Paul Samuelson’s critique and equilibrium concepts in evolutionary game theory

We present two new notions of evolutionary stability, the truly evolutionarily stable state (TESS ) and the generalized evolutionarily stable equilibrium (GESE ). The GESE generalizes the evolutionarily stable equilibrium (ESE ) of Joosten [1996]. An ESE attracts all nearby trajectories monotonically, i.e., the Euclidean distance decreasing steadily in time. For a GESE this property should holds for at least one metric. The TESS generalizes the evolutionarily stable strategy (ESS ) of Maynard Smith & Price [1973]. A TESS attracts nearby trajectories too, but the behavior of the dynamics nearby must be similar to the behavior of the replicator dynamics near an ESS. Both notions are de…ned on the dynamics and immediately imply asymptotical stability for the dynamics at hand, i.e., the equilibrium attracts all trajectories su¢ ciently nearby. We consider this the relevant and conceptually right approach in de…ning evolutionary equilibria, rather than de…ning a static equilibrium notion and search for appropriate dynamics guaranteeing its dynamic stability. Moreover, the GESE and the TESS take similar positions as the ESE and ESE do in relation to other equilibrium and …xed point concepts in general. Key words: evolutionary stability, evolutionary game theory. JEL-Codes: A12; C62; C72; C73; D83 Just a few days before the completion of this paper, Paul Samuelson, one of the earlyday greats in economics and winner of The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, passed away. In Samuelson [1941] he proved an advocate of de…ning stability of an economic equilibrium in terms of the dynamics of a system of price adjustment, instead of the prevailing practice to de…ne stability properties of economic equilibria on the underlying economic system à la Hicks [1939]. See also Negishi [1962] for a similar evaluation and interpretation of Samuelson’s critique. The concepts to be presented in our contribution, do withstand the general point made by Samuelson’s critique and we have argued before (Joosten [1996, 2006]) and will argue again, that the ESS concept of Maynard Smith & Price [1973] fails to do so. yFELab and School of Management & Governance, University of Twente, POB 217, 7500 AE Enschede, The Netherlands. Email: [email protected]