Evolutionary game theoretic approach to Bertrand Duopoly Model using adaptive dynamics

The Bertrand Duopoly (BD) is a competition of rms aiming to achieve dominance in a certain market. In this competition, the Nash equilibrium is attained when rms set prices equal to the unit cost of the product. This results to the Bertrand paradox in which rms attain zero profit in contrast to realistic goals of achieving positive profits. In this study, an evolutionary game dynamics is applied to the BD model to resolve this paradox. The rst step is to develop a smoothed continuous function for the Bertrand Duopoly game which will address the probability of consumers choosing the products with the higher price. Using this smoothed Bertrand Duopoly model, we need to apply the adaptive dynamics to deter- mine the existence of evolutionary stable strategy (ESS) which will guarantee that the prices will converge to the equilibrium price, and not equal to the unit cost. To validate this result, we utilize an agent-based simulation through Net- Logo software, as a tool allowing the user to control the actions of agents