Andreas Flache and Rainer Hegselmann (2001)
Do Irregular Grids make a Difference? Relaxing the Spatial Regularity Assumption in Cellular Models of Social Dynamics
Journal of Artificial Societies and Social
Simulation vol. 4, no. 4
<http://jasss.soc.surrey.ac.uk/4/4/6.html>
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Received: 4-July-01 Accepted: 10-Sep-01 Published: 31-Oct-01
Figure 1: Neighbourhood definitions in a cellular automaton. |
Figure 2: A Voronoi diagram. |
Figure 3: Cells with 3, 4, 5, and 8 next neighbours in a Voronoi graph. |
Figure 4: Neighborhood of two individuals in the rectangular cellular world (r=300, cellsize 50 × 50). |
Figure 5: Neighbourhoods of three different cells in an irregular grid. Range r = 100, 11 × 11 cells. Focal cells are blue, neighbour cells are red. |
Figure 6: Non-linear effects of neighbourhood size r in a rectangular torus world. Expectation level E = 30%, Initial cluster of 9 × 9 = 81 contributors, 50 × 50 torus grid. |
Figure 7: Effects of expectation level E, cluster size h and neighbourhood radius r on final state of TFT-dynamics. Rectangular grid. |
Figure 8: Effects of expectation level E, Cluster size h and neighbourhood radius r on final state of TFT-dynamics. Rectangular grid with varying location of cluster. |
Figure 9: Path dependence in the growth of a cluster in an irregular grid. Colours indicate relative frequency of co-operation in this cell in the endstate, based on 100 replications. Dark blue = 100%, Blue = 83%, Red = 0%). Initial cluster size h = 190, expectation level E = 0.3, neighbourhood size r = 30. |
Figure 10: Fraction of co-operators in the endstate as a function of the relative density of the initial cluster. E = 0.3, r = 50 and h = 30. Results are based on 2500 different cluster locations and 50 replications per condition. |
Table 1: Payoffs of the constituent support game. |
[3]
Figure 11: Change in propensities for co-operation in dyadic support game simulated with the Bush Mosteller stochastic learning algorithm and high learning rate l = 1. |
Figure 12: Migration windows with radius 3 (red) and 5 (yellow) around focal cell (black) in 3 different grid structures. |
Table 2: Main assumptions of simulation of migration dynamics. |
Figure 13: Dynamics of co-operation and migration in three different grid structures. |
Figure 14: Structures of 3 × 3 clusters in three different grids. |
^{2}For the mathematical details see Okabe et al. (1992). Details of our implementation of Voronoi-diagrams and some sample data sets can be found on our website.
^{3}With exception of the four corner members of the cluster, who change into defectors, because they see only 25% co-operation around them, less than their expectation level E. However, the conversion of these four cells is more than outweighed by the changes from defection to co-operation.
^{4}With this assumption, the support game used here is different from the support game used in Hegselmann and Flache (1998). The main difference is that bilateral help is always possible in the game we use here, though the degree varies for different neediness class combinations. By contrast, the support game of Hegselmann and Flache (1998) allows in each period only unilateral help. As a consequence, that game imposes imperfect information which greatly complicates the game theoretical analysis. Obviously, there are different plausible approaches to model what in the daily life is simply called mutual help.
^{5}This implementation of the learning dynamics is different from an alternative formalization of the "power law of practice" that was proposed by Erev et al (1999). Flache and Macy (2001) showed that for parameter values corresponding to those used in the present paper, both models generate very similar behaviour. However, the authors also reveal subtle but important differences that occur in other regions of the parameter space, in particular when actors¬í aspiration levels are high or their learning rates are low.
^{6}To recall, we define a next neighbour as a cell that has a common border with the focal cell. For the rectangular world, this imposes a von Neumann neighborhood structure.
^{7}Simultaneously, the opponent adapts her corresponding attractiveness assessment. The sequence in which dyads are selected is randomly chosen.
^{8}More precisely, we assume that a particular player i assesses the attractiveness payoff attained in interactions with others after the t'th encounter as: . The symbol u_{it} indicates the payoff attained at the t'th encounter and a_{i,t-1} is the estimate of others' attractiveness before the t'th encounter occurred. The parameter d in this function indicates the rate of decay . Only the most recent payoff is taken into account, when d = 0, while d = 1 indicates that the attractiveness assessment is a weighted mean of the payoffs i attained in all previous encounters. In the simulations reported here, we assumed d = 0.5. Note that this function guarantees that a_{it} is always within the range of valid payoffs [-0.25,1].
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