Why evolution does not always lead to an optimal proto-language.An approach based on the replicator dynamics
Sender–receiver models in the style of Lewis (1969), Hurford (1989), or Nowak and Krakauer (1999) can be used to explain meaning of signals in situations of cooperative interaction. Importantly, meaning here is not an ex–ante concept, but arises as an equilibrium property of a game. A strategy of this game is a pair of a sender and a receiver matrix, where the sender matrix links events that possibly become the object of communication to signals, and the receiver matrix links potentially received signals to events. A Nash equilibrium strategy of this game can be interpreted as a so–called proto–language, that is, a set of event–signals relations that facilitate communication over a finite number of events. A typical property of this game is that it admits a multiplicity of Nash equilibrium components, where two (or more) events share the use of one signal or where two (or more) signals are associated with the same event, leading to a situation where some of the potential of communication if left unexploited. W¨arneryd (1993) as well as Trapa and Nowak (2000) show that only the strict Nash strategies, where each event is bijectively linked to one signal and where the inverse of this mapping is used to associate signals with events, which therefore guarantee the full potential of communication, are evolutionarily stable. Evolutionary stability implies asymptotic stability in the replicator dynamics. Interestingly, simulations with this model in the style of a replicator dynamics as reported in Nowak and Krakauer (1999) typically give rise to a suboptimal proto–language, where more than one event is linked to the same signal whereas another signal remains idle. In view of W¨arneryd (1993) and Trapa and Nowak (2000) this raises the following questions: Does this reflect generic behavior of the replicator dynamics for this model? And, if so, what are the properties of a strategy that can protect itself from being driven out by this dynamics despite the fact that it cannot be evolutionarily stable? This paper gives answers to these questions in terms of neutral stability and its dynamic consequences. It, first, provides a complete characterization of neutrally stable strategies for this game, showing that in such a situation, indeed, there can be two (or more) events that are linked to the same signal or two (or more) signals that are linked to the same event, as long as the degree of ambiguity is not too high. Second, it analyzes the long–run behavior of the replicator dynamics of this model. This essentially derives from neutral stability together with the symmetry properties of this game. Building on a result by Bomze (2002), which establishes equivalence of neutral stability and Lyapunov stability in the replicator dynamics for doubly symmetric games with pairwise interaction, it can be shown that the replicator dynamics of this model does not necessarily lead to an optimal proto–language, but that it can be trapped in situations of ambiguous event–signal relations, where some of the potential of communication is left unexploited.
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