Part III. Spatial interactions on a grid
III-1. Spatial chaos in the Prisoner’s Dilemma
1. Goal
The goal of this chapter is to learn how to build agent-based models with spatial structure. In models with spatial structure, agents do not interact with all other agents with the same probability, but they interact preferentially with those who are nearby.[1]
More generally, populations where some pairs of agents are more likely to interact with each other than with others are called structured populations. This contrasts with the models developed in the previous Part, where all members of the population were equally likely to interact with each other.[2] The dynamics of an evolutionary process under random matching can be very different from the dynamics of the same process in a structured population. In social dilemmas in particular, population structure can play a crucial role (Gotts et al. (2003), Hauert (2002,[3] 2006), Roca et al. (2009a, 2009b)).[4]
2. Motivation. Cooperation in spatial settings
In the previous Part, we saw that if agents play the Prisoner’s Dilemma in a population where all members are equally likely to interact with each other, then defection prevails. Here we want to explore whether adding spatial structure may affect that observation. Could cooperation be sustained if we removed the unrealistic assumption that all members of the population are equally likely to interact with each other? To shed some light on this question, in this chapter we will implement a model analyzed by Nowak and May (1992, 1993).
3. Description of the model
In this model, there is a population of agents arranged on a 2-dimensional lattice of “patches”. There is one agent in each patch. The size of the lattice, i.e. the number of patches in each of the two dimensions, can be set by the user. Each patch has eight neighboring patches (i.e. the eight cells which surround it), except for the patches at the boundary, which have five neighbors if they are on a side, or three neighbors if they are at one of the four corners.
Agents repeatedly play a symmetric 2-player 2-strategy game, where the two possible strategies are labeled C (for Cooperate) and D (for Defect). The payoffs of the game are determined using four parameters: CC-payoff, CD-payoff, DC-payoff, and DD-payoff, where XY-payoff denotes the payoff obtained by an X-player who meets a Y-player.
The initial percentage of C-players in the population is initial-%-of-C-players, and they are randomly distributed in the grid. From then onwards, the following sequence of events –which defines a tick– is repeatedly executed:
- Every agent plays the game with all his neighbors (once with each neighbor) and with himself (Moore neighborhood). The total payoff for the player is the sum of the payoffs in these encounters.
- All agents simultaneously revise their strategy according to the imitate the best neighbor decision rule, which reads as follows:
Consider the set of all your neighbors plus yourself; then