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Evolutionary dynamics of hyperbolic language

Fig 4

Original meaning of words is lost with increasing misalignment.

(left): A natural measure for long-term language change is with what probability speakers use word x for state 0, y for 1/2, and z for 1 at the end of the game, since this is the fully-informative language which constitutes the initial condition for all players. This can be written as a “percentage change” in the form 1 − (S_ijδ_ji)/3, where is the population-averaged speaking matrix and is the population-averaged listening matrix (and Einstein summation conventions have been implemented). (Right): A similar metric for the listeners. is understood to be the 3 × 5 matrix with and all other entries equal to zero. Both panels shown are for the asymmetric game, however the symmetric game shows nearly identical results, see Fig I in S1 File. Note that the upper bounds of 2/3 (left) and 4/5 (right) reflect two different situations—in the limit of high μ, each word becomes equiprobable for each state (probability 1/3 for speakers, so Tr(S) = 1, and 1/5 for listeners, so ); in the limit of high Γ, one-word languages come to dominate (Fig 2), so one diagonal element of S is always equal to 1 and the rest to 0.

doi: https://doi.org/10.1371/journal.pcbi.1010872.g004