On the Evolution of Continuous Types Under Replicator and Gradient Dynamics: Two Examples

Abstract

This paper illustrates techniques for assessing the dynamic stability of games where a continuum of types might be present by re-analyzing two models under incomplete information, the Lohman et al. (Unpublished manuscript, 2001) public goods game and the Kopel et al. (J Econ Dyn Control 48:394–409, 2014) Cournot duopoly game. The evolution of continuous types follows either replicator dynamics Oechssler and Riedel (Econ Theory 17:141–162, 2001; J Econ Theory 107:223–252 2002) or gradient dynamics Friedman and Ostrov (J Math Econ 46:691–707, 2010; J Econ Theory 148:743–777, 2013). The techniques rely on a system of partial differential equations. Numerical solutions obtained through replicator and gradient dynamics highlight the differences and the similarities that arise under both approaches. In the public goods game, the dynamic system affects the stationary distribution of types while in the Cournot duopoly model, the types evolve to a single mass point regardless of the dynamics used. Lastly, these techniques allow us to endogenize the distribution of player types.