Vainstein & Arenzon, 2001, Disordered environments in spatial games

Ref: Vainstein, M. H., & Arenzon, J. J. (2001). Disordered environments in spatial games. Physical Review E, 64, 051905.

This is the simple case for a spatial PD with cellular automata, where some of the spaces are empty. Thus, this is the special case of the game considered in their later paper (Vainstein et al., 2007), where mobility, m = 0. They again follow Nowak and May (1992, 1993) in using a modified payoff matrix: R = 1, P = S = 0, T = b > 1. According to them, there are actually at least 2 papers where Nowak and others considered cases with a diluted population, but they claim that those treatments were brief and didn't consider many interesting aspects of the dynamics.

The other parameters: is the population density, and is the density of cooperators. There is also the fraction of active sites - those that have just changed strategy from time t - 1 to time t: .

Results
When is very small, agents do not interact, and stay in their initial settings. As increases, dyads are likely, any of which are CD will become DD. As approaches 0.3, the dynamics become more interesting, and cooperators can do all right. The curve of becomes dependent on the value of b.
In the interesting case where b is set so that neither strategy will dominate in the fullspace case, (b = 1.4 for the von Neumann space considered here), there is a transition point around where the number of active sites increases sharply.
A number of different initial values of were considered between .2 and .8. For , each went to a different asymptotic fraction of cooperators, but these all merged when .
They define the persistence as the number of sites that do not change strategy between an initial time and time t. After an initial decrease, the persistence tends to stabilize at a value dependent on . There is a sharp decline right around , indicating that this region is very sensitive to small changes in population density.

Conclusions:
The system is able to sustain cooperation even under lattice disorder (already suggested by Nowak et al., 1994). Two things unnoticed by Nowak et al were (1) the dynamical transition as a function of quenched disorder, and (2) Disorder may enhance cooperation.

Comments
Disorder may enhance cooperation only for certain values of b and only if the initial frequency of cooperators is already high. Cooperators almost never increase from their initial frequency, except for certain values near or at . This is still an important result concerning the spatial PD with cellular automata (CA).
This is a much more thorough investigation than that presented for the mobile case (Vainstein et al., 2007). It demonstrates interesting dynamics of the lattice CA spatial PD.