The evolution of the mixed conjectures in the rent-extraction game

This paper adopts an evolutionary perspective on the rent-extraction model with conjectural variations (CV) allowing for mixed-strategies. We analyze the dynamics of the model with n CVs under the replicator equation. We find that the end points of the evolutionary dynamics include the pure-strategy consistent CVs. However, there are also mixed-strategy equilibria that occur: these are on the boundaries between the basins of attraction of the pure-strategy sinks. Further, we develop a more general notion of consistency which applies to mixed-strategy equilibria. In a three conjecture example, by conducting a global dynamics analysis, we prove that in contrast to the pure-strategy equilibria, the mixed-strategy equilibria are not ESS: under the replicator dynamics, there are three or four mixed equilibria that may either be totally unstable (both eigenvalues positive), or saddle-stable (one stable eigenvalue). There also exist heteroclinic orbits that link equilibria together. Whilst only the pure-strategies can be fully consistent, we find a lower bound for the probability that mixed strategy conjectures will be ex post consistent.