Numerical approximation of the replicator equations for the Nash bargaining game.

Abstract

There is a variety of phenomena that take place around us. A good mathematical tool to analyse and understand the behaviour of these phenomena, is to approximate them numerically. That is, first we model the phenomenon itself, then we computationally reproduce it, so that we can imitate the phenomenon as many times and in any way we need. That gives us the opportunity, to check the behaviour of the model for different parameters in order to figure out in which way, the phenomenon is affected by them. The phenomenon that is numerically approximated in this work, comes from the field of game theory and it is known as the Nash bargaining game. Its mathematical model, consists of two time dependent partial integrodifferential equations that interact with each other, also known as the replicator equations for the Nash bargaining game. Both fixed grid method and adaptive moving grid method, have been used to numerically approx- imate the model. The central difference scheme, has been used to numerically approximate the 2nd order space derivatives, while the approximation of all the integrals, has been based on the trape- zoidal rule for both fixed and adaptive moving grid methods. For the approximation of the 1st order time derivative, Euler forward numerical scheme has been used for the fixed grid method and Euler backward scheme for adaptive moving grid. The results that have been achieved during this work, show that the behaviour of the model, depends mostly on the parameters ε1, ε2. For the case where ε1 = ε2, the initial curves are moving towards to each other until they have the same shape and position, where they become stable and as ε1; ε2 get closer to zero, although the curves preserve their volumes, they become steeper and steeper. While for ε1 ≠ ε2, we see that the curves converge to stationary solutions, that are no longer concentrated at the same position. In this case, the ratio r = ε_1/ε_2 , seems to affect the behaviour of the model. Finally, it seems that, the initial values of the model, do not affect the result. The algorithms that have been used are built on Matlab for the fixed grid method, while for the adaptive moving method fortran 77 was used. The computer that was used for the runs has processor Intel Core 2 Duo T7250 (2.0 GHz).