Part III. Spatial interactions on a grid
III-4. Other types of neighborhoods and other decision rules
1. Goal
Our goal in this chapter is to extend the model we have created in the previous chapter by adding two features that are crucial to assess the impact of space on evolutionary models:
- The possibility to model different types of neighborhoods of arbitrary size. Besides Moore neighborhoods, we will implement Von Neumann neighborhoods, and both of them of any size.
- The possibility to model other decision rules besides the imitate the best neighbor rule. In particular, we will implement the imitative pairwise-difference rule, the imitate if better rule, the imitative positive proportional rule, and the Fermi rule.[1]
2. Motivation. The impact of decision rules
In chapter III-2 we explored the impact of different assumptions on the robustness of cooperation in spatial settings. However, one assumption we did not change was the decision rule (aka update rule). Roca et al. (2009a, 2009b) conducted an impressive simulation study of the effect of spatial structure in 2×2 games, and discovered that the decision rule can play a major role. In particular, they found that the imitate the best neighbor rule (which Roca et al. (2009a, 2009b) call “unconditional imitation”) favors cooperation in the Prisoner’s Dilemma more than any other decision rule they studied.
In what concerns the Prisoner’s Dilemma, the above results also prove that the promotion of cooperation in this game is not robust against changes in the update rule, because the beneficial effect of spatial lattices practically disappears for rules different from unconditional imitation, when seen in the wider scope of the ST plane. Roca et al. (2009b, p. 9)
In this chapter we are going to implement several decision rules. This will allow us to replicate Roca et al.’s (2009a, 2009b) results… and many others (see the proposed exercises).
3. Description of the model
The model we are going to develop here is a generalization of the model implemented in the previous chapter. In particular, we are going to add the following four parameters:
- neighborhood-type. This parameter is used to define the agents’ neighborhood (both for playing and for strategy updating).[2] The parameter can have two possible values: “Moore” for a Moore neighborhood, or “Von Neumann” for a Von Neumann neighborhood.
- neighborhood-range. This parameter determines the range of the neighborhood. A patch’s Moore neighborhood of range r consists of the patches within Chebyshev distance r. A patch’s Von Neumann neighborhood of range r is composed of the patches within Manhattan distance r. Note that in our descriptions of decision rules below, we do not consider a patch to be a neighbor of itself unless otherwise stated.
- decision-rule. This parameter determines the decision rule that agents will follow. It will be implemented with a chooser, with five possible values:
- “best-neighbor” (Nowak and May, 1992, 1993). This is t