Dynamic stability and basins of attraction in the Sir Philip Sidney game

Abstract

We study the handicap principle in terms of the Sir Philip Sidney game. The handicap principle asserts that cost is required to allow for honest signalling in the face of conflicts of interest. We show that the significance of the handicap principle can be challenged from two new directions. Firstly, both the costly signalling equilibrium and certain states of no communication are stable under the replicator dynamics (i.e. standard evolutionary dynamics); however, the latter states are more likely in cases where honest signalling should apply. Secondly, we prove the existence and stability of polymorphisms where players mix between being honest and being deceptive and where signalling costs can be very low. Neither the polymorphisms nor the states of no communication are evolutionarily stable, but they turn out to be more important for standard evolutionary dynamics than the costly signalling equilibrium.

Figure 1.

An extensive form representation of the Sir Philip Sidney game. The dotted lines represent the donor's information sets, i.e. the donor is unable to distinguish between the decision nodes connected by the dotted lines. The terminal nodes show the sender's and the donor's payoffs, respectively.

Figure 2.

A phase portrait of one boundary face of the space of population states (face K) of the Sir Philip Sidney game illustrating the motion around the polymorphic equilibrium. Close to the unique rest point, interior trajectories (of the full state space) converge to K. Notice the resemblance between this phase portrait and the phase portrait of the well-known game of matching pennies under the replicator dynamics.

Figure 3.

Simulation results showing the basin of attraction for the boundary face containing the hybrid equilibrium for different values of m and c. Here, k = 0.54, a = 31/32, b = 9/32 and d = 1/2. Plus symbol, c = 0; multiplication symbol, c = 0.05; asterisk, c = 0.1; unfilled square, c = 0.15; filled square, c = 0.2.

Figure 4.

Simulation results showing the cumulative size of the basins of attraction for the two signalling equilibria as k and m vary. m is set so as to maximize the basins of attraction of signalling given the specified k. As before, the simulation uses the discrete time replicator dynamics when a = 31/32, b = 9/32 and d = 1/2. Plus symbol, c = 0.20; multiplication symbol, c = 0.15; asterisk, c = 0.10; unfilled square, c = 0.05; filled square, c = 0.

Figure 5.

Simulation results showing the cumulative size of the basins of attraction for the two signalling equilibria when k = 1 for several differing values of c. The simulation uses the discrete time replicator dynamics when a = 31/32, b = 9/32 and d = 1/2. Plus symbol, c = 0.001; multiplication symbol, c = 0.005; asterisk, c = 0.01.