that the equilibrium selection problem has been solved!
^{16} Details will be demonstrated in another article.
^{17} For an analysis of different types of simulations see Hartmann (1996), Kliemt (1996) and Troitzsch (1998).
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