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### Urn model of collective decisions in swarms

Supplementary material for:
 [J12][.pdf .bib] Heiko Hamann Towards Swarm Calculus: Urn Models of Collective Decisions and Universal Properties of Swarm Performance Swarm Intelligence, 7(2-3):145-172, 2013

 [C26][.pdf .bib cited by] Heiko Hamann Towards Swarm Calculus: Universal Properties of Swarm Performance and Collective Decisions ANTS 2012, Brussels, Belgium, Marco Dorigo et al. (eds.), LNCS 7461, pp. 168-179, Springer-Verlag, 2012

The idea is to define a simple but general model of collective decisions in swarms. We define an urn model:

Say r is the ratio of red marbles in the urn.
1) With probability r we draw a red marble and put it back,
3) we define the probability of positive feedback by
$$P(s)=\varphi\sin(\pi s)$$ (1st plot in the applet below),
for having drawn a red marble in step 1) we have P(r).
4a) in case of positive feedback we replace a blue marble from the urn with a red one
4b) in case of negative feedback we replace another red marble from the urn with a blue one

Based on P(s) we can calculate the expected change of the fraction of - say - red marbles s by
$$\Delta s(s)=4(P(s)-0.5)(s-0.5)$$ (2nd plot in the applet below).

Based on this expected change we can define a Markov process, define a transition matrix, and then determine the eigenvectors of this matrix to obtain the expected steady state of the probability density of s (3rd plot in applet). powered by NetLogo