An Investigation about Classical and Modern Game Theory: A Case Study of Evolutionary and New Game Theory | Original Article
This study begins with some basic terminology, introducing elementary game theoretic notions such as strategy, best reply, Nash equilibrium pairs etc. Players who use strategies which are in Nash equilibrium have no incentive to deviate unilaterally. Next, a population viewpoint is introduced. The simplest description of such an evolution is based on the replicator equation. The relation between Nash equilibria and rest points of the replicator equation are investigated, which leads to a short proof of the existence of Nash equilibria. We then study mixed strategies and evolutionarily stable strategies. This introductory section continues with a brief discussion of other game dynamics, such as the best reply dynamics, and ends with the simplest extension of replicator dynamics to asymmetric games. Evolutionary game theory has grown into an active area of research that bridges concepts from biology, evolution, non-linear dynamics, and game theory. The mechanisms necessary to conduct an evolutionary analysis of games are presented. Relations between evolutionary stable strategies and Nash equilibria are considered. Replicator dynamics are developed and applied to three relevant games. The analysis of example games is used to illustrate the weaknesses and strengths of the theory.