Meloni et al. (2009) Effects of mobility on prisoner's dilemma players

Ref: Meloni, S., et al. (2009). Effects of mobility in a population of prisoner's dilemma players. Physical Review E, 79, 067101. 

This paper is not particularly insightful or well-written, but is a rare study of the PD game with mobile agents.

They mention evolutionary game theory, and then talk about population structure, in particular scale-free networks and random (Erdos-Renyi) networks.
Recent work shows cooperation is enhanced on scale-free networks. This is because cooperators are fixed in highly-connected nodes, allowing positive assortment.

Models where individuals are mobile have not been well-explored. They mention 4 studies (all of which I am familiar with) on mobile agents. They also note that these have been limited to movement on a 2-D lattice.

They consider agents moving unconstrained (no collisions) in a 2-D continuous plane. They claim: Our results show that cooperation is actually promoted provided that players do not move too fast and that cooperation is not too expensive. Additionally, at variance with other cases, the dynamics of the system exhibits only two stable attractors—those in which the whole population plays with one of the two possible strategies.

The model: 
N agents move in a square plane of size L with periodic boundaries and play a game on the "instantaneous network of contacts".
Motion. All agents move with a constant speed, v, the same for all agents. At each time step they turn a random angle chosen from a uniform distribution between +/- 180 degree.
Network of interactions. At each time step, the network of agent i is all individuals within a threshold radius r. Thus, an instantaneous network is set up.
Evolutionary dynamics. At each time step, all agents play the PD game with each of their instantaneous neighbors. They use classic Nowak payoffs of R = 1, P = S = 0, and T = b > 1. After play,  each agent picks a random neighbor, and adopts its strategy (if it did better) with a probability proportional to the payoff difference.

They started with 50% cooperators, population density p. Time steps go as follows: (1) agents perform a movement, (2) a network is established, (3) the PD games are played and strategies are updated. The process is repeated until a stationary state is reached.

Results:
If v = 0, then you get a random graph, and you can dynamics oscillations and a stable population of cooperators. If v > 0, the system always goes to either all defectors or all cooperators, and much more quickly than it does for v = 0.
Density. At very low population densities, defectors always win, because agents are very spread out and cooperators could not form clusters. Cooperators do well at intermediate densities. At high densities, cooperators start to lose, because the population is so dense that it resembles a "well-mixed" population with random mixing.
The main result: Cooperators do well as long as b is not too high (i.e, the temptation is low enough), and, more interestingly, as long as the velocity of movement is low enough (Figure 3). Note that the network radius is r = 1. Thus, there needs to be some correlation between the network at time t and t + 1. In other words, if v is too high, the population starts resembling a well-mixed one.

The threshold value of b for cooperation to succeed falls as v increases, until it reaches a point where cooperation cannot survive for any b > 1 (Figure 4.