Game Theory: Spatial Prisoner's Dilemna
This script is a simulation of Martin A. Nowak and Robert M. May paper about "Evolutionary Games and Spatial Chaos" 1992.
It is created by N. Giakoumoglou, M. Demetriou and P. Manouselis for a presentation in Game Theory course in May 2020.
1. The Spatial Prisoner's Dilemna
In spatial prisoner's dileman there are two players those who always cooperate, C, and those who always defect, D. We place those players on a two dimensional lattice (grid), each lattice site is occupied either by a C or a D. In each round of the game (each generation), the players play the PD game with nearest neighboring sites and with one's own site (thus we define these sites as a territory – a 3x3 grid). The score for each player is the sum of the payoffs in these encounters with neighbors. At the start of the next generation, each lattice-site is occupied by the player with the highest score among the previous owner and the immediate neighbors. Boundaries are fixed but we can also define the lattice as a torus. Conclusions we will deduct remain true if players interact only with the four orthogonal neighbors in square lattices or self-interactions are included.
2. The Prisoner's Dilemna Game
The PD can be formulated in tabular form as follows, where T > R > P ≥ S.
3. Chaos in the Spatial PD game
The dynamical behavior of the system depends on the parameter b:
- (b > 1.8) 2x2 or larger cluster of D will continue to grow at the corners.
- (b < 1.8) big D cluster will shrink
- (b < 2) 2x2 or larger cluster of C will continue to grow
- (b > 2) C clusters do not grow
- (2 > b > 1) C clusters can grow in regions of D and vice versa
Chaos persists in shifting patterns C → D, D → C, D → D, C → C.