Costly signalling theory and dishonest signalling

We analyze the model of costly signalling theory and show that dishonest signalling is still a possible outcome even for costly indices that cannot be faked. We assume that signallers pay the cost for sending a signal and that the cost correlates negatively with signaller's quality q and correlates positively with signal's strength s. We show that for any given function f with continuous derivative, there is a cost function t(s, q) increasing in s and decreasing in q so that when the signaller of quality q optimizes the strength of the signal, it will send the signal of strength f(q). In particular, optimal signals can follow any given function f. Our results can explain the curvilinear relationship between the strength of signals and physical condition of three-spined stickleback (Gasterosteus aculeatus).