Analysis of an information-theoretic model for communication

We study the cost-minimization problem posed by Ferrer i Cancho and Sole in their model of communication that aimed at explaining the origin of Zipf's law (2003 Proc. Nat. Acad. Sci. 100 788). Direct analysis shows that the minimum cost is min{lambda, 1 - lambda}, where lambda determines the relative weights of speaker's and hearer's costs in the total, as shown in several previous works using different approaches. The nature and multiplicity of the minimizing solution change discontinuously at lambda = 1/2, being qualitatively different for lambda < 1/2, lambda > 1/2, and lambda = 1/2. Zipf's law is found only in a vanishing fraction of the minimum-cost solutions at lambda = 1/2 and therefore is not explained by this model. Imposing the further condition of equal costs yields distributions substantially closer to Zipf's law ones, but significant differences persist. We also investigate the solutions reached with the previously used minimization algorithm and find that they correctly recover global minimum states at the transition.